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+The Project Gutenberg eBook, The Phase Rule and Its Applications, by
+Alexander Findlay
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+
+
+
+Title: The Phase Rule and Its Applications
+
+
+Author: Alexander Findlay
+
+
+
+Release Date: November 27, 2010  [eBook #34457]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+
+***START OF THE PROJECT GUTENBERG EBOOK THE PHASE RULE AND ITS
+APPLICATIONS***
+
+
+E-text prepared by Juliet Sutherland, Keith Edkins, and the Online
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+
+
+Transcriber's note:
+
+      Page numbers enclosed by curly braces (example: {25}) have been
+      incorporated to facilitate the use of the Table of Contents.
+
+      In chemical formulas an underscore is used to indicate that
+      the following number enclosed by curly braces is a subscript.
+      For example, the formula of water is indicated by H_{2}O.
+
+      A carat character followed by a number enclosed by curly
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+
+      A few typographical errors have been corrected and are listed
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+
+
+
+
+THE PHASE RULE AND ITS APPLICATIONS
+
+by
+
+ALEX. FINDLAY, M.A., PH.D., D.SC.
+
+       *       *       *       *       *
+
+TEXT-BOOKS OF PHYSICAL CHEMISTRY.
+
+EDITED BY SIR WILLIAM RAMSAY, K.C.B., F.R.S., D.SC.
+
+       *       *       *       *       *
+
+    STOICHIOMETRY. By SYDNEY YOUNG, D.Sc., F.R.S., Professor of Chemistry
+    in the University of Dublin; together with an INTRODUCTION TO THE STUDY
+    OF PHYSICAL CHEMISTRY by Sir WILLIAM RAMSAY, K.C.B., F.R.S., Editor of
+    the Series. Crown 8vo. 7s. 6d.
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+    Introduction to the Series by Sir WILLIAM RAMSAY, K.C.B., F.R.S., D.Sc.
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+
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+    CHANGE, CATALYSIS AND EXPLOSIONS. BY J. W. MELLOR, D.Sc. (N.Z.), B.Sc.
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+
+    THE PHASE RULE AND ITS APPLICATIONS. By ALEX. FINDLAY, M.A., Ph.D.,
+    D.Sc., Lecturer and Demonstrator in Chemistry, University of
+    Birmingham. With 134 Figures in the Text. Crown 8vo. 5s.
+
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+
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+
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+
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+
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+
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+39 PATERNOSTER ROW, LONDON
+NEW YORK, BOMBAY, AND CALCUTTA
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+       *       *       *       *       *
+
+
+THE PHASE RULE AND ITS APPLICATIONS
+
+BY
+
+ALEX. FINDLAY, M.A., PH.D., D.SC.
+
+Lecturer on Physical Chemistry, University of Birmingham
+
+With One Hundred and Thirty-Four Figures in the Text
+
+Third Impression
+
+
+
+
+
+
+
+Longmans, Green, and Co. 39 Paternoster Row, London
+New York, Bombay, and Calcutta
+1908
+
+All rights reserved
+
+
+
+
+DEDICATED
+
+TO
+
+FRANCIS ROBERT JAPP, LL.D., F.R.S.
+
+PROFESSOR OF CHEMISTRY, UNIVERSITY OF ABERDEEN,
+
+IN GRATITUDE FOR EARLY TRAINING
+
+AND ADVICE
+
+
+
+{vii}
+
+PREFACE TO THE SECOND EDITION.
+
+During the two years which have elapsed since the first edition of this
+book appeared, the study of chemical equilibria has been prosecuted with
+considerable activity, and valuable additions have been made to our
+knowledge in several departments of this subject. In view of the scope of
+the present work, it has been, of course, impossible to incorporate all
+that has been done; but several new sections have been inserted, notably
+those on the study of basic salts; the interpretation of cooling curves,
+and the determination of the composition of solid phases without analysis;
+the equilibria between iron, carbon monoxide, and carbon dioxide, which are
+of importance in connection with the processes occurring in the blast
+furnace; and the Phase Rule study of the ammonia-soda process. I have also
+incorporated a short section on the reciprocal salt-pair barium
+carbonate--potassium sulphate, which had been written for the German
+edition of this book by the late Professor W. Meyerhoffer. The section on
+the iron-carbon alloys, which in the first edition was somewhat
+unsatisfactory, has been rewritten.
+
+A. F.
+
+_September, 1906._
+
+
+
+{viii}
+
+PREFACE
+
+Although we are indebted to the late Professor Willard Gibbs for the first
+enunciation of the Phase Rule, it was not till 1887 that its practical
+applicability to the study of Chemical Equilibria was made apparent. In
+that year Roozeboom disclosed the great generalization, which for upwards
+of ten years had remained hidden and unknown save to a very few, by
+stripping from it the garb of abstract Mathematics in which it had been
+clothed by its first discoverer. The Phase Rule was thus made generally
+accessible; and its adoption by Roozeboom as the basis of classification of
+the different cases of chemical equilibrium then known established its
+value, not only as a means of co-ordinating the large number of isolated
+cases of equilibrium and of giving a deeper insight into the relationships
+existing between the different systems, but also as a guide in the
+investigation of unknown systems.
+
+While the revelation of the principle embedded in the Phase Rule is
+primarily due to Roozeboom, it should not be forgotten that, some years
+previously, van't Hoff, in ignorance of the work of Willard Gibbs, had
+enunciated his "law of the incompatibility of condensed systems," which in
+some respects coincides with the Phase Rule; and it is only owing to the
+more general applicability of the latter that the very {ix} important
+generalization of van't Hoff has been somewhat lost sight of.
+
+The exposition of the Phase Rule and its applications given in the
+following pages has been made entirely non-mathematical, the desire having
+been to explain as clearly as possible the principles underlying the Phase
+Rule, and to illustrate their application to the classification and
+investigation of equilibria, by means of a number of cases actually
+studied. While it has been sought to make the treatment sufficiently
+elementary to be understood by the student just commencing the study of
+chemical equilibria, an attempt has been made to advance his knowledge to
+such a stage as to enable him to study with profit the larger works on the
+subject, and to follow with intelligence the course of investigation in
+this department of Physical Chemistry. It is also hoped that the volume may
+be of use, not only to the student of Physical Chemistry, or of the other
+branches of that science, but also to the student of Metallurgy and of
+Geology, for whom an acquaintance with at least the principles of the Phase
+Rule is becoming increasingly important.
+
+In writing the following account of the Phase Rule, it is scarcely
+necessary to say that I have been greatly indebted to the larger works on
+Chemical Equilibria by Ostwald ("Lehrbuch"), Roozeboom ("Die Heterogenen
+Gleichgewichte"), and Bancroft ("The Phase Rule"); and in the case of the
+first-named, to the inspiration also of personal teaching. My indebtedness
+to these and other authors I have indicated in the following pages.
+
+In conclusion, I would express my thanks to Sir William Ramsay, whose
+guidance and counsel have been constantly {x} at my disposal; and to my
+colleagues, Dr. T. Slater Price and Dr. A. McKenzie, for their friendly
+criticism and advice. To Messrs. J. N. Friend, M.Sc., and W. E. S. Turner,
+B.Sc., I am also indebted for their assistance in reading the proof-sheets.
+
+A. F.
+
+_November, 1903._
+
+
+
+
+{xi}
+
+CONTENTS
+
+                                                                       PAGE
+
+  CHAPTER I
+
+  INTRODUCTION                                                            1
+
+  General, I. Homogeneous and heterogeneous equilibrium,
+  5. Real and apparent equilibrium, 5.
+
+  CHAPTER II
+
+  THE PHASE RULE                                                          7
+
+  Phases, 8. Components, 10. Degree of freedom. Variability
+  of a system, 14. The Phase Rule, 16. Classification of systems
+  according to the Phase Rule, 17. Deduction of the Phase
+  Rule, 18.
+
+  CHAPTER III
+
+  TYPICAL SYSTEMS OF ONE COMPONENT                                       21
+
+  A. _Water._ Equilibrium between liquid and vapour. Vaporization
+  curve, 21. Upper limit of vaporization curve, 23.
+  Sublimation curve of ice, 24. Equilibrium between ice and
+  water. Curve of fusion, 25. Equilibrium between ice, water,
+  and vapour. The triple point, 27. Bivariant systems of water,
+  29. Supercooled water. Metastable state, 30. Other systems
+  of the substance water, 32. B. _Sulphur_, 33. Polymorphism, 33.
+  Sulphur, 34. Triple point--Rhombic and monoclinic sulphur
+  and vapour. Transition point, 34. Condensed systems, 36.
+  Suspended transformation, 37. Transition curve--Rhombic
+  and monoclinic sulphur, 37. Triple point--Monoclinic sulphur,
+  liquid, and vapour. Melting point of monoclinic sulphur, 38.
+  Triple point--Rhombic and monoclinic sulphur and liquid, 38.
+  Triple point--Rhombic sulphur, liquid, and vapour. Metastable
+  triple point, 38. Fusion curve of rhombic sulphur, 39.
+  Bivariant systems, 39. C. _Tin_, 41. Transition point, 41.
+  {xii}
+  Enantiotropy and monotropy, 44. D. _Phosphorus_, 46. Enantiotropy
+  combined with monotropy, 51. E. _Liquid Crystals_, 51.
+  Phenomena observed, 51. Nature of liquid crystals, 52. Equilibrium
+  relations in the case of liquid crystals, 53.
+
+  CHAPTER IV
+
+  GENERAL SUMMARY                                                        55
+
+  Triple point, 55. Theorems of van't Hoff and of Le Chatelier,
+  57. Changes at the triple point, 58. Triple point solid--solid--vapour,
+  62. Sublimation and vaporization curves,
+  63. Fusion curve--Transition curve, 66. Suspended transformation.
+  Metastable equilibria, 69. Velocity of transformation,
+  70. Law of successive reactions, 73.
+
+  CHAPTER V
+
+  SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION                   76
+
+  Different systems of two components, 77. PHENOMENA OF
+  DISSOCIATION. Bivariant systems, 79. Univariant systems,
+  80. Ammonia compounds of metal chlorides, 82. Salts with
+  water of crystallization, 85. Efflorescence, 86. Indefiniteness
+  of the vapour pressure of a hydrate, 87. Suspended transformation,
+  89. Range of existence of hydrates, 90. Constancy
+  of vapour pressure and the formation of compounds, 90.
+  Measurement of the vapour pressure of hydrates, 91.
+
+  CHAPTER VI
+
+  SOLUTIONS                                                              92
+
+  Definition, 92. SOLUTIONS OF GASES IN LIQUIDS, 93.
+  SOLUTIONS OF LIQUIDS IN LIQUIDS, 95. Partial or limited
+  miscibility, 96. Phenol and water, 97. Methylethylketone
+  and water, 100. Triethylamine and water, 101. General form
+  of concentration-temperature curve, 101. Pressure-concentration
+  diagram, 102. Complete miscibility, 104. Pressure-concentration
+  diagram, 104.
+
+  CHAPTER VII
+
+  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
+  VOLATILE                                                              106
+
+  General, 106. The saturated solution, 108. Form of the
+  solubility curve, 108. A. ANHYDROUS SALT AND WATER.
+  {xiii}
+  The solubility curve, 111. Suspended transformation and
+  supersaturation, 113. Solubility curve at higher temperatures,
+  114. (1) _Complete miscibility of the fused components._ Ice as
+  solid phase, 116. Cryohydrates, 117. Changes at the quadruple
+  point, 119. Freezing mixtures, 120. (2) _Partial miscibility of
+  the fused components._ Supersaturation, 124. Pressure-temperature
+  diagram, 126. Vapour pressure of solid--solution--vapour,
+  126. Other univariant systems, 127. Bivariant systems, 129.
+  Deliquescence, 130. Separation of salt on evaporation, 130.
+  General summary, 131.
+
+  CHAPTER VIII
+
+  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
+  VOLATILE                                                              133
+
+  B. HYDRATED SALT AND WATER, (1) _The compounds
+  formed do not have a definite melting point._ Concentration-temperature
+  diagram, 133. Sodium sulphate and water, 134.
+  Suspended transformation, 137. Dehydration by means of
+  anhydrous sodium sulphate, 138. Pressure-temperature diagram,
+  138. (2) _The compounds formed have a definite melting point._
+  Solubility curve of calcium chloride hexahydrate, 145.
+  Pressure-temperature diagram, 149. The indifferent point, 150.
+  The hydrates of ferric chloride, 151. Suspended transformation,
+  155. Evaporation of solutions at constant temperature, 155.
+  Inevaporable solutions, 157. Illustration, 158.
+
+  CHAPTER IX
+
+  EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS                            161
+
+  General, 161. Iodine and chlorine, 161. Concentration-temperature
+  diagram, 162. Pressure-temperature diagram, 165.
+  Bivariant systems, 167. Sulphur dioxide and water, 169.
+  Pressure-temperature diagram, 170. Bivariant systems, 173.
+
+  CHAPTER X
+
+  SOLID SOLUTIONS. MIXED CRYSTALS                                       175
+
+  General, 175. Solution of gases in solids, 176. Palladium
+  and hydrogen, 178. Solutions of solids in solids. Mixed
+  crystals, 180. Formation of mixed crystals of isomorphous
+  substances, 182. I. The two components can form an unbroken
+  series of mixed crystals. (_a_) _The freezing points of all mixtures
+  lie between the freezing points of the pure components._ Examples,
+  183. Melting-point curve, 183. (_b_) _The freezing-point curve passes
+  through a maximum._ Example, 186. (_c_) _The freezing-point
+  curve passes through a minimum._ Example, 188. Fractional
+  {xiv}
+  crystallization of mixed crystals, 188. II. The two components
+  do not form a continuous series of mixed crystals. (_a_) _The
+  freezing-point curve exhibits a transition point_, 190. Example,
+  190. (_b_) _The freezing-point curve exhibits a eutectic point_, 191.
+  Examples, 192. Changes in mixed crystals with the temperature,
+  192.
+
+  CHAPTER XI
+
+  EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES                                195
+
+  Temperature-concentration diagram, 196. Transformation
+  of the unstable into the stable form, 201. Examples, 203.
+  _Benzaldoximes_, 203. _Acetaldehyde and paraldehyde_, 204.
+
+  CHAPTER XII
+
+  SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF
+  TWO COMPONENTS                                                        207
+
+  Summary of the different systems of two components, 208.
+  (1) _Organic compounds_, 212. (2) _Optically active substances_,
+  213. Examples, 216. Transformations, 217. (3) _Alloys_, 220.
+  Iron--carbon alloys, 223. Determination of the composition of
+  compounds without analysis, 228. Formation of minerals, 232.
+
+  CHAPTER XIII
+
+  SYSTEMS OF THREE COMPONENTS                                           234
+
+  General, 234. Graphic representation, 235.
+
+  CHAPTER XIV
+
+  SOLUTIONS OF LIQUIDS IN LIQUIDS                                       240
+
+  1. _The three components form only one pair of partially
+  miscible liquids_, 240. Retrograde solubility, 245. The influence
+  of temperature, 247. 2. _The three components can form two
+  pairs of partially miscible liquids_, 249. 3. _The three components
+  form three pairs of partially miscible liquids_, 251.
+
+  CHAPTER XV
+
+  PRESENCE OF SOLID PHASES                                              253
+
+  A. The ternary eutectic point, 253. Formation of compounds,
+  255. B. Equilibria at higher temperatures. Formation
+  of double salts, 258. Transition point, 258. Vapour pressure.
+  {xv}
+  Quintuple point, 261. Solubility curves at the transition point,
+  264. Decomposition of the double salt by water, 267. Transition
+  interval, 270. Summary, 271.
+
+  CHAPTER XVI
+
+  ISOTHERMAL CURVES AND THE SPACE MODEL                                 272
+
+  Non-formation of double salts, 272. Formation of double
+  salt, 273. Transition interval, 277. Isothermal evaporation,
+  278. Crystallization of double salt from solutions containing
+  excess of one component, 280. Formation of mixed crystals,
+  281. Application to the characterization of racemates, 282.
+  _Representation in space._ Space model for carnallite, 284.
+  Summary and numerical data, 287. Ferric chloride--hydrogen
+  chloride--water, 290. Ternary systems, 291. The isothermal
+  curves, 294. Basic Salts, 296. Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O, 298.
+  Basic mercury salts, 301. Indirect determination of the composition
+  of the solid phase, 302.
+
+  CHAPTER XVII
+
+  ABSENCE OF LIQUID PHASE                                               305
+
+  Iron, carbon monoxide, carbon dioxide, 305.
+
+  CHAPTER XVIII
+
+  SYSTEMS OF FOUR COMPONENTS                                            312
+
+  Reciprocal salt-pairs. Choice of components, 313. Transition
+  point, 314. Formation of double salts, 315. Transition
+  interval, 315. Graphic representation, 316. Example, 317.
+  Ammonia-soda process, 320. Preparation of barium nitrite, 327.
+  Barium carbonate and potassium sulphate, 328.
+
+  APPENDIX
+
+  EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT                    331
+
+  I. The dilatometric method, 331. II. Measurement of
+  the vapour pressure, 334. III. Solubility measurements, 335.
+  IV. Thermometric method, 337. V. Optical method, 338.
+  VI. Electrical methods, 338.
+
+  NAME INDEX                                                            341
+
+  SUBJECT INDEX                                                         345
+
+       *       *       *       *       *
+
+
+{1}
+
+THE PHASE RULE
+
+CHAPTER I
+
+INTRODUCTION
+
+General.--Before proceeding to the more systematic treatment of the Phase
+Rule, it may, perhaps, be not amiss to give first a brief forecast of the
+nature of the subject we are about to study, in order that we may gain some
+idea of what the Phase Rule is, of the kind of problem which it enables us
+to solve, and of the scope of its application.
+
+It has long been known that if water is placed in a closed, exhausted
+space, vapour is given off and a certain pressure is created in the
+enclosing vessel. Thus, when water is placed in the Torricellian vacuum of
+the barometer, the mercury is depressed, and the amount of depression
+increases as the temperature is raised. But, although the pressure of the
+vapour increases as the temperature rises, its value at any given
+temperature is constant, no matter whether the amount of water present or
+the volume of the vapour is great or small; if the pressure on the vapour
+is altered while the temperature is maintained constant, either the water
+or the vapour will ultimately disappear; the former by evaporation, the
+latter by condensation. At any given temperature within certain limits,
+therefore, water and vapour can exist permanently in contact with one
+another--or, as it is said, be in equilibrium with one another--only when
+the pressure has a certain definite value. The same law of constancy of
+vapour pressure at a given {2} temperature, quite irrespective of the
+volumes of liquid and vapour,[1] holds good also in the case of alcohol,
+ether, benzene, and other pure liquids. It is, therefore, not unnatural to
+ask the question, Does it hold good for all liquids? Is it valid, for
+example, in the case of solutions?
+
+We can find the answer to these questions by studying the behaviour of a
+solution--say, a solution of common salt in water--when placed in the
+Torricellian vacuum. In this case, also, it is observed that the pressure
+of the vapour increases as the temperature is raised, but the pressure is
+no longer independent of the volume; as the volume increases, the pressure
+slowly diminishes. If, however, solid salt is present in contact with the
+solution, then the pressure again becomes constant at constant temperature,
+even when the volume of the vapour is altered. As we see, therefore,
+solutions do not behave in the same way as pure liquids.
+
+Moreover, on lowering the temperature of water, a point is reached at which
+ice begins to separate out; and if heat be now added to the system or
+withdrawn from it, no change will take place in the temperature or vapour
+pressure of the latter until either the ice or the water has
+disappeared.[2] Ice, water, and vapour, therefore, can be in equilibrium
+with one another only at one definite temperature and one definite
+pressure.
+
+In the case of a solution of common salt, however, we may have ice in
+contact with the solution at different temperatures and pressures. Further,
+it is possible to have a solution in equilibrium not only with anhydrous
+salt (NaCl), but also with the hydrated salt (NaCl, 2H_{2}O), as well as
+with ice, and the question, therefore, arises: Is it possible to state in a
+general manner the conditions under which such different systems can exist
+in equilibrium; or to obtain some insight {3} into the relations which
+exist between pure liquids and solutions? As we shall learn, the Phase Rule
+enables us to give an answer to this question.
+
+The preceding examples belong to the class of so-called "physical"
+equilibria, or equilibria depending on changes in the physical state. More
+than a hundred years ago, however, it was shown by Wenzel and Berthollet
+that "chemical" equilibria can also exist; that chemical reactions do not
+always take place completely in one direction as indicated by the usual
+chemical equation, but that before the reacting substances are all used up
+the reaction ceases, and there is a condition of equilibrium between the
+reacting substances and the products of reaction. As an example of this,
+there may be taken the process of lime-burning, which depends on the fact
+that when calcium carbonate is heated, carbon dioxide is given off and
+quicklime is produced. If the carbonate is heated in a closed vessel it
+will be found, however, not to undergo entire decomposition. When the
+pressure of the carbon dioxide reaches a certain value (which is found to
+depend on the temperature), decomposition ceases, and calcium carbonate
+exists side by side with calcium oxide and carbon dioxide. Moreover, at any
+given temperature the pressure is constant and independent of the amount of
+carbonate or oxide present, or of the volume of the gas; _nor does the
+addition of either of the products of dissociation, carbon dioxide or
+calcium oxide, cause any change in the equilibrium_. Here, then, we see
+that, although there are three different substances present, and although
+the equilibrium is no longer due to physical, but to chemical change, it
+nevertheless obeys the same law as the vapour pressure of a pure volatile
+liquid, such as water.
+
+It might be supposed, now, that this behaviour would be shown by other
+dissociating substances, _e.g._ ammonium chloride. When this substance is
+heated it dissociates into ammonia and hydrogen chloride, and at any given
+temperature the pressure of these gases is constant,[3] and is independent
+of the amounts of solid and gas present. So far, therefore, ammonium
+chloride behaves like calcium carbonate. If, however, one of the {4}
+products of dissociation be added to the system, it is found that the
+pressure is no longer constant at a given temperature, but varies with the
+amount of gas, ammonia or hydrogen chloride, which is added. In the case of
+certain dissociating substances, therefore, addition of one of the products
+of dissociation alters the equilibrium, while in other cases it does not.
+With the help of the Phase Rule, however, a general interpretation of this
+difference of behaviour can be given--an interpretation which can be
+applied not only to the two cases cited, but to all cases of dissociation.
+
+Again, it is well known that sulphur exists in two different crystalline
+forms, octahedral and prismatic, each of which melts at a different
+temperature. The problem here is, therefore, more complicated than in the
+case of ice, for there is now a possibility not only of one solid form, but
+of two different forms of the same substance existing in contact with
+liquid. What are the conditions under which these two forms can exist in
+contact with liquid, either singly or together, and under what conditions
+can the two solid forms exist together without the presence of liquid
+sulphur? To these questions an answer can also be given with the help of
+the Phase Rule.
+
+These cases are, however, comparatively simple; but when we come, for
+instance, to study the conditions under which solutions are formed, and
+especially when we inquire into the solubility relations of salts capable
+of forming, perhaps, a series of crystalline hydrates; and when we seek to
+determine the conditions under which these different forms can exist in
+contact with the solution, the problem becomes more complicated, and the
+necessity of some general guide to the elucidation of the behaviour of
+these different systems becomes more urgent.
+
+It is, now, to the study of such physical and chemical equilibria as those
+above-mentioned that the Phase Rule finds application; to the study, also,
+of the conditions regulating, for example, the formation of alloys from
+mixtures of the fused metals, or of the various salts of the Stassfurt
+deposits; the behaviour of iron and carbon in the formation of steel and
+the {5} separation of different minerals from a fused rock-mass.[4] With
+the help of the Phase Rule we can group together into classes the large
+number of different isolated cases of systems in equilibrium; with its aid
+we are able to state, in a general manner at least, the conditions under
+which a system can be in equilibrium, and by its means we can gain some
+insight into the relations existing between different kinds of systems.
+
+Homogeneous and Heterogeneous Equilibrium.--Before passing to the
+consideration of this generalization, it will be well to first make mention
+of certain restrictions which must be placed on its treatment, and also of
+the limitations to which it is subject. If a system is uniform throughout
+its whole extent, and possesses in every part identical physical properties
+and chemical composition, it is called _homogeneous_. Such is, for example,
+a solution of sodium chloride in water. An equilibrium occurring in such a
+homogeneous system (such as the equilibrium occurring in the formation of
+an ester in alcoholic solution) is called _homogeneous equilibrium_. If,
+however, the system consists of parts which have different physical
+properties, perhaps also different chemical properties, and which are
+marked off and separated from one another by bounding surfaces, the system
+is said to be _heterogeneous_. Such a system is formed by ice, water, and
+vapour, in which the three portions, each in itself homogeneous, can be
+mechanically separated from one another. When equilibrium exists between
+different, physically distinct parts, it is known as _heterogeneous
+equilibrium_. It is, now, with heterogeneous equilibria, with the
+conditions under which a heterogeneous system can exist, that we shall deal
+here.
+
+Further, we shall not take into account changes of equilibrium due to the
+action of electrical, magnetic, or capillary forces, or of gravity; but
+shall discuss only those which are due to changes of pressure, temperature,
+and volume (or concentration).
+
+Real and Apparent Equilibrium.--In discussing equilibria, also, a
+distinction must be drawn between real and {6} apparent equilibria. In the
+former case there is a state of rest which undergoes continuous change with
+change of the conditions (_e.g._ change of temperature or of pressure), and
+for which the chief criterion is that _the same condition of equilibrium is
+reached from whichever side it is approached_. Thus in the case of a
+solution, if the temperature is maintained constant, the same concentration
+will be obtained, no matter whether we start with an unsaturated solution
+to which we add more solid, or with a supersaturated solution from which we
+allow solid to crystallize out; or, in the case of water in contact with
+vapour, the same vapour pressure will be obtained, no matter whether we
+heat the water up to the given temperature or cool it down from a higher
+temperature. In this case, water and vapour are in _real_ equilibrium. On
+the other hand, water in contact with hydrogen and oxygen at the ordinary
+temperature is a case only of _apparent_ equilibrium; on changing the
+pressure and temperature continuously within certain limits there is no
+continuous change observed in the relative amounts of the two gases. On
+heating beyond these limits there is a sudden and not a continuous change,
+and the system no longer regains its former condition on being cooled to
+the ordinary temperature. In all such cases the system may be regarded as
+undergoing change and as tending towards a state of true or real
+equilibrium, but with such slowness that no change is observed.
+
+Although the case of water in contact with hydrogen and oxygen is an
+extreme one, it must be borne in mind that the condition of true
+equilibrium may not be reached instantaneously or even with measurable
+velocity, and in all cases it is necessary to be on one's guard against
+mistaking apparent (or false) for real (or true) equilibrium. The
+importance of this will be fully illustrated in the sequel.
+
+       *       *       *       *       *
+
+
+{7}
+
+CHAPTER II
+
+THE PHASE RULE
+
+Although the fact that chemical reactions do not take place completely in
+one direction, but proceed only to a certain point and there make a halt,
+was known in the last quarter of the eighteenth century (Wenzel, 1777;
+Berthollet, 1799); and although the opening and subsequent decades of the
+following century brought many further examples of such equilibria to our
+knowledge, it was not until the last quarter of the nineteenth century that
+a theorem, general in its application and with foundations weakened by no
+hypothetical assumptions as to the nature or constitution of matter, was
+put forward by Willard Gibbs;[5] a generalization which serves at once as a
+golden rule by which the condition of equilibrium of a system can be
+tested, and as a guide to the similarities and dissimilarities existing in
+different systems.
+
+Before that time, certainly, attempts had been made to bring the different
+known cases of equilibria--chemical and physical--under general laws. From
+the very first, both Wenzel[6] and Berthollet[7] recognized the influence
+exercised by the _mass_ of the substances on the equilibrium of the system.
+It was reserved, however, for Guldberg and Waage, by their more general
+statement and mathematical treatment of the Law of Mass Action,[8] to
+inaugurate the period of quantitative study of equilibria. The law which
+these investigators enunciated {8} served satisfactorily to summarize the
+conditions of equilibrium in many cases both of homogeneous and, with the
+help of certain assumptions and additions, of heterogeneous equilibrium. By
+reason, however, of the fact that it was developed on the basis of the
+kinetic and molecular theories, and involved, therefore, certain
+hypothetical assumptions as to the nature and condition of the substances
+taking part in the equilibrium, the law of mass action failed, as it
+necessarily must, when applied to those systems in which neither the number
+of different molecular aggregates nor the degree of their molecular
+complexity was known.
+
+Ten years after the law of mass action was propounded by Guldberg and
+Waage, Willard Gibbs,[9] Professor of Physics in Yale University, showed
+how, in a perfectly general manner, free from all hypothetical assumptions
+as to the molecular condition of the participating substances, all cases of
+equilibrium could be surveyed and grouped into classes, and how
+similarities in the behaviour of apparently different kinds of systems, and
+differences in apparently similar systems, could be explained.
+
+As the basis of his theory of equilibria, Gibbs adopted the laws of
+thermodynamics,[10] a method of treatment which had first been employed by
+Horstmann.[11] In deducing the law of equilibrium, Gibbs regarded a system
+as possessing only three independently variable factors[12]--temperature,
+pressure, and the concentration of the components of the system--and he
+enunciated the general theorem now usually known as the _Phase Rule_, by
+which he defined the conditions of equilibrium as a relationship between
+the number of what are called the phases and the components of the system.
+
+Phases.--Before proceeding farther we shall first consider what exactly is
+meant by the terms _phase_ and _component_. We have already seen (p. 5)
+that a heterogeneous system is made {9} up of different portions, each in
+itself homogeneous, but marked off in space and separated from the other
+portions by bounding surfaces. These homogeneous, physically distinct and
+mechanically separable portions are called _phases_. Thus ice, water, and
+vapour, are three phases of the same chemical substance--water. A phase,
+however, whilst it must be physically and chemically homogeneous, need not
+necessarily be chemically simple. Thus, a gaseous mixture or a solution may
+form a phase; but a heterogeneous mixture of solid substances constitutes
+as many phases as there are substances present. Thus when calcium carbonate
+dissociates under the influence of heat, calcium oxide and carbon dioxide
+are formed. There are then _two_ solid phases present, viz. calcium
+carbonate and oxide, and one gas phase, carbon dioxide.
+
+The _number of phases_ which can exist side by side may vary greatly in
+different systems. In all cases, however, there can be but one gas or
+vapour phase on the account of the fact that all gases are miscible with
+one another in all proportions. In the case of liquid and solid phases the
+number is indefinite, since the above property does not apply to them. The
+number of phases which can be formed by any given substance or group of
+substances also differs greatly, and in general increases with the number
+of participating substances. Even in the case of a single substance,
+however, the number may be considerable; in the case of sulphur, for
+example, at least eight different solid phases are known (_v._ Chap. III.).
+
+It is of importance to bear in mind that equilibrium is _independent of the
+amounts_ of the phases present.[13] Thus it is a familiar fact that the
+pressure of a vapour in contact with a {10} liquid (_i.e._ the pressure of
+the saturated vapour) is unaffected by the amounts, whether relative or
+absolute, of the liquid and vapour; also the amount of a substance
+dissolved by a liquid is independent of the amount of solid in contact with
+the solution. It is true that deviations from this general law occur when
+the amount of liquid or the size of the solid particles is reduced beyond a
+certain point,[14] owing to the influence of surface energy; but we have
+already (p. 5) excluded such cases from consideration.
+
+Components.--Although the conception of phases is one which is readily
+understood, somewhat greater difficulty is experienced when we come to
+consider what is meant by the term _component_; for the components of a
+system are not synonymous with the chemical elements or compounds present,
+_i.e._ with the _constituents_ of the system, although both elements and
+compounds may be components. By the latter term there are meant only those
+constituents the concentration of which can undergo _independent_ variation
+in the different phases, and it is only with these that we are concerned
+here.[15]
+
+To understand the meaning of this term we shall consider briefly some cases
+with which the reader will be familiar, and at the outset it must be
+emphasized that the Phase Rule is concerned merely with those constituents
+which take part in the state of real equilibrium (p. 5); for it is only to
+the final state, not to the processes by which that state is reached, that
+the Phase Rule applies.
+
+Consider now the case of the system water--vapour or ice--water--vapour.
+The number of constituents taking part in the equilibrium here is only one,
+viz. the chemical substance, water. Hydrogen and oxygen, the constituents
+of water, are not to be regarded as components, because, in the first
+place, they are {11} not present in the system in a state of real
+equilibrium (p. 6); in the second place, they are combined in definite
+proportions to form water, and their amounts, therefore, cannot be varied
+independently. A variation in the amount of hydrogen necessitates a
+definite variation in the amount of oxygen.
+
+In the case, already referred to, in which hydrogen and oxygen are present
+along with water at the ordinary temperature, we are not dealing with a
+condition of true equilibrium. If, however, the temperature is raised to a
+certain point, a state of true equilibrium between hydrogen, oxygen, and
+water-vapour will be possible. In this case hydrogen and oxygen will be
+components, because now they do take part in the equilibrium; also, they
+need no longer be present in definite proportions, but excess of one or the
+other may be added. Of course, if the restriction be arbitrarily made that
+the free hydrogen and oxygen shall be present always and only in the
+proportions in which they are combined to form water, there will be, as
+before, only one component, water. From this, then, we see that a change in
+the conditions of the experiment (in the present case a rise of
+temperature) may necessitate a change in the number of the components.
+
+It is, however, only in the case of systems of more than one component that
+any difficulty will be found; for only in this case will a choice of
+components be possible. Take, for instance, the dissociation of calcium
+carbonate into calcium oxide and carbon dioxide. At each temperature, as we
+have seen, there is a definite state of equilibrium. When equilibrium has
+been established, there are three different substances present--calcium
+carbonate, calcium oxide, and carbon dioxide; and these are the
+constituents of the system between which equilibrium exists. Now, although
+these constituents take part in the equilibrium, they are not all to be
+regarded as components, for they are not mutually independent. On the
+contrary, the different phases are related to one another, and if two of
+these are taken, the composition of the third is defined by the equation
+
+  CaCO_{3} = CaO + CO_{2}
+
+{12} Now, in deciding the number of components in any given system, not
+only must the constituents chosen be capable of independent variation, but
+a further restriction is imposed, and we obtain the following rule: _As the
+components of a system there are to be chosen the_ smallest number _of
+independently variable constituents by means of which the composition of
+each phase participating in the state of equilibrium can be expressed in
+the form of a chemical equation._
+
+Applying this rule to the case under consideration, we see that of the
+three constituents present when the system is in a state of equilibrium,
+only two, as already stated, are independently variable. It will further be
+seen that in order to express the composition of each phase present, two of
+these constituents are necessary. The system is, therefore, one of _two
+components_, or a system of the second order.
+
+When, now, we proceed to the actual choice of components, it is evident
+that any two of the constituents can be selected. Thus, if we choose as
+components CaCO_{3} and CaO, the composition of each phase can be expressed
+by the following equations:--
+
+  CaCO_{3} = CaCO_{3} + 0CaO
+       CaO = CaO + 0CaCO_{3}
+    CO_{2} = CaCO_{3} - CaO
+
+As we see, then, both zero and negative quantities of the components have
+been introduced; and similar expressions would be obtained if CaCO_{3} and
+CO_{2} were chosen as components. The matter can, however, be simplified
+and the use of negative quantities avoided if CaO and CO_{2} are chosen;
+and it is, therefore, customary to select these as the components.
+
+While it is possible in the case of systems of the second order to choose
+the two components in such a way that the composition of each phase can be
+expressed by positive quantities of these, such a choice is not always
+possible when dealing with systems of a higher order (containing three or
+four components).
+
+From the example which has just been discussed, it might {13} appear as if
+the choice of the components was rather arbitrary. On examining the point,
+however, it will be seen that the arbitrariness affects only the _nature_,
+not the _number_, of the components; a choice could be made with respect to
+which, not to how many, constituents were to be regarded as components. As
+we shall see presently, however, it is only the number, not the nature of
+the components that is of importance.
+
+After the discussion of the conditions which the substances chosen as
+components must satisfy, another method may be given by which the number of
+components present in a system can be determined. Suppose a system
+consisting of several phases in equilibrium, and the composition of each
+phase determined by analysis. If each phase present, regarded as a whole,
+has the same composition, the system contains only one component, or is of
+the first order. If two phases must be mixed in suitable quantities in
+order that the composition of a third phase may be obtained, the system is
+one of two components or of the second order; and if three phases are
+necessary to give the composition of a fourth coexisting phase, the system
+is one of three components, or of the third order.[16]
+
+Although the examples to be considered in the sequel will afford sufficient
+illustration of the application of the rules given above, one case may
+perhaps be discussed to show the application of the method just given for
+determining the number of components.
+
+Consider the system consisting of Glauber's salt in equilibrium with
+solution and vapour. If these three phases are analyzed, the composition of
+the solid will be expressed by Na_{2}SO_{4}, 10H_{2}O; that of the solution
+by Na_{2}SO_{4} + _x_H_{2}O, while the vapour phase will be H_{2}O. The
+system evidently cannot be a one-component system, for the phases have not
+all the same composition. By varying the amounts of two phases, however
+(_e.g._ Na_{2}SO_{4}, 10H_{2}O and H_{2}O), the composition of the third
+phase--the solution--can be obtained. The system is, therefore, one of _two
+components_.
+
+But sodium sulphate can also exist in the anhydrous form and as the hydrate
+Na_{2}SO_{4}, 7H_{2}O. In these cases there may {14} be chosen as
+components Na_{2}SO_{4} and H_{2}O, and Na_{2}SO_{4}, 7H_{2}O and H_{2}O
+respectively. In both cases, therefore, there are two components. But the
+two systems (Na_{2}SO_{4}, 10H_{2}O--H_{2}O, and Na_{2}SO_{4},
+7H_{2}O--H_{2}O) can be regarded as special cases of the system
+Na_{2}SO_{4}--H_{2}O, and these two components will apply to all systems
+made up of sodium sulphate and water, no matter whether the solid phase is
+anhydrous salt or one of the hydrates. In all three cases, of course, the
+_number_ of components is the same; but by choosing Na_{2}SO_{4} and H_{2}O
+as components, the possible occurrence of negative quantities of components
+in expressing the composition of the phases is avoided; and, further, these
+components apply over a much larger range of experimental conditions.
+Again, therefore, we see that, although the number of the components of a
+system is definite, a certain amount of liberty is allowed in the choice of
+the substances; and we also see that the choice will be influenced by the
+conditions of experiment.
+
+Summing up, now, we may say--
+
+(1) The components are to be chosen from among the constituents which are
+present when the system is in a state of true equilibrium, and which take
+part in that equilibrium.
+
+(2) As components are to be chosen the _smallest number_ of such
+constituents necessary to express the composition of each phase
+participating in the equilibrium, zero and negative quantities of the
+components being permissible.
+
+(3) In any given system the _number_ of the components is definite, but may
+alter with alteration of the conditions of experiment. A certain freedom of
+choice, however, is allowed in the (qualitative, not quantitative)
+selection of the components, the choice being influenced by considerations
+of simplicity, suitability, or generality of application.[17]
+
+Degree of Freedom. Variability of a System.--It is well known that in
+dealing with a certain mass of gas or vapour, _e.g._ water vapour, if only
+one of the independently variable factors--temperature, pressure, and
+concentration (or volume)--is fixed, the state of the gas or vapour is
+undefined; while occupying the same volume (the concentration, therefore,
+remaining {15} unchanged), the temperature and the pressure may be altered;
+at a given temperature, a gas can exist under different pressures and
+occupy different volumes, and under any given pressure the temperature and
+volume may vary. If, however, two of the factors are arbitrarily fixed,
+then the third factor can only have a certain definite value; at any given
+values of temperature and pressure a given mass of gas can occupy only a
+definite volume.
+
+Suppose, however, that the system consists of water in contact with vapour.
+The condition of the system then becomes perfectly defined on arbitrarily
+giving one of the variables a certain value. If the temperature is fixed,
+the pressure under which water and water vapour can coexist is also
+determined; and conversely, if a definite pressure is chosen, the
+temperature is also defined. Water and vapour can coexist under a given
+pressure only at a definite temperature.
+
+Finally, let the water and vapour be cooled down until ice begins to
+separate out. So soon as the third phase, ice, appears, the state of the
+system as regards temperature and pressure of the vapour is perfectly
+defined, and none of the variables can be arbitrarily changed without
+causing the disappearance of one of the phases, ice, water, or vapour.
+
+We see, therefore, that in the case of some systems two, in other cases,
+only one of the independent variables (temperature, pressure,
+concentration) can be altered without destroying the nature of the system;
+while in other systems, again, these variables have all fixed and definite
+values. We shall therefore define the number of degrees of freedom[18] of a
+system as the _number of the variable factors, temperature, pressure, and
+concentration of the components, which must be arbitrarily fixed in order
+that the condition of the system may be perfectly defined_. From what has
+been said, therefore, we shall describe a gas or vapour as having two
+degrees of freedom; the system water--vapour as having only one; and the
+system ice--water--vapour as having no degrees of freedom. We may also
+speak of the {16} _variability_ or _variance_ of a system, and describe a
+system as being invariant, univariant, bivariant, multivariant,[19]
+according as the number of degrees of freedom is nought, one, two, or more
+than two.
+
+A knowledge of its variability is, therefore, of essential importance in
+studying the condition and behaviour of a system, and it is the great merit
+of the Phase Rule that _the state of a system is defined entirely by the
+relation existing between the number of the components and the phases
+present_, no account being taken of the molecular complexity of the
+participating substances, nor any assumption made with regard to the
+constitution of matter. It is, further, as we see, quite immaterial whether
+we are dealing with "physical" or "chemical" equilibrium; in principle,
+indeed, no distinction need be drawn between the two classes, although it
+is nevertheless often convenient to make use of the terms, in spite of a
+certain amount of indefiniteness which attaches to them--an indefiniteness,
+indeed, which attaches equally to the terms "physical" and "chemical"
+process.[20]
+
+The Phase Rule.--The Phase Rule of Gibbs, which defines the condition of
+equilibrium by the relation between the number of coexisting phases and the
+components, may be stated as follows: A system consisting of n components
+can exist in _n_ + 2 phases only when the temperature, pressure, and
+concentration have fixed and definite values; if there are _n_ components
+in _n_ + 1 phases, equilibrium can exist while one of the factors varies,
+and if there are only _n_ phases, two of the varying factors may be
+arbitrarily fixed. This rule, the application of which, it is hoped, will
+become clear in the sequel, may be very concisely and conveniently
+summarized in the form of the equation--
+
+  P + F = C + 2, or F = C + 2 - P
+
+where P denotes the number of the phases, F the degrees of freedom, and C
+the number of components. From the second form of the equation it can be
+readily seen that the greater the number of the phases, the fewer are the
+degrees of freedom. With increase in the number of the phases, therefore,
+the {17} condition of the system becomes more and more defined, or less and
+less variable.
+
+Classification of Systems according to the Phase Rule.--We have already
+learned in the introductory chapter that systems which are apparently quite
+different in character may behave in a very similar manner. Thus it was
+stated that the laws which govern the equilibrium between water and its
+vapour are quite analogous to those which are obeyed by the dissociation of
+calcium carbonate into carbon dioxide and calcium oxide; in each case a
+certain temperature is associated with a definite pressure, no matter what
+the relative or absolute amounts of the respective substances are. And
+other examples were given of systems which were apparently similar in
+character, but which nevertheless behaved in a different manner. The
+relations between the various systems, however, become perfectly clear and
+intelligible in the light of the Phase Rule. In the case first mentioned,
+that of water in equilibrium with its vapour, we have one
+component--water--present in two phases, _i.e._ in two physically distinct
+forms, viz. liquid and vapour. According to the Phase Rule, therefore,
+since C = 1, and P = 2, the degree of freedom F is equal to 1 + 2 - 2 = 1;
+the system possesses one degree of freedom, as has already been stated. But
+in the case of the second system mentioned above there are two components,
+viz. calcium oxide and carbon dioxide (p. 12), and three phases, viz. two
+solid phases, CaO and CaCO_{3}, and the gaseous phase, CO_{2}. The number
+of degrees of freedom of the system, therefore, is 2 + 2 - 3 = 1; this
+system, therefore, also possesses one degree of freedom. We can now
+understand why these two systems behave in a similar manner; both are
+univariant, or possess only one degree of freedom. We shall therefore
+expect a similar behaviour in the case of all univariant systems, no matter
+how dissimilar the systems may outwardly appear. Similarly, all bivariant
+systems will exhibit analogous behaviour; and generally, systems possessing
+the same degree of freedom will show a like behaviour. In accordance with
+the Phase Rule, therefore, we may classify the different systems which may
+be found into invariant, univariant, bivariant, multivariant, {18}
+according to the relation which obtains between the number of the
+components and the number of coexisting phases; and we shall expect that in
+each case the members of any particular group will exhibit a uniform
+behaviour. By this means we are enabled to obtain an insight into the
+general behaviour of any system, so soon as we have determined the number
+of the components and the number of the coexisting phases.
+
+The adoption of the Phase Rule for the purposes of classification has been
+of great importance in studying changes in the equilibrium existing between
+different substances; for not only does it render possible the grouping
+together of a large number of isolated phenomena, but the guidance it
+affords has led to the discovery of new substances, has given the clue to
+the conditions under which these substances can exist, and has led to the
+recognition of otherwise unobserved resemblances existing between different
+systems.
+
+Deduction of the Phase Rule.--In the preceding pages we have restricted
+ourselves to the statement of the Phase Rule, without giving any indication
+of how it has been deduced. At the close of this chapter, therefore, the
+mathematical deduction of the generalization will be given, but in brief
+outline only, the reader being referred to works on Thermodynamics for a
+fuller treatment of the subject.[21]
+
+All forms of energy can be resolved into two factors, the _capacity_ factor
+and the _intensity_ factor; but for the production of equilibrium, only the
+intensity factor is of importance. Thus, if two bodies having the same
+temperature are brought in contact with each other, they will be in
+equilibrium as regards heat energy, no matter what may be the amounts of
+heat (capacity factor) contained in either, because the intensity
+factor--the temperature--is the same. But if the temperature of the two
+bodies is different, _i.e._ if the intensity factor of heat energy is
+different, the two bodies will no longer be in equilibrium; but heat will
+pass from the hotter to the colder until both have the same temperature.
+
+As with heat energy, so with chemical energy. If we have a substance
+existing in two different states, or in two different {19} phases of a
+system, equilibrium can occur only when the intensity factor of chemical
+energy is the same. This intensity factor may be called the _chemical
+potential_; and we can therefore say that a system will be in equilibrium
+when the chemical potential of each component is the same in all the phases
+in which the component occurs. Thus, for example, ice, water, and vapour
+have, at the triple point, the same chemical potential.
+
+The potential of a component in any phase depends not only on the
+composition of the phase, but also on the temperature and the pressure (or
+volume). If, therefore, we have a system of C components existing in P
+phases, then, in order to fix the composition of unit mass of each phase,
+it is necessary to know the masses of (C - 1) components in each of the
+phases. As regards the composition, therefore, each phase possesses (C - 1)
+variables. Since there are P phases, it follows that, as regards
+composition, the whole system possesses P(C - 1) variables. Besides these
+there are, however, two other variables, viz. temperature and pressure, so
+that altogether a system of C components in P phases possesses P(C - 1) + 2
+variables.
+
+In order to define the state of the system completely, it will be necessary
+to have as many equations as there are variables. If, therefore, there are
+fewer equations than there are variables, then, according to the deficiency
+in the number of the equations, one or more of the variables will have an
+undefined value; and values must be assigned to these variables before the
+system is entirely defined. The number of these undefined values gives us
+the variability or the degree of freedom of the system.
+
+The equations by which the system is to be defined are obtained from the
+relationship between the potential of a component and the composition of
+the phase, the temperature and the pressure. Further, as has already been
+stated, equilibrium occurs when the potential of each component is the same
+in the different phases in which it is present. If, therefore, we choose as
+standard one of the phases in which all the components occur, then in any
+other phase in equilibrium with {20} it, the potential of each component
+must be the same as in the standard phase. For each phase in equilibrium
+with the standard phase, therefore, there will be a definite equation of
+state for each component in the phase; so that, if there are P phases, we
+obtain for each component (P - 1) equations; and for C components,
+therefore, we obtain C(P - 1) equations.
+
+But we have seen above that there are P(C - 1) + 2 variables, and as we
+have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2
+- P variables undefined. That is to say, the degree of freedom (F) of a
+system consisting of C components in P phases is--
+
+  F = C + 2 - P
+
+       *       *       *       *       *
+
+
+{21}
+
+CHAPTER III
+
+TYPICAL SYSTEMS OF ONE COMPONENT
+
+A. _Water._
+
+For the sake of rendering the Phase Rule more readily intelligible, and at
+the same time also for the purpose of obtaining examples by which we may
+illustrate the general behaviour of systems, we shall in this chapter
+examine in detail the behaviour of several well-known systems consisting of
+only one component.
+
+The most familiar examples of equilibria in a one-component system are
+those furnished by the three phases of water, viz. ice, water, water
+vapour. The system consists of one component, because all three phases have
+the same chemical composition, represented by the formula H_{2}O. As the
+criterion of equilibrium we shall choose a definite pressure, and shall
+study the variation of the pressure with the temperature; and for the
+purpose of representing the relationships which we obtain we shall employ a
+temperature-pressure diagram, in which the temperatures are measured as
+abscissæ and the pressures as ordinates. In such a diagram invariant
+systems will be represented by points; univariant systems by lines, and
+bivariant systems by areas.
+
+Equilibrium between Liquid and Vapour. Vaporization Curve.--Consider in the
+first place the conditions for the coexistence of liquid and vapour.
+According to the Phase Rule (p. 16), a system consisting of one component
+in two phases has one degree of freedom, or is univariant. We should
+therefore expect that it will be possible for liquid water to coexist with
+water vapour at different values of temperature and {22} pressure, but that
+if we arbitrarily fix one of the variable factors, pressure, temperature,
+or volume (in the case of a given mass of substance), the state of the
+system will then be defined. If we fix, say, the temperature, then the
+pressure will have a definite value; or if we adopt a certain pressure, the
+liquid and vapour can coexist only at a certain definite temperature. Each
+temperature, therefore, will correspond to a definite pressure; and if in
+our diagram we join by a continuous line all the points indicating the
+values of the pressure corresponding to the different temperatures, we
+shall obtain a curve (Fig. 1) representing the variation of the pressure
+with the temperature. This is the curve of vapour pressure, or the
+_vaporization curve_ of water.
+
+[Illustration: FIG. 1.]
+
+Now, the results of experiment are quite in agreement with the requirements
+of the Phase Rule, and at any given temperature the system water--vapour
+can exist in equilibrium only under a definite pressure.
+
+The vapour pressure of water at different temperatures has been subjected
+to careful measurement by Magnus,[22] Regnault,[23] Ramsay and Young,[24]
+Juhlin,[25] Thiesen and Scheel,[26] and others. In the following table the
+values of the vapour pressure from -10° to +100° are those calculated from
+the measurements of Regnault, corrected by the measurements of Wiebe and
+Thiesen and Scheel;[27] those from 120° to 270° were determined {23} by
+Ramsay and Young, while the values of the critical pressure and temperature
+are those determined by Battelli.[28]
+
+  VAPOUR PRESSURE OF WATER.
+
+  -------------+-----------------+--------------+--------------------
+               |                 |              |
+  Temperature. | Pressure in cm. | Temperature. |   Pressure in cm.
+               |    mercury.     |              |      mercury.
+  -------------+-----------------+--------------+--------------------
+               |                 |              |
+      -10°     |     0.213       |     120°     |   148.4
+        0°     |     0.458[29]   |     130°     |   201.9
+      +20°     |     1.752       |     150°     |   356.8
+       40°     |     5.516       |     200°     |  1162.5
+       60°     |    14.932       |     250°     |  2973.4
+       80°     |    35.54        |     270°     |  4110.1
+      100°     |    76.00        |     364.3°   | 14790.4 (194.6 atm.)
+               |                 | (critical    | (critical pressure).
+               |                 | temperature) |
+  -------------+-----------------+--------------+--------------------
+
+The pressure is, of course, independent of the relative or absolute volumes
+of the liquid and vapour; on increasing the volume at constant temperature,
+a certain amount of the liquid will pass into vapour, and the pressure will
+regain its former value. If, however, the pressure be permanently
+maintained at a value different from that corresponding to the temperature
+employed, then either all the liquid will pass into vapour, or all the
+vapour will pass into liquid, and we shall have either vapour alone or
+liquid alone.
+
+Upper Limit of Vaporization Curve.--On continuing to add heat to water
+contained in a closed vessel, the pressure of the vapour will gradually
+increase. Since with increase of pressure the density of the vapour must
+increase, and since with rise of temperature the density of the liquid must
+decrease, a point will be reached at which the density of liquid and vapour
+become identical; the system ceases to be heterogeneous, and passes into
+one homogeneous phase. The temperature at which this occurs is called the
+_critical temperature_. To this temperature there will, of course,
+correspond a certain definite pressure, called the _critical pressure_. The
+curve representing the {24} equilibrium between liquid and vapour must,
+therefore, end abruptly at the critical point. At temperatures above this
+point no pressure, however great, can cause the formation of the liquid
+phase; at temperatures above the critical point the vapour becomes a gas.
+In the case of water, the critical temperature is 364.3°, and the critical
+pressure 194.6 atm.; at the point representing these conditions the
+vapour-pressure curve of water must cease.
+
+Sublimation Curve of Ice.--Vapour is given off not only by liquid water,
+but also by solid water, or ice. That this is so is familiar to every one
+through the fact that ice or snow, even at temperatures below the melting
+point, gradually disappears in the form of vapour. Even at temperatures
+considerably lower than 0°, the vapour pressure of ice, although small, is
+quite appreciable; and it is possible, therefore, to have ice and vapour
+coexisting in equilibrium. When we inquire into the conditions under which
+such a system can exist, we see again that we are dealing with a univariant
+system--one component existing in two phases--and that, therefore, just as
+in the case of the system water and vapour, there will be for each
+temperature a certain definite pressure of the vapour, and this pressure
+will be independent of the relative or absolute amounts of the solid or
+vapour present, and will depend solely on the temperature. Further, just as
+in the case of the vapour pressure of water, the condition of equilibrium
+between ice and water vapour will be represented by a line or curve showing
+the change of pressure with the temperature. Such a curve, representing the
+conditions of equilibrium between a solid and its vapour, is called a
+_sublimation curve_. At temperatures represented by any point on this
+curve, the solid (ice) will sublime or pass into vapour without previously
+fusing. Since ice melts at 0° (_vide infra_), the sublimation curve must
+end at that temperature.
+
+The following are the values of the vapour pressure of ice between 0° and
+-50°.[30]
+
+{25}
+
+  VAPOUR PRESSURE OF ICE.
+
+  ---------------------------------------------------------------
+  Temperature. | Pressure in mm. | Temperature. | Pressure in mm.
+               |     mercury.    |              |     mercury.
+  -------------+-----------------+--------------+----------------
+     -50°      |     0.050       |    -8°       |     2.379
+     -40°      |     0.121       |    -6°       |     2.821
+     -30°      |     0.312       |    -4°       |     3.334
+     -20°      |     0.806       |    -2°       |     3.925
+     -15°      |     1.279       |     0°       |     4.602
+     -10°      |     1.999       |              |
+  ----------------------------------------------------------------
+
+Equilibrium between Ice and Water. Curve of Fusion.--There is still another
+univariant system of the one component water, the existence of which, at
+definite values of temperature and pressure, the Phase Rule allows us to
+predict. This is the system solid--liquid. Ice on being heated to a certain
+temperature melts and passes into the liquid state; and since this system
+solid--liquid is univariant, there will be for each temperature a certain
+definite pressure at which ice and water can coexist or be in equilibrium,
+independently of the amounts of the two phases present. Since now the
+temperature at which the solid phase is in equilibrium with the liquid
+phase is known as the melting point or point of fusion of the solid, the
+curve representing the temperatures and pressures at which the solid and
+liquid are in equilibrium will represent the change of the melting point
+with the pressure. Such a curve is called the _curve of fusion_, or the
+melting-point curve.
+
+It was not until the middle of the nineteenth century that this connection
+between the pressure and the melting point, or the change of the melting
+point with the pressure, was observed. The first to recognize the existence
+of such a relationship was James Thomson,[31] who in 1849 showed that from
+theoretical considerations such a relationship must exist, and predicted
+that in the case of ice the melting point would be lowered by pressure.
+This prediction was fully confirmed by his brother, W. Thomson[32] (Lord
+Kelvin), who found that under a pressure {26} of 8.1 atm. the melting point
+of ice was -0.059°; under a pressure of 16.8 atm. the melting point was
+-0.129°.
+
+The experiments which were first made in this connection were more of a
+qualitative nature, but in recent years careful measurements of the
+influence of pressure on the melting point of ice have been made more
+especially by Tammann,[33] and the results obtained by him are given in the
+following table and represented graphically in Fig. 2.
+
+  FUSION PRESSURE OF ICE.
+
+  ---------------------------------------------------------------------
+                  | Pressure in kilogms. per | Change of melting point for
+  Temperature.    |  sq. cm.[34]             | an increase of pressure of
+                  |                          | 1 kilogm. per sq. cm.
+  ---------------------------------------------------------------------
+      -0°         |        1                 |
+      -2.5°       |      336                 |       0.0074°
+      -5°         |      615                 |       0.0090°
+      -7.5°       |      890                 |       0.0091°
+      -10.0°      |     1155                 |       0.0094°
+      -12.5°      |     1410                 |       0.0100°
+      -15.0°      |     1625                 |       0.0116°
+      -17.5°      |     1835                 |       0.0119°
+      -20.0°      |     2042                 |       0.0121°
+      -22.1°      |     2200                 |       0.0133°
+  ---------------------------------------------------------------------
+
+From the numbers in the table and from the figure we see that as the
+pressure is increased the melting point of ice is lowered; but we also
+observe that a very large change of pressure is required in order to
+produce a very small change in the melting point. The curve, therefore, is
+very steep. Increase of pressure by one atmosphere lowers the melting point
+by only 0.0076°,[35] or an increase of pressure of 135 atm. is required to
+produce a lowering of the melting point of 1°. We see further that the
+fusion curve bends slightly as the pressure is increased, which signifies
+that the variation of {27} the melting point with the pressure changes; at
+-15°, when the pressure is 1625 kilogm. per sq. cm., increase of pressure
+by 1 kilogm. per sq. cm. lowers the melting point by 0.012°. This curvature
+of the fusion curve we shall later (Chap. IV.) see to be an almost
+universal phenomenon.
+
+[Illustration: FIG. 2.]
+
+[Illustration: FIG. 3.]
+
+Equilibrium between Ice, Water, and Vapour. The Triple Point.--On examining
+the vapour-pressure curves of ice and water (Fig. 3), we see that at a
+temperature of about 0° and under a pressure of about 4.6 mm. mercury, the
+two curves cut. At this point liquid water and solid ice are each in
+equilibrium with vapour at the same pressure. Since this is so, they must,
+of course, be in equilibrium {28} with one another, as experiment also
+shows. At this point, therefore, ice, water, and vapour can be in
+equilibrium, and as there are three phases present, the point is called a
+_triple point_.[36]
+
+The triple point, however, does not lie exactly at 0° C., for this
+temperature is defined as the melting point of ice under atmospheric
+pressure. At the triple point, however, the pressure is equal to the vapour
+pressure of ice and water, and this pressure, as we see from the tables on
+pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less than in the
+previous case. Now, we have just seen that a change of pressure of 1 atm.
+corresponds to a change of the melting point of 0.0076°; the melting point
+of ice, therefore, when under the pressure of its own vapour, will be very
+nearly +0.0076°, and the pressure of the vapour will be very slightly
+greater than 4.579 mm., which is the pressure at 0° (p. 21). The difference
+is, however, slight, and may be neglected here. At the temperature, then,
+of +0.0076°, and under a pressure of 4.6 mm. of mercury, ice, water, and
+vapour will be in equilibrium; the point in our diagram representing this
+particular temperature and pressure is, therefore, the triple point of the
+system ice--water--vapour.
+
+Since at the triple point we have three phases of one component, the system
+at this point is invariant--it possesses no degrees of freedom. If the
+temperature is changed, the system will undergo alteration in such a way
+that one of the phases will disappear, and a univariant system will result;
+if heat be added, ice will melt, and we shall have left water and vapour;
+if heat be abstracted, water will freeze, and we shall have left ice and
+vapour; if, when the temperature is altered, the pressure is kept constant,
+then we shall ultimately obtain only one phase (see Chap. IV.).
+
+The triple point is not only the point of intersection of the vaporization
+and sublimation curves, but it is also the end-point of the fusion curve.
+The fusion curve, as we have seen, is the curve of equilibrium between ice
+and water; and since at the triple point ice and water are each in
+equilibrium with {29} vapour of the same pressure, they must, of course,
+also be in equilibrium with one another.
+
+[Illustration: FIG. 4.]
+
+Bivariant Systems of Water.--If we examine Fig. 4, we see that the curves
+OA, OB, OC, which represent diagrammatically the conditions under which
+water and vapour, ice and vapour, and water and ice are in equilibrium,
+form the boundaries of three "fields," or areas, I., II., III. These areas,
+now, represent the conditions for the existence of the single phases,
+solid, liquid, and vapour respectively. At temperatures and pressures
+represented by any point in the field I., solid only can exist as a stable
+phase. Since we have here one component in only one phase, the system is
+bivariant, and at any given temperature, therefore, ice can exist under a
+series of pressures; and under any given pressure, at a series of
+temperatures, these pressures and temperatures being limited only by the
+curves OB, OC. Similarly also with the areas II. and III.
+
+We see, further, that the different areas are the regions of stability of
+the phase common to the two curves by which the area is enclosed.[37] Thus,
+the phase common to the two systems {30} represented by BO (ice and
+vapour), and OA (water and vapour) is the vapour phase; and the area BOA is
+therefore the area of the vapour phase. Similarly, BOC is the area of the
+ice phase, and COA the area of the water phase.
+
+Supercooled Water. Metastable State.--When heated under the ordinary
+atmospheric pressure, ice melts when the temperature reaches 0°, and it has
+so far not been found possible to raise the temperature of ice above this
+point without liquefaction taking place. On the other hand, it has long
+been known that water can be cooled below zero without solidification
+occurring. This was first discovered in 1724 by Fahrenheit,[38] who found
+that water could be exposed to a temperature of -9.4° without solidifying;
+so soon, however, as a small particle of ice was brought in contact with
+the water, crystallization commenced. Superfused or supercooled
+water--_i.e._ water cooled below 0°--is unstable only in respect of the
+solid phase; so long as the presence of the solid phase is carefully
+avoided, the water can be kept for any length of time without solidifying,
+and the system supercooled water and vapour behaves in every way like a
+stable system. A system, now, which in itself is stable, and which becomes
+instable only in contact with a particular phase, is said to be
+_metastable_, and the region throughout which this condition exists is
+called the metastable region. Supercooled water, therefore, is in a
+metastable condition. If the supercooling be carried below a certain
+temperature, solidification takes place spontaneously without the addition
+of the solid phase; the system then ceases to be metastable, and becomes
+_instable_.
+
+Not only has water been cooled to temperatures considerably below the
+melting point of ice, but the vapour pressure of the supercooled water has
+been measured. It is of interest and importance, now, to see what
+relationship exists between the vapour pressure of ice and that of
+supercooled water at the same temperature. This relationship is clearly
+shown by the numbers in the following table,[39] and is represented in Fig.
+3, {31} p. 27., and diagrammatically in Fig. 4, the vapour pressures of
+supercooled water being represented by the curve OA', which is the unbroken
+continuation of AO.
+
+  VAPOUR PRESSURE OF ICE AND OF SUPERCOOLED WATER.
+
+  ---------------------------------------------------------------------
+                 |       Pressure in mm. mercury.
+                 ------------------------------------------------------
+  Temperature.   |               |               |
+                 |     Water.    |     Ice.      |    Difference.
+  ---------------------------------------------------------------------
+         0°      |     4.618     |     4.602     |    0.016[40]
+        -2°      |     3.995     |     3.925     |    0.070
+        -4°      |     3.450     |     3.334     |    0.116
+        -8°      |     2.558     |     2.379     |    0.179
+       -10°      |     2.197     |     1.999     |    0.198
+       -15°      |     1.492     |     1.279     |    0.213
+       -20°      |     1.005     |     0.806     |    0.199
+  ---------------------------------------------------------------------
+
+At all temperatures below 0° (more correctly +0.0076°), at which
+temperature water and ice have the same vapour pressure, the vapour
+pressure of supercooled water is _greater_ than that of ice at the same
+temperature.
+
+From the relative positions of the curves OB and OA (Fig. 4) we see that at
+all temperatures above 0°, the (metastable) sublimation curve of ice, if it
+could be obtained, would be higher than the vaporization curve of water.
+This shows, therefore, that at 0° a "break" must occur in the curve of
+states, and that in the neighbourhood of this break the curve above that
+point must ascend less rapidly than the curve below the break. Since,
+however, the differences in the vapour pressures of supercooled water and
+of ice are very small, the change in the direction of the vapour-pressure
+curve on passing from ice to water was at first not observed, and Regnault
+regarded the sublimation curve as passing continuously into {32} the
+vaporization curve. The existence of a break was, however, shown by James
+Thomson[41] and by Kirchhoff[42] to be demanded by thermo-dynamical
+considerations, and the prediction of theory was afterwards realized
+experimentally by Ramsay and Young in their determinations of the vapour
+pressure of water and ice, as well as in the case of other substances.[43]
+
+From what has just been said, we can readily understand why ice and water
+cannot exist in equilibrium below 0°. For, suppose we have ice and water in
+the same closed space, but not in contact with one another, then since the
+vapour pressure of the supercooled water is higher than that of ice, the
+vapour of the former must be supersaturated in contact with the latter;
+vapour must, therefore, condense on the ice; and in this way there will be
+a slow distillation from the water to the ice, until at last all the water
+will have disappeared, and only ice and vapour remain.[44]
+
+Other Systems of the Substance Water.--We have thus far discussed only
+those systems which are constituted by the three phases--ice, water, and
+water vapour. It has, however, been recently found that at a low
+temperature and under a high pressure ordinary ice can pass into two other
+crystalline varieties, called by Tammann[45] ice II. and ice III., ordinary
+ice being ice I. According to the Phase Rule, now, since each of these
+solid forms constitutes a separate phase (p. 9), it will be possible to
+have the following (and more) systems of water, in addition to those
+already studied, viz. water, ice I., ice II.; water, ice I., ice III.;
+water, ice II., ice III., forming invariant systems and existing in
+equilibrium only at a definite triple point; further, water, ice II.;
+water, ice III.; ice I., ice II.; ice I., ice III.; ice II., ice III.,
+forming univariant systems, existing, therefore, at definite corresponding
+values of {33} temperature and pressure; and lastly, the bivariant systems,
+ice II. and ice III. Several of these systems have been investigated by
+Tammann. The triple point for water, ice I., ice III., lies at -22°, and a
+pressure of 2200 kilogms. per sq. cm. (2130 atm.), as indicated in Fig. 2,
+p. 27.[46] In contrast with the behaviour of ordinary ice, the temperature
+of equilibrium in the case of water--ice II., and water--ice III., is
+_raised_ by increase of pressure.
+
+B. _Sulphur._
+
+Polymorphism.--Reference has just been made to the fact that ice can exist
+not only in the ordinary form, but in at least two other crystalline
+varieties. This phenomenon, the existence of a substance in two or more
+different crystalline forms, is called _polymorphism_. Polymorphism was
+first observed by Mitscherlich[47] in the case of sodium phosphate, and
+later in the case of sulphur. To these two cases others were soon added, at
+first of inorganic, and later of organic substances, so that polymorphism
+is now recognized as of very frequent occurrence indeed.[48] These various
+forms of a substance differ not only in crystalline shape, but also in
+melting point, specific gravity, and other physical properties. In the
+liquid state, however, the differences do not exist.
+
+According to our definition of phases (p. 9), each of these polymorphic
+forms constitutes a separate phase of the particular substance. As is
+readily apparent, the number of possible systems formed of one component
+may be considerably increased when that component is capable of existing in
+different crystalline forms. We have, therefore, to inquire what are the
+conditions under which different polymorphic forms can coexist, either
+alone or in presence of the liquid and vapour phase. For the purpose of
+illustrating the general behaviour of such systems, we shall study the
+systems formed by the different crystalline forms of sulphur, tin, and
+benzophenone.
+
+{34}
+
+Sulphur exists in two well-known crystalline forms--rhombic, or octahedral,
+and monoclinic, or prismatic sulphur. Of these, the former melts at 114.5°;
+the latter at 120°.[49] Further, at the ordinary temperature, rhombic
+sulphur can exist unchanged, whereas, on being heated to temperatures
+somewhat below the melting point, it passes into the prismatic variety. On
+the other hand, at temperatures above 96°, prismatic sulphur can remain
+unchanged, whereas at the ordinary temperature it passes slowly into the
+rhombic form.
+
+If, now, we examine the case of sulphur with the help of the Phase Rule, we
+see that the following systems are theoretically possible:--
+
+  I. _Bivariant Systems: One component in one phase._
+    (_a_) Rhombic sulphur.
+    (_b_) Monoclinic sulphur.
+    (_c_) Sulphur vapour.
+    (_d_) Liquid sulphur.
+
+  II. _Univariant Systems: One component in two phases._
+    (_a_) Rhombic sulphur and vapour.
+    (_b_) Monoclinic sulphur and vapour.
+    (_c_) Rhombic sulphur and liquid.
+    (_d_) Monoclinic sulphur and liquid.
+    (_e_) Rhombic and monoclinic sulphur.
+    (_f_) Liquid and vapour.
+
+  III. _Invariant Systems: One component in three phases._
+    (_a_) Rhombic and monoclinic sulphur and vapour.
+    (_b_) Rhombic sulphur, liquid and vapour.
+    (_c_) Monoclinic sulphur, liquid and vapour.
+    (_d_) Rhombic and monoclinic sulphur and liquid.
+
+[Illustration: FIG. 5.]
+
+Triple Point--Rhombic and Monoclinic Sulphur and Vapour. Transition
+Point.--In the case of ice, water and vapour, we saw that at the triple
+point the vapour pressures of ice and water are equal; below this point,
+ice is stable; above this point, water is stable. We saw, further, that
+below 0° the vapour pressure of the stable system is lower than that of the
+metastable, and therefore that at the triple point there is a break in the
+vapour pressure curve of such a kind that above {35} the triple point the
+vapour-pressure curve ascends more slowly than below it. Now, although the
+vapour pressure of solid sulphur has not been determined, we can
+nevertheless consider that it does possess a certain, even if very small,
+vapour pressure,[50] and that at the temperature at which the vapour
+pressures of rhombic and monoclinic sulphur become equal, we can have these
+two solid forms existing in equilibrium with the vapour. Below that point
+only one form, that with the lower vapour pressure, will be stable; above
+that point only the other form will be stable. On passing through the
+triple point, therefore, there will be a change of the one form into the
+other. This point is represented in our diagram (Fig. 5) by the point O,
+the two curves AO and OB representing diagrammatically the vapour pressures
+of rhombic and monoclinic sulphur respectively. If the vapour phase is
+absent and the system maintained under a constant pressure, _e.g._ {36}
+atmospheric pressure, there will also be a definite temperature at which
+the two solid forms are in equilibrium, and on passing through which
+complete and reversible transformation of one form into the other occurs.
+This temperature, which refers to equilibrium in absence of the vapour
+phase, is known as the _transition temperature_ or _inversion temperature_.
+
+Were we dependent on measurements of pressure and temperature, the
+determination of the transition point might be a matter of great
+difficulty. When we consider, however, that the other physical properties
+of the solid phases, _e.g._ the density, undergo an abrupt change on
+passing through the transition point, owing to the transformation of one
+form into the other, then any method by which this abrupt change in the
+physical properties can be detected may be employed for determining the
+transition point. A considerable number of such methods have been devised,
+and a description of the most important of these is given in the Appendix.
+
+In the case of sulphur, the transition point of rhombic into monoclinic
+sulphur was found by Reicher[51] to lie at 95.5°. Below this temperature
+the octahedral, above it the monoclinic, is the stable form.
+
+Condensed Systems.--We have already seen that in the change of the melting
+point of water with the pressure, a very great increase of the latter was
+necessary in order to produce a comparatively small change in the
+temperature of equilibrium. This is a characteristic of all systems from
+which the vapour phase is absent, and which are composed only of solid and
+liquid phases. Such systems are called _condensed systems_,[52] and in
+determining the temperature of equilibrium of such systems, practically the
+same point will be obtained whether the measurements are carried out under
+atmospheric pressure or under the pressure of the vapour of the solid or
+liquid phases. The transition point, therefore, as determined in open
+vessels at atmospheric pressure, will differ only by a very slight amount
+from the triple point, or point at which the two solid or liquid phases are
+in equilibrium under the pressure of their vapour. {37} The determination
+of the transition point is thereby greatly simplified.
+
+Suspended Transformation.--In many respects the transition point of two
+solid phases is analogous to the melting point of a solid, or point at
+which the solid passes into a liquid. In both cases the change of phase is
+associated with a definite temperature and pressure in such a way that
+below the point the one phase, above the point the other phase, is stable.
+The transition point, however, differs in so far from a point of fusion,
+that while it is possible to supercool a liquid, no definite case is known
+where the solid has been heated above the triple point without passing into
+the liquid state. Transformation, therefore, is suspended only on one side
+of the melting point. In the case of two solid phases, however, the
+transition point can be overstepped in both directions, so that each phase
+can be obtained in the metastable condition. In the case of supercooled
+water, further, we saw that the introduction of the stable, solid phase
+caused the speedy transformation of the metastable to the stable condition
+of equilibrium; but in the case of two solid phases the change from the
+metastable to the stable modification may occur with great slowness, even
+in presence of the stable form. This tardiness with which the stable
+condition of equilibrium is reached greatly increases in many cases the
+difficulty of accurately determining the transition point. The phenomena of
+suspended transformation will, however, receive a fuller discussion later
+(p. 68).
+
+Transition Curve--Rhombic and Monoclinic Sulphur.--Just as we found the
+melting point of ice to vary with the pressure, so also do we find that
+change of pressure causes an alteration in the transition point. In the
+case of the transition point of rhombic into monoclinic sulphur, increase
+of pressure by 1 atm. raises the transition point by 0.04°-0.05°.[53] The
+transition curve, or curve representing the change of the transition point
+with pressure, will therefore slope to the right away from the pressure
+axis. This is curve OC (Fig. 5).
+
+{38}
+
+Triple Point--Monoclinic Sulphur, Liquid, and Vapour. Melting Point of
+Monoclinic Sulphur.--Above 95.5°, monoclinic sulphur is, as we have seen,
+the stable form. On being heated to 120°, under atmospheric pressure, it
+melts. This temperature is, therefore, the point of equilibrium between
+monoclinic sulphur and liquid sulphur under atmospheric pressure. Since we
+are dealing with a condensed system, this temperature may be regarded as
+very nearly that at which the solid and liquid are in equilibrium with
+their vapour, _i.e._ the triple point, solid (monoclinic)--liquid--vapour.
+This point is represented in the diagram by B.
+
+Triple Point--Rhombic and Monoclinic Sulphur and Liquid.--In contrast with
+that of ice, the fusion point of monoclinic sulphur is _raised_ by increase
+of pressure, and the fusion curve, therefore, slopes to the right. The
+transition curve of rhombic and monoclinic sulphur, as we have seen, also
+slopes to the right, and more so than the fusion curve of monoclinic
+sulphur. There will, therefore, be a certain pressure and temperature at
+which the two curves will cut. This point lies at 151°, and a pressure of
+1320 kilogm. per sq. cm., or about 1288 atm.[54] It, therefore, forms
+another triple point, the existence of which had been predicted by
+Roozeboom,[55] at which rhombic and monoclinic sulphur are in equilibrium
+with liquid sulphur. It is represented in our diagram by the point C.
+_Beyond this point monoclinic sulphur ceases to exist in a stable
+condition._ At temperatures and pressures above this triple point, rhombic
+sulphur will be the stable modification, and this fact is of mineralogical
+interest, because it explains the occurrence in nature of well-formed
+rhombic crystals. Under ordinary conditions, prismatic sulphur separates
+out on cooling fused sulphur, but at temperatures above 151° and under
+pressures greater than 1288 atm., the rhombic form would be produced.[56]
+
+Triple Point--Rhombic Sulphur, Liquid, and Vapour. Metastable Triple
+Point.--On account of the slowness with {39} which transformation of one
+form into the other takes place on passing the transition point, it has
+been found possible to heat rhombic sulphur up to its melting point
+(114.5°). At this temperature, not only is rhombic sulphur in a metastable
+condition, but the liquid is also metastable, its vapour pressure being
+greater than that of solid monoclinic sulphur. This point is represented in
+our diagram by the point b.
+
+From the relative positions of the metastable melting point of rhombic
+sulphur and the stable melting point of monoclinic sulphur at 120°, we see
+that, of the two forms, the metastable form has the lower melting point.
+This, of course, is valid only for the relative stability in the
+neighbourhood of the melting point; for we have already learned that at
+lower temperatures rhombic sulphur is the stable, monoclinic sulphur the
+metastable (or unstable) form.
+
+Fusion Curve of Rhombic Sulphur.--Like any other melting point, that of
+rhombic sulphur will be displaced by increase of pressure; increase of
+pressure raises the melting point, and we can therefore obtain a metastable
+fusion curve representing the conditions under which rhombic sulphur is in
+equilibrium with liquid sulphur. This metastable fusion curve must pass
+through the triple point for rhombic sulphur--monoclinic sulphur--liquid
+sulphur, and on passing this point it becomes a stable fusion curve. The
+continuation of this curve, therefore, above 151° forms the stable fusion
+curve of rhombic sulphur (curve CD).
+
+These curves have been investigated at high pressures by Tammann, and the
+results are represented according to scale in Fig. 6,[57] _a_ being the
+curve for monoclinic sulphur and liquid; _b_, that for rhombic sulphur and
+liquid; and _c_, that for rhombic and monoclinic sulphur.
+
+Bivariant Systems.--Just as in the case of the diagram of states of water,
+the areas in Fig. 5 represent the conditions for the stable existence of
+the single phases: rhombic sulphur in the area to the left of AOCD;
+monoclinic sulphur in the area OBC; liquid sulphur in the area EBCD;
+sulphur vapour below the curves AOBE. As can be seen from the diagram, {40}
+the existence of monoclinic sulphur is limited on all sides, its area being
+bounded by the curves OB, OC, BC. At any point outside this area,
+monoclinic sulphur can exist only in a metastable condition.
+
+[Illustration: FIG. 6.]
+
+Other crystalline forms of sulphur have been obtained,[58] so that the
+existence of other systems of the one-component sulphur besides those
+already described is possible. Reference will be made to these later
+(p. 51).
+
+{41}
+
+C. _Tin._
+
+Another substance capable of existing in more than one crystalline form, is
+the metal tin, and although the general behaviour, so far as studied, is
+analogous to that of sulphur, a short account of the two varieties of tin
+may be given here, not only on account of their metallurgical interest, but
+also on account of the importance which the phenomena possess for the
+employment of this metal in everyday life.
+
+After a winter of extreme severity in Russia (1867-1868), the somewhat
+unpleasant discovery was made that a number of blocks of tin, which had
+been stored in the Customs House at St. Petersburg, had undergone
+disintegration and crumbled to a grey powder.[59] That tin undergoes change
+on exposure to extreme cold was known, however, before that time, even as
+far back as the time of Aristotle, who spoke of the tin as "melting."[60]
+Ludicrous as that term may now appear, Aristotle nevertheless unconsciously
+employed a strikingly accurate analogy, for the conditions under which
+ordinary white tin passes into the grey modification are, in many ways,
+quite analogous to those under which a substance passes from the solid to
+the liquid state. The knowledge of this was, however, beyond the wisdom of
+the Greek philosopher.
+
+For many years there existed considerable confusion both as to the
+conditions under which the transformation of white tin into its allotropic
+modification occurs, and to the reason of the change. Under the guidance of
+the Phase Rule, however, the confusion which obtained has been cleared
+away, and the "mysterious" behaviour of tin brought into accord with other
+phenomena of transformation.[61]
+
+Transition Point.--Just as in the case of sulphur, so also in the case of
+tin, there is a transition point above which the {42} one form, ordinary
+white tin, and below which the other form, grey tin, is the stable variety.
+In the case of this metal, the transition point was found by Cohen and van
+Eyk, who employed both the dilatometric and the electrical methods
+(Appendix) to be 20°. Below this temperature, grey tin is the stable form.
+But, as we have seen in the case of sulphur, the change of the metastable
+into the stable solid phase occurs with considerable slowness, and this
+behaviour is found also in the case of tin. Were it not so, we should not
+be able to use this metal for the many purposes to which it is applied in
+everyday life; for, with the exception of a comparatively small number of
+days in the year, the temperature of our climate is below 20°, and _white
+tin is, therefore, at the ordinary temperature, in a metastable condition_.
+The change, however, into the stable form at the ordinary temperature,
+although slow, nevertheless takes place, as is shown by the partial or
+entire conversion of articles of tin which have lain buried for several
+hundreds of years.
+
+On lowering the temperature, the velocity with which the transformation of
+the tin occurs is increased, and Cohen and van Eyk found that the
+temperature of maximum velocity is about -50°. Contact with the stable form
+will, of course, facilitate the transformation.
+
+The change of white tin into grey takes place also with increased velocity
+in presence of a solution of tin ammonium chloride (pink salt), which is
+able to dissolve small quantities of tin. In presence of such a solution
+also, it was found that the temperature at which the velocity of
+transformation was greatest was raised to 0°. At this temperature, white
+tin in contact with a solution of tin ammonium chloride, and the grey
+modification, undergoes transformation to an appreciable extent in the
+course of a few days.
+
+Fig. 7 is a photograph of a piece of white tin undergoing transformation
+into the grey variety.[62] The bright surface of the tin becomes covered
+with a number of warty masses, formed of the less dense grey form, and the
+number and size of these continue to grow until the whole of the white tin
+has passed {43} into a grey powder. On account of the appearance which is
+here seen, this transformation of tin has been called by Cohen the "tin
+plague."
+
+[Illustration: FIG. 7.]
+
+{44}
+
+Enantiotropy and Monotropy.--In the case of sulphur and tin, we have met
+with two substances existing in polymorphic forms, and we have also learned
+that these forms exhibit a definite transition point at which their
+relative stability is reversed. Each form, therefore, possesses a definite
+range of stable existence, and is capable of undergoing transformation into
+the other, at temperatures above or below that of the transition point.
+
+Another class of dimorphous substances is, however, met with as, for
+instance, in the case of the well-known compounds iodine monochloride and
+benzophenone. Each crystalline form has its own melting point, the
+dimorphous forms of iodine monochloride melting at 13.9° and 27.2°,[63] and
+those of benzophenone at 26° and 48°.[64] This class of substance differs
+from that which we have already studied (_e.g._ sulphur and tin), in that
+at all temperatures up to the melting point, only one of the forms is
+stable, the other being metastable. There is, therefore, no transition
+point, and transformation of the crystalline forms can be observed _only in
+one direction_. These two classes of phenomena are distinguished by the
+names _enantiotropy_ and _monotropy_; enantiotropic substances being such
+that the change of one form into the other is a reversible process (_e.g._
+rhombic sulphur into monoclinic, and monoclinic sulphur into rhombic), and
+monotropic substances, those in which the transformation of the crystalline
+forms is irreversible.
+
+[Illustration: FIG. 8.]
+
+[Illustration: FIG. 9.]
+
+These differences in the behaviour can be explained very well in many cases
+by supposing that in the case of enantiotropic substances the transition
+point lies below the melting point, while in the case of monotropic
+substances, it lies above the melting point.[65] These conditions would be
+represented by the Figs. 8 and 9.
+
+In these two figures, O_{3} is the transition point, O_{1} and O_{2} the
+melting points of the metastable and stable forms {45} respectively. From
+Fig. 9 we see that the crystalline form I. at all temperatures up to its
+melting point is metastable with respect to the form II. In such cases the
+transition point could be reached only at higher pressures.
+
+Although, as already stated, this explanation suffices for many cases, it
+does not prove that in all cases of monotropy the transition point is above
+the melting point of the two forms. It is also quite possible that the
+transition point may lie below the melting points;[66] in this case we have
+what is known as _pseudomonotropy_. It is possible that graphite and
+diamond,[67] perhaps also the two forms of phosphorus, stand in the
+relation of pseudomonotropy (_v._ p. 49).
+
+The disposition of the curves in Figs. 8 and 9 also explains the phenomenon
+sometimes met with, especially in organic chemistry, that the substance
+first melts, then solidifies, and remelts at a higher temperature. On again
+determining the melting point after re-solidification, only the higher
+melting point is obtained.
+
+The explanation of such a behaviour is, that if the determination of the
+melting point is carried out rapidly, the point O_{1}, the melting point of
+the metastable solid form, may be realized. At this temperature, however,
+the liquid is metastable with respect to the stable solid form, and if the
+temperature is {46} not allowed to rise above the melting point of the
+latter, the liquid may solidify. The stable solid modification thus
+obtained will melt only at a higher temperature.
+
+D. _Phosphorus._
+
+An interesting case of a monotropic dimorphous substance is found in
+phosphorus, which occurs in two crystalline forms; white phosphorus
+belonging to the regular system, and red phosphorus belonging to the
+hexagonal system. From determinations of the vapour pressures of liquid
+white phosphorus, and of solid red phosphorus,[68] it was found that the
+vapour pressure of red phosphorus was considerably lower than that of
+liquid white phosphorus at the same temperature, the values obtained being
+given in the following table.
+
+  VAPOUR PRESSURES OF WHITE AND RED PHOSPHORUS.
+
+  -------------------------------------------------------------------------
+    Vapour pressure of liquid white phosphorus.    | Vapour pressure of red
+                                                   |     phosphorus.
+  -------------------------------------------------+-----------------------
+  Temperature.| Pressure | Temperature.|  Pressure | Temperature.| Pressure
+              |  in cm.  |             |   in atm. |             |  in atm.
+  ------------+----------+-------------+-----------+-------------+---------
+    165°      |  12      |    360°     |    3.2    |    360°     |   0.1
+    180°      |  20.4    |    440°     |    7.5    |    440°     |   1.75
+    200°      |  26.6    |    494°     |   18.0    |    487°     |   6.8
+    219°      |  35.9    |    503°     |   21.9    |    510°     |  10.8
+    230°      |  51.4    |    511°     |   26.2    |    531°     |  16.0
+    290°      |  76.0    |     --      |     --    |    550°     |  31.0
+    --        |   --     |     --      |     --    |    577°     |  56.0
+  -------------------------------------------------------------------------
+
+These values are also represented graphically in Fig. 10.
+
+[Illustration: FIG. 10.]
+
+At all temperatures above about 260°, transformation of the white into the
+red modification takes place with appreciable velocity, and this velocity
+increases as the temperature is raised. Even at lower temperatures, _e.g._
+at the ordinary temperature, the velocity of transformation is increased
+under the influence {47} of light,[69] or by the presence of certain
+substances, _e.g._ iodine,[70] just as the velocity of transformation of
+white tin into the grey modification was increased by the presence of a
+solution of tin ammonium chloride (p. 40). At the ordinary temperature,
+therefore, white phosphorus must be considered as the less stable
+(metastable) form, for although it can exist in contact with red phosphorus
+for a long period, its vapour pressure, as we have seen, is greater than
+that of the red modification, and also, its solubility in different
+solvents is greater[71] than that of the red modification; as we shall find
+later, the solubility of the metastable form is always greater than that of
+the stable.
+
+The relationships which are met with in the case of phosphorus can be best
+represented by the diagram, Fig. 11.[72]
+
+In this figure, BO_{1} represents the conditions of equilibrium of the
+univariant system red phosphorus and vapour, which ends at O_{1}, the
+melting point of red phosphorus. By heating in capillary tubes of hard
+glass, Chapman[73] found that red phosphorus melts at the melting point of
+potassium iodide, _i.e._ about 630°,[74] but the pressure at this
+temperature is unknown.
+
+At O_{1}, then, we have the triple point, red phosphorus, liquid, and
+vapour, and starting from it, we should have the {48} vaporization curve of
+liquid phosphorus, O_{1}A, and the fusion curve of red phosphorus, O_{1}F.
+Although these have not been determined, the latter curve must, from
+theoretical considerations (_v._ p. 58), slope slightly to the right;
+_i.e._ increase of pressure raises the melting point of red phosphorus.
+
+[Illustration: FIG. 11.]
+
+When white phosphorus is heated to 44°, it melts. At this point, therefore,
+we shall have another triple point, white phosphorus--liquid--vapour; the
+pressure at this point has been calculated to be 3 mm.[75] This point is
+the intersection of three curves, viz. sublimation curve, vaporization
+curve, and the fusion curve of white phosphorus. The fusion curve, O_{2}E,
+has been determined by Tammann[76] and by G. A. Hulett,[77] and it was
+found that increase of pressure by 1 atm. raises the melting point by
+0.029°. The sublimation curve of white phosphorus has not yet been
+determined.
+
+As can be seen from the table of vapour pressures (p. 46), the vapour
+pressure of white phosphorus has been determined up to 500°; at
+temperatures above this, however, the velocity with which transformation
+into red phosphorus takes place is so great as to render the determination
+of the vapour pressure {49} at higher temperatures impossible. Since,
+however, the difference between white phosphorus and red phosphorus
+disappears in the liquid state, the vapour pressure curve of white
+phosphorus must pass through the point O_{1}, the melting point of red
+phosphorus, and must be continuous with the curve O_{1}A, the vapour
+pressure curve of liquid phosphorus (_vide infra_). Since, as Fig. 10
+shows, the vapour pressure curve of white phosphorus ascends very rapidly
+at higher temperatures, the "break" between BO_{1} and O_{1}A must be very
+slight.
+
+As compared with monotropic substances like benzophenone, phosphorus
+exhibits the peculiarity that transformation of the metastable into the
+stable modification takes place with great slowness; and further, the time
+required for the production of equilibrium between red phosphorus and
+phosphorus vapour is great compared with that required for establishing the
+same equilibrium in the case of white phosphorus. This behaviour can be
+best explained by the assumption that change in the molecular complexity
+(polymerization) occurs in the conversion of white into red phosphorus, and
+when red phosphorus passes into vapour (depolymerization).[78]
+
+This is borne out by the fact that measurements of the vapour density of
+phosphorus vapour at temperatures of 500° and more, show it to have the
+molecular weight represented by P_{4},[79] and the same molecular weight
+has been found for phosphorus in solution.[80] On the other hand, it has
+recently been shown by R. Schenck,[81] that the molecular weight of red
+phosphorus is at least P_{8}, and very possibly higher.
+
+In the case of phosphorus, therefore, it is more than possible that we are
+dealing, not simply with two polymorphic {50} forms of the same substance,
+but with polymeric forms, and that there is no transition point at
+temperatures above the absolute zero, unless we assume the molecular
+complexity of the two forms to become the same. The curve for red
+phosphorus would therefore lie below that of white phosphorus, for the
+vapour pressure of the polymeric form, if produced from the simpler form
+with evolution of heat, must be lower than that of the latter. A transition
+point would, of course, become possible if the sign of the heat effect in
+the transformation of the one modification into the other should change.
+If, further, the liquid which is produced by the fusion of red phosphorus
+at 630° under high pressure also exists in a polymeric form, greater than
+P_{4}, then the metastable vaporization curve of white phosphorus would not
+pass through the melting point of red phosphorus, as was assumed above.[82]
+
+We have already seen in the case of water (p. 31) that the vapour pressure
+of supercooled water is greater than that of ice, and that therefore it is
+possible, theoretically at least, by a process of distillation, to transfer
+the water from one end of a closed tube to the other, and to there condense
+it as ice. On account of the very small difference between the vapour
+pressure of supercooled water and ice, this distillation process has not
+been experimentally realized. In the case of phosphorus, however, where the
+difference in the vapour pressures is comparatively great, it has been
+found possible to distil white phosphorus from one part of a closed tube to
+another, and to there condense it as red phosphorus; and since the vapour
+pressure of red phosphorus at 350° is less than the vapour pressure of
+white phosphorus at 200°, it is possible to carry out the distillation from
+a _colder_ part of the tube to a _hotter_, by having white phosphorus at
+the former and red phosphorus at the latter. Such a process of distillation
+has been carried out by Troost and Hautefeuille between 324° and 350°.[83]
+
+Relationships similar to those found in the case of phosphorus are also met
+with in the case of cyanogen and {51} paracyanogen, which have been studied
+by Chappuis,[84] Troost and Hautefeuille,[85] and Dewar,[86] and also in
+the case of other organic substances.
+
+Enantiotropy combined with Monotropy.--Not only can polymorphic substances
+exhibit enantiotropy or monotropy, but, if the substance is capable of
+existing in more than two crystalline forms, both relationships may be
+found, so that some of the forms may be enantiotropic to one another, while
+the other forms exhibit only monotropy. This behaviour is seen in the case
+of sulphur, which can exist in as many as eight different crystalline
+varieties. Of these only monoclinic and rhombic sulphur exhibit the
+relationship of enantiotropy, _i.e._ they possess a definite transition
+point, while the other forms are all metastable with respect to rhombic and
+monoclinic sulphur, and remain so up to the melting point; that is to say,
+they are monotropic modifications.[87]
+
+E. _Liquid Crystals._
+
+Phenomena observed.--In 1888 it was discovered by Reinitzer[88] that the
+two substances, cholesteryl acetate and cholesteryl benzoate, possess the
+peculiar property of melting sharply at a definite temperature to milky
+liquids; and that the latter, on being further heated, suddenly become
+clear, also at a definite temperature. Other substances, more especially
+_p_-azoxyanisole and _p_-azoxyphenetole, were, later, found to possess the
+same property of having apparently a double melting point.[89] On cooling
+the clear liquids, the reverse series of changes occurred.
+
+The turbid liquids which were thus obtained were found to possess not only
+the usual properties of liquids (such as the {52} property of flowing and
+of assuming a perfectly spherical shape when suspended in a liquid of the
+same density), but also those properties which had hitherto been observed
+only in the case of solid crystalline substances, viz. the property of
+double refraction and of giving interference colours when examined by
+polarized light; the turbid liquids are _anisotropic_. To such liquids, the
+optical properties of which were discovered by O. Lehmann,[90] the name
+_liquid crystals_, or crystalline liquids, was given.
+
+Nature of Liquid Crystals.--During the past ten years the question as to
+the nature of liquid crystals has been discussed by a number of
+investigators, several of whom have contended strongly against the idea of
+the term "liquid" being applied to the crystalline condition; and various
+attempts have been made to prove that the turbid liquids are in reality
+heterogeneous and are to be classed along with emulsions.[91] This view was
+no doubt largely suggested by the fact that the anisotropic liquids were
+turbid, whereas the "solid" crystals were clear. Lehmann found, however,
+that, when examined under the microscope, the "simple" liquid crystals were
+also clear,[92] the apparent turbidity being due to the aggregation of a
+number of differently oriented crystals, in the same way as a piece of
+marble does not appear transparent although composed of transparent
+crystals.[93]
+
+Further, no proof of the heterogeneity of liquid crystals has yet been
+obtained, but rather all chemical and physical investigations indicate that
+they are homogeneous.[94] No separation {53} of a solid substance from the
+milky, anisotropic liquids has been effected; the anisotropic liquid is in
+some cases less viscous than the isotropic liquid formed at a higher
+temperature; and the temperature of liquefaction is constant, and is
+affected by pressure and admixture with foreign substances exactly as in
+the case of a pure substance.[95]
+
+[Illustration: FIG. 12.]
+
+Equilibrium Relations in the Case of Liquid Crystals.--Since, now, we have
+seen that we are dealing here with substances in two crystalline forms
+(which we may call the solid and liquid[96] crystalline form), which
+possess a definite transition point, at which, transformation of the one
+form into the other occurs in both directions, we can represent the
+conditions of equilibrium by a diagram in all respects similar to that
+employed in the case of other enantiotropic substances, _e.g._ sulphur
+(p. 35).
+
+{54}
+
+In Fig. 12 there is given a diagrammatic representation of the
+relationships found in the case of _p_-azoxyanisole.[97]
+
+Although the vapour pressure of the substance in the solid, or liquid
+state, has not been determined, it will be understood from what we have
+already learned, that the curves AO, OB, BC, representing the vapour
+pressure of solid crystals, liquid crystals, isotropic liquid, must have
+the relative positions shown in the diagram. Point O, the transition point
+of the solid into the liquid crystals, lies at 118.27°, and the change of
+the transition point with the pressure is +0.032° pro 1 atm. The transition
+curve OE slopes, therefore, slightly to the right. The point B, the melting
+point of the liquid crystals, lies at 135.85°, and the melting point is
+raised 0.0485° pro 1 atm. The curve BD, therefore, also slopes to the
+right, and more so than the transition curve. In this respect azoxyanisole
+is different from sulphur.
+
+The areas bounded by the curves represent the conditions for the stable
+existence of the four single phases, solid crystals, liquid crystals,
+isotropic liquid and vapour.
+
+The most important substances hitherto found to form liquid crystals
+are[98]:--
+
+  ----------------------------------+------------+--------
+                                    |            |
+          Substance.                | Transition | Melting
+                                    |   point.   | point.
+  ----------------------------------+------------+--------
+                                    |            |
+  Cholesteryl benzoate              |   145.5°   |  178.5°
+  Azoxyanisole                      |   118.3°   |  135.9°
+  Azoxyphenetole                    |   134.5°   |  168.1°
+  Condensation product from         |            |
+   benzaldehyde and benzidine       |   234°     |  260°
+  Azine of _p_-oxyethylbenzaldehyde |   172°     |  196°
+  Condensation product from         |            |
+   _p_-tolylaldehyde and benzidine  |   231°     |   --
+  _p_-Methoxycinnamic acid          |   169°     |  185°
+  ----------------------------------+------------+--------
+
+       *       *       *       *       *
+
+
+{55}
+
+CHAPTER IV
+
+GENERAL SUMMARY
+
+In the preceding pages we have learned how the principles of the Phase Rule
+can be applied to the elucidation of various systems consisting of one
+component. In the present chapter it is proposed to give a short summary of
+the relationships we have met with, and also to discuss more generally how
+the Phase Rule applies to other one-component systems. On account of the
+fact that beginners are sometimes inclined to expect too much of the Phase
+Rule; to expect, for example, that it will inform them as to the exact
+behaviour of a substance, it may here be emphasized that the Phase Rule is
+a general rule; it informs us only as to the general conditions of
+equilibrium, and leaves the determination of the definite, numerical data
+to experiment.
+
+Triple Point.--We have already (p. 28) defined a triple point in a
+one-component system, as being that pressure and temperature at which three
+phases coexist in equilibrium; it represents, therefore, an invariant
+system (p. 16). At the triple point also, three curves cut, viz. the curves
+representing the conditions of equilibrium of the three univariant systems
+formed by the combination of the three phases in pairs. The most common
+triple point of a one-component system is, of course, the triple point,
+solid, liquid, vapour (S-L-V), but other triple points[99] are also
+possible when, as in the case of {56} sulphur or benzophenone, polymorphic
+forms occur. Whether or not all the triple points can be experimentally
+realized will, of course, depend on circumstances. We shall, in the first
+place, consider only the triple point S-L-V.
+
+As to the general arrangement of the three univariant curves around the
+triple point, the following rules may be given. (1) The prolongation of
+each of the curves beyond the triple point must lie between the other two
+curves. (2) The middle position at one and the same temperature in the
+neighbourhood of the triple point is taken by that curve (or its metastable
+prolongation) which represents the two phases of most widely differing
+specific volume.[100] That is to say, if a line of constant temperature is
+drawn immediately above or below the triple point so as to cut the three
+curves--two stable curves and the metastable prolongation of the third--the
+position of the curves at that temperature will be such that the middle
+position is occupied by that curve (or its metastable prolongation) which
+represents the two phases of most widely differing specific volume.
+
+Now, although these rules admit of a considerable variety of possible
+arrangements of curves around the triple point,[101] only two of these have
+been experimentally obtained in the case of the triple point
+solid--liquid--vapour. At present, therefore, we shall consider only these
+two cases (Figs. 13 and 14).
+
+[Illustration: FIG. 13.]
+
+[Illustration: FIG. 14.]
+
+An examination of these two figures shows that they satisfy the rules laid
+down. Each of the curves on being prolonged passes between the other two
+curves. In the case of substances of the first type (Fig. 13), the specific
+volume of the solid is greater than that of the liquid (the substance
+contracts on fusion); the difference of specific volume will, therefore, be
+greatest between liquid and vapour. The curve, therefore, for liquid and
+vapour (or its prolongation) must lie between the other two curves; this is
+seen from the figure to be the case. Similarly, the rule is satisfied by
+the arrangement of curves in Fig. 14, where the difference of specific
+volumes is {57} greatest between the solid and vapour. In this case the
+curve S-V occupies the intermediate position.
+
+As we see, the two figures differ from one another only in that the fusion
+curve OC in one case slopes to the right away from the pressure axis, thus
+indicating that the melting point is raised by increase of pressure; in the
+other case, to the left, indicating a lowering of the melting point with
+the pressure. These conditions are found exemplified in the case of sulphur
+and ice (pp. 29 and 35). We see further from the two figures, that O in
+Fig. 13 gives the highest temperature at which the solid can exist, for the
+curve for solid--liquid slopes back to regions of lower temperature; in
+Fig. 14, O gives the lowest temperature at which the liquid phase can exist
+as stable phase.[102]
+
+Theorems of van't Hoff and of Le Chatelier.--So far we have studied only
+the conditions under which various systems exist in equilibrium; and we now
+pass to a consideration of the changes which take place in a system when
+the external conditions of temperature and pressure are altered. For all
+such changes there exist two theorems, based on the laws of thermodynamics,
+by means of which the alterations in a system can be qualitatively
+predicted.[103] The first of these, usually {58} known as van't Hoff's _law
+of movable equilibrium_,[104] states: When the temperature of a system in
+equilibrium is raised, that reaction takes place which is accompanied by
+absorption of heat; and, conversely, when the temperature is lowered, that
+reaction occurs which is accompanied by an evolution of heat.
+
+The second of the two theorems refers to the effect of change of pressure,
+and states:[105] When the pressure on a system in equilibrium is increased,
+that reaction takes place which is accompanied by a diminution of volume;
+and when the pressure is diminished, a reaction ensues which is accompanied
+by an increase of volume.
+
+The demonstration of the universal applicability of these two theorems is
+due chiefly to Le Chatelier, who showed that they may be regarded as
+consequences of the general law of action and reaction. For this reason
+they are generally regarded as special cases of the more general law, known
+as the _theorem of Le Chatelier_, which may be stated in the words of
+Ostwald, as follows:[106] _If a system in equilibrium is subjected to a
+constraint by which the equilibrium is shifted, a reaction takes place
+which opposes the constraint, _i.e._ one by which its effect is partially
+destroyed._
+
+This theorem of Le Chatelier is of very great importance, for it applies to
+all systems and changes of the condition of equilibrium, whether physical
+or chemical; to vaporization and fusion; to solution and chemical action.
+In all cases, whenever changes in the external condition of a system in
+equilibrium are produced, processes also occur within the system which tend
+to counteract the effect of the external changes.
+
+_Changes at the Triple Point._--If now we apply this theorem to equilibria
+at the triple point S-L-V, and ask what changes will occur in such a system
+when the external conditions of pressure and temperature are altered, the
+general answer to the question will be: So long as the three phases are
+present, no {59} change in the temperature or pressure of the system can
+occur, but _only changes in the relative amounts of the phases_; that is to
+say, the effect on the system of change in the external conditions is
+opposed by the reactions or changes which take place within the system
+(according to the theorems of van't Hoff and Le Chatelier). We now proceed
+to discuss what these changes are, and shall consider first the effect of
+alteration of the temperature at constant volume and constant pressure, and
+then the effect of alteration of the pressure both when the temperature
+remains constant and when it varies.
+
+When the volume is kept constant, the effect of the addition of heat to a
+system at the triple point S-L-V differs somewhat according as there is an
+increase or diminution of volume when the solid passes into the liquid
+state. In the former and most general case (Fig. 14), addition of heat will
+cause a certain amount of the solid phase to melt, whereby the heat which
+is added becomes latent; the temperature of the system therefore does not
+rise. Since, however, the melting of the solid is accompanied by an
+increase of volume, whereby an increase of pressure would result, a certain
+portion of the vapour must condense to liquid, in order that the pressure
+may remain constant. The total effect of addition of heat, therefore, is to
+cause both solid and vapour to pass into liquid, _i.e._ there occurs the
+change S + V --> L. It will, therefore, depend on the relative quantities
+of solid and vapour, which will disappear first. If the solid disappears
+first, then we shall pass to the system L-V; if vapour disappears first, we
+shall obtain the system S-L. Withdrawal of heat causes the reverse change,
+L --> S + V; at all temperatures below the triple point the liquid is
+unstable or metastable (p. 30).
+
+When fusion is accompanied by a diminution of volume (_e.g._ ice, Fig. 13),
+then, since the melting of the solid phase would decrease the total volume,
+_i.e._ would lower the pressure, a certain quantity of the solid must also
+pass into vapour in order that the pressure may be maintained constant. On
+addition of heat, therefore, there occurs the reaction S --> L + V;
+withdrawal of heat causes the reverse change L + V --> S. Above the
+temperature of the triple point the {60} solid cannot exist; below the
+triple point both systems, S-L and S-V, can exist, and it will therefore
+depend on the relative amounts of liquid and vapour which of these two
+systems is obtained on withdrawing heat from the system at constant volume.
+
+The same changes in the phases occur when heat is added or withdrawn at
+constant pressure, so long as the three phases are present. Continued
+addition of heat, however, at constant pressure will ultimately cause the
+formation of the bivariant system vapour alone; continued withdrawal of
+heat will ultimately cause the formation of solid alone. This will be
+readily understood from Fig. 15. The dotted line D'OD is a line of constant
+pressure; on adding heat, the system passes along the line OD into the
+region of vapour; on heat being withdrawn, the system passes along OD' into
+the area of solid.
+
+[Illustration: FIG. 15.]
+
+Similar changes are produced when the volume of the system is altered.
+Alteration of volume may take place either while transference of heat to or
+from the system is cut off (adiabatic change), or while such transference
+may occur (isothermal change). In the latter case, the temperature of the
+system will remain constant; in the former case, since at the triple point
+the pressure must be constant so long as the three phases are present,
+increase of volume must be compensated by the evaporation of liquid. This,
+however, would cause the temperature to fall (since communication of heat
+from the outside is supposed to be cut off), and a portion of the liquid
+must therefore freeze. In this way the latent heat of evaporation is
+counterbalanced by the latent heat of fusion. As the result of increase of
+volume, therefore, the process occurs L --> S + V. Diminution of volume,
+without transference of heat, will bring about the opposite change, S + V
+--> L. In the former case there is ultimately obtained the univariant
+system S-V; in the latter case there will be {61} obtained either S-L or
+L-V according as the vapour or solid phase disappears first.
+
+This argument holds good for both types of triple point shown in Figs. 13
+and 14 (p. 57). A glance at these figures will show that increase of volume
+(diminution of pressure) will lead ultimately to the system S-V, for at
+pressures lower than that of the triple point, the liquid phase cannot
+exist. Decrease of volume (increase of pressure), on the other hand, will
+lead either to the system S-L or L-V, because these systems can exist at
+pressures higher than that of the triple point. If the vapour phase
+disappears and we pass to the curve S-L, continued diminution of volume
+will be accompanied by a fall in temperature in the case of systems of the
+first type (Fig. 13), and by a rise in temperature in the case of systems
+of the second type (Fig. 14).
+
+[Illustration: FIG. 16.]
+
+[Illustration: FIG. 17.]
+
+Lastly, if the temperature is maintained constant, _i.e._ if heat can pass
+into or out of the system, then on changing the volume the same changes in
+the phases will take place as described above until one of the phases has
+disappeared. Continued increase of volume (decrease of pressure) will then
+cause the disappearance of a second phase, the system passing along the
+dotted line OE' (Figs. 16, 17), so that ultimately there remains only the
+vapour phase. Conversely, diminution of volume (increase of pressure) will
+ultimately lead either to solid (Fig. 16) or to liquid alone (Fig. 17), the
+system passing along the dotted line OE. {62}
+
+In discussing the alterations which may take place at the triple point with
+change of temperature and pressure, we have considered only the triple
+point S-L-V. The same reasoning, however, applies, _mutatis mutandis_, to
+all other triple points, so that if the specific volumes of the phases are
+known, and the sign of the heat effects which accompany the transformation
+of one phase into the other, it is possible to predict (by means of the
+theorem of Le Chatelier) the changes which will be produced in the system
+by alteration of the pressure and temperature.
+
+In all cases of transformation at the triple point, it should be noted that
+all _three phases are involved in the change_,[107] and not two only; the
+fact that in the case, say, of the transformation from solid to liquid, or
+liquid to solid, at the melting point with change of temperature, only
+these two phases appear to be affected, is due to there generally being a
+large excess of the vapour phase present and to the prior disappearance
+therefore of the solid or liquid phase.
+
+In the case of triple points at which two solid phases are in equilibrium
+with liquid, other arrangements of the curves around the triple point are
+found. It is, however, unnecessary to give a general treatment of these
+here, since the principles which have been applied to the triple point
+S-L-V can also be applied to the other triple points.[108]
+
+Triple Point Solid--Solid--Vapour.--The triple point solid--solid--vapour
+is one which is of considerable importance. Examples of such a triple point
+have already been given in sulphur and tin, and a list of other substances
+capable of yielding two solid phases is given below. The triple point S-S-V
+is not precisely the same as the transition point, but is very nearly so.
+The transition point is the temperature at which the relative stability of
+the two solid phases undergoes change, when the vapour phase is absent and
+the pressure is 1 atm.; whereas at the triple point the pressure is that of
+the system itself. The transition point, therefore, bears the same relation
+to the triple point S-S-V as the melting point to the triple point S-L-V.
+
+{63}
+
+In the following table is given a list of the most important polymorphous
+substances, and the temperatures of the transition point.[109]
+
+  ------------------------------------+-------------
+                                      |
+              Substance.              |  Transition
+                                      | temperature.
+  ------------------------------------+-------------
+                                      |
+  Ammonium nitrate--                  |
+   [beta]-rhombic --> [alpha]-rhombic |     35°
+   [alpha]-rhombic --> rhombohedral   |     83°
+   Rhombohedral --> regular           |    125°
+  Mercuric iodide                     |    126°
+  Potassium nitrate                   |    129°
+  Silver iodide                       |    145°
+  Silver nitrate                      |    160°
+  Sulphur                             |     95.5°
+  Tetrabrommethane                    |     46.8°
+  Thallium nitrate--                  |
+   Rhombic --> rhombohedral           |     80°
+   Rhombohedral --> regular           |    142.5°
+  Thallium picrate                    |     46°
+  Tin                                 |     20°
+  ------------------------------------+-------------
+
+Sublimation and Vaporization Curves.--We have already seen, in the case of
+ice and liquid water, that the vapour pressure increases as the temperature
+rises, the increase of pressure per degree being greater the higher the
+temperature. The sublimation and vaporization curves, therefore, are not
+straight lines, but are bent, the convex side of the curve being towards
+the temperature axis in the ordinary _pt_-diagram.
+
+In the case of sulphur and of tin, we assumed vapour to be given off by the
+solid substance, although the pressure of the vapour has not hitherto been
+measured. The assumption, however, is entirely justified, not only on
+theoretical grounds, but also because the existence of a vapour pressure
+has been observed in the case of many solid substances at temperatures much
+below the melting point,[110] and in some cases, _e.g._ camphor,[111] the
+vapour pressure is considerable.
+
+{64}
+
+As the result of a large number of determinations, it has been found that
+all vapour pressure curves have the same general form alluded to above.
+Attempts have also been made to obtain a general expression for the
+quantitative changes in the vapour pressure with change of temperature, but
+without success. Nevertheless, the _qualitative_ changes, or the general
+direction of the curves, can be predicted by means of the theorem of Le
+Chatelier.
+
+As we have already learned (p. 16), the Phase Rule takes no account of the
+molecular complexity of the substances participating in an equilibrium. A
+dissociating substance, therefore, in contact with its vaporous products of
+dissociation (_e.g._ ammonium chloride in contact with ammonia and hydrogen
+chloride), will likewise constitute a univariant system of one component,
+provided the composition of the vapour phase as a whole is the same as that
+of the solid or liquid phase (p. 13). For all such substances, therefore,
+the conditions of equilibrium will be represented by a curve of the same
+general form as the vapour pressure curve of a non-dissociating
+substance.[112] The same behaviour is also found in the case of substances
+which polymerize on passing into the solid or liquid state (_e.g._ red
+phosphorus). Where such changes in the molecular state occur, however, the
+time required for equilibrium to be established is, as a rule, greater than
+when the molecular state is the same in both phases.
+
+From an examination of Figs. 13 and 14, it will be easy to predict the
+effect of change of pressure and temperature on the univariant systems S-V
+or L-V. If the volume is kept constant, addition of heat will cause an
+increase of pressure, the system S-V moving along the curve AO until at the
+triple point the liquid phase is formed, and the system L-V moving along
+the curve OB; so long as two phases are present, the condition of the
+system must be represented by these two curves. Conversely, withdrawal of
+heat will cause condensation of vapour, and therefore diminution of
+pressure; the system will therefore move along the vaporization or
+sublimation curve to lower temperatures and pressures, so long as the
+system remains univariant.
+
+{65}
+
+If transference of heat to or from the system is prevented, increase of
+volume (diminution of pressure) will cause the system L-V to pass along the
+curve BO; liquid will pass into vapour and the temperature will fall.[113]
+At O solid may appear, and the temperature of the system will then remain
+constant until the liquid phase has disappeared (p. 57); the system will
+then follow the curve OA until the solid phase disappears, and we are
+ultimately left with vapour. On the other hand, diminution of volume
+(increase of pressure) will cause condensation of vapour, and the system
+S-V will pass along the curve AO to higher temperatures and pressures; at O
+the solid will melt, and the system will ultimately pass to the curve OB or
+to OC (p. 57).
+
+Addition or withdrawal of heat at constant pressure, and increase or
+diminution of the pressure at constant temperature, will cause the system
+to pass along lines parallel to the temperature and the pressure axis
+respectively; the working out of these changes may be left to the reader,
+guided by what has been said on pp. 60 and 61.
+
+The sublimation curve of all substances, so far as yet found, has its upper
+limit at the melting point (triple point), although the possibility of the
+existence of a superheated solid is not excluded. The lower limit is,
+theoretically at least, at the absolute zero, provided no new phase, _e.g._
+a different crystalline modification, is formed. If the sublimation
+pressure of a substance is greater than the atmospheric pressure at any
+temperature below the point of fusion, then the substance will _sublime
+without melting_ when heated in an open vessel; and fusion will be possible
+only at a pressure higher than the atmospheric. This is found, for example,
+in the case of red phosphorus (p. 47). If, however, the sublimation
+pressure of a substance at its triple point S-L-V is less than one
+atmosphere, then the substance will melt when heated in an open vessel.
+
+In the case of the vaporization curve, the upper limit lies at the critical
+point where the liquid ceases to exist;[114] the {66} lower limit is
+determined by the range of the metastable state of the supercooled liquid.
+
+The interpolation and extrapolation of vapour-pressure curves is rendered
+very easy by means of a relationship which Ramsay and Young[115] found to
+exist between the vapour-pressure curves of different substances. It was
+observed that in the case of closely related substances, the ratio of the
+absolute temperatures corresponding to equal vapour pressures is constant,
+_i.e._ T_{1}/T'_{1} = T_{2}/T'_{2}. When the two substances are not closely
+related, it was found that the relationship could be expressed by the
+equation T_{1}/T'_{1} = T_{2}/T'_{2} + _c_(_t_' - _t_) where _c_ is a
+constant having a small positive or negative value, and _t_' and _t_ are
+the temperatures at which one of the substances has the two values of the
+vapour pressure in question. By means of this equation, if the
+vapour-pressure curve of one substance is known, the vapour-pressure curve
+of any other substance can be calculated from the values at any two
+temperatures of the vapour pressure of that substance.
+
+Fusion Curve--Transition Curve.--The fusion curve represents the conditions
+of equilibrium between the solid and liquid phase; it shows the change of
+the melting point of a substance with change of pressure.
+
+As shown in Figs. 13 and 14, the fusion curve is inclined either towards
+the pressure axis or away from it; that is, increase of pressure can either
+lower or raise the melting point. It is easy to predict in a qualitative
+manner the different effect of pressure on the melting point in the two
+cases mentioned, if we consider the matter in the light of the theorem of
+Le Chatelier (p. 58). Water, on passing into ice, expands; therefore, if
+the pressure on the system ice--water be increased, a reaction will take
+place which is accompanied by a diminution in volume, _i.e._ the ice will
+melt. Consequently, a lower temperature will be required in order to
+counteract the effect of increase of pressure; or, in other words, the
+melting point will {67} be lowered by pressure.[116] In the second case,
+the passage of the liquid to the solid state is accompanied by a diminution
+of volume; the effect of increase of pressure will therefore be the reverse
+of that in the previous case.
+
+If the value of the heat of fusion and the alteration of volume
+accompanying the change of state are known, it is possible to calculate
+_quantitatively_ the effect of pressure.[117]
+
+We have already seen (p. 25) that the effect of pressure on the melting
+point of a substance was predicted as the result of theoretical
+considerations, and was first proved experimentally in the case of ice.
+Soon after, Bunsen[118] showed that the melting point of other substances
+is also affected by pressure; and in more recent years, ample experimental
+proof of the change of the melting point with the pressure has been
+obtained. The change of the melting point is, however, small; as a rule,
+increase of pressure by 1 atm. changes the melting point by about 0.03°,
+but in the case of water the change is much less (0.0076°), and in the case
+of camphor much more (0.13°). In other words, if we take the mean case, an
+increase of pressure of more than 30 atm. is required to produce a change
+in the melting point of 1°.
+
+Investigations which were made of the influence of pressure on the
+melting-point, showed that up to pressures of several hundred atmospheres
+the fusion curve is a straight line.[119] Tammann[120] has, however, found
+that on increasing the pressure the fusion curve no longer remains
+straight, but bends towards the pressure axis, so that, on sufficiently
+increasing the pressure, a maximum temperature might at length be reached.
+This maximum has, so far, however, not been attained, although the melting
+point curves of various substances have been studied up to pressures of
+4500 atm. This is to be accounted for partly {68} by the fact that the
+probable maximum temperature in the case of most substances lies at very
+great pressures, and also by the fact that other solid phases make their
+appearance, as, for example, in the case of ice (p. 32).
+
+As to the upper limit of the fusion curve, the view has been expressed[121]
+that just as in the case of liquid and vapour, so also in the case of solid
+and liquid, there exists a critical point at which the solid and the liquid
+phase become identical. Experimental evidence, however, does not appear to
+favour this view.[122]
+
+The _transition point_, like the melting point, is also influenced by the
+pressure, and in this case also it is found that pressure may either raise
+or lower the transition point, so that the transition curve may be inclined
+either away from or towards the pressure axis. The direction of the
+transition curve can also be predicted if the change of volume accompanying
+the passage of one form into the other is known. In the case of sulphur, we
+saw that the transition point is raised by increase of pressure; in the
+case of the transition of rhombohedral into [alpha]-rhombic form of
+ammonium nitrate, however, the transition point is lowered by pressure, as
+shown by the following table.[123]
+
+  -------------+----------
+               |
+  Temperature. | Pressure.
+  -------------+----------
+               |
+     85.85°    |   1 atm.
+     84.38°    | 100  "
+     83.03°    | 200  "
+     82.29°    | 250  "
+  -------------+----------
+
+So far as investigations have been carried out, it appears that in most
+cases the transition curve is practically a straight line.
+
+It has, however, been found in the case of Glauber's salt, that with
+increase of pressure the transition curve passes through a point of maximum
+temperature, and exhibits, therefore, a form similar to that assumed by
+Tammann for the fusion curve.[124]
+
+{69}
+
+Suspended Transformation. Metastable Equilibria.--Hitherto we have
+considered only systems in stable equilibrium. We have, however, already
+seen, in the case of water, that on cooling the liquid down to the triple
+point, solidification did not necessarily take place, although the
+conditions were such as to allow of its formation. Similarly, we saw that
+rhombic sulphur can be heated above the transition point, and monoclinic
+sulphur can be obtained at temperatures below the transition point,
+although in both cases transformation into a more stable form is possible;
+the system becomes metastable.
+
+The same reluctance to form a new phase is observed also in the phenomena
+of superheating of liquids, and in the "hanging" of mercury in barometers,
+in which case the vapour phase is not formed. In general, then, we may say
+that _a new phase will not necessarily be formed immediately the system
+passes into such a condition that the existence of that phase is possible_;
+but rather, instead of the system undergoing transformation so as to pass
+into the most stable condition under the existing pressure and temperature,
+this transformation will be "suspended" or delayed, and the system will
+become metastable. Only in the case of the formation of the liquid from the
+solid phase, in a one-component system, has this reluctance to form a new
+phase not been observed.
+
+_To ensure the formation of the new phase, it is necessary to have that
+phase present._ The presence of the solid phase will prevent the
+supercooling of the liquid; and the presence of the vapour phase will
+prevent the superheating of the liquid. However, even in the presence of
+the more stable phase, transformation of the metastable phase occurs with
+very varying velocity; in some cases so quickly as to appear almost
+instantaneous; while in other cases, the change takes place so slowly as to
+require hundreds of years for its achievement. It is this slow rate of
+transformation that renders the existence of metastable forms possible,
+when in contact with the more stable phase. Thus, for example, although
+calcite is the most stable form of calcium carbonate at the ordinary
+temperature,[125] the less stable {70} modification, aragonite,
+nevertheless exists under the ordinary conditions in an apparently very
+stable state.
+
+As to the amount of the new phase required to bring about the
+transformation of the metastable phase, quantitative measurements have been
+carried out only in the case of the initiation of crystallization in a
+supercooled liquid.[126] As the result of these investigations, it was
+found that, in the case of superfused salol, the very small amount of 1 ×
+10^{-7} gm. of the solid phase was sufficient to induce crystallization.
+Crystallization of a supercooled liquid, however, can be initiated only by
+a "nucleus" of the same substance in the solid state, or, as has also been
+found, by a nucleus of an isomorphous solid phase; it is not brought about
+by the presence of any chance solid.
+
+Velocity of Transformation.--Attention has already been drawn to the
+sluggishness with which reciprocal transformation of the polymorphic forms
+of a substance may occur. In the case of tin, for example, it was found
+that the white modification, although apparently possessing permanence, is
+in reality in a metastable state, under the ordinary conditions of
+temperature and pressure. This great degree of stability is due to the
+tardiness with which transformation into the grey form occurs.
+
+What was found in the case of tin, is met with also in the case of all
+transformations in the solid state, but the velocity of the change is less
+in some cases than in others, and appears to decrease with increase of the
+valency of the element.[127] To this fact van't Hoff attributes the great
+permanence of many really unstable (or metastable) carbon compounds.
+
+Reference has been made to the fact that the velocity of transformation can
+be accelerated by various means. One of the most important of these is the
+employment of a liquid which has a solvent action on the solid phases. Just
+as we have seen that at any given temperature the less stable form has the
+higher vapour pressure, but that at the transition point the vapour
+pressure of both forms becomes identical, so also it can be proved
+theoretically, and be shown experimentally, that {71} at a given
+temperature the solubility of the less stable form is greater than that of
+the more stable, but that at the transition point the solubility of the two
+forms becomes identical.[128]
+
+If, then, the two solid phases are brought into contact with a solvent, the
+less stable phase will dissolve more abundantly than the more stable; the
+solution will therefore become supersaturated with respect to the latter,
+which will be deposited. A gradual change of the less stable form,
+therefore, takes place through the medium of the solvent. In this way the
+more rapid conversion of white tin into grey in presence of a solution of
+tin ammonium chloride (p. 42) is to be explained. Although, as a rule,
+solvents accelerate the transformation of one solid phase into the other,
+they may also have a retarding influence on the velocity of transformation,
+as was found by Reinders in the case of mercuric iodide.[129]
+
+The velocity of inversion, also, is variously affected by different
+solvents, and in some cases, at least, it appears to be slower the more
+viscous the solvent;[130] indeed, Kastle and Reed state that yellow
+crystals of mercuric iodide, which, ordinarily, change with considerable
+velocity into the red modification, have been preserved for more than a
+year under vaseline.
+
+Temperature, also, has a very considerable influence on the velocity of
+transformation. The higher the temperature, and the farther it is removed
+from the equilibrium point (transition point), the greater is the velocity
+of change. Above the transition point, these two factors act in the same
+direction, and the velocity of transformation will therefore go on
+increasing indefinitely the higher the temperature is raised. Below the
+transition point, however, the two factors act in opposite directions, and
+the more the temperature is lowered, the more is the effect of removal from
+the equilibrium point counteracted. A point will therefore be reached at
+which the velocity is a maximum. Reduction of the temperature {72} below
+this point causes a rapid falling off in the velocity of change. The point
+of maximum velocity, however, is not definite, but may be altered by
+various causes. Thus, Cohen found that in the case of tin, the point of
+maximum velocity was altered if the metal had already undergone
+transformation; and also by the presence of different liquids.[131]
+
+Lastly, the presence of small quantities of different substances--catalytic
+agents or catalyzers--has a great influence on the velocity of
+transformation. Thus, _e.g._, the conversion of white to red phosphorus is
+accelerated by the presence of iodine (p. 47).
+
+Greater attention, however, has been paid to the study of the velocity of
+crystallization of a supercooled liquid, the first experiments in this
+direction having been made by Gernez[132] on the velocity of
+crystallization of phosphorus and sulphur. Since that time, the velocity of
+crystallization of other supercooled liquids has been investigated; such as
+acetic acid and phenol by Moore;[133] supercooled water by Tumlirz;[134]
+and a number of organic substances by Tammann,[135] Friedländer and
+Tammann,[136] and by Bogojawlenski.[137]
+
+In measuring the velocity of crystallization, the supercooled liquids were
+contained in narrow glass tubes, and the time required for the
+crystallization to advance along a certain length of the tube was
+determined, the velocity being expressed in millimetres per minute. The
+results which have so far been obtained may be summarized as follows. For
+any given degree of supercooling of a substance, the velocity of
+crystallization is constant. As the degree of supercooling increases, the
+velocity of crystallization also increases, until a certain point is
+reached at which the velocity is a maximum, which has a definite
+characteristic value for each substance. This maximum velocity remains
+constant over a certain range of {73} temperature; thereafter, the velocity
+diminishes fairly rapidly, and, with sufficient supercooling, may become
+zero. The liquid then passes into a glassy mass, which will remain
+(practically) permanent even in contact with the crystalline solid.
+
+In ordinary glass we have a familiar example of a liquid which has been
+cooled to a temperature at which crystallization takes place with very
+great slowness. If, however, glass is heated, a temperature is reached,
+much below the melting point of the glass, at which crystallization occurs
+with appreciable velocity, and we observe the phenomenon of
+devitrification.[138]
+
+When the velocity of crystallization is studied at temperatures above the
+maximum point, it is found that the velocity is diminished by the addition
+of foreign substances; and in many cases, indeed, it has been found that
+the diminution is the same for equimolecular quantities of different
+substances. It would hence appear possible to utilize this behaviour as a
+method for determining molecular weights.[139] The rule is, however, by no
+means a universal one. Thus it has been found by F. Dreyer,[140] in
+studying the velocity of crystallization of formanilide, that the
+diminution in the velocity produced by equivalent amounts of different
+substances is not the same, but that the foreign substances exercise a
+specific influence. Further, von Pickardt's rule does not hold when the
+foreign substance forms mixed crystals (Chap. X.) with the crystallizing
+substance.[141]
+
+Law of Successive Reactions.--When sulphur vapour is cooled at the ordinary
+temperature, it first of all condenses to drops of liquid, which solidify
+in an amorphous form, and only after some time undergo crystallization; or,
+when phosphorus vapour is condensed, white phosphorus is first formed, and
+not the more stable form--red phosphorus. It has also been observed that
+even at the ordinary temperature (therefore much below the transition
+point) sulphur may crystallize out from solution in benzene, alcohol,
+carbon disulphide, and other {74} solvents, in the prismatic form, the less
+stable prismatic crystals then undergoing transformation into the rhombic
+form;[142] a similar behaviour has also been observed in the transformation
+of the monotropic crystalline forms of sulphur.[143]
+
+Many other examples might be given. In organic chemistry, for instance, it
+is often found that when a substance is thrown out of solution, it is first
+deposited as a liquid, which passes later into the more stable crystalline
+form. In analysis, also, rapid precipitation from concentrated solution
+often causes the separation of a less stable and more soluble amorphous
+form.
+
+On account of the great frequency with which the prior formation of the
+less stable form occurs, Ostwald[144] has put forward the _law of
+successive reactions_, which states that when a system passes from a less
+stable condition it does not pass directly into the most stable of the
+possible states; but into the next more stable, and so step by step into
+the most stable. This law explains the formation of the metastable forms of
+monotropic substances, which would otherwise not be obtainable. Although it
+is not always possible to observe the formation of the least stable form,
+it should be remembered that that may quite conceivably be due to the great
+velocity of transformation of the less stable into the more stable form.
+From what we have learned about the velocity of transformation of
+metastable phases, we can understand that rapid cooling to a low
+temperature will tend to preserve the less stable form; and, on account of
+the influence of temperature in increasing the velocity of change, it can
+be seen that the formation of the less stable form will be more difficult
+to observe in superheated than in supercooled systems. The factors,
+however, which affect the readiness with which {75} the less stable
+modification is produced, appear to be rather various.[145]
+
+Although a number of at least apparent exceptions to Ostwald's law have
+been found, it may nevertheless be accepted as a very useful generalization
+which sums up very frequently observed phenomena.
+
+       *       *       *       *       *
+
+
+{76}
+
+CHAPTER V
+
+SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION
+
+In the preceding pages we have studied the behaviour of systems consisting
+of only one component, or systems in which all the phases, whether solid,
+liquid, or vapour, had the same chemical composition (p. 13). In some
+cases, as, for example, in the case of phosphorus and sulphur, the
+component was an elementary substance; in other cases, however, _e.g._
+water, the component was a compound. The systems which we now proceed to
+study are characterized by the fact that the different phases have no
+longer all the same chemical composition, and cannot, therefore, according
+to definition, be considered as one-component systems.
+
+In most cases, little or no difficulty will be experienced in deciding as
+to the _number_ of the components, if the rules given on pp. 12 and 13 are
+borne in mind. If the composition of all the phases, each regarded as a
+whole, is the same, the system is to be regarded as of the first order, or
+a one-component system; if the composition of the different phases varies,
+the system must contain more than one component. If, in order to _express_
+the composition of all the phases present when the system is in
+equilibrium, two of the constituents participating in the equilibrium are
+necessary and sufficient, the system is one of two components. Which two of
+the possible substances are to be regarded as components will, however, be
+to a certain extent a matter of arbitrary choice.
+
+The principles affecting the choice of components will best be learned by a
+study of the examples to be discussed in the sequel. {77}
+
+Different Systems of Two Components.--Applying the Phase Rule
+
+  P + F = C + 2
+
+to systems of two components, we see that in order that the system may be
+invariant, there must be four phases in equilibrium together; two
+components in three phases constitute a univariant, two components in two
+phases a bivariant system. In the case of systems of one component, the
+highest degree of variability found was two (one component in one phase);
+but, as is evident from the formula, there is a higher degree of freedom
+possible in the case of two-component systems. Two components existing in
+only one phase constitute a tervariant system, or a system with three
+degrees of freedom. In addition to the pressure and temperature, therefore,
+a third variable factor must be chosen, and as such there is taken the
+_concentration of the components_. In systems of two components, therefore,
+not only may there be change of pressure and temperature, as in the case of
+one-component systems, but the concentration of the components in the
+different phases may also alter; a variation which did not require to be
+considered in the case of one-component systems.
+
+[Illustration: FIG. 18.]
+
+Since a two-component system may undergo three possible {78} independent
+variations, we should require for the graphic representation of all the
+possible conditions of equilibrium a system of three co-ordinates in space,
+three axes being chosen, say, at right angles to one another, and
+representing the three variables--pressure, temperature, and concentration
+of components (Fig. 18). A curve (_e.g._ AB) in the plane containing the
+pressure and temperature axes would then represent the change of pressure
+with the temperature, the concentration remaining unaltered (_pt_-diagram);
+one in the plane containing the pressure and concentration axes (_e.g._ AF
+or DF), the change of pressure with the concentration, the temperature
+remaining constant (_pc_-diagram), while in the plane containing the
+concentration and the temperature axes, the simultaneous change of these
+two factors at constant pressure would be represented (_tc_-diagram). If
+the points on these three curves are joined together, a surface, ABDE, will
+be formed, and any line on that surface (_e.g._ FG, or GH, or GI) would
+represent the simultaneous variation of the three factors--pressure,
+temperature, concentration. Although we shall at a later point make some
+use of these solid figures, we shall for the present employ the more
+readily intelligible plane diagram.
+
+The number of different systems which can be formed from two components, as
+well as the number of the different phenomena which can there be observed,
+is much greater than in the case of one component. In the case of no two
+substances, however, have all the possible relationships been studied; so
+that for the purpose of gaining an insight into the very varied behaviour
+of two-component systems, a number of different examples will be discussed,
+each of which will serve to give a picture of some of the relationships.
+
+Although the strict classification of the different systems according to
+the Phase Rule would be based on the variability of the systems, the study
+of the many different phenomena, and the correlation of the comparatively
+large number of different systems, will probably be rendered easiest by
+grouping these different phenomena into classes, each of these classes
+being studied with the help of one or more typical examples. The order of
+treatment adopted here is, of course, quite arbitrary; {79} but has been
+selected from considerations of simplicity and clearness.
+
+PHENOMENA OF DISSOCIATION.
+
+Bivariant Systems.--As the first examples of the equilibria between a
+substance and its products of dissociation, we shall consider very briefly
+those cases in which there is one solid phase in equilibrium with vapour.
+Reference has already been made to such systems in the case of ammonium
+chloride. On being heated, ammonium chloride dissociates into ammonia and
+hydrogen chloride. Since, however, in that case the vapour phase has the
+same total composition as the solid phase, viz. NH_{3} + HCl = NH_{4}Cl,
+the system consists of only one component existing in two phases; it is
+therefore univariant, and to each temperature there will correspond a
+definite vapour pressure (dissociation pressure).[146]
+
+If, however, excess of one of the products of dissociation be added, the
+system becomes one of two components.
+
+In the first place, analysis of each of the two phases yields as the
+composition of each, solid: NH_{4}Cl (= NH_{3} + HCl); vapour: _m_NH_{3} +
+_n_HCl. Obviously the smallest number of substances by which the
+composition of the two phases can be expressed is two; that is, the number
+of components is two. What, then, are the components? The choice lies
+between NH_{3} + HCl, NH_{4}Cl + NH_{3}, and NH_{4}Cl + HCl; for the three
+substances, ammonium chloride, ammonia, hydrogen chloride, are the only
+ones taking part in the equilibrium of the system.
+
+Of these three pairs of components, we should obviously choose as the most
+simple NH_{3} and HCl, for we can then represent the composition of the two
+phases as the _sum_ of the two components. If one of the other two possible
+pairs of components be chosen, we should have to introduce negative
+quantities of one of the components, in order to represent the composition
+of the vapour phase. Although it must be allowed that the introduction of
+negative quantities of a component in such cases is quite permissible,
+still it will be {80} better to adopt the simpler and more direct choice,
+whereby the composition of each of the phases is represented as a sum of
+two components in varying proportions (p. 12).
+
+If, therefore, we have a solid substance, such as ammonium chloride, which
+dissociates on volatilization, and if the products of dissociation are
+added in varying amounts to the system, we shall have, in the sense of the
+Phase Rule, a _two-component system existing in two phases_. Such a system
+will possess two degrees of freedom. At any given temperature, not only the
+pressure, but also the composition, of the vapour-phase, _i.e._ the
+concentration of the components, can vary. Only after one of these
+independent variables, pressure or composition, has been arbitrarily fixed
+does the system become univariant, and exhibit a definite, constant
+pressure at a given temperature.
+
+Now, although the Phase Rule informs us that at a given temperature change
+of composition of the vapour phase will be accompanied by change of
+pressure, it does not cast any light on the relation between these two
+variables. This relationship, however, can be calculated theoretically by
+means of the Law of Mass Action.[147] From this we learn that in the case
+of a substance which dissociates into equivalent quantities of two gases,
+the product of the partial pressures of the gases is constant at a given
+temperature.
+
+This has been proved experimentally in the case of ammonium hydrosulphide,
+ammonium cyanide, phosphonium bromide, and other substances.[148]
+
+Univariant Systems.--In order that a system of two components shall possess
+only one degree of freedom, three phases must be present. Of such systems,
+there are seven possible, viz. S-S-S, S-S-L, S-S-V, L-L-L, S-L-L, L-L-V,
+S-L-V; S denoting solid, L liquid, and V vapour. In the present chapter we
+shall consider only the systems S-S-V, _i.e._ those systems in which there
+are two solid phases and a vapour phase present.
+
+{81}
+
+As an example of this, we may first consider the well-known case of the
+dissociation of calcium carbonate. This substance on being heated
+dissociates into calcium oxide, or quick-lime, and carbon dioxide, as shown
+by the equation CaCO_{3} <--> CaO + CO_{2}. In accordance with our
+definition (p. 9), we have here two solid phases, the carbonate and the
+quick-lime, and one vapour phase; the system is therefore univariant. To
+each temperature, therefore, there will correspond a certain, definite
+maximum pressure of carbon dioxide (dissociation pressure), and this will
+follow the same law as the vapour pressure of a pure liquid (p. 21). More
+particularly, it will be independent of the relative or absolute amounts of
+the two solid phases, and of the volume of the vapour phase. If the
+temperature is maintained constant, increase of volume will cause the
+dissociation of a further amount of the carbonate until the pressure again
+reaches its maximum value corresponding to the given temperature.
+Diminution of volume, on the other hand, will bring about the combination
+of a certain quantity of the carbon dioxide with the calcium oxide until
+the pressure again reaches its original value.
+
+The dissociation pressure of calcium carbonate was first studied by
+Debray,[149] but more exact measurements have been made by Le
+Chatelier,[150] who found the following corresponding values of temperature
+and pressure:--
+
+  -------------+-------------------------
+               |
+  Temperature. | Pressure in cm. mercury.
+  -------------+-------------------------
+               |
+      547°     |             2.7
+      610°     |             4.6
+      625°     |             5.6
+      740°     |            25.5
+      745°     |            28.9
+      810°     |            67.8
+      812°     |            76.3
+      865°     |           133.3
+  -------------+-------------------------
+
+From this table we see that it is only at a temperature of about 812° that
+the pressure of the carbon dioxide becomes equal to atmospheric pressure.
+In a vessel open to {82} the air, therefore, the complete decomposition of
+the calcium carbonate would not take place below this temperature by the
+mere heating of the carbonate. If, however, the carbon dioxide is removed
+as quickly as it is formed, say by a current of air, then the entire
+decomposition can be made to take place at a much lower temperature. For
+the dissociation equilibrium of the carbonate depends only on the partial
+pressure of the carbon dioxide, and if this is kept small, then the
+decomposition can proceed, even at a temperature below that at which the
+pressure of the carbon dioxide is less than atmospheric pressure.
+
+Ammonia Compounds of Metal Chlorides.--Ammonia possesses the property of
+combining with various substances, chiefly the halides of metals, to form
+compounds which again yield up the ammonia on being heated. Thus, for
+example, on passing ammonia over silver chloride, absorption of the gas
+takes place with formation of the substances AgCl,3NH_{3} and
+2AgCl,3NH_{3}, according to the conditions of the experiment. These were
+the first known substances belonging to this class, and were employed by
+Faraday in his experiments on the liquefaction of ammonia. Similar
+compounds have also been obtained by the action of ammonia on silver
+bromide, iodide, cyanide, and nitrate; and with the halogen compounds of
+calcium, zinc, and magnesium, as well as with other salts. The behaviour of
+the ammonia compounds of silver chloride is typical for the compounds of
+this class, and may be briefly considered here.
+
+It was found by Isambert[151] that at temperatures below 15°, silver
+chloride combined with ammonia to form the compound AgCl,3NH_{3}, while at
+temperatures above 20° the compound 2AgCl,3NH_{3} was produced. On heating
+these substances, ammonia was evolved, and the pressure of this gas was
+found in the case of both compounds to be constant at a given temperature,
+but was greater in the case of the former than in the case of the latter
+substance; the pressure, further, was independent of the amount decomposed.
+The behaviour of these two substances is, therefore, exactly analogous to
+that shown by calcium carbonate, and the explanation is also similar.
+
+{83}
+
+Regarded from the point of view of the Phase Rule, we see that we are here
+dealing with two components, AgCl and NH_{3}. On being heated, the
+compounds decompose according to the equations:--
+
+  2(AgCl,3NH_{3}) <--> 2AgCl,3NH_{3} + 3NH_{3}.
+    2AgCl,3NH_{3} <--> 2AgCl + 3NH_{3}.
+
+There are, therefore, three phases, viz. AgCl,3NH_{3}; 2AgCl,3NH_{3}, and
+NH_{3}, in the one case; and 2AgCl,3NH_{3}; AgCl, and NH_{3} in the other.
+These two systems are therefore univariant, and to each temperature there
+must correspond a definite pressure of dissociation, quite irrespective of
+the amounts of the phases present. Similarly, if, at constant temperature,
+the volume is increased (or if the ammonia which is evolved is pumped off),
+the pressure will remain constant so long as two solid phases, AgCl,3NH_{3}
+and 2AgCl,3NH_{3}, are present, _i.e._ until the compound richer in ammonia
+is completely decomposed, when there will be a sudden fall in the pressure
+to the value corresponding to the system 2AgCl,3NH_{3}--AgCl--NH_{3}. The
+pressure will again remain constant at constant temperature, until all the
+ammonia has been pumped off, when there will again be a sudden fall in the
+pressure to that of the system formed by solid silver chloride in contact
+with its vapour.
+
+The reverse changes take place when the pressure of the ammonia is
+gradually increased. If the volume is continuously diminished, the pressure
+will first increase until it has reached a certain value; the compound
+2AgCl,3NH_{3} can then be formed, and the pressure will now remain constant
+until all the silver chloride has disappeared. The pressure will again
+rise, until it has reached the value at which the compound AgCl,3NH_{3} can
+be formed, when it will again remain constant until the complete
+disappearance of the lower compound. _There is no gradual change of
+pressure_ on passing from one system to another; but the changes are
+abrupt, as is demanded by the Phase Rule, and as experiment has
+conclusively proved.[152]
+
+The dissociation pressures of the two compounds of silver {84} chloride and
+ammonia, as determined by Isambert,[153] are given in the following
+table:--
+
+  -------------------------+-------------------------
+                           |
+          AgCl,3NH_{3}.    |        2AgCl,3NH_{3}.
+  -------------+-----------+--------------+----------
+               |           |              |
+  Temperature. | Pressure. | Temperature. | Pressure.
+  -------------+-----------+--------------+----------
+               |           |              |
+         0°    |  29.3 cm. |     20.0°    |   9.3 cm.
+      10.6°    |  50.5 "   |     31.0°    |  12.5 "
+      17.5°    |  65.5 "   |     47.0°    |  26.8 "
+      24.0°    |  93.7 "   |     58.5°    |  52.8 "
+      28.0°    | 135.5 "   |     69.0°    |  78.6 "
+      34.2°    | 171.3 "   |     71.5°    |  94.6 "
+      48.5°    | 241.4 "   |     77.5°    | 119.8 "
+      51.5°    | 413.2 "   |     83.5°    | 159.3 "
+      54.0°    | 464.1 "   |     86.1°    | 181.3 "
+               |           |     88.5°    | 201.3 "
+  -------------+-----------+--------------+----------
+
+The conditions for the formation of these two compounds, by passing ammonia
+over silver chloride, to which reference has already been made, will be
+readily understood from the above tables. In the case of the triammonia
+mono-chloride, the dissociation pressure becomes equal to atmospheric
+pressure at a temperature of about 20°; above this temperature, therefore,
+it cannot be formed by the action of ammonia at atmospheric pressure on
+silver chloride. The triammonia dichloride can, however, be formed, for its
+dissociation pressure at this temperature amounts to only 9 cm., and
+becomes equal to the atmospheric pressure only at a temperature of about
+68°; and this temperature, therefore, constitutes the limit above which no
+combination can take place between silver chloride and ammonia under
+atmospheric pressure.
+
+Attention may be here drawn to the fact, to which reference will also be
+made later, that _two_ solid phases are necessary in order that the
+dissociation pressure at a given temperature shall be definite; _and for
+the exact definition of this pressure it is necessary to know, not merely
+what is the substance undergoing dissociation, but also what is the solid
+product of dissociation formed_. For the definition of the equilibrium, the
+latter is as important as the former. We shall presently find proof of this
+in the case {85} of an analogous class of phenomena, viz. the dissociation
+of salt hydrates.
+
+Salts with Water of Crystallization.--In the case of the dehydration of
+crystalline salts containing water of crystallization, we meet with
+phenomena which are in all respects similar to those just studied. A salt
+hydrate on being heated dissociates into a lower hydrate (or anhydrous
+salt) and water vapour. Since we are dealing with two components--salt and
+water[154]--in three phases, viz. hydrate _a_, hydrate _b_ (or anhydrous
+salt), and vapour, the system is univariant, and to each temperature there
+will correspond a certain, definite vapour pressure (the dissociation
+pressure), which will be independent of the relative or absolute amounts of
+the phases, _i.e._ of the amount of hydrate which has already undergone
+dissociation or dehydration.
+
+[Illustration: FIG. 19.]
+
+The constancy of the dissociation pressure had been proved experimentally
+by several investigators[155] a number of years before the theoretical
+basis for its necessity had been given. In the case of salts capable of
+forming more than one hydrate, we should obtain a series of dissociation
+curves (_pt_-curves), as in the case of the different hydrates of copper
+sulphate. In Fig. 19 there are represented diagrammatically the
+vapour-pressure curves of the following univariant systems of copper
+sulphate and water:--
+
+  Curve OA: CuSO_{4},5H_{2}O <--> CuSO_{4},3H_{2}O + 2H_{2}O.
+  Curve OB: CuSO_{4},3H_{2}O <--> CuSO_{4},H_{2}O + 2H_{2}O.
+  Curve OC: CuSO_{4},H_{2}O   <--> CuSO_{4} + H_{2}O.
+
+Let us now follow the changes which take place on {86} increasing the
+pressure of the aqueous vapour in contact with anhydrous copper sulphate,
+the temperature being meanwhile maintained constant. If, starting from the
+point D, we slowly add water vapour to the system, the pressure will
+gradually rise, without formation of hydrate taking place; for at pressures
+below the curve OC only the anhydrous salt can exist. At E, however, the
+hydrate CuSO_{4},H_{2}O will be formed, and as there are now three phases
+present, viz. CuSO_{4}, CuSO_{4},H_{2}O, and vapour, the system becomes
+_univariant_; and since the temperature is constant, the pressure must also
+be constant. Continued addition of vapour will result merely in an increase
+in the amount of the hydrate, and a decrease in the amount of the anhydrous
+salt. When the latter has entirely disappeared, _i.e._ has passed into
+hydrated salt, the system again becomes _bivariant_, and passes along the
+line EF; the pressure gradually increases, therefore, until at F the
+hydrate 3H_{2}O is formed, and the system again becomes univariant; the
+three phases present are CuSO_{4},H_{2}O, CuSO_{4},3H_{2}O, vapour. The
+pressure will remain constant, therefore, until the hydrate 1H_{2}O has
+disappeared, when it will again increase till G is reached; here the
+hydrate 5H_{2}O is formed, and the pressure once more remains constant
+until the complete disappearance of the hydrate 3H_{2}O has taken place.
+
+Conversely, on dehydrating CuSO_{4},5H_{2}O at constant temperature, we
+should find that the pressure would maintain the value corresponding to the
+dissociation pressure of the system
+CuSO_{4},5H_{2}O--CuSO_{4},3H_{2}O--vapour, until all the hydrate 5H_{2}O
+had disappeared; further removal of water would then cause the pressure to
+fall _abruptly_ to the pressure of the system
+CuSO_{4},3H_{2}O--CuSO_{4},H_{2}O--vapour, at which value it would again
+remain constant until the tri-hydrate had passed into the monohydrate, when
+a further sudden diminution of the pressure would occur. This behaviour is
+represented diagrammatically in Fig. 20, the values of the pressure being
+those at 50°.
+
+Efflorescence.--From Fig. 19 we are enabled to predict the conditions under
+which a given hydrated salt will effloresce when exposed to the air. We
+have just learned that copper {87} sulphate pentahydrate, for example, will
+not be formed unless the pressure of the aqueous vapour reaches a certain
+value; and that conversely, if the vapour pressure falls below the
+dissociation pressure of the pentahydrate, this salt will undergo
+dehydration. From this, then, it is evident that a crystalline salt hydrate
+will effloresce when exposed to the air, if the partial pressure of the
+water vapour in the air is lower than the dissociation pressure of the
+hydrate. At the ordinary temperature the dissociation pressure of copper
+sulphate is less than the pressure of water vapour in the air, and
+therefore copper sulphate does not effloresce. In the case of sodium
+sulphate decahydrate, however, the dissociation pressure is greater than
+the normal vapour pressure in a room, and this salt therefore effloresces.
+
+[Illustration: FIG. 20.]
+
+Indefiniteness of the Vapour Pressure of a Hydrate.--Reference has already
+been made (p. 84), in the case of the ammonia compounds of the metal
+chlorides, to the importance of the solid product of dissociation for the
+definition of the dissociation pressure. Similarly also in the case of a
+hydrated salt. A salt hydrate in contact with vapour constitutes only a
+bivariant system, and can exist therefore at different values of
+temperature and pressure of vapour, as is seen from the diagram, Fig. 19.
+Anhydrous copper sulphate can exist in contact with water vapour at all
+values of temperature and pressure lying in the field below the curve OC;
+and the hydrate CuSO_{4},H_{2}O can exist in contact with vapour at all
+values of temperature and pressure in the field BOC. Similarly, each of the
+other hydrates can exist in contact with vapour at different values of
+temperature and pressure.
+
+From the Phase Rule, however, we learn that, in order that at a given
+temperature the pressure of a two-component system {88} may be constant,
+there must be three phases present. Strictly, therefore, we can speak only
+of the vapour pressure of a _system_; and since, in the cases under
+discussion, the hydrates dissociate into a solid and a vapour, any
+statement as to the vapour pressure of a hydrate has a definite meaning
+_only when the second solid phase produced by the dissociation is given_.
+The everyday custom of speaking of the vapour pressure of a hydrated salt
+acquires a meaning only through the assumption, tacitly made, that the
+second solid phase, or the solid produced by the dehydration of the
+hydrate, is the _next lower_ hydrate, where more hydrates than one exist.
+That a hydrate always dissociates in such a way that the next lower hydrate
+is formed is, however, by no means certain; indeed, cases have been met
+with where apparently the anhydrous salt, and not the lower hydrate (the
+existence of which was possible), was produced by the dissociation of the
+higher hydrate.[156]
+
+That a salt hydrate can exhibit different vapour pressures according to the
+solid product of dissociation, can not only be proved theoretically, but it
+has also been shown experimentally to be a fact. Thus CaCl_{2},6H_{2}O can
+dissociate into water vapour and either of two lower hydrates, each
+containing four molecules of water of crystallization, and designated
+respectively as CaCl_{2},4H_{2}O[alpha], and CaCl_{2},4H_{2}O[beta].
+Roozeboom[157] has shown that the vapour pressure which is obtained differs
+according to which of these two hydrates is formed, as can be seen from the
+following figures:--
+
+  -------------+----------------------------------------------------------
+               |                  Pressure of System.
+  Temperature. +-----------------------------+----------------------------
+               | CaCl_{2},6H_{2}O; CaCl_{2}, | CaCl_{2},6H_{2}O; CaCl_{2},
+               | 4H_{2}O[alpha]; vapour.     | 4H_{2}O[beta]; vapour.
+  -------------+-----------------------------+----------------------------
+       -15°    |         0.027 cm.           |         0.022 cm.
+         0     |         0.092  "            |         0.076  "
+       +10     |         0.192  "            |         0.162  "
+        20     |         0.378  "            |         0.315  "
+        25     |         0.508  "            |         0.432  "
+        29.2   |            --               |         0.567  "
+        29.8   |         0.680  "            |            --
+  -------------+-----------------------------+---------------------------
+
+{89}
+
+By reason of the non-recognition of the importance of the solid
+dissociation product for the definition of the dissociation pressure of a
+salt hydrate, many of the older determinations lose much of their value.
+
+Suspended Transformation.--Just as in systems of one component we found
+that a new phase was not necessarily formed when the conditions for its
+existence were established, so also we find that even when the vapour
+pressure is lowered below the dissociation pressure of a system,
+dissociation does not necessarily occur. This is well known in the case of
+Glauber's salt, first observed by Faraday. Undamaged crystals of
+Na_{2}SO_{4},10H_{2}O could be kept unchanged in the open air, although the
+vapour pressure of the system Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
+is greater than the ordinary pressure of aqueous vapour in the air. That is
+to say, the possibility of the formation of the new phase Na_{2}SO_{4} was
+given; nevertheless this new phase did not appear, and the system therefore
+became metastable, or unstable with respect to the anhydrous salt. When,
+however, a trace of the new phase--the anhydrous salt--was brought in
+contact with the hydrate, transformation occurred; the hydrate effloresced.
+
+The possibility of suspended transformation or the non-formation of the new
+phases must also be granted in the case where the vapour pressure is raised
+above that corresponding to the system hydrate--anhydrous salt (or lower
+hydrate)--vapour; in this case the formation of the higher hydrate becomes
+a possibility, but not a certainty. Although there is no example of this
+known in the case of hydrated salts, the suspension of the transformation
+has been observed in the case of the compounds of ammonia with the metal
+chlorides (p. 82). Horstmann,[158] for example, found that the pressure of
+ammonia in contact with 2AgCl,3NH_{3} could be raised to a value higher
+than the dissociation pressure of AgCl,3NH_{3} without this compound being
+formed. We see, therefore, that even when the existence of the higher
+compound in contact with the lower became possible, the higher compound was
+not immediately formed.
+
+Range of Existence of Hydrates.--In Fig. 19 the vapour {90} pressure curves
+of the different hydrates of copper sulphate are represented as maintaining
+their relative positions throughout the whole range of temperatures. But
+this is not necessarily the case. It is possible that at some temperature
+the vapour pressure curve of a lower hydrate may cut that of a higher
+hydrate. At temperatures above the point of intersection, the lower hydrate
+would have a higher vapour pressure than the higher hydrate, and would
+therefore be metastable with respect to the latter. The range of stable
+existence of the lower hydrate would therefore end at the point of
+intersection. This appears to be the case with the two hydrates of sodium
+sulphate, to which reference will be made later.[159]
+
+Constancy of Vapour Pressure and the Formation of Compounds.--We have seen
+in the case of the salt hydrates that the continued addition of the vapour
+phase to the system caused an increase in the pressure until at a definite
+value of the pressure a hydrate is formed; the pressure then becomes
+constant, and remains so, until one of the solid phases has disappeared.
+Conversely, on withdrawing the vapour phase, the pressure remained constant
+so long as any of the dissociating compound was present, independently of
+the degree of the decomposition (p. 86). This behaviour, now, has been
+employed for the purpose of determining whether or not definite chemical
+compounds are formed. Should compounds be formed between the vapour phase
+and the solid, then, on continued addition or withdrawal of the vapour
+phase, it will be found that the vapour pressure remains constant for a
+certain time, and will then suddenly assume a new value, at which it will
+again remain constant. By this method, Ramsay[160] found that no definite
+hydrates were formed in the case of ferric and aluminium oxides, but that
+two are formed in the case of lead oxide, viz. 2PbO,H_{2}O and 3PbO,H_{2}O.
+
+The method has also been applied to the investigation of the so-called
+palladium hydride,[161] and the results obtained appear to show that no
+compound is formed. Reference will, however, be made to this case later
+(Chap. X.).
+
+{91}
+
+Measurement of the Vapour Pressure of Hydrates.--For the purpose of
+measuring the small pressures exerted by the vapour of salt hydrates, use
+is very generally made of a differential manometer called the
+_Bremer-Frowein tensimeter_.[162]
+
+This apparatus has the form shown in Fig. 21. It consists of a U-tube, the
+limbs of which are bent close together, and placed in front of a millimetre
+scale. The bend of the tube is filled with oil or other suitable liquid,
+_e.g._ bromonaphthalene. If it is desired to measure the dissociation
+pressure of, say, a salt hydrate, concentrated sulphuric acid is placed in
+the flask _e_, and a quantity of the hydrate, well dried and powdered,[163]
+in the bulb d. The necks of the bulbs _d_ and _e_ are then sealed off.
+Since, as we have learned, suspended transformation may occur, it is
+advisable to first partially dehydrate the salt, in order to ensure the
+presence of the second solid product of dissociation; the value of the
+dissociation pressure being independent of the degree of dissociation of
+the hydrate (p. 86). The small bulbs _d_ and _e_ having been filled, the
+apparatus is placed on its side, so as to allow the liquid to run from the
+bend of the tube into the bulbs _a_ and _b_; it is then exhausted through
+_f_ by means of a mercury pump, and sealed off. The apparatus is now placed
+in a perpendicular position in a thermostat, and kept at constant
+temperature until equilibrium is established. Since the vapour pressure on
+the side containing the sulphuric acid may be regarded as zero, the
+difference in level of the two surfaces of liquid in the U-tube gives
+directly the dissociation pressure of the hydrate in terms of the
+particular liquid employed; if the density of the latter is known, the
+pressure can then be calculated to cm. of mercury.
+
+[Illustration: FIG. 21.]
+
+       *       *       *       *       *
+
+
+{92}
+
+CHAPTER VI
+
+SOLUTIONS
+
+Definition.--In all the cases which have been considered in the preceding
+pages, the different phases--with the exception of the vapour
+phase--consisted of a single substance of definite composition, or were
+definite chemical individuals.[164] But this invariability of the
+composition is by no means imposed by the Phase Rule; on the contrary, we
+shall find in the examples which we now proceed to study, that the
+participation of phases of variable composition in the equilibrium of a
+system is in no way excluded. To such phases of variable composition there
+is applied the term _solution_. A solution, therefore, is to be defined as
+_a homogeneous mixture, the composition of which can undergo continuous
+variation within certain limits_; the limits, namely, of its
+existence.[165]
+
+From this definition we see that the term solution is not restricted to any
+particular physical state of substances, but includes within its range not
+only the liquid, but also the gaseous and solid states. We may therefore
+have solutions of gases in liquids, and of gases in solids; of liquids in
+liquids or in solids; of solids in liquids, or of solids in solids.
+Solutions of gases in gases are, of course, also possible; since, however,
+gas solutions never give rise to more than one phase, their {93} treatment
+does not come within the scope of the Phase Rule, which deals with
+heterogeneous equilibria.
+
+It should also be emphasized that the definition of solution given above,
+neither creates nor recognizes any distinction between solvent and
+dissolved substance (solute); and, indeed, a too persistent use of these
+terms and the attempt to permanently label the one or other of two
+components as the solvent or the solute, can only obscure the true
+relationships and aggravate the difficulty of their interpretation. In all
+cases it should be remembered that we are dealing with equilibria between
+two components (we confine our attention in the first instance to such),
+the solution being constituted of these components in variable and varying
+amounts. The change from the case where the one component is in great
+excess (ordinarily called the solvent) to that in which the other component
+predominates, may be quite gradual, so that it is difficult or impossible
+to say at what point the one component ceases to be the solvent and becomes
+the solute. The adoption of this standpoint need not, however, preclude one
+from employing the conventional terms solvent and solute in ordinary
+language, especially when reference is made only to some particular
+condition of equilibrium of the system, when the concentration of the two
+components in the solution is widely different.
+
+SOLUTIONS OF GASES IN LIQUIDS.
+
+As the first class of solutions to which we shall turn our attention, there
+may be chosen the solutions of gases in liquids, or the equilibria between
+a liquid and a gas. These equilibria really constitute a part of the
+equilibria to be studied more fully in Chapter VIII.; but since the
+two-phase systems formed by the solutions of gases in liquids are among the
+best-known of the two-component systems, a short section may be here
+allotted to their treatment.
+
+When a gas is passed into a liquid, absorption takes place to a greater or
+less extent, and a point is at length reached when the liquid absorbs no
+more of the gas; a condition of equilibrium is attained, and the liquid is
+said to be saturated {94} with the gas. In the light of the Phase Rule,
+now, such a system is bivariant (two components in two phases); and two of
+the variable factors, pressure, temperature, and concentration of the
+components, must therefore be chosen in order that the condition of the
+system may be defined. If the concentration and the temperature are fixed,
+then the pressure is also defined; or under given conditions of temperature
+and pressure, the concentration of the gas in the solution must have a
+definite value. If, however, the temperature alone is fixed, the
+concentration and the pressure can alter; a fact so well known that it does
+not require to be further insisted on.
+
+As to the way in which the solubility of a gas in a liquid varies with the
+pressure, the Phase Rule of course does not state; but guidance on this
+point is again yielded by the theorem of van't Hoff and Le Chatelier. Since
+the absorption of a gas is in all cases accompanied by a diminution of the
+total volume, this process must take place with increase of pressure. This,
+indeed, is stated in a quantitative manner in the law of Henry, according
+to which the amount of a gas absorbed is proportional to the pressure. But
+this law must be modified in the case of gases which are very readily
+absorbed; the _direction of change_ of concentration with the pressure
+will, however, still be in accordance with the theorem of Le Chatelier.
+
+If, on the other hand, the pressure is fixed, then the concentration will
+vary with the temperature; and since the absorption of gases is in all
+cases accompanied by the evolution of heat, the solubility is found, in
+accordance with the theorem of Le Chatelier, to diminish with rise of
+temperature.
+
+In considering the changes of pressure accompanying changes of
+concentration and temperature, a distinction must be drawn between the
+total pressure and the partial pressure of the dissolved gas, in cases
+where the solvent is volatile. In these cases, the law of Henry applies not
+to the total pressure of the vapour, but only to the partial pressure of
+the dissolved gas. {95}
+
+SOLUTIONS OF LIQUIDS IN LIQUIDS.
+
+When mercury and water are brought together, the two liquids remain side by
+side without mixing. Strictly speaking, mercury undoubtedly dissolves to a
+certain extent in the water, and water no doubt dissolves, although to a
+less extent, in the mercury; the amount of substance passing into solution
+is, however, so minute, that it may, for all practical purposes, be left
+out of account, so long as the temperature does not rise much above the
+ordinary.[166] On the other hand, if alcohol and water be brought together,
+complete miscibility takes place, and one homogeneous solution is obtained.
+Whether water be added in increasing quantities to pure alcohol, or pure
+alcohol be added in increasing amount to water, at no point, at no degree
+of concentration, is a system obtained containing more than one liquid
+phase. At the ordinary temperature, water and alcohol can form only two
+phases, liquid and vapour. If, however, water be added to ether, or if
+ether be added to water, solution will not occur to an indefinite extent;
+but a point will be reached when the water or the ether will no longer
+dissolve more of the other component, and a further addition of water on
+the one hand, or ether on the other, will cause the formation of two liquid
+layers, one containing excess of water, the other excess of ether. We
+shall, therefore, expect to find all grades of miscibility, from almost
+perfect immiscibility to perfect miscibility, or miscibility in all
+proportions. In cases of perfect immiscibility, the components do not
+affect one another, and the system therefore remains unchanged. Such cases
+do not call for treatment here. We have to concern ourselves here only with
+the second and third cases, viz. with cases of complete and of partial
+miscibility. There is no essential difference between the two classes, for,
+as we shall see, {96} the one passes into the other with change of
+temperature. The formal separation into two groups is based on the
+miscibility relations at ordinary temperatures.
+
+Partial or Limited Miscibility.--In accordance with the Phase Rule, a pure
+liquid in contact with its vapour constitutes a univariant system. If,
+however, a small quantity of a second substance is added, which is capable
+of dissolving in the first, a bivariant system will be obtained; for there
+are now two components and, as before, only two phases--the homogeneous
+liquid solution and the vapour. At constant temperature, therefore, both
+the composition of the solution and the pressure of the vapour can undergo
+change; or, if the composition of the solution remains unchanged, the
+pressure and the temperature can alter. If the second (liquid) component is
+added in increasing amount, the liquid will at first remain homogeneous,
+and its composition and pressure will undergo a continuous change; when,
+however, the concentration has reached a definite value, solution no longer
+takes place; two liquid phases are produced. Since there are now three
+phases present, two liquids and vapour, the system is univariant; at a
+given temperature, therefore, the concentration of the components in the
+two liquid phases, as well as the vapour pressure, must have definite
+values. Addition of one of the components, therefore, cannot alter the
+concentrations or the pressure, but can only cause a change in the relative
+amounts of the phases.
+
+The two liquid phases can be regarded, the one as a solution of the
+component I. in component II., the other as a solution of component II. in
+component I. If the pressure is maintained constant, then to each
+temperature there will correspond a definite concentration of the
+components in the two liquid phases; and addition of excess of one will
+merely alter the relative amounts of the two solutions. As the temperature
+changes, the composition of the two solutions will change, and there will
+therefore be obtained two solubility curves, one showing the solubility of
+component I. in component II., the other showing the solubility of
+component II. in component I. Since heat may be either evolved or absorbed
+when one liquid dissolves in another, the solubility may diminish or
+increase {97} with rise of temperature. The two solutions which at a given
+temperature correspond to one another are known as _conjugate solutions_.
+
+The solubility relations of partially miscible liquids have been studied by
+Guthrie,[167] and more especially by Alexejeff[168] and by Rothmund.[169] A
+considerable variety of curves have been obtained, and we shall therefore
+discuss only a few of the different cases which may be taken as typical of
+the rest.
+
+Phenol and Water.--When phenol is added to water at the ordinary
+temperature, solution takes place, and a homogeneous liquid is produced.
+When, however, the concentration of the phenol in the solution has risen to
+about 8 per cent., phenol ceases to be dissolved; and a further addition of
+it causes the formation of a second liquid phase, which consists of excess
+of phenol and a small quantity of water. In ordinary language it may be
+called a solution of water in phenol. If now the temperature is raised,
+this second liquid phase will disappear, and a further amount of phenol
+must be added in order to produce a separation of the liquid into two
+layers. In this way, by increasing the amount of phenol and noting the
+temperature at which the two layers disappear, the so-called solubility
+curve of phenol in water can be obtained. By noting the change of the
+solubility with the temperature in this manner, it is found that at all
+temperatures below 68.4°, the addition of more than a certain amount of
+phenol causes the formation of two layers; at temperatures above this,
+however, two layers cannot be formed, no matter how much phenol is added.
+At temperatures above 68.4°, therefore, water and phenol are miscible in
+all proportions.
+
+On the other hand, if water is added to phenol at the ordinary temperature,
+a liquid is produced which consists chiefly of phenol, and on increasing
+the amount of water beyond a certain point, two layers are formed. On
+raising the temperature these two layers disappear, and a homogeneous
+solution is again obtained. The phenomena are exactly analogous to those
+already described. Since, now, in the second {98} case the concentration of
+the phenol in the solution gradually decreases, while in the former case it
+gradually increases, a point must at length be reached at which the
+composition of the two solutions becomes the same. On mixing the two
+solutions, therefore, one homogeneous liquid will be obtained. But the
+point at which two phases become identical is called a critical point, so
+that, in accordance with this definition, the temperature at which the two
+solutions of phenol and water become identical may be called the _critical
+solution temperature_, and the concentration at this point may be called
+the _critical concentration_.
+
+[Illustration: FIG. 22.]
+
+From what has been said above, it will be seen that at any temperature
+below the critical solution temperature, two conjugate solutions containing
+water and phenol in different concentration can exist together, one
+containing excess of water, the other excess of phenol. The following table
+gives the composition of the two layers, and the values are represented
+graphically in Fig. 22.[170]
+
+  PHENOL AND WATER.
+
+  C_{1} is the percentage amount of phenol in the first layer.
+  C_{2}         "          "          "           second layer.
+  -------------+--------+--------
+  Temperature. |  C_{1}.|  C_{2}.
+  -------------+--------+--------
+      20°      |   8.5  |  72.2
+      30°      |   8.7  |  69.9
+      40°      |   9.7  |  66.8
+      50°      |  12.0  |  62.7
+      55°      |  14.2  |  60.0
+      60°      |  17.5  |  56.2
+      65°      |  22.7  |  49.7
+      68.4°    |  36.1  |  36.1
+  -------------+--------+--------
+
+{99}
+
+The critical solution temperature for phenol and water is 68.4°, the
+critical concentration 36.1 per cent. of phenol. At all temperatures above
+68.4°, only homogeneous solutions of phenol and water can be obtained;
+water and phenol are then miscible in all proportions.
+
+At the critical solution point the system exists in only two phases--liquid
+and vapour. It ought, therefore, to possess two degrees of freedom. The
+restriction is, however, imposed that the composition of the two liquid
+phases, coexisting at a point infinitely near to the critical point,
+becomes the same, and this disposes of one of the degrees of freedom. The
+system is therefore univariant; and at a given temperature the pressure
+will have a definite value. Conversely, if the pressure is fixed (as is the
+case when the system is under the pressure of its own vapour), then the
+temperature will also be fixed; that is, the critical solution temperature
+has a definite value depending only on the substances. If the vapour phase
+is omitted, the temperature will alter with the pressure; in this case,
+however, as in the case of other condensed systems, the effect of pressure
+is slight.
+
+From Fig. 22 it is easy to predict the effect of bringing together water
+and phenol in any given quantities at any temperature. Start with a
+solution of phenol and water having the composition represented by the
+point _x_. If to this solution phenol is added at constant temperature, it
+will dissolve, and the composition of the solution will gradually change,
+as shown by the dotted line _xy_. When, however, the concentration has
+reached the value represented by the point _y_, two liquid layers will be
+formed, the one solution having the composition represented by _y_, the
+other that represented by _y'_. The system is now univariant, and on
+further addition of phenol, the composition of the two liquid phases will
+remain unchanged, but their relative amounts will alter. The phase richer
+in phenol will increase in amount; that richer in water will decrease, and
+ultimately disappear, and there will remain the solution _y'_. Continued
+addition of phenol will then lead to the point _x'_, there being now only
+one liquid phase present.
+
+Since the critical solution point represents the highest temperature at
+which two liquid phases consisting of phenol and {100} water can exist
+together, these two substances can be brought together in any amount
+whatever at temperatures higher than 68.4°, without the formation of two
+layers. It will therefore be possible to pass from a system represented by
+_x_ to one represented by _x'_, without at any time two liquid phases
+appearing. Starting with _x_, the temperature is first raised above the
+critical solution temperature; phenol is then added until the concentration
+reaches the point _x__{2}. On allowing the temperature to fall, the system
+will then pass into the condition represented by _x'_.
+
+[Illustration: FIG. 23.]
+
+Methylethylketone and Water.--In the case just described, the solubility of
+each component in the other increased continuously with the temperature.
+There are, however, cases where a maximum or minimum of solubility is
+found, _e.g._ methylethylketone and water. The curve which represents the
+equilibria between these two substances is given in Fig. 23, the
+concentration values being contained in the following table:[171]--
+
+  METHYLETHYLKETONE AND WATER.
+
+  --------------+-----------------+-----------------
+  Temperature.  | C_{1} per cent. |  C_{2} per cent.
+  --------------+-----------------+-----------------
+       -10°     |      34.5       |     89.7
+       +10°     |      26.1       |     90.0
+        30°     |      21.9       |     89.9
+        50°     |      17.5       |     89.0
+        70°     |      16.2       |     85.7
+        90°     |      16.1       |     84.8
+       110°     |      17.7       |     80.0
+       130°     |      21.8       |     71.9
+       140°     |      26.0       |     64.0
+       151.8°   |      44.2       |     44.2
+  --------------+-----------------+-----------------
+
+{101}
+
+These numbers and Fig. 23 show clearly the occurrence of a minimum in the
+solubility of the ketone in water, and also a minimum (at about 10°) in the
+solubility of water in methylethylketone. Minima of solubility have also
+been found in other cases.
+
+[Illustration: FIG. 24.]
+
+Triethylamine and Water.--Although in most of the cases studied the
+solubility of one liquid in another increases with rise of temperature,
+this is not so in all cases. Thus, at temperatures below 18°, triethylamine
+and water mix together in all proportions; but, on raising the temperature,
+the homogeneous solution becomes turbid and separates into two layers. In
+this case, therefore, the critical solution temperature is found in the
+direction of lower temperature, not in the direction of higher.[172] This
+behaviour is clearly shown by the graphic representation in Fig. 24, and
+also by the numbers in the following table:--
+
+  TRIETHYLAMINE AND WATER.
+
+  -------------+-----------------+----------------
+  Temperature. | C_{1} per cent. |  C_{2} per cent.
+  -------------+-----------------+----------------
+       70°     |      1.6        |     --
+       50°     |      2.9        |     --
+       30°     |      5.6        |     96
+       25°     |      7.3        |     95.5
+       20°     |     15.5        |     73
+      ±18.5°   |    ±30          |    ±30
+  -------------+-----------------+----------------
+
+General Form of Concentration-Temperature Curve.--From the preceding
+figures it will be seen that the general {102} form of the solubility curve
+is somewhat parabolic in shape; in the case of triethylamine and water, the
+closed end of the curve is very flat. Since for all liquids there is a
+point (critical point) at which the liquid and gaseous states become
+identical, and since all gases are miscible in all proportions, it follows
+that there must be some temperature at which the liquids become perfectly
+miscible. In the case of triethylamine and water, which has just been
+considered, there must therefore be an upper critical solution temperature,
+so that the complete solubility relations would be represented by a closed
+curve of an ellipsoidal aspect. An example of such a curve is furnished by
+nicotine and water. At temperatures below 60° and above 210°, nicotine and
+water mix in all proportions.[173] Although it is possible that this is the
+general form of the curve for all pairs of liquids, there are as yet
+insufficient data to prove it.
+
+With regard to the closed end of the curve it may be said that it is
+continuous; the critical solution point is not the intersection of two
+curves, for such a break in the continuity of the curve could occur only if
+there were some discontinuity in one of the phases. No such discontinuity
+exists. The curve is, therefore, not to be considered as two solubility
+curves cutting at a point; it is a curve of equilibrium between two
+components, and so long as the phases undergo continuous change, the curve
+representing the equilibrium must also be continuous. As has already been
+emphasized, a distinction between solvent and solute is merely conventional
+(p. 93).
+
+Pressure-Concentration Diagram.--In considering the pressure-concentration
+diagram of a system of two liquid components, a distinction must be drawn
+between the total pressure of the system and the partial pressures of the
+components. On studying the total pressure of a system, it is found that
+two cases can be obtained.[174]
+
+So long as there is only one liquid phase, the system is bivariant. The
+pressure therefore can change with the concentration and the temperature.
+If the temperature is maintained {103} constant, the pressure will vary
+only with the concentration, and this variation can therefore be
+represented by a curve. If, however, two liquid phases are formed, the
+system becomes univariant: and if one of the variables, say the
+temperature, is arbitrarily fixed, the system no longer possesses any
+degree of freedom. _When two liquid phases are formed, therefore, the
+concentrations and the vapour pressure have definite values, which are
+maintained so long as the two liquid phases are present_; the temperature
+being supposed constant.
+
+In Fig. 25 is given a diagrammatic representation of the two kinds of
+pressure-concentration curves which have so far been obtained. In the one
+case, the vapour pressure of the invariant system (at constant temperature)
+lies higher than the vapour pressure of either of the pure components; a
+phenomenon which is very generally found in the case of partially miscible
+liquids, _e.g._ ether and water.[175] Accordingly, by the addition of water
+to ether, or of ether to water, there is an increase in the _total_ vapour
+pressure of the system.
+
+[Illustration: FIG. 25.]
+
+With regard to the second type, the vapour pressure of the systems with two
+liquid phases lies between that of the two single components. An example of
+this is found in sulphur dioxide and water.[176] On adding sulphur dioxide
+to water there is an increase of the total vapour pressure; but on adding
+water to liquid sulphur dioxide, the total vapour pressure is diminished.
+
+The case that the vapour pressure of the system with two {104} liquid
+phases is _less_ than that of each of the components is not possible.
+
+With regard to the _partial pressure_ of the components, the behaviour is
+more uniform. The partial pressure of one component is in all cases lowered
+by the addition of the other component, the diminution being approximately
+proportional to the amount added. If two liquid phases are present, the
+partial pressure of the components, as well as the total pressure, is
+constant, and is the same for both phases. That is to say, in the case of
+the two liquids, saturated solution of water in ether, and of ether in
+water, the partial pressure of the ether in the vapour in contact with the
+one solution is the same as that in the vapour over the other
+solution.[177]
+
+Complete Miscibility.--Although the phenomena of complete miscibility are
+here treated under a separate heading, it must not be thought that there is
+any essential difference between those cases where the liquids exhibit
+limited miscibility and those in which only one homogeneous solution is
+formed. As has been already pointed out, the solubility relations alter
+with the temperature; and liquids which at one temperature can dissolve in
+one another only to a limited extent, are found at some other temperature
+to possess the property of complete miscibility. Conversely, we may expect
+that liquids which at one temperature, say at the ordinary temperature, are
+miscible in all proportions, will be found at some other temperature to be
+only partially miscible. Thus, for example, it was found by Guthrie that
+ethyl alcohol and carbon disulphide, which are miscible in all proportions
+at the ordinary temperature, possess only limited miscibility at
+temperatures below -14.4°.[178] Nevertheless, it is doubtful if the
+critical solution temperature is in all cases experimentally realizable.
+
+Pressure-Concentration Diagram.--Since, in the cases of complete
+miscibility of two liquid components, there are never more than two phases
+present, the system must always be bivariant; and two of the variables
+pressure, temperature or concentration of the components, must be
+arbitrarily chosen {105} before the system becomes defined. For this reason
+the Phase Rule affords only a slight guidance in the study of such
+equilibria; and we shall therefore not enter in detail into the behaviour
+of these homogeneous mixtures. All that the Phase Rule can tell us in
+connection with these solutions, is that at constant temperature the vapour
+pressure of the solution varies with the composition of the liquid phase;
+and if the composition of the liquid phase remains unchanged, the pressure
+also must remain unchanged. This constancy of composition is exhibited not
+only by pure liquids, but also by liquid solutions in all cases where the
+vapour pressure of the solution reaches a maximum or minimum value. This is
+the case, for example, with mixtures of constant boiling point.[179]
+
+       *       *       *       *       *
+
+
+{106}
+
+CHAPTER VII
+
+SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE
+
+General.--When a solid is brought into contact with a liquid in which it
+can dissolve, a certain amount of it passes into solution; and the process
+continues until the concentration reaches a definite value independent of
+the amount of solid present. A condition of equilibrium is established
+between the solid and the solution; the solution becomes _saturated_. Since
+the number of components is two, and the number of phases three, viz.
+solid, liquid solution, vapour, the system is univariant. If, therefore,
+one of the factors, pressure, temperature, or concentration of the
+components (in the solution[180]), is arbitrarily fixed, the state of the
+system becomes perfectly defined. Thus, at any given temperature, the
+vapour pressure of the system and the concentration of the components have
+a definite value. If the temperature is altered, the vapour pressure and
+also, in general, the concentration will undergo change. Likewise, if the
+pressure varies, while the system is isolated so that no heat can pass
+between it and its surroundings, the concentration and the temperature must
+also undergo variation until they attain values corresponding to the
+particular pressure.
+
+That the temperature has an influence, sometimes a very considerable
+influence, on the amount of substance passing into solution, is
+sufficiently well known; the effect of pressure, although less apparent, is
+no less certain. If at any given temperature the volume of the vapour phase
+is diminished, {107} vapour will condense to liquid, in order that the
+pressure may remain constant, and so much of the solid will pass into
+solution that the concentration may remain unchanged; for, so long as the
+three phases are present, the state of the system cannot alter. If,
+however, one of the phases, _e.g._ the vapour phase, disappears, the system
+becomes bivariant; at any given temperature, therefore, there may be
+different values of concentration and pressure.
+
+The direction in which change of concentration will occur with change of
+pressure can be predicted by means of the theorem of Le Chatelier, if it is
+known whether solution is accompanied by increase or diminution of the
+total volume. If diminution of the total volume of the system occurs on
+solution, increase of pressure will increase the solubility; in the reverse
+case, increase of pressure will diminish the solubility.
+
+This conclusion has also been verified by experiment, as is shown by the
+following figures.[181]
+
+  ---------------------------------------------------------------
+                    |Change of       |  Solubility (at 18°) (grams salt
+                    |volume by       |  in 1 gram of solution).
+                    |dissolving 1 gm.|---------------------------
+       Salt.        |of salt in the  |          |
+                    |saturated       | Pressure | Pressure
+                    |solution.       | = 1 atm. | = 500 atm.
+  ------------------+----------------+----------+----------------
+  Sodium chloride   |     -0.07      |  0.264   |   0.270
+  Ammonium chloride |     +0.10      |  0.272   |   0.258
+  Alum              |     -0.067     |  0.115   |   0.142
+                    |                |          |(_p_ = 400 atm.)
+  -------------------------------------------------------------
+
+As can be seen, a large increase of the pressure brings about a no more
+than appreciable alteration of the solubility; a result which is due, as in
+the case of the alteration of the fusion point with the pressure, to the
+small change in volume accompanying solution or increase of pressure. For
+all practical purposes, therefore, the solubility as determined under
+atmospheric pressure may be taken as equal to the true {108} solubility,
+that is, the solubility when the system is under the pressure of its own
+vapour.
+
+The Saturated Solution.--From what has been said above, it will be seen
+that the condition of saturation of a solution can be defined only with
+respect to a certain solid phase; if no solid is present, the system is
+undefined, for it then consists of only two phases, and is therefore
+bivariant. Under such circumstances not only can there be at one given
+temperature solutions of different concentration, all containing less of
+one of the components than when that component is present in the solid
+form, but there can also exist solutions containing more of that component
+than corresponds to the equilibrium when the solid is present. In the
+former case the solutions are _unsaturated_, in the latter case they are
+_supersaturated with respect to a certain solid phase_; in themselves, the
+solutions are stable, and are neither unsaturated nor supersaturated.
+Further, if the solid substance can exist in different allotropic
+modifications, the particular form of the substance which is in equilibrium
+with the solution must be known, in order that the statement of the
+solubility may be definite; for each form has its own solubility, and, as
+we shall see presently, the less stable form has the greater solubility
+(cf. p. 47). In all determinations of the solubility, therefore, not only
+must the concentration of the components in the solution be determined, but
+equal importance should be attached to the characterisation of the solid
+phase present.
+
+In this connection, also, one other point may be emphasised. For the
+production of the equilibrium between a solid and a liquid, time is
+necessary, and this time not only varies with the state of division of the
+solid and the efficiency of the stirring, but is also dependent on the
+nature of the substance.[182] Considerable care must therefore be taken
+that sufficient time is allowed for equilibrium to be established. Such
+care is more especially needful when changes may occur in the solid phase,
+and neglect of it has greatly diminished the value of many of the older
+determinations of solubility.
+
+Form of the Solubility Curve.--The solubility curve--that {109} is, the
+curve representing the change of concentration of the components in the
+solution with the temperature--differs markedly from the curve of vapour
+pressure (p. 63), in that it possesses no general form, but may vary in the
+most diverse manner. Not only may the curve have an almost straight and
+horizontal course, or slope or curve upwards at varying angles; but it may
+even slope downwards, corresponding to a decrease in the solubility with
+rise of temperature; may exhibit maxima or minima of solubility, or may, as
+in the case of some hydrated salts, pass through a point of maximum
+temperature. In the latter case the salt may possess two values of
+solubility at the same temperature. We shall consider these cases in the
+following chapter.
+
+[Illustration: FIG. 26.]
+
+The great variety of form shown by solubility curves is at once apparent
+from Fig. 26, in which the solubility curves of various substances (not,
+however, drawn to scale) are reproduced.[183]
+
+Varied as is the form of the solubility curve, its _direction_,
+nevertheless, can be predicted by means of the theorem of van't Hoff and Le
+Chatelier; for in accordance with that theorem (p. 57) increase of
+solubility with the temperature must occur in those cases where the process
+of solution is accompanied by an _absorption_ of heat; and a decrease in
+the solubility with rise of temperature will be found in cases where
+solution occurs with _evolution_ of heat. Where there is no heat effect
+accompanying solution, {110} change of temperature will be without
+influence on the solubility; and if the sign of the heat of solution
+changes, the direction of the solubility curve must also change, _i.e._
+must show a maximum or minimum point. This has in all cases been verified
+by experiment.[184]
+
+In applying the theorem of Le Chatelier to the course of the solubility
+curve, it should be noted that by heat of solution there is meant, not the
+heat effect produced on dissolving the salt in a large amount of solvent
+(which is the usual signification of the expression), but the heat which is
+absorbed or evolved when the salt is dissolved in the almost saturated
+solution (the so-called last heat of solution). Not only does the heat
+effect in the two cases have a different value, but it may even have a
+different sign. A striking example of this is afforded by cupric chloride,
+as the following figures show:[185]--
+
+  -----------------------------------------------------------
+  Number of gram-molecules of        |
+  CuCl_{2}, 2H_{2}O dissolved in 198 |       Heat effect.
+  gram-molecules of water.           |
+  -----------------------------------+-----------------------
+              1                      |          +37 K
+              2.02                   |          +66 "
+              4.15                   |         +105 "
+              7.07                   |         +117 "
+              9.95                   |         +117 "
+             11                      |          +91 "
+             18.8                    |          -10 "
+             19.6                    |          -31 "
+             24.75                   |         -198 "
+  ------------------------------------------------------------
+
+In the above table the positive sign indicates evolution of heat, the
+negative sign, absorption of heat; and the values of the heat effect are
+expressed in centuple calories. Judging from the heat effect produced on
+dissolving cupric chloride in a large bulk of water, we should predict that
+the solubility of that salt would diminish with rise of temperature; as a
+matter of fact, it increases. This is in accordance with the fact that
+{111} the last heat of solution is _negative_ (as expressed above), _i.e._
+solution of the salt in the almost saturated solution is accompanied by
+absorption of heat. We are led to expect this from the fact that the heat
+of solution changes sign from positive to negative as the concentration
+increases; experiment also showed it to be the case.
+
+Despite its many forms, it should be particularly noted that the solubility
+curve of any substance is _continuous_, so long as the solid phase, or
+solid substance in contact with the solution, remains unchanged. If any
+"break" or discontinuous change in the direction of the curve occurs, it is
+a sign that the _solid phase has undergone alteration_. Conversely, if it
+is known that a change takes place in the solid phase, a break in the
+solubility curve can be predicted. We shall presently meet with examples of
+this.[186]
+
+A.--ANHYDROUS SALT AND WATER.
+
+The Solubility Curve.--In studying the equilibria in those systems of two
+components in which the liquid phase is a solution or phase of varying
+composition, we shall in the present chapter limit the discussion to those
+cases where no compounds are formed, but where the components crystallise
+out in the pure state. Since some of the best-known examples of such
+systems are yielded by the solutions of anhydrous salts in water, we shall
+first of all briefly consider some of the results which have been obtained
+with them.
+
+For the most part the solubility curves have been studied only at
+temperatures lying between 0° and 100°, the solid phase in contact with the
+solution being the anhydrous salt. For the representation of these
+equilibria, the concentration-temperature {112} diagram is employed, the
+concentration being expressed as the number of grams of the salt dissolved
+in 100 grams of water, or as the number of gram-molecules of salt in 100
+gram-molecules of water. The curves thus obtained exhibit the different
+forms to which reference has already been made. So long as the salt remains
+unchanged the curve will be continuous, but if the salt alters its form,
+then the solubility curve will show a break.
+
+[Illustration: FIG. 27.]
+
+Now, we have already seen in Chapter III. that certain substances are
+capable of existing in various crystalline forms, and these forms are so
+related to one another that at a given temperature the relative stability
+of each pair of polymorphic forms undergoes change. Since each crystalline
+variety of a substance must have its own solubility, there must be a break
+in the solubility curve at the temperature of transition of the two
+enantiotropic forms. At this point the two solubility curves must cut, for
+since the two forms are in equilibrium with respect to their vapour, they
+must also be in equilibrium with respect to their solutions. From the table
+on p. 63 it is seen that potassium nitrate, ammonium nitrate, silver
+nitrate, thallium nitrate, thallium picrate, are capable of existing in two
+or more different enantiotropic crystalline forms, the range of stability
+of these forms being limited by definite temperatures (transition
+temperature). Since the transition point is not altered by a solvent
+(provided the latter is not absorbed by the solid phase), we should find on
+studying the solubility of these substances in water that the solubility
+curve would exhibit a change in direction at the temperature of transition.
+As a matter of fact this has been verified, more especially in the case of
+ammonium nitrate[187] {113} and thallium picrate.[188] The following table
+contains the values of the solubility of ammonium nitrate obtained by
+Müller and Kaufmann, the solubility being expressed in gram-molecules
+NH_{4}NO_{3} in 100 gram-molecules of water. In Fig. 27 these results are
+represented graphically. The equilibrium point was approached both from the
+side of unsaturation and of supersaturation, and the condition of
+equilibrium was controlled by determinations of the density of the
+solution.
+
+  SOLUBILITY OF AMMONIUM NITRATE.
+
+  ------------------------------------------------------------
+   Temperature. | Solubility. | Temperature. | Solubility.
+  --------------+-------------+--------------+----------------
+      12.2°     |    34.50    |    32.7°     |    57.90
+      20.2°     |    43.30    |    34.0°     |    58.89
+      25.05°    |    48.19    |    35.0°     |    59.80
+      28.0°     |    51.86    |    36.0°     |    61.00
+      30.0°     |    54.40    |    37.5°     |    62.90
+      30.2°     |    54.61    |    38.0°     |    63.60
+      31.9°     |    57.20    |    39.0°     |    65.09
+      32.1°     |    57.60    |    40.0°     |    66.80
+  ------------------------------------------------------------
+
+From the graphic representation of the solubility given in Fig. 27, there
+is seen to be a distinct change in the direction of the curve at a
+temperature of 32°; and this break in the curve corresponds to the
+transition of the [beta]-rhombic into the [alpha]-rhombic form of ammonium
+nitrate (p. 63).
+
+Suspended Transformation and Supersaturation.--As has already been learned,
+the transformation of the one crystalline form into the other does not
+necessarily take place immediately the transition point has been passed;
+and it has therefore been found possible in a number of cases to follow the
+solubility curve of a given crystalline form beyond the point at which it
+ceases to be the most stable modification. Now, it will be readily seen
+from Fig. 27 that if the two solubility curves be prolonged beyond the
+point of intersection, the solubility of the less stable form is greater
+than that of the more stable. A solution, therefore, which is saturated
+with respect to the less stable form, _i.e._ which is in equilibrium with
+that form, is _supersaturated with respect to the more stable
+modification_. If, {114} therefore, a small quantity of the more stable
+form is introduced into the solution, the latter must deposit such an
+amount of the more stable form that the concentration of the solution
+corresponds to the solubility of the stable form at the particular
+temperature. Since, however, the solution is now _unsaturated_ with respect
+to the less stable variety, the latter, if present, must pass into
+solution; and the two processes, deposition of the stable and solution of
+the metastable form, must go on until the latter form has entirely
+disappeared and a saturated solution of the stable form is obtained. There
+will thus be a conversion, through the medium of the solvent, of the less
+stable into the more stable modification. This behaviour is of practical
+importance in the determination of transition points (_v._ Appendix).
+
+From the above discussion it will be seen how important is the statement of
+the solid phase for the definition of saturation and supersaturation.[189]
+
+Solubility Curve at Higher Temperatures.--On passing to the consideration
+of the solubility curves at higher temperatures, two chief cases must be
+distinguished.
+
+    (1) The two components in the fused state can mix in all proportions.
+
+    (2) The two components in the fused state cannot mix in all
+    proportions.
+
+1. _Complete Miscibility of the Fused Components._
+
+[Illustration: FIG. 28.]
+
+The best example of this which has been studied, so far as anhydrous salts
+and water are concerned, is that of silver nitrate and water. The
+solubility of this salt at temperatures {115} above 100° has been studied
+chiefly by Etard[190] and by Tilden and Shenstone.[191] The values obtained
+by Etard are given in the following table, and represented graphically in
+Fig. 28.
+
+  SOLUBILITY OF SILVER NITRATE.
+
+  ---------------------------------------------------
+      Temperature.    |Parts of dry salt in 100 parts
+                      |   of solution.
+  --------------------+------------------------------
+          -7°         |         46.2
+          -1°         |         52.1
+          +5°         |         56.3
+          10°         |         61.2
+          20°         |         67.8
+          40.5°       |         76.8
+          73°         |         84.0
+         135°         |         92.8
+         182°         |         96.9
+  ---------------------------------------------------
+
+In this figure the composition of the solution is expressed in parts of
+silver nitrate in 100 parts by weight of the solution, so that 100 per
+cent. represents pure silver nitrate. As can be seen, the solubility
+increases with the temperature. At a temperature of about 160° there should
+be a break in the curve due to change of crystalline form (p. 63). Such a
+change in the direction of the solubility curve, however, does not in any
+way alter the essential nature of the relationships discussed here, and may
+for the present be left out of account. On following the solubility curve
+of silver nitrate to higher temperatures, therefore, the concentration of
+silver nitrate in the solution gradually increases, until at last, at a
+temperature of 208°,[192] the melting point of pure silver nitrate is
+reached, and the concentration of the water has become zero. The curve
+throughout its whole extent represents the equilibrium between silver
+nitrate, solution, and vapour. Conversely, starting with pure silver
+nitrate in contact with the fused salt, addition of water will lower the
+melting point, _i.e._ will lower the temperature at which the solid salt
+can exist in contact with the liquid; {116} and the depression will be all
+the greater the larger the amount of water added. As the concentration of
+the water in the liquid phase is increased, therefore, the system will pass
+back along the curve from higher to lower temperatures, and from greater to
+smaller concentrations of silver nitrate in the liquid phase. The curve in
+Fig. 28 may, therefore, be regarded either as the solubility curve of
+silver nitrate in water, or as the freezing point curve for silver nitrate
+in contact with a solution consisting of that salt and water.
+
+As the temperature of the saturated solution falls, silver nitrate is
+deposited, and on lowering the temperature sufficiently a point will at
+last be reached at which ice also begins to separate out. Since there are
+now four phases co-existing, viz. silver nitrate, ice, solution, vapour,
+the system is invariant, and the point is a _quadruple point_. This
+quadruple point, therefore, forms the lower limit of the solubility curve
+of silver nitrate. Below this point the solution becomes metastable.
+
+Ice as Solid Phase.--Ice melts or is in equilibrium with water at a
+temperature of 0°. The melting point, will, however, be lowered by the
+solution of silver nitrate in the water; and the greater the concentration
+of the salt in the solution the greater will be the depression of the
+temperature of equilibrium. On continuing the addition of silver nitrate, a
+point will at length be reached at which the salt is no longer dissolved,
+but remains in the solid form along with the ice. We again obtain,
+therefore, the invariant system ice--salt--solution--vapour. The
+temperature at which this invariant system can exist has been found by
+Middelberg[193] to be -7.3°, the solution at this point containing 47.1 per
+cent. of silver nitrate.
+
+The same general behaviour will be found in the case of all other systems
+of two components belonging to this class; that is, in the case of systems
+from which the components crystallise out in the pure state, and in which
+the fused components are miscible in all proportions. In all such cases,
+therefore, the solubility curves (curves of equilibrium) can be represented
+diagrammatically as in Fig. 29. In this figure OA represents the solubility
+curve of the salt, and OB the freezing {117} point curve of ice. O is the
+quadruple point at which the invariant system exists, and may be regarded
+as the point of intersection of the solubility curve with the
+freezing-point curve. Since this point is fixed, the condition of the
+system as regards temperature, vapour pressure, and concentration of the
+components (or composition of the solution), is perfectly definite. From
+the way, also, in which the condition is attained, it is evident that the
+quadruple point is the lowest temperature that can be obtained with
+mixtures of the two components in presence of vapour. It is known as the
+_cryohydric point_, or, generally, the _eutectic point_.[194]
+
+[Illustration: FIG. 29.]
+
+Cryohydrates.[195]--On cooling a solution of common salt in water to a
+temperature of -3°, Guthrie observed that the hydrate NaCl,2H_{2}O
+separated out. This salt continued to be deposited until at a temperature
+of -22° opaque crystals made their appearance, and the liquid passed into
+the solid state without change of temperature. A similar behaviour was
+found by Guthrie in the case of a large number of other salts, a
+temperature below that of the melting point of ice being reached at which
+on continued withdrawal of heat, the solution solidified at a constant
+temperature. When the system had attained this minimum temperature, it was
+found that the composition of the solid and the liquid phases was the same,
+and remained unchanged throughout the period of solidification. This is
+shown by the following figures, which give the composition of different
+samples of the solid phase deposited from the solution at constant
+temperature.[196]
+
+{118}
+
+  ---------------------------------
+  No. | Temperature of  |   NaCl.
+      | solidification. | Per cent.
+  ----|-----------------|----------
+   1  |  -21° to -22°   |   23.72
+   2  |      -22°       |   23.66
+   3  |      -22°       |   23.73
+   4  |      -23°       |   23.82
+   5  |      -23°       |   23.34
+   6  |      -23°       |   23.35
+  ---------------------------------
+             Mean           23.6
+  ---------------------------------
+
+Conversely, a mixture of ice and salt containing 23.6 per cent. of sodium
+chloride will melt at a definite and constant temperature, and exhibit,
+therefore, a behaviour supposed to be characteristic of a pure chemical
+compound. This, then, combined with the fact that the solid which was
+deposited was crystalline, and that the same constant temperature was
+attained, no matter with what proportions of water and salt one started,
+led Guthrie to the belief that the solids which thus separated at constant
+temperature were definite chemical compounds, to which he gave the general
+name _cryohydrate_. A large number of such cryohydrates were prepared and
+analysed by Guthrie, and a few of these are given in the following table,
+together with the temperature of the cryohydric point:[197]--
+
+  CRYOHYDRATES.
+
+  ------------------------------------------------------------------
+         Salt.       | Cryohydric point. | Percentage of anhydrous
+                     |                   | salt in the cryohydrate.
+  ------------------------------------------------------------------
+  Sodium bromide     |        -24°       |          41.33
+  Sodium chloride    |        -22°       |          23.60
+  Potassium iodide   |        -22°       |          52.07
+  Sodium nitrate     |        -17.5°     |          40.80
+  Ammonium sulphate  |        -17°       |          41.70
+  Ammonium chloride  |        -15°       |          19.27
+  Sodium iodide      |        -15°       |          59.45
+  Potassium bromide  |        -13°       |          32.15
+  Potassium chloride |        -11.4°     |          20.03
+  Magnesium sulphate |         -5°       |          21.86
+  Potassium nitrate  |         -2.6°     |          11.20
+  Sodium sulphate    |         -0.7°     |           4.55
+  ------------------------------------------------------------------
+
+{119}
+
+The chemical individuality of these cryohydrates was, however, called in
+question by Pfaundler,[198] and disproved by Offer,[199] who showed that in
+spite of the constancy of the melting point, the cryohydrates had the
+properties, not of definite chemical compounds, but of mixtures; the
+arguments given being that the heat of solution and the specific volume are
+the same for the cryohydrate as for a mixture of ice and salt of the same
+composition; and it was further shown that the cryohydrate had not a
+definite crystalline form, but separated out as an opaque mass containing
+the two components in close juxtaposition. The heterogeneous nature of
+cryohydrates can also be shown by a microscopical examination.
+
+At the cryohydric point, therefore, we are not dealing with a single solid
+phase, but with two solid phases, ice and salt; the cryohydric point,
+therefore, as already stated, is a quadruple point and represents an
+invariant system.
+
+Although on cooling a solution to the cryohydric point, separation of ice
+may occur, it will not necessarily take place; the system may become
+metastable. Similarly, separation of salt may not take place immediately
+the cryohydric point is reached. It will, therefore, be possible to follow
+the curves BO and AO beyond the quadruple point,[200] which is thereby
+clearly seen to be the point of intersection of the solubility curve of the
+salt and the freezing-point curve of ice. At this point, also, the curves
+of the univariant systems ice--salt--vapour and ice--salt--solution
+intersect.
+
+Changes at the Quadruple Point.--Since the invariant system
+ice--salt--solution--vapour can exist only at a definite temperature,
+addition or withdrawal of heat must cause the disappearance of one of the
+phases, whereby the system will become univariant. So long as all four
+phases are present the temperature, pressure, and concentration of the
+components in the solution must remain constant. When, therefore, heat is
+added to or withdrawn from the system, mutually compensatory changes will
+take place within the system whereby the {120} condition of the latter is
+preserved. These changes can in all cases be foreseen with the help of the
+theorem of van't Hoff and Le Chatelier; and, after what was said in Chap.
+IV., need only be briefly referred to here. In the first place, addition of
+heat will cause ice to melt, and the concentration of the solution will be
+thereby altered; salt must therefore dissolve until the original
+concentration is reached, and the heat of fusion of ice will be
+counteracted by the heat of solution of the salt. Changes of volume of the
+solid and liquid phases must also be taken into account; an alteration in
+the volume of these phases being compensated by condensation or
+evaporation. All four phases will therefore be involved in the change, and
+the final state of the system will be dependent on the amounts of the
+different phases present; the ultimate result of addition or withdrawal of
+heat or of change of pressure at the quadruple point will be one of the
+four univariant systems: ice--solution--vapour; salt--solution--vapour;
+ice--salt--vapour; ice--salt--solution. If the vapour phase disappear,
+there will be left the univariant system ice--salt--solution, and the
+temperature at which this system can exist will alter with the pressure.
+Since in this case the influence of pressure is comparatively slight, the
+temperature of the quadruple point will differ only slightly from that of
+the cryohydric point as determined under atmospheric pressure.
+
+Freezing Mixtures.--Not only will the composition of a univariant system
+undergo change when the temperature is varied, but, conversely, if the
+_composition_ of the system is caused to change, corresponding changes of
+temperature must ensue. Thus, if ice is added to the univariant system
+salt--solution--vapour, the ice must melt and the temperature fall; and if
+sufficient ice is added, the temperature of the cryohydric point must be at
+length reached, for it is only at this temperature that the four phases
+ice--salt--solution--vapour can coexist. Or, on the other hand, if salt is
+added to the system ice--solution--vapour, the concentration of the
+solution will increase, ice must melt, and the temperature must thereby
+fall; and this process also will go on until the cryohydric point is
+reached. In both cases ice melts and there is a change in the {121}
+composition of the solution; in the former case, salt will be
+deposited[201] because the solubility diminishes as the temperature falls;
+in the latter, salt will pass into solution. This process may be
+accompanied either by an evolution or, more generally, by absorption of
+heat; in the former case the effect of the addition of ice will be
+partially counteracted; in the latter case it will be augmented.
+
+These principles are made use of in the preparation of _freezing mixtures_.
+The lowest temperature which can be reached by means of these (under
+atmospheric pressure) is the cryohydric point. This temperature-minimum is,
+however, not always attained in the preparation of a freezing mixture, and
+that for various reasons. The chief of these are radiation and the heat
+absorbed in cooling the solution produced. The lower the temperature falls,
+the more rapid does the radiation become; and the rate at which the
+temperature sinks decreases as the amount of solution increases. Both these
+factors counteract the effect of the latent heat of fusion and the heat of
+solution, so that a point is reached (which may lie considerably above the
+cryohydric point) at which the two opposing influences balance. The
+absorption of heat by the solution can be diminished by allowing the
+solution to drain off as fast as it is produced; and the effect of
+radiation can be partially annulled by increasing the rate of cooling. This
+can be done by the more intimate mixing of the components. Since, under
+atmospheric pressure, the temperature of the cryohydric point is constant,
+the cryohydrates are very valuable for the production of baths of constant
+low temperature.
+
+2. _Partial Miscibility of the Fused Components._
+
+On passing to the study of the second class of systems of two components
+belonging to this group, namely, those in which the fused components are
+not miscible in all proportions, we find that the relationships are not
+quite so simple as {122} in the case of silver nitrate and water. In the
+latter case, only one liquid phase was possible; in the cases now to be
+studied, two liquid phases can be formed, and there is a marked
+discontinuity in the solubility curve on passing from the cryohydric point
+to the melting point of the second (non-volatile) component.
+
+Paratoluidine dissolves in water, and the solubility increases as the
+temperature rises.[202] At 44.2°, however, paratoluidine in contact with
+water melts, and two liquid phases are formed, viz. a solution of water in
+fused paratoluidine and a solution of fused paratoluidine in water. We
+have, therefore, the phenomenon of _melting under the solvent_. This
+melting point will, of course, be lower than the melting point of the pure
+substance, because the solid is now in contact with a solution, and, as we
+have already seen, addition of a foreign substance lowers the melting
+point. Such cases of melting under the solvent are by no means rare, and a
+review of the relationships met with may, therefore, be undertaken here. As
+an example, there may be chosen the equilibrium between succinic nitrile,
+C_{2}H_{4}(CN)_{2} and water, which has been fully studied by
+Schreinemakers.[203]
+
+[Illustration: FIG. 30.]
+
+If to the system ice--water at 0° succinic nitrile is added, the
+temperature will fall; and continued addition of the nitrile will lead at
+last to the cryohydric point _b_ (Fig. 30), at which solid nitrile, ice,
+solution, and vapour can coexist. The temperature of the cryohydric point
+is -1.2°, and the composition of the solution is 1.29 mol. of nitrile in
+100 mol. of solution. From _a_ to _b_ the solid phase in contact with the
+solution is ice. {123} If the temperature be now raised so as to cause the
+disappearance of the ice, and the addition of nitrile be continued, the
+concentration of the nitrile in the solution will increase as represented
+by the curve _bc_. At the point _c_ (18.5°), when the concentration of the
+nitrile in the solution has increased to 2.5 molecules per cent., the
+nitrile melts and two liquid phases are formed; the concentration of the
+nitrile in these two phases is given by the points _c_ and _c'_. As there
+are now four phases present, viz. solid nitrile, solution of fused nitrile
+in water, solution of water in fused nitrile, and vapour, the system is
+_invariant_. Since at this point the concentration, temperature, and
+pressure are completely defined, addition or withdrawal of heat can only
+cause a change in the relative amounts of the phases, _but no variation of
+the concentrations_ of the respective phases. As a matter of fact,
+continued addition of nitrile and addition of heat will cause an increase
+in the amount of the liquid phase containing excess of nitrile (_i.e._ the
+solution of water in fused nitrile), whereas the other liquid phase, the
+solution of fused nitrile in water, will gradually disappear. When it has
+completely disappeared, the system will be represented by the point _c'_,
+where the molecular concentration of nitrile is now 75 per cent., and again
+becomes univariant, the three phases being solid nitrile, liquid phase
+containing excess of nitrile, and vapour; and as the amount of the water is
+diminished the temperature of equilibrium rises, until at 54° the melting
+point of the pure nitrile is reached.
+
+Return now to the point c. At this point there exists the invariant system
+solid nitrile, two liquid phases, vapour. If heat be added, the solid
+nitrile will disappear, and there will be left the univariant system,
+consisting of two liquid phases and vapour.[204] Such a system will exhibit
+relationships similar to those already studied in the previous chapter. As
+the temperature rises, the mutual solubility of the two fused components
+becomes greater, until at _d_ (55.5°) the critical solution temperature is
+reached, and the fused components become miscible in all proportions.
+
+At all temperatures and concentrations lying to the right {124} of the
+curve _abcdc'e_ there can be only one liquid phase; in the field _cdc'_
+there are two liquid phases.
+
+From the figure it will be easy to see what will be the result of bringing
+together succinic nitrile and water at different temperatures and in
+different amounts. Since _b_ is the lowest temperature at which liquid can
+exist in stable equilibrium with solid, ice and succinic nitrile can be
+mixed in any proportions at temperatures below _b_ without undergoing
+change. Between _b_ and _c_ succinic nitrile will be dissolved until the
+concentration reaches the value on the curve _bc_, corresponding to the
+given temperature. On adding the nitrile to water at temperatures between
+_c_ and _d_, it will dissolve until a concentration lying on the curve _cd_
+is attained; at this point two liquid phases will be formed, and further
+addition of nitrile will cause the one liquid phase (that containing excess
+of nitrile) to increase, while the other liquid phase will decrease, until
+it finally disappears and there is only one liquid phase left, that
+containing excess of nitrile. This can dissolve further quantities of the
+nitrile, and the concentration will increase until the curve _c'e_ is
+reached, when the concentration will remain unchanged, and addition of
+solid will merely increase the amount of the solid phase.
+
+If a solution represented by any point in the field lying below the curve
+_bcd_ is heated to a temperature above _d_, the critical solution
+temperature, then the concentration of the nitrile can be increased to any
+desired amount without at any time two liquid phases making their
+appearance; the system can then be cooled down to a temperature represented
+by any point between the curves _dc'e_. In this way it is possible to pass
+continuously from a solution containing excess of one component to
+solutions containing excess of the other, as represented by the dotted line
+_xxxx_ (_v._ p. 100). At no point is there formation of two liquid phases.
+
+Supersaturation.--Just as suspended transformation is rarely met with in
+the passage from the solid to the liquid state, so also it is found in the
+case of the melting of substances under the solvent that suspended fusion
+does not occur; but that when the temperature of the invariant point is
+reached at which, therefore, the formation of two liquid layers is
+possible, {125} these two liquid layers, as a matter of fact, make their
+appearance. Suspended transformation can, however, take place from the side
+of the liquid phase, just as water or other liquid can be cooled below the
+normal freezing point without solidification occurring. The question,
+therefore, arises as to the relative solubilities of the solid and the
+supercooled liquid at the same temperature.
+
+[Illustration: FIG. 31.]
+
+The answer to this question can at once be given from what we have already
+learned (p. 113), if we recollect that at temperatures below the point of
+fusion under the solvent, the solid form, at temperatures above that point,
+the liquid form, is the more stable; at this temperature, therefore, the
+relative stability of the solid and liquid forms changes. Since, as we have
+already seen, the less stable form has the greater solubility, it follows
+that the supercooled liquid, being the less stable form, must have the
+greater solubility. This was first proved experimentally by Alexejeff[205]
+in the case of benzoic acid and water, the solubility curves for which are
+given in Fig. 31. As can be seen from the figure, the prolongation of the
+curve for liquid--liquid, which represents the solubility of the
+supercooled liquid benzoic acid, lies above that for the solubility of the
+{126} solid benzoic acid in water; the solution saturated with respect to
+the supercooled liquid is therefore supersaturated with respect to the
+solid form. A similar behaviour has been found in the case of other
+substances.[206]
+
+Pressure-Temperature Diagram.--Having considered the changes which occur in
+the concentration of the components in a solution with the temperature, we
+may conclude the discussion of the equilibrium between a salt and water by
+studying the variation of the vapour pressure.
+
+Since in systems of two components the two phases, solution and vapour,
+constitute a bivariant system, the vapour pressure is undefined, and may
+have different values at the same temperature, depending on the
+concentration. In order that there may be for each temperature a definite
+corresponding pressure of the vapour, a third phase must be present. This
+condition is satisfied by the system solid--liquid (solution)--vapour; that
+is, by the saturated solution (p. 108). In the case of a saturated
+solution, therefore, the pressure of the vapour at any given temperature is
+constant.
+
+Vapour Pressure of Solid--Solution--Vapour.--It has long been known that
+the addition of a non-volatile solid to a liquid in which it is soluble
+lowers the vapour pressure of the solvent; and the diminution of the
+pressure is approximately proportional to the amount of substance dissolved
+(Law of Babo). The vapour-pressure curve, therefore, of a solution of a
+salt in water must lie below that for pure water. Further, in the case of a
+pure liquid, the vaporization curve is a function only of the temperature
+(p. 63), whereas, in the case of a solution, the pressure varies both with
+the temperature and the _concentration_. These two factors, however, act in
+opposite directions; for although the vapour pressure in all cases
+increases as the temperature rises, increase of concentration, as we have
+seen, lowers the vapour pressure. Again, since the concentration itself
+varies with the temperature, two cases have to be considered, viz. where
+the concentration increases with rise of {127} temperature, and where the
+concentration diminishes with rise of temperature.
+
+The relations which are found here will be best understood with the help of
+Fig. 32.[207] In this figure, OB represents the sublimation curve of ice,
+and BC the vaporization curve of water; the curve for the solution must lie
+below this, and must cut the sublimation curve of ice at some temperature
+below the melting point. The point of intersection A is the cryohydric
+point. If the solubility increases with rise of temperature, the increase
+of the vapour pressure due to the latter will be partially annulled. Since
+at first the effect of increase of temperature more than counteracts the
+depressing action of increase of concentration, the vapour pressure will
+increase on raising the temperature above the cryohydric point. If the
+elevation of temperature is continued, however, to the melting point of the
+salt, the effect of increasing concentration makes itself more and more
+felt, so that the vapour-pressure curve of the solution falls more and more
+below that of the pure liquid, and the pressure will ultimately become
+equal to that of the pure salt; that is to say, practically equal to zero.
+The curve will therefore be of the general form AMF shown in Fig. 32. If
+the solubility should diminish with rise of temperature, the two factors,
+temperature and concentration, will act in the same direction, and the
+vapour-pressure curve will rise relatively more rapid than that of the pure
+liquid; since, however, the pure salt is ultimately obtained, the
+vapour-pressure curve must in this case also finally approach the value
+zero.
+
+[Illustration: FIG. 32.]
+
+Other Univariant Systems.--Besides the univariant system {128}
+salt--solution--vapour already considered, three others are possible, viz.
+ice--solution--vapour, ice--salt--solution, and ice--salt--vapour.
+
+The fusion point of a substance is lowered, as we have seen, by the
+addition of a foreign substance, and the depression is all the greater the
+larger the quantity of substance added. The vapour pressure of the water,
+also, is lowered by the solution in it of other substances, so that the
+vapour pressure of the system ice--solution--vapour must decrease as the
+temperature falls from the fusion point of ice to the cryohydric point.
+This curve is represented by BA (Fig. 32), and is coincident with the
+sublimation curve of ice.
+
+This, at first sight, strange fact will be readily understood when we
+consider that since ice and solution are together in equilibrium with the
+same vapour, they must have the same vapour pressure. For suppose at any
+given temperature equilibrium to have been established in the system
+ice--solution--vapour, removal of the ice will not alter this equilibrium.
+Suppose, now, the ice and the solution placed under a bell-jar so that they
+have a common vapour, but are not themselves in contact; then, if they do
+not have the same vapour pressure, distillation must take place and the
+solution will become more dilute or more concentrated. Since, at the
+completion of this process, the ice and solution are now in equilibrium
+when they are not in contact, they must also be in equilibrium when they
+are in contact (p. 32). But if distillation has taken place the
+concentration of the solution must have altered, so that the ice will now
+be in equilibrium with a solution of a different concentration from before.
+But according to the Phase Rule ice cannot at one and the same temperature
+be in equilibrium with two solutions of different concentration, for the
+system ice--solution--vapour is univariant, and at any given temperature,
+therefore, not only the pressure but also the _concentration of the
+components in the solution must be constant_. Distillation could not,
+therefore, take place from the ice to the solution or _vice versâ_; that is
+to say, the solution and the ice must have the same vapour pressure--the
+sublimation pressure of ice. The reason of the coincidence is the
+non-volatility of the salt: had {129} the salt a measurable vapour pressure
+itself, the sublimation curve of ice and the curve for
+ice--solution--vapour would no longer fall together.
+
+The curve AO represents the pressures of the system ice--salt--vapour. This
+curve will also be coincident with the sublimation curve of ice, on account
+of the non-volatility of the salt.
+
+The equilibria of the fourth univariant system ice--salt--solution are
+represented by AE. Since this is a condensed system, the effect of a small
+change of temperature will be to cause a large change of pressure, as in
+the case of the fusion point of a pure substance. The direction of this
+curve will depend on whether there is an increase or diminution of volume
+on solidification; but the effect in any given case can be predicted with
+the help of the theorem of Le Chatelier.
+
+Since the cryohydric point is a quadruple point in a two-component system,
+it represents an invariant system. The condition of the system is,
+therefore, completely defined; the four phases, ice, salt, solution,
+vapour, can co-exist only when the temperature, pressure, and concentration
+of the solution have constant and definite values. Addition or withdrawal
+of heat, therefore, can cause no alteration of the condition of the system
+except a variation of the relative amounts of the phases. Addition
+of heat at constant volume will ultimately lead to the system
+salt--solution--vapour or the system ice--solution--vapour, according as
+ice or salt disappears first. This is readily apparent from the diagram
+(Fig. 32), for the systems ice--salt--solution and ice--salt--vapour can
+exist only at temperatures below the cryohydric point (provided the curve
+for ice--salt--solution slopes towards the pressure axis).
+
+Bivariant Systems.--Besides the univariant systems already discussed,
+various bivariant systems are possible, the conditions for the existence of
+which are represented by the different areas of Fig. 32. They are as
+follows:--
+
+  _Area._             _System._
+
+  OAMF     Salt--vapour.
+  CBAMF    Solution--vapour; salt--solution.
+  EABD     Salt--solution; ice--solution.
+  EAO      Ice--salt.
+
+{130}
+
+Deliquescence.--As is evident from Fig. 32, salt can exist in contact with
+water vapour at pressures under those represented by OAMF. If, however, the
+pressure of the vapour is increased until it reaches a value lying on this
+curve at temperatures above the cryohydric point, solution will be formed;
+for the curve AMF represents the equilibria between salt--solution--vapour.
+From this, therefore, it is clear that if the pressure of the aqueous
+vapour in the atmosphere is greater than that of the saturated solution of
+a salt, that salt will, on being placed in the air, form a solution; it
+will _deliquesce_.
+
+Separation of Salt on Evaporation.--With the help of Fig. 32 it is possible
+to state in a general manner whether or not salt will be deposited when a
+solution is evaporated under a constant pressure.[208]
+
+The curve AMF (Fig. 32) is the vapour-pressure curve of the saturated
+solutions of the salt, _i.e._ it represents, as we have seen, the maximum
+vapour pressure at which salt can exist in contact with solution and
+vapour. The dotted line _aa_ represents atmospheric pressure. If, now, an
+unsaturated solution, the composition of which is represented by the point
+_x_, is heated in an open vessel, the temperature will rise, and the vapour
+pressure of the solution will increase. The system will, therefore, pass
+along a line represented diagrammatically by _xx'_. At the point _x'_ the
+vapour pressure of the system becomes equal to 1 atm.; and as the vessel is
+open to the air, the pressure cannot further rise; the solution boils. If
+the heating is continued, water passes off, the concentration increases,
+and the boiling point rises. The system will therefore pass along the line
+_x'm_, until at the point _m_ solid salt separates out (provided
+supersaturation is excluded). The system is now univariant, and continued
+heating will no longer cause an alteration of the concentration; as water
+passes off, solid salt will be deposited, and the solution will evaporate
+to dryness.
+
+If, however, the atmospheric pressure is represented not by _aa_ but by
+_bb_, then, as Fig. 32 shows, the maximum vapour {131} pressure of the
+system salt--solution--vapour never reaches the pressure of 1 atm. Further,
+since the curve _bb_ lies in the area of the bivariant system
+solution--vapour there can at no point be a separation of the solid form;
+for the system solid--solution--vapour can exist only along the curve AMF.
+
+On evaporating the solution of a salt in an open vessel, therefore, salt
+can be deposited only if at some temperature the pressure of the saturated
+solution is equal to the atmospheric pressure. This is found to be the case
+with most salts. In the case of aqueous solutions of sodium and potassium
+hydroxide, however, the vapour pressure of the saturated solution never
+reaches the value of 1 atm., and on evaporating these solutions, therefore,
+in an open vessel, there is no separation of the solid. Only a homogeneous
+fused mass is obtained. If, however, the evaporation be carried out under a
+pressure which is lower than the maximum pressure of the saturated
+solution, separation of the solid substance will be possible.
+
+General Summary.--The systems which have been discussed in the present
+chapter contained water as one of their components, and an anhydrous salt
+as the other. It will, however, be clear that the relationships which were
+found in the case of these will be found also in other cases where it is a
+question of the equilibria between two components, which crystallize out in
+the pure state, and only one of which possesses a measurable vapour
+pressure. A similar behaviour will, for example, be found in the case of
+many pairs of organic substances; and in all cases the equilibria will be
+represented by a diagram of the general appearance of Fig. 29 or Fig. 30.
+That is to say: Starting from the fusion point of component I., the system
+will pass, by progressive addition of component II., to regions of lower
+temperature, until at last the cryohydric or eutectic point is reached. On
+further addition of component II., the system will pass to regions of
+higher temperature, the solid phase now being component II. If the fused
+components are miscible with one another in all proportions a continuous
+curve will be obtained leading up to the point of fusion of component II.
+Slight changes of direction, it is true, due to changes in the crystalline
+form, may be found along this curve, {132} but throughout its whole course
+there will be but one liquid phase. If, on the other hand, the fused
+components are not miscible in all proportions, then the second curve will
+exhibit a marked discontinuity, and two liquid phases will make their
+appearance.
+
+       *       *       *       *       *
+
+
+{133}
+
+CHAPTER VIII
+
+SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE
+
+B.--HYDRATED SALT AND WATER.
+
+In the preceding chapter we discussed the behaviour of systems formed of
+two components, only one of which was volatile, in those cases where the
+two components separated from solution in the pure state. In the present
+chapter we shall consider those systems in which combination between the
+components can occur with the formation of definite compounds; such as are
+found in the case of crystalline salt hydrates. Since a not inconsiderable
+amount of study has been devoted to the systems formed by hydrated salts
+and water, systems which are of great chemical interest and importance, the
+behaviour of these will first call for discussion in some detail, and it
+will be found later that the relationships which exist in such systems
+appear also in a large number of other two-component systems.
+
+The systems belonging to this group may be divided into two classes
+according as the compounds formed possess a definite melting point, _i.e._
+form a liquid phase of the same composition, or do not do so. We shall
+consider the latter first.
+
+1. _The Compounds formed do not have a Definite Melting Point._
+
+Concentration-Temperature Diagram.--In the case of salts which can form
+crystalline hydrates, the temperature-concentration diagram, representing
+the equilibria of the {134} different possible systems, must necessarily be
+somewhat more complicated than where no such combination of the components
+occurs. For, as has already been pointed out, each substance has its own
+solubility curve; and there will therefore be as many solubility curves as
+there are solid phases possible, _the curve for each particular solid phase
+being continuous so long as it remains unchanged in contact with the
+solution_. As an example of the relationships met with in such cases, we
+shall first of all consider the systems formed of sodium sulphate and
+water.
+
+[Illustration: FIG. 33.]
+
+Sodium Sulphate and Water.--At the ordinary temperatures, sodium sulphate
+crystallises from water with ten molecules of water of crystallisation,
+forming Glauber's salt. On determining the solubility of this salt in
+water, it is found that the solubility increases as the temperature rises,
+the values of the solubility, represented graphically by the curve AC (Fig.
+33), being given in the following table.[209] The numbers denote grams of
+sodium sulphate, calculated as anhydrous salt, dissolved by 100 grams of
+water.
+
+  SOLUBILITY OF Na_{2}SO_{4},10H_{2}O.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+       0°      |     5.02
+      10°      |     9.00
+      15°      |    13.20
+      18°      |    16.80
+      20°      |    19.40
+      25°      |    28.00
+      30°      |    40.00
+      33°      |    50.76
+      34°      |    55.00
+  --------------------------
+
+{135}
+
+On continuing the investigation at higher temperatures, it was found that
+the solubility no longer increased, but _decreased with rise of
+temperature_. At the same time, it was observed that the solid phase was
+now different from that in contact with the solution at temperatures below
+33°; for whereas in the latter case the solid phase was sodium sulphate
+decahydrate, at temperatures above 33° the solid phase was the anhydrous
+salt. The course of the solubility curve of anhydrous sodium sulphate is
+shown by BD, and the values of the solubility are given in the following
+table:--[210]
+
+  SOLUBILITY OF ANHYDROUS SODIUM SULPHATE.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+    18°        |    53.25
+    20°        |    52.76
+    25°        |    51.53
+    30°        |    50.37
+    33°        |    49.71
+    34°        |    49.53
+    36°        |    49.27
+    40.15°     |    48.78
+    50.40°     |    46.82
+  --------------------------
+
+As is evident from the figure, the solubility curve which is obtained when
+anhydrous sodium sulphate is present as the solid phase, cuts the curve
+representing the solubility of the decahydrate, at a temperature of about
+33°.
+
+If a solution of sodium sulphate which has been saturated at a temperature
+of about 34° be cooled down to a temperature below 17°, while care is taken
+that the solution is protected against access of particles of Glauber's
+salt, crystals of a second hydrate of sodium sulphate, having the
+composition Na_{2}SO_{4},7H_{2}O, separate out. On determining the
+composition of the solutions in equilibrium with this hydrate at different
+temperatures, the following values were obtained, these values being
+represented by the curve FE (Fig. 33):--
+
+{136}
+
+  SOLUBILITY OF Na_{2}SO_{4},7H_{2}O.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+       0°      |    19.62
+      10°      |    30.49
+      15°      |    37.43
+      18°      |    41.63
+      20°      |    44.73
+      25°      |    52.94
+      26°      |    54.97
+  --------------------------
+
+Since, as has already been stated, each solid substance has its own
+solubility curve, there are three separate curves to be considered in the
+case of sodium sulphate and water. Where two curves cut, the solution must
+be saturated with respect to two solid phases; at the point B, therefore,
+the point of intersection of the solubility curve of anhydrous sodium
+sulphate with that of the decahydrate, the solution must be saturated with
+respect to these two solid substances. But a system of two components
+existing in four phases, anhydrous salt--hydrated salt--solution--vapour,
+is invariant; and this invariability will remain even if only three phases
+are present, provided that one of the factors, pressure, temperature, or
+concentration of components retains a constant value. This is the case when
+solubilities are determined in open vessels; the pressure is then equal to
+atmospheric pressure. Under these circumstances, then, the system,
+anhydrous sodium sulphate--decahydrate--solution, will possess no degree of
+freedom, and can exist, therefore, only at one definite temperature and
+when the solution has a certain definite composition. The temperature of
+this point is 32.482° on a mercury thermometer, or 32.379° on the hydrogen
+thermometer.[211]
+
+{137}
+
+Suspended Transformation.--Although it is possible for the anhydrous salt
+to make its appearance at the temperature of the quadruple point, it will
+not necessarily do so; and it is therefore possible to follow the
+solubility curve of sodium sulphate decahydrate to a higher temperature.
+Since, however, the solubility of the decahydrate at temperatures above the
+quadruple point is greater than that of the anhydrous salt, the solution
+which is _saturated_ with respect to the former will be _supersaturated_
+with respect to the latter. On bringing a small quantity of the anhydrous
+salt in contact with the solution, therefore, anhydrous salt will be
+deposited; and all the hydrated salt present will ultimately undergo
+conversion into the anhydrous salt, through the medium of the solution. In
+this case, as in all cases, the solid phase, which is the most stable at
+the temperature of the experiment, has at that temperature the least
+solubility.
+
+Similarly, the solubility curve of anhydrous sodium sulphate has been
+followed to temperatures below 32.5°. Below this temperature, however, the
+solubility of this salt is greater than that of the decahydrate, and the
+saturated solution of the anhydrous salt will therefore be supersaturated
+for the decahydrate, and will deposit this salt if a "nucleus" is added to
+the solution. From this we see that at temperatures above 32.5° the
+anhydrous salt is the stable form, while the decahydrate is unstable (or
+metastable); at temperatures below 32.5° the decahydrate is stable. This
+temperature, therefore, is the _transition temperature_ for decahydrate and
+anhydrous salt.
+
+From Fig. 33 we see further that the solubility curve of the anhydrous salt
+(which at all temperatures below 32.5° is metastable) is cut by the
+solubility curve of the heptahydrate; and this point of intersection (at a
+temperature of 24.2°) must be the _transition point_ for heptahydrate and
+anhydrous salt. Since at all temperatures the solubility of the
+heptahydrate is greater than that of the decahydrate, the former hydrate
+must be metastable with respect to the latter; so that throughout its whole
+course the solubility curve of the heptahydrate {138} represents only
+metastable equilibria. Sodium sulphate, therefore, forms only one stable
+hydrate, the decahydrate.
+
+The solubility relations of sodium sulphate illustrate very clearly the
+importance of the solid phase for the definition of saturation and
+supersaturation. Since the solubility curve of the anhydrous salt has been
+followed backwards to a temperature of about 18°, it is readily seen, from
+Fig. 33, that at a temperature of, say, 20° three different _saturated_
+solutions of sodium sulphate are possible, according as the anhydrous salt,
+the heptahydrate or the decahydrate, is present as the solid phase. Two of
+these solutions, however, would be metastable and _supersaturated with
+respect to the decahydrate_.
+
+Further, the behaviour of sodium sulphate and water furnishes a very good
+example of the fact that a "break" in the solubility curve occurs when, and
+only when, the solid phase undergoes change. So long as the decahydrate,
+for example, remained unaltered in contact with the solution, the
+solubility curve was continuous; but when the anhydrous salt appeared in
+the solid phase, a distinct change in the direction of the solubility curve
+was observed.
+
+Dehydration by Means of Anhydrous Sodium Sulphate.--The change in the
+relative stability of sodium sulphate decahydrate and anhydrous salt in
+presence of water at a temperature of 32.5° explains why the latter salt
+cannot be employed for dehydration purposes at temperatures above the
+transition point. The dehydrating action of the anhydrous salt depends on
+the formation of the decahydrate; but since at temperatures above 33° the
+latter is unstable, and cannot be formed in presence of the anhydrous salt,
+this salt cannot, of course, effect a dehydration above that temperature.
+
+Pressure-Temperature Diagram.--The consideration of the
+pressure-temperature relations of the two components, sodium sulphate and
+water, must include not only the vapour pressure of the saturated
+solutions, but also that of the crystalline hydrates. The vapour pressures
+of salt hydrates have already been treated in a general manner (Chap. V.),
+so that it is only necessary here to point out the connection between the
+two classes of systems. {139}
+
+In most cases the vapour pressure of a salt hydrate, _i.e._ the vapour
+pressure of the system hydrate--anhydrous salt (or lower hydrate)--vapour,
+is at all temperatures lower than that of the system anhydrous salt (or
+lower hydrate)--solution--vapour. This, however, is not a necessity; and
+cases are known where the vapour pressure of the former system is, under
+certain circumstances, equal to or higher than that of the latter. An
+example of this is found in sodium sulphate decahydrate.
+
+On heating Na_{2}SO_{4},10H_{2}O, a point is reached at which the
+dissociation pressure into anhydrous salt and water vapour becomes equal to
+the vapour pressure of the saturated solution of the anhydrous salt, as is
+apparent from the following measurements;[212] the differences in pressure
+being expressed in millimetres of a particular oil.
+
+  Temperature:      29.0°   30.83°   31.79°   32.09°   32.35°   32.6°
+  Difference of
+  pressure:         23.8    10.8      5.6      3.6      1.6      0
+
+At 32.6°, therefore, the vapour pressures of the two systems
+
+  Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
+  Na_{2}SO_{4}--solution--vapour
+
+are equal; at this temperature the four phases, Na_{2}SO_{4},10H_{2}O;
+Na_{2}SO_{4}; solution; vapour, can coexist. From this it is evident that
+when sodium sulphate decahydrate is heated to 32.6°, the two new phases
+anhydrous salt and solution will be formed (suspended transformation being
+supposed excluded), and the hydrate will appear to undergo _partial
+fusion_; and during the process of "melting" the vapour pressure and
+temperature will remain constant.[213] This is, however, not a true but a
+so-called _incongruent_ melting point; for the composition of the liquid
+phase is not the same as that of the solid. As has already been pointed out
+(p. 137), we are dealing here with the _transition point_ of the
+decahydrate and anhydrous salt, _i.e._ with the reaction
+Na_{2}SO_{4},10H_{2}O <--> Na_{2}SO_{4} + 10H_{2}O.
+
+Since at the point of partial fusion of the decahydrate four {140} phases
+can coexist, the point is a quadruple point in a two-component system, and
+the system at this point is therefore invariant. The temperature of this
+point is therefore perfectly definite, and on this account the proposal has
+been made to adopt this as a fixed point in thermometry.[214] The
+temperature is, of course, practically the same as that at which the two
+solubility curves intersect (p. 112). If, however, the vapour phase
+disappears, the system becomes univariant, and the equilibrium temperature
+undergoes change with change of pressure. The transition curve has been
+determined by Tammann,[215] and shown to pass through a point of maximum
+temperature.
+
+[Illustration: FIG. 34.]
+
+The vapour pressure of the different systems of sodium sulphate and water
+can best be studied with the help of the diagram in Fig. 34.[216] The curve
+ABCD represents the vapour-pressure curve of the saturated solution of
+anhydrous sodium sulphate. GC is the pressure curve of decahydrate +
+anhydrous salt, which, as we have seen, cuts the curve ABCD at the
+transition temperature, 32.6°. Since at this point the solution is
+saturated with respect to both the anhydrous salt and the decahydrate, the
+vapour-pressure curve of the saturated solution of the latter must also
+pass through the point C.[217] As at temperatures below this point the
+solubility of the decahydrate is less than that of the anhydrous salt, the
+vapour pressure of the solution will, in accordance with Babo's law
+(p. 126), be higher than that of the solution of the anhydrous salt; which
+was also found experimentally to be the case (curve HC).
+
+{141}
+
+In connection with the vapour pressure of the saturated solutions of the
+anhydrous salt and the decahydrate, attention must be drawn to a
+conspicuous deviation from what was found to hold in the case of
+one-component systems in which a vapour phase was present (p. 31). There,
+it was seen that the vapour pressure of the more stable system was always
+_lower_ than that of the less stable; in the present case, however, we find
+that this is no longer so. We have already learned that at temperatures
+below 32.5° the system decahydrate--solution--vapour is more stable than
+the system anhydrous salt--solution--vapour; but the vapour pressure of the
+latter system is, as has just been stated, lower than that of the former.
+At temperatures above the transition point the vapour pressure of the
+saturated solution of the decahydrate will be lower than that of the
+saturated solution of the anhydrous salt.
+
+This behaviour depends on the fact that the less stable form is the more
+soluble, and that the diminution of the vapour pressure increases with the
+amount of salt dissolved.
+
+With regard to sodium sulphate heptahydrate the same considerations will
+hold as in the case of the decahydrate. Since at 24° the four phases
+heptahydrate, anhydrous salt, solution, vapour can coexist, the
+vapour-pressure curves of the systems hydrate--anhydrous salt--vapour
+(curve EB) and hydrate--solution--vapour (curve FB) must cut the pressure
+curve of the saturated solution of the anhydrous salt at the above
+temperature, as represented in Fig. 34 by the point B. This constitutes,
+therefore, a second quadruple point, which is, however, metastable.
+
+From the diagram it is also evident that the dissociation pressure of the
+heptahydrate is higher than that of the decahydrate, although it contains
+less water of crystallization. The system heptahydrate--anhydrous
+salt--vapour must be metastable with respect to the system
+decahydrate--anhydrous salt--vapour, and will pass into the latter.[218]
+Whether or not there is a temperature at which the vapour-pressure curves
+of the two systems intersect, and below which the heptahydrate becomes the
+more stable form, is not known.
+
+{142}
+
+In the case of sodium sulphate there is only one stable hydrate. Other
+salts are known which exhibit a similar behaviour; and we shall therefore
+expect that the solubility relationships will be represented by a diagram
+similar to that for sodium sulphate. A considerable number of such cases
+have, indeed, been found,[219] and in some cases there is more than one
+metastable hydrate. This is found, for example, in the case of nickel
+iodate,[220] the solubility curves for which are given in Fig. 35. As can
+be seen from the figure, suspended transformation occurs, the solubility
+curves having in some cases been followed to a considerable distance beyond
+the transition point. One of the most brilliant examples, however, of
+suspended transformation in the case of salt hydrates, and the sluggish
+transition from the less stable to the more stable form, is found in the
+case of the hydrates of calcium chromate.[221]
+
+[Illustration: FIG. 35.]
+
+In the preceding cases, the dissociation-pressure curve of the hydrated
+salt cuts the vapour-pressure curve of the saturated {143} solution of the
+anhydrous salt. It can, however, happen that the dissociation-pressure
+curve of one hydrate cuts the solubility curve, not of the anhydrous salt,
+but of a lower hydrate; in this case there will be more than one stable
+hydrate, each having a stable solubility curve; and these curves will
+intersect at the temperature of the transition point. Various examples of
+this behaviour are known, and we choose for illustration the solubility
+relationships of barium acetate and its hydrates[222] (Fig. 36).
+
+[Illustration: FIG. 36.]
+
+At temperatures above 0°, barium acetate can form two stable hydrates, a
+trihydrate and a monohydrate. The solubility of the trihydrate increases
+very rapidly with rise of temperature, and has been determined up to 26.1°.
+At temperatures above 24.7°, however, the trihydrate is metastable with
+respect to the monohydrate; for at this temperature the solubility curve of
+the latter hydrate cuts that of the former. This is, therefore, the
+transition temperature for the trihydrate and monohydrate. The solubility
+curve of the monohydrate succeeds that of the trihydrate, and exhibits a
+conspicuous point of minimum solubility at about 30°. Below 24.7° the {144}
+monohydrate is the less stable hydrate, but its solubility has been
+determined to a temperature of 22°. At 41° the solubility curve of the
+monohydrate intersects that of the anhydrous salt, and this is therefore
+the transition temperature for the monohydrate and anhydrous salt. Above
+this temperature the anhydrous salt is the stable solid phase. Its
+solubility curve also passes through a minimum.
+
+The diagram of solubilities of barium acetate not only illustrates the way
+in which the solubility curves of the different stable hydrates of a salt
+succeed one another, but it has also an interest and importance from
+another point of view. In Fig. 36 there is also shown a faintly drawn curve
+which is continuous throughout its whole course. This curve represents the
+solubility of barium acetate as determined by Krasnicki.[223] Since,
+however, three different solid phases can exist under the conditions of
+experiment, it is evident, from what has already been stated (p. 111), that
+the different equilibria between barium acetate and water could not be
+represented by one _continuous_ curve.
+
+Another point which these experiments illustrate and which it is of the
+highest importance to bear in mind is, that in making determinations of the
+solubility of salts which are capable of forming hydrates, it is not only
+necessary to determine the composition of the solution, but _it is of equal
+importance to determine the composition of the solid phase in contact with
+it_. In view of the fact, also, that the solution equilibrium is in many
+cases established with comparative slowness, it is necessary to confirm the
+point of equilibrium, either by approaching it from higher as well as from
+lower temperatures, or by actually determining the rate with which the
+condition of equilibrium is attained. This can be accomplished by actual
+weighing of the dissolved salt or by determinations of the density of the
+solution, as well as by other methods.
+
+{145}
+
+2. _The Compounds formed have a Definite Melting Point._
+
+In the cases which have just been considered we saw that the salt hydrates
+on being heated did not undergo complete fusion, but that a solid was
+deposited consisting of a lower hydrate or of the anhydrous salt. It has,
+however, been long known that certain crystalline salt hydrates (_e.g._
+sodium thiosulphate, Na_{2}S_{2}O_{3},5H_{2}O, sodium acetate,
+NaC_{2}H_{3}O_{2},3H_{2}O) melt completely in their water of
+crystallization, and yield a liquid of the _same composition_ as the
+crystalline salt. In the case of sodium thiosulphate pentahydrate the
+temperature of liquefaction is 56°; in the case of sodium acetate
+trihydrate, 58°. These two salts, therefore, have a definite melting point.
+For the purpose of studying the behaviour of such salt hydrates, we shall
+choose not the cases which have just been mentioned, but two others which
+have been more fully studied, viz. the hydrates of calcium chloride and of
+ferric chloride.
+
+Solubility Curve of Calcium Chloride Hexahydrate.[224]--Although calcium
+chloride forms several hydrates, each of which possesses its own
+solubility, it is nevertheless the solubility curve of the hexahydrate
+which will chiefly interest us at present, and we shall therefore first
+discuss that curve by itself.
+
+[Illustration: FIG. 37.]
+
+The solubility of this salt has been determined from the cryohydric point,
+which lies at about -55°, up to the melting point of the salt.[225] The
+solubility increases with rise of temperature, as is shown by the figures
+in the following table, and by the (diagrammatic) curve AB in Fig. 37. In
+the table, the numbers under the heading "solubility" denote the number of
+grams of CaCl_{2} dissolved in 100 grams {146} of water; those under the
+heading "composition," the number of gram-molecules of water in the
+solution to one gram-molecule of CaCl_{2}.
+
+  SOLUBILITY OF CALCIUM CHLORIDE HEXAHYDRATE.
+
+  -----------------------------------------
+  Temperature. | Solubility. | Composition.
+  -----------------------------------------
+      -55°     |     42.5    |    14.5
+      -25°     |     50.0    |    12.3
+      -10°     |     55.0    |    11.2
+        0°     |     59.5    |    10.37
+       10°     |     65.0    |     9.49
+       20°     |     74.5    |     8.28
+       25°     |     82.0    |     7.52
+       28.5°   |     90.5    |     6.81
+       29.5°   |     95.5    |     6.46
+       30.2°   |    102.7    |     6.00
+       29.6°   |    109.0    |     5.70
+       29.2°   |    112.8    |     5.41
+  -----------------------------------------
+
+So far as the first portion of the curve is concerned, it resembles the
+most general type of solubility curve. In the present case the solubility
+is so great and increases so rapidly with rise of temperature, that a point
+is reached at which the water of crystallization of the salt is sufficient
+for its complete solution. This temperature is 30.2°; and since the
+composition of the solution is the same as that of the solid salt, viz. 1
+mol. of CaCl_{2} to 6 mols. of water, this temperature must be the melting
+point of the hexahydrate. At this point the hydrate will fuse or the
+solution will solidify without change of temperature and without change of
+composition. Such a melting point is called a _congruent_ melting point.
+
+But the solubility curve of calcium chloride hexahydrate differs markedly
+from the other solubility curves hitherto considered in that it possesses a
+_retroflex portion_, represented in the figure by BC. As is evident from
+the figure, therefore, calcium chloride hexahydrate exhibits the peculiar
+and, as it was at first thought, impossible behaviour that it can be in
+equilibrium at one and the same temperature with two different solutions,
+one of which contains more, the other less, water than the solid hydrate;
+for it must be remembered that {147} throughout the whole course of the
+curve ABC the solid phase present in equilibrium with the solution is the
+hexahydrate.
+
+Such a behaviour, however, on the part of calcium chloride hexahydrate will
+appear less strange if one reflects that the melting point of the hydrate
+will, like the melting point of other substances, be lowered by the
+addition of a second substance. If, therefore, water is added to the
+hydrate at its melting point, the temperature at which the solid hydrate
+will be in equilibrium with the liquid phase (solution) will be lowered; or
+if, on the other hand, anhydrous calcium chloride is added to the hydrate
+at its melting point (or what is the same thing, if water is removed from
+the solution), the temperature at which the hydrate will be in equilibrium
+with the liquid will also be lowered; _i.e._ the hydrate will melt at a
+lower temperature. In the former case we have the hydrate in equilibrium
+with a solution containing more water, in the latter case with a solution
+containing less water than is contained in the hydrate itself.
+
+It has already been stated (p. 109) that the solubility curve (in general,
+the equilibrium curve) is continuous so long as the solid phase remains
+unchanged; and we shall therefore expect that the curve ABC will be
+continuous. Formerly, however, it was considered by some that the curve was
+not continuous, but that the melting point is the point of intersection of
+two curves, a solubility curve and a fusion curve. Although the earlier
+solubility determinations were insufficient to decide this point
+conclusively, more recent investigation has proved beyond doubt that the
+curve is continuous and exhibits no break.[226]
+
+{148}
+
+Although in taking up the discussion of the equilibria between calcium
+chloride and water, it was desired especially to call attention to the form
+of the solubility curve in the case of salt hydrates possessing a definite
+melting point, nevertheless, for the sake of completeness, brief mention
+may be made of the other systems which these two components can form.
+
+[Illustration: FIG. 38.]
+
+Besides the hexahydrate, the solubility curve of which has already been
+described, calcium chloride can also crystallize in two different forms,
+each of which contains four molecules {149} of water of crystallization;
+these are distinguished as [alpha]-tetrahydrate, and [beta]-tetrahydrate.
+Two other hydrates are also known, viz. a dihydrate and a monohydrate. The
+solubility curves of these different hydrates are given in Fig. 38.
+
+On following the solubility curve of the hexahydrate from the ordinary
+temperature upwards, it is seen that at a temperature of 29.8° represented
+by the point H, it cuts the solubility curve of the [alpha]-tetrahydrate.
+This point is therefore a quadruple point at which the four phases
+hexahydrate, [alpha]-tetrahydrate, solution, and vapour can coexist. It is
+also the transition point for these two hydrates. Since, at temperatures
+above 29.8°, the [alpha]-tetrahydrate is the stable form, it is evident
+from the data given before (p. 146), as also from Fig. 38, that the portion
+of the solubility curve of the hexahydrate lying above this temperature
+represents _metastable_ equilibria. The realization of the metastable
+melting point of the hexahydrate is, therefore, due to suspended
+transformation. At the transition point, 29.8°, the solubility of the
+hexahydrate and [alpha]-tetrahydrate is 100.6 parts of CaCl_{2} in 100
+parts of water.
+
+The retroflex portion of the solubility curve of the hexahydrate extends to
+only 1° below the melting point of the hydrate. At 29.2° crystals of a new
+hydrate, [beta]-tetrahydrate, separate out, and the solution, which now
+contains 112.8 parts of CaCl_{2} to 100 parts of water, is saturated with
+respect to the two hydrates. Throughout its whole extent the solubility
+curve EDF of the [beta]-tetrahydrate represents metastable equilibria. The
+upper limit of the solubility curve of [beta]-tetrahydrate is reached at
+38.4° (F), the point of intersection with the curve for the dihydrate.
+
+Above 29.8° the stable hydrate is the [alpha]-tetrahydrate; and its
+solubility curve extends to 45.3° (K), at which temperature it cuts the
+solubility curve of the dihydrate. The curve of the latter hydrate extends
+to 175.5° (L), and is then succeeded by the curve for the monohydrate. The
+solubility curve of the anhydrous salt does not begin until a temperature
+of about 260°. The whole diagram, therefore, shows a succession of stable
+hydrates, a metastable hydrate, a metastable melting point and retroflex
+solubility curve. {150}
+
+Pressure-Temperature Diagram.--The complete study of the equilibria between
+the two components calcium chloride and water would require the discussion
+of the vapour pressure of the different systems, and its variation with the
+temperature. For our present purpose, however, such a discussion would not
+be of great value, and will therefore be omitted here; in general, the same
+relationships would be found as in the case of sodium sulphate (p. 138),
+except that the rounded portion of the solubility curve of the hexahydrate
+would be represented by a similar rounded portion in the pressure
+curve.[227] As in the case of sodium sulphate, the transition points of the
+different hydrates would be indicated by breaks in the curve of pressures.
+Finally, mention may again be made of the difference of the pressure of
+dissociation of the hexahydrate according as it becomes dehydrated to the
+[alpha]- or the [beta]-tetrahydrate (p. 88).
+
+The Indifferent Point.--We have already seen that at 30.2° calcium chloride
+hexahydrate melts congruently, and that, provided the pressure is
+maintained constant, addition or withdrawal of heat will cause the complete
+liquefaction or solidification, without the temperature of the system
+undergoing change. This behaviour, therefore, is similar to, but is not
+quite the same as the fusion of a simple substance such as ice; and the
+difference is due to the fact that in the case of the hexahydrate the
+emission of vapour by the liquid phase causes an alteration in the
+composition of the latter, owing to the non-volatility of the calcium
+chloride; whereas in the case of ice this is, of course, not so.
+
+Consider, however, for the present that the vapour phase is absent, and
+that we are dealing with the two-phase system solid--solution. Then, since
+there are two components, the system is bivariant. For any given value of
+the pressure, therefore, we should expect that the system could exist at
+different temperatures; which, indeed, is the case. It has, however,
+already been noted that when the composition of the liquid phase becomes
+the same as that of the solid, the system then behaves as a _univariant_
+system; for, at a given pressure, the system solid--solution can exist only
+at _one_ temperature, change of temperature producing complete
+transformation in {151} one or other direction. _The variability of the
+system has therefore been diminished._
+
+This behaviour will perhaps be more clearly understood when one reflects
+that since the composition of the two phases is the same, the system may be
+regarded as being formed of _one component_, just as the system NH_{4}Cl
+<--> NH_{3} + HCl was regarded as being composed of one component when the
+vapour had the same total composition as the solid (p. 13). One component
+in two phases, however, constitutes a univariant system, and we can
+therefore see that calcium chloride hexahydrate in contact with solution of
+the same composition will constitute a univariant system. The temperature
+of equilibrium will, however, vary with the pressure;[228] if the latter is
+constant, the temperature will also be constant.
+
+A point such as has just been referred to, which represents the special
+behaviour of a system of two (or more) components, in which the composition
+of two phases becomes identical, is known as an _indifferent point_,[229]
+and it has been shown[230] that at a given pressure the temperature in the
+indifferent point is the _maximum_ or _minimum_ temperature possible at the
+particular pressure[231] (cf. critical solution temperature). At such a
+point a system loses one degree of freedom, or behaves like a system of the
+next lower order.
+
+The Hydrates of Ferric Chloride.--A better illustration of the formation of
+compounds possessing a definite melting point, and of the existence of
+retroflex solubility curves, is afforded by the hydrates of ferric
+chloride, which not only possess definite points of fusion, but these
+melting points are stable. A very brief description of the relations met
+with will suffice.[232]
+
+{152}
+
+Ferric chloride can form no less than four stable hydrates, viz.
+Fe_{2}Cl_{6},12H_{2}O, Fe_{2}Cl_{6},7H_{2}O, Fe_{2}Cl_{6},5H_{2}O, and
+Fe_{2}Cl_{6},4H_{2}O, and each of these hydrates possesses a definite,
+stable melting point. On analogy with the behaviour of calcium chloride,
+therefore, we shall expect that the solubility curves of these different
+hydrates will exhibit a series of _temperature maxima_; the points of
+maximum temperature representing systems in which the composition of the
+solid and liquid phases is the same. A graphical representation of the
+solubility relations is given in Fig. 39, and the composition of the
+different saturated solutions which can be formed is given in the following
+tables, the composition being expressed in molecules of Fe_{2}Cl_{6} to 100
+molecules of water. The figures printed in thick type refer to transition
+and melting points.
+
+[Illustration: FIG. 39.]
+
+{153}
+
+  COMPOSITION OF THE SATURATED SOLUTIONS OF FERRIC CHLORIDE AND ITS
+      HYDRATES.
+
+  (_The name placed at the head of each table is the solid phase._)
+
+  ICE.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     ±-55°     |    ±2.75
+      -40°     |     2.37
+      -27.5°   |     1.90
+      -20.5°   |     1.64
+      -10°     |     1.00
+        0°     |     0
+  ---------------------------
+
+  Fe_{2}Cl_{6},12H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     -55°      |    ±2.75
+     -41°      |     2.81
+     -27°      |     2.98
+       0°      |     4.13
+      10°      |     4.54
+      20°      |     5.10
+      30°      |     5.93
+      35°      |     6.78
+      36.5°    |     7.93
+      37°      |     8.33
+      36°      |     9.29
+      33°      |    10.45
+      30°      |    11.20
+      27·4°    |    12.15
+      20°      |    12.83
+      10°      |    13.20
+       8°      |    13.70
+  ---------------------------
+
+  Fe_{2}Cl_{6},7H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+      20°      |    11.35
+      27·4°    |    12.15
+      32°      |    13.55
+      32.5°    |    14.29
+      30°      |    15.12
+      25°      |    15.54
+  ---------------------------
+
+  Fe_{2}Cl_{6},5H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+      20°      |    11.35
+      12°      |    12.87
+      20°      |    13.95
+      27°      |    14.85
+      30°      |    15.12
+      35°      |    15.64
+      50°      |    17.50
+      55°      |    19.15
+      56°      |    20.00
+      55°      |    20.32
+  ---------------------------
+
+  Fe_{2}Cl_{6},4H_{2}O
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+      20°      |    11.35
+      50°      |    19.96
+      55°      |    20.32
+      60°      |    20.70
+      69°      |    21.53
+      72.5°    |    23.35
+      73.5°    |    25.00
+      72.5°    |    26.15
+      70°      |    27.90
+      66°      |    29.20
+  ---------------------------
+
+  Fe_{2}Cl_{6} (ANHYDROUS).
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+      20°      |    11.35
+      66°      |    29.20
+      70°      |    29.42
+      75°      |    28.92
+      80°      |    29.20
+     100°      |    29.75
+  ---------------------------
+
+The lowest portion of the curve, AB, represents the equilibria between ice
+and solutions containing ferric chloride. It represents, in other words,
+the lowering of the fusion point of ice by addition of ferric chloride. At
+the point B (-55°), the cryohydric point (p. 117) is reached, at which the
+solution is in equilibrium with ice and ferric chloride dodecahydrate. As
+{154} has already been shown, such a point represents an invariant system;
+and the liquid phase will, therefore, solidify to a mixture of ice and
+hydrate without change of temperature. If heat is added, ice will melt and
+the system will pass to the curve BCDN, which is the solubility curve of
+the dodecahydrate. At C (37°), the point of maximum temperature, the
+hydrate melts completely. The retroflex portion of this curve can be
+followed backwards to a temperature of 8°, but below 27.4° (D), the
+solutions are supersaturated with respect to the heptahydrate; point D is
+the eutectic point for dodecahydrate and heptahydrate. The curve DEF is the
+solubility curve of the heptahydrate, E being the melting point, 32.5°. On
+further increasing the quantity of ferric chloride, the temperature of
+equilibrium is lowered until at F (30°) another eutectic point is reached,
+at which the heptahydrate and pentahydrate can co-exist with solution. Then
+follow the solubility curves for the pentahydrate, the tetrahydrate, and
+the anhydrous salt; G (56°) is the melting point of the former hydrate, J
+(73.5°) the melting point of the latter. H and K, the points at which the
+curves intersect, represent eutectic points; the temperature of the former
+is 55°, that of the latter 66°. The dotted portions of the curves represent
+metastable equilibria.
+
+As is seen from the diagram, a remarkable series of solubility curves is
+obtained, each passing through a point of maximum temperature, the whole
+series of curves forming an undulating "festoon." To the right of the
+series of curves the diagram represents unsaturated solutions; to the left,
+supersaturated.
+
+If an unsaturated solution, the composition of which is represented by a
+point in the field to the right of the solubility curves, is cooled down,
+the result obtained will differ according as the composition of the
+solution is the same as that of a cryohydric point, or of a melting point,
+or has an intermediate value. Thus, if a solution represented by _x__{1} is
+cooled down, the composition will remain unchanged as indicated by the
+horizontal dotted line, until the point D is reached. At this point,
+dodecahydrate and heptahydrate will separate out, and the liquid will
+ultimately solidify completely to a mixture or "conglomerate" of these two
+hydrates; the temperature of {155} the system remaining constant until
+complete solidification has taken place. If, on the other hand, a solution
+of the composition _x__{3} is cooled down, ferric chloride dodecahydrate
+will be formed when the temperature has fallen to that represented by C,
+and the solution will completely solidify, without alteration of
+temperature, with formation of this hydrate. In both these cases,
+therefore, a point is reached at which complete solidification occurs
+without change of temperature.
+
+Somewhat different, however, is the result when the solution has an
+intermediate composition, as represented by _x__{2} or _x__{4}. In the
+former case the dodecahydrate will first of all separate out, but on
+further withdrawal of heat the temperature will fall, the solution will
+become relatively richer in ferric chloride, owing to separation of the
+hydrate, and ultimately the eutectic point D will be reached, at which
+complete solidification will occur. Similarly with the second solution.
+Ferric chloride dodecahydrate will first be formed, and the temperature
+will gradually fall, the composition of the solution following the curve CB
+until the cryohydric point B is reached, when the whole will solidify to a
+conglomerate of ice and dodecahydrate.
+
+Suspended Transformation.--Not only can the upper branch of the solubility
+curve of the dodecahydrate be followed backwards to a temperature of 8°, or
+about 19° below the temperature of transition to the heptahydrate; but
+suspended transformation has also been observed in the case of the
+heptahydrate and the pentahydrate. To such an extent is this the case that
+the solubility curve of the latter hydrate has been followed downwards to
+its point of intersection with the curve for the dodecahydrate. This point
+of intersection, represented in Fig. 39 by M, lies at a temperature of
+about 15°; and at this temperature, therefore, it is possible for the two
+solid phases dodecahydrate and pentahydrate to coexist, so that M is a
+eutectic point for the dodecahydrate and the pentahydrate. It is, however,
+a metastable eutectic point, for it lies in the region of supersaturation
+with respect to the heptahydrate; and it can be realized only because of
+the fact that the latter hydrate is not readily formed.
+
+Evaporation of Solutions at Constant Temperature.--On {156} evaporating
+dilute solutions of ferric chloride at constant temperature, a remarkable
+series of changes is observed, which, however, will be understood with the
+help of Fig. 40. Suppose an unsaturated solution, the composition of which
+is represented by the point _x__{1}, is evaporated at a temperature of
+about 17° - 18°. As water passes off, the composition of the solution will
+follow the dotted line of constant temperature, until at the point where it
+cuts the curve BC the solid hydrate Fe_{2}Cl_{6},12H_{2}O separates out. As
+water continues to be removed, the hydrate must be deposited (in order that
+the solution shall remain saturated), until finally the solution dries up
+to the hydrate. As dehydration proceeds, the heptahydrate can be formed,
+and the dodecahydrate will finally pass into the heptahydrate; and this, in
+turn, into the pentahydrate.
+
+[Illustration: FIG. 40.]
+
+But the heptahydrate is not always formed by the dehydration of the
+dodecahydrate, and the behaviour on evaporation is therefore somewhat
+perplexing at first sight. After the solution has dried to the
+dodecahydrate, as explained above, further removal of water causes
+liquefaction, and the system is now represented by the point of
+intersection at _a_; at this point the solid hydrate is in equilibrium with
+a solution containing relatively more ferric chloride. If, therefore,
+evaporation is continued, the solid hydrate must _pass into solution_ in
+order that the composition of the latter may remain unchanged, so that
+ultimately a liquid will again be obtained. A very slight further
+dehydration will bring the solution into the state represented by _b_, at
+which the pentahydrate is formed, and the solution will at last disappear
+and leave this hydrate alone.
+
+Without the information to be obtained from the curves in Figs. 39 and 40,
+the phenomena which would be observed on carrying out the evaporation at a
+temperature of about 31 - 32° {157} would be still more bewildering. The
+composition of the different solutions formed will be represented by the
+perpendicular line _x__{2}12345. Evaporation will first cause the
+separation of the dodecahydrate, and then total disappearance of the liquid
+phase. Then liquefaction will occur, and the system will now be represented
+by the point 2, in which condition it will remain until the solid hydrate
+has disappeared. Following this there will be deposition of the
+heptahydrate (point 3), with subsequent disappearance of the liquid phase.
+Further dehydration will again cause liquefaction, when the concentration
+of the solution will be represented by the point 4; the heptahydrate will
+ultimately disappear, and then will ensue the deposition of the
+pentahydrate, and complete solidification will result. On evaporating a
+solution, therefore, of the composition _x__{2}, the following series of
+phenomena will be observed: solidification to dodecahydrate; liquefaction;
+solidification to heptahydrate; liquefaction; solidification to
+pentahydrate.[233]
+
+Although ferric chloride and water form the largest and best-studied series
+of hydrates possessing definite melting points, examples of similar
+hydrates are not few in number; and more careful investigation is
+constantly adding to the list.[234] In all these cases the solubility curve
+will show a point of maximum temperature, at which the hydrate melts, and
+will end, above and below, in a cryohydric point. Conversely, if such a
+curve is found in a system of two components, we can argue that a definite
+compound of the components possessing a definite melting point is formed.
+
+Inevaporable Solutions.--If a saturated solution in contact with two
+hydrates, or with a hydrate and anhydrous salt is heated, the temperature
+and composition of the solution will, of course, remain unchanged so long
+as the two solid phases are present, for such a system is invariant. In
+addition to this, however, the _quantity_ of the solution will also remain
+unchanged, the water which evaporates being supplied by the higher hydrate.
+The same phenomenon is also observed in the case of cryohydric points when
+ice is a solid phase; so long as the latter is present, evaporation will be
+accompanied {158} by fusion of the ice, and the quantity of solution will
+remain constant. Such solutions are called _inevaporable_.[235]
+
+[Illustration: FIG. 41.]
+
+Illustration.--In order to illustrate the application of the principles of
+the Phase Rule to the study of systems formed by a volatile and a
+non-volatile component, a brief description may be given of the behaviour
+of sulphur dioxide and potassium iodide, which has formed the subject of a
+recent investigation. After it had been found[236] that liquid sulphur
+dioxide has the property of dissolving potassium iodide, and that the
+solutions thus obtained present certain peculiarities of behaviour, the
+question arose as to whether or not compounds are formed between the
+sulphur dioxide and the potassium iodide, and if so, what these compounds
+are. To find an answer to this question, Walden and Centnerszwer[237] made
+a complete investigation of the solubility curves (equilibrium curves) of
+these two components, the investigation extending from the freezing point
+to the critical point of sulphur dioxide. For convenience of reference, the
+results which they obtained are represented diagrammatically in Fig. 41.
+The freezing point (A) of pure sulphur dioxide was found to be -72.7°.
+Addition of potassium iodide lowered the freezing point, but the maximum
+depression obtained was very small, and was reached when the concentration
+of the potassium iodide in the solution was only 0.336 mols. per cent.
+Beyond this point, an increase in the concentration of the iodide was
+accompanied by an elevation of the freezing point, the change of the
+freezing point with the concentration being represented by the curve BC.
+The solid {159} which separated from the solutions represented by BC was a
+bright _yellow_ crystalline substance. At the point C (-23.4°) a
+temperature-maximum was reached; and as the concentration of the potassium
+iodide was continuously increased, the temperature of equilibrium first
+fell and then slowly rose, until at +0.26° (E) a second temperature-maximum
+was registered. On passing the point D, the solid which was deposited from
+the solution was a _red_ crystalline substance. On withdrawing sulphur
+dioxide from the system, the solution became turbid, and the temperature
+remained constant. The investigation was not pursued farther at this point,
+the attention being then directed to the equilibria at higher temperatures.
+
+When a solution of potassium iodide in liquid sulphur dioxide containing
+1.49 per cent. of potassium iodide was heated, solid (potassium iodide) was
+deposited at a temperature of 96.4°. Solutions containing more than about 3
+per cent. of the iodide separated, on being heated, into two layers, and
+the temperature at which the liquid became heterogeneous fell as the
+concentration was increased; a temperature-minimum being obtained with
+solutions containing 12 per cent. of potassium iodide. On the other hand,
+solutions containing 30.9 per cent. of the iodide, on being heated,
+deposited potassium iodide; while a solution containing 24.5 per cent. of
+the salt first separated into two layers at 89.3°, and then, on cooling,
+solid was deposited and one of the liquid layers disappeared.
+
+Such are, in brief, the results of experiment; their interpretation in the
+light of the Phase Rule is the following:--
+
+The curve AB is the freezing-point curve of solid sulphur dioxide in
+contact with solutions of potassium iodide. BCD is the solubility curve of
+the yellow crystalline solid which is deposited from the solutions. C, the
+temperature-maximum, is the melting point of this _yellow_ solid, and the
+composition of the latter must be the same as that of the solution at this
+point (p. 145), which was found to be that represented by the formula
+KI,14SO_{2}. B is therefore the eutectic point, at which solid sulphur
+dioxide and the compound KI,14SO_{2} can exist together in equilibrium with
+solution and vapour. The curve DE is the solubility curve of the _red_
+crystalline solid, and the {160} point E, at which the composition of
+solution and solid is the same, is the melting point of the solid. The
+composition of this substance was found to be KI,4SO_{2}.[238] D is,
+therefore, the eutectic point at which the compounds KI,14SO_{2} and
+KI,4SO_{2} can coexist in equilibrium with solution and vapour. The curve
+DE does not exhibit a retroflex portion; on the contrary, on attempting to
+obtain more concentrated solutions in equilibrium with the compound
+KI,4SO_{2}, a new solid phase (probably potassium iodide) was formed. Since
+at this point there are four phases in equilibrium, viz. the compound
+KI,4SO_{2}, potassium iodide, solution, and vapour, the system is
+invariant. E is, therefore, the _transition point_ for KI,4SO_{2} and KI.
+
+Passing to higher temperatures, FG is the solubility curve of potassium
+iodide in sulphur dioxide; at G two liquid phases are formed, and the
+system therefore becomes invariant (cf. p. 121). The curve GHK is the
+solubility curve for two partially miscible liquids; and since complete
+miscibility occurs on _lowering_ the temperature, the curve is similar to
+that obtained with triethylamine and water (p. 101). K is also an invariant
+point at which potassium iodide is in equilibrium with two liquid phases
+and vapour.
+
+The complete investigation of the equilibria between sulphur dioxide and
+potassium iodide, therefore, shows that these two components form the
+compounds KI,14SO_{2} and KI,4SO_{2}; and that when solutions having a
+concentration between those represented by the points G and K are heated,
+separation into two layers occurs. The temperatures and concentrations of
+the different characteristic points are as follows:--
+
+  -------------------------------------------------------------
+                              |                | Composition of
+              Point.          |  Temperature.  |  the solution
+                              |                |  per cent. KI.
+  -------------------------------------------------------------
+  A (m.p. of SO_{2})          |        -72.7°  |       --
+  B (eutectic point)          |        --      |      0.86
+  C (m.p. of KI,14SO_{2})     |        -23.4°  |     17.63
+  E (m.p. of KI,4SO_{2})      |         +0.26° |     39.33
+  G (KI + two liquid phases)  | (about) 88°    |     24.0
+  H (critical solution point) |         77.3°  |     12
+  K (KI + two liquid phases)  | (about) 88°    |      2.7
+  -------------------------------------------------------------
+
+       *       *       *       *       *
+
+
+{161}
+
+CHAPTER IX
+
+EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS
+
+General.--In the two preceding chapters certain restrictions were imposed
+on the discussion of the equilibria between two components; but in the
+present chapter the restriction that only one of the components is volatile
+will be allowed to fall, and the general behaviour of two volatile[239]
+components, each of which is capable of forming a liquid solution with the
+other, will be studied. As we shall see, however, the removal of the
+previous restriction produces no alteration in the general aspect of the
+equilibrium curves for concentration and temperature, but changes to some
+extent the appearance of the pressure-temperature diagram. The latter would
+become still more complicated if account were taken not only of the total
+pressure but also of the partial pressures of the two components in the
+vapour phase; this complication, however, will not be introduced in the
+present discussion.[240] In this chapter we shall consider the systems
+formed by the two components iodine and chlorine, and sulphur dioxide and
+water.
+
+Iodine and Chlorine.--The different systems furnished by iodine and
+chlorine, rendered classical by the studies of Stortenbeker,[241] form a
+very complete example of equilibria in a two-component system. We shall
+first of all consider the {162} relations between concentration and
+temperature, with the help of the accompanying diagram, Fig. 42.
+
+[Illustration: FIG. 42.]
+
+Concentration-Temperature Diagram.--In this diagram the temperatures are
+taken as the abscissæ, and the composition of the solution, expressed in
+atoms of chlorine to one atom of iodine,[242] is represented by the
+ordinates. In the diagram, A represents the melting point of pure iodine,
+114°. If chlorine is added to the system, a solution of chlorine in liquid
+iodine is obtained, and the temperature at which solid iodine is in
+equilibrium with the liquid solution will be all the lower the greater the
+concentration of the chlorine. We therefore obtain the curve ABF, which
+represents the composition of the solution {163} with which solid iodine is
+in equilibrium at different temperatures. This curve can be followed down
+to 0°, but at temperatures below 7.9° (B) it represents metastable
+equilibria. At B iodine monochloride can be formed, and if present the
+system becomes invariant; B is therefore a quadruple point at which the
+four phases, iodine, iodine monochloride, solution, and vapour, can
+coexist. Continued withdrawal of heat at this point will therefore lead to
+the complete solidification of the solution to a mixture or conglomerate of
+iodine and iodine monochloride, while the temperature remains constant
+during the process. B is the eutectic point for iodine and iodine
+monochloride.
+
+Just as we found in the case of aqueous salt solutions that at temperatures
+above the cryohydric or eutectic point, two different solutions could
+exist, one in equilibrium with ice, the other in equilibrium with the salt
+(or salt hydrate), so in the case of iodine and chlorine there can be two
+solutions above the eutectic point B, one containing a lower proportion of
+chlorine in equilibrium with iodine, the other containing a higher
+proportion of chlorine in equilibrium with iodine monochloride. The
+composition of the latter solution is represented by the curve BCD. As the
+concentration of chlorine is increased, the temperature at which there is
+equilibrium between iodine monochloride and solution rises until a point is
+reached at which the composition of the solution is the same as that of the
+solid. At this point (C), iodine monochloride melts. Addition of one of the
+components will lower the temperature of fusion, and a continuous
+curve,[243] exhibiting a retroflex portion as in the case of
+CaCl_{2},6H_{2}O, will be obtained. At temperatures below its melting
+point, therefore, iodine monochloride can be in equilibrium with two
+different solutions.
+
+The upper portion of this curve, CD, can be followed downwards to a
+temperature of 22.7°. At this temperature iodine trichloride can separate
+out, and a second quadruple {164} point (D) is obtained. This is the
+eutectic point for iodine monochloride and iodine trichloride.
+
+By addition of heat and increase in the amount of chlorine, the iodine
+monochloride disappears, and the system passes along the curve DE, which
+represents the composition of the solutions in equilibrium with solid
+iodine trichloride. The concentration of chlorine in the solution increases
+as the temperature is raised, until at the point E, where the solution has
+the same composition as the solid, the maximum temperature is reached; the
+iodine trichloride melts. On increasing still further the concentration of
+chlorine in the solution, the temperature of equilibrium falls, and a
+continuous curve, similar to that for the monochloride, is obtained. The
+upper branch of this curve has been followed down to a temperature of 30°,
+the solution at this point containing 99.6 per cent. of chlorine.[244] The
+very rounded form of the curve is due to the trichloride being largely
+dissociated in the liquid state.
+
+One curve still remains to be considered. As has already been mentioned,
+iodine monochloride can exist in two crystalline forms, only one of which,
+however, is stable at temperatures below the melting point; the two forms
+are _monotropic_ (p. 44). The stable form which melts at 27.2°, is called
+the [alpha]-form, while the less stable variety, melting at 13.9°, is known
+as the [beta]-form. If, now, the presence of [alpha]-ICl is excluded, it is
+possible to obtain the [beta]-form, and to study the conditions of
+equilibrium between it and solutions of iodine and chlorine, from the
+eutectic point F to the melting point G. As the [beta]-ICl becomes less
+stable in presence of excess of chlorine, it has not been possible to study
+the retroflex portion of the curve represented by the dotted continuation
+of FG.
+
+The following table gives some of the numerical data from which Fig. 42 was
+constructed.[245]
+
+{165}
+
+  IODINE AND CHLORINE.
+
+  I. _Invariant systems._
+
+  -------------------------------------------------------------------------
+          |          |                  Phases present.
+   Temper-| Pressure.+--------------------+-----------------+--------------
+   ature. |          |      Solid.        |    Liquid.      |   Vapour.
+  --------+----------+--------------------+-----------------+--------------
+   7.9°   | 11 mm.   | I_{2},[alpha]-ICl  | I[wavy]Cl_{0.66}| I + Cl_{0.92}
+   0.9°   |   --     | I_{2},[beta]-ICl   | I[wavy]Cl_{0.72}|     --
+   22.7°  | 42 mm.   | [alpha]-ICl,ICl_{3}| I[wavy]Cl_{1.19}| I + Cl_{1.75}
+   [-102° | <1 atm.  | ICl_{3},Cl_{2}     | I[wavy]Cl_{m}   | I + Cl_{n}]
+  --------+----------+--------------------+-----------------+--------------
+
+  II. _Melting points._
+
+  A. Iodine,[246] 114.15° (pressure 89.8 mm.).
+  C. [alpha]-Iodine monochloride, 27.2° (pressure 37 mm.).
+  E. Iodine trichloride, 101° (pressure 16 atm.).
+  G. [beta]-Iodine monochloride, 13.9°.
+
+Since the vapour pressure at the melting point of iodine trichloride
+amounts to 16 atm., the experiments must of course be carried out
+in closed vessels. At 63.7° the vapour pressure of the system
+trichloride--solution--vapour is equal to 1 atm.
+
+Pressure-Temperature Diagram.--In this diagram there are represented the
+values of the vapour pressure of the saturated solutions of chlorine and
+iodine. To give a complete picture of the relations between pressure,
+temperature, and concentration, a solid model would be required, with three
+axes at right angles to one another along which could be measured the
+values of pressure, temperature, and concentration of the components in the
+solution. Instead of this, however, there may be employed the accompanying
+projection figure[247] (Fig. 43), the lower portion of which shows the
+projection of the equilibrium curve on the surface containing the
+concentration and temperature axes, while the upper portion is the
+projection on the plane containing the pressure and temperature axes. The
+lower portion is therefore a concentration-temperature diagram; {166} the
+upper portion, a pressure-temperature diagram. The corresponding points of
+the two diagrams are joined by dotted lines.
+
+[Illustration: FIG. 43.]
+
+Corresponding to the point C, the melting point of pure iodine, there is
+the point C_{1}, which represents the vapour pressure of iodine at its
+melting point. At this point three curves cut: 1, the sublimation curve of
+iodine; 2, the vaporization curve of fused iodine; 3, C_{1}B_{1}, the
+vapour-pressure curve of the saturated solutions in equilibrium with solid
+iodine. Starting, therefore, with the system solid iodine--liquid iodine,
+addition of chlorine will cause the temperature of equilibrium to fall
+continuously, while the vapour pressure will first increase, pass through a
+maximum and then fall continuously {167} until the eutectic point, B
+(B_{1}), is reached.[248] At this point the system is invariant, and the
+pressure will therefore remain constant until all the iodine has
+disappeared. As the concentration of the chlorine increases in the manner
+represented by the curve B_f_H, the pressure of the vapour also increases
+as represented by the curve B_{1}_f__{1}H_{1}. At H_{1}, the eutectic point
+for iodine monochloride and iodine trichloride, the pressure again remains
+constant until all the monochloride has disappeared. As the concentration
+of the solution passes along the curve HF, the pressure of the vapour
+increases as represented by the curve H_{1}F_{1}; F_{1} represents the
+pressure of the vapour at the melting point of iodine trichloride. If the
+concentration of the chlorine in the solution is continuously increased
+from this point, the vapour pressure first increases and then decreases,
+until the eutectic point for iodine trichloride and solid chlorine is
+reached (D_{1}). Curves Cl_{2} solid and Cl_{2} liquid represent the
+sublimation and vaporization curves of chlorine, the melting point of
+chlorine being -102°.
+
+Although complete measurements of the vapour pressure of the different
+systems of pure iodine to pure chlorine have not been made, the
+experimental data are nevertheless sufficient to allow of the general form
+of the curves being indicated with certainty.
+
+Bivariant Systems.--To these, only a brief reference need be made. Since
+there are two components, two phases will form a bivariant system. The
+fields in which these systems can exist are shown in Fig. 43 and Fig. 44,
+which is a more diagrammatic representation of a portion of Fig. 43.
+
+    I. Iodine--vapour.
+   II. Solution--vapour.
+  III. Iodine trichloride--vapour.
+  IV. Iodine monochloride--vapour.
+
+[Illustration: FIG. 44.]
+
+The conditions for the existence of these systems will probably be best
+understood from Fig. 44. Since the curve B'A' {168} represents the
+pressures under which the system iodine--solution--vapour can exist,
+increase of volume (diminution of pressure) will cause the volatilization
+of the solution, and the system iodine--vapour will remain. If, therefore,
+we start with a system represented by _a_, diminution of pressure at
+constant temperature will lead to the condition represented by _x_. On the
+other hand, increase of pressure at _a_ will lead to the condensation of a
+portion of the vapour phase. Since, now, the concentration of chlorine in
+the vapour is greater than in the solution, condensation of vapour would
+increase the concentration of chlorine in the solution; a certain amount of
+iodine must therefore pass into solution in order that the composition of
+the latter shall remain unchanged.[249] If, therefore, the volume of vapour
+be sufficiently great, continued diminution of volume will ultimately lead
+to the disappearance of all the iodine, and there will remain only solution
+and vapour (field II.). As the diminution of volume is continued, the
+vapour pressure and the concentration of the chlorine in the solution will
+increase, until when the pressure has reached the value _b_, iodine
+monochloride can separate out. The system, therefore, again becomes
+univariant, and at constant temperature the pressure and composition of the
+phases must remain unchanged. Diminution of volume will therefore not
+effect an increase of pressure, but a condensation of the vapour; and since
+this is richer in chlorine than the {169} solution, solid iodine
+monochloride must separate out in order that the concentration of the
+solution remain unchanged.[250] As the result, therefore, we obtain the
+bivariant system iodine monochloride--vapour.
+
+A detailed discussion of the effect of a continued increase of pressure
+will not be necessary. From what has already been said and with the help of
+Fig. 44, it will readily be understood that this will lead successively to
+the univariant system (_c_), iodine monochloride--solution--vapour; the
+bivariant system solution--vapour (field II.); the univariant system (_d_),
+iodine trichloride--solution--vapour; and the bivariant system _x'_, iodine
+trichloride--vapour. If the temperature of the experiment is above the
+melting point of the monochloride, then the systems in which this compound
+occurs will not be formed.
+
+Sulphur Dioxide and Water.--In the case just studied we have seen that the
+components can combine to form definite compounds possessing stable melting
+points. The curves of equilibrium, therefore, resemble in their general
+aspect those of calcium chloride and water, or of ferric chloride and
+water. In the case of sulphur dioxide and water, however, the melting point
+of the compound formed cannot be realized, because transition to another
+system occurs; retroflex concentration-temperature curves are therefore not
+found here, but the curves exhibit breaks or sudden changes in direction at
+the transition points, as in the case of the systems formed by sodium
+sulphate and water. The case of sulphur dioxide and water is also of
+interest from the fact that two liquid phases can be formed.
+
+The phases which occur are--Solid: ice, sulphur dioxide hydrate,
+SO_{2},7H_{2}O. Liquid: two solutions, the one containing excess of sulphur
+dioxide, the other excess of water, and represented by the symbols SO_{2}
+[wavy] _x_H_{2}O (solution I.), and H_{2}O [wavy] _y_SO_{2} (solution II.).
+Vapour: a mixture of sulphur dioxide and water vapour in varying
+proportions. Since there are two components, sulphur dioxide and water, the
+number of {170} possible systems is considerable. Only the following,
+however, have been studied:--
+
+  I. _Invariant Systems: Four co-existing phases._
+    (_a_) Ice, hydrate, solution, vapour.
+    (_b_) Hydrate, solution I., solution II., vapour.
+
+  II. _Univariant Systems: Three co-existing phases._
+    (_a_) Hydrate, solution I., vapour.
+    (_b_) Hydrate, solution II., vapour.
+    (_c_) Solution I., solution II., vapour.
+    (_d_) Hydrate, solution I., solution II.
+    (_e_) Hydrate, ice, vapour.
+    (_f_) Ice, solution II., vapour.
+    (_g_) Ice, hydrate, solution II.
+
+  III. _Bivariant Systems: Two co-existing phases._
+    (_a_) Hydrate, solution I.
+    (_b_) Hydrate, solution II.
+    (_c_) Hydrate, vapour.
+    (_d_) Hydrate, ice.
+    (_e_) Solution I., solution II.
+    (_f_) Solution I., vapour.
+    (_g_) Solution I., ice.
+    (_h_) Solution II., vapour.
+    (_i_) Solution II., ice.
+    (_j_) Ice, vapour.
+
+[Illustration: FIG. 45.]
+
+Pressure-Temperature Diagram.[251]--If sulphur dioxide is passed into water
+at 0°, a solution will be formed and the temperature at which ice can
+exist in equilibrium with this solution will fall more and more as the
+concentration of the sulphur dioxide increases. At -2.6°, however, a
+cryohydric point is reached at which solid hydrate separates out,
+and the system becomes invariant. The curve AB (Fig. 45) therefore
+represents the pressure of the system ice--solution II.--vapour, and B
+represents the temperature and pressure at which the invariant system
+ice--hydrate--solution II.--vapour can exist. At this point the temperature
+is -2.6°, and the pressure 21.1 cm. If heat is withdrawn from this
+system, the solution will ultimately {171} solidify to a mixture of
+ice and hydrate, and there will be obtained the univariant system
+ice--hydrate--vapour. The vapour pressure of this system has been
+determined down to a temperature of -9.5°, at which temperature the
+pressure amounts to 15 cm. The pressures for this system are represented by
+the curve BC. If at the point B the volume is diminished, the pressure must
+remain constant, but the relative amounts of the different phases will
+undergo change. If suitable quantities of these are present, diminution of
+volume will ultimately lead to the total condensation of the vapour phase,
+and there will remain the univariant system ice--hydrate--solution. The
+temperature of equilibrium of this system will alter with the pressure,
+but, as in the case of the melting point of a simple substance, great
+differences of pressure will cause only comparatively small changes in the
+temperature of equilibrium. The change of the cryohydric point with the
+pressure is represented by the line BE; the actual values have not been
+determined, but the curve must slope towards the pressure axis because
+fusion is accompanied by diminution of volume, as in the case of pure ice.
+{172}
+
+A fourth univariant system can be formed at B. This is the system
+hydrate--solution II.--vapour. The conditions for the existence of
+this system are represented by the curve BF, which may therefore be
+regarded as the vapour-pressure curve of the saturated solution of
+sulphur dioxide heptahydrate in water. Unlike the curve for iodine
+trichloride--solution--vapour, this curve cannot be followed to the melting
+point of the hydrate. Before this point is reached, a second liquid phase
+appears, and an invariant system consisting of hydrate--solution
+I.--solution II.--vapour is formed. We have here, therefore, the phenomenon
+of melting under the solution as in the case of succinic nitrile and water
+(p. 122). This point is represented in the diagram by F; the temperature at
+this point is 12.1°, and the pressure 177.3 cm. The range of stable
+existence of the hydrate is therefore from -2.6° to 12.1°; nevertheless,
+the curve FB has been followed down to a temperature of -6°, at which point
+ice formed spontaneously.
+
+So long as the four phases hydrate, two liquid phases, and vapour are
+present, the condition of the system is perfectly defined. By altering the
+conditions, however, one of the phases can be made to disappear, and a
+univariant system will then be obtained. Thus, if the vapour phase is made
+to disappear, the univariant system solution I.--solution II.--hydrate,
+will be left, and the temperature at which this system is in equilibrium
+will vary with the pressure. This is represented by the curve FI; under a
+pressure of 225 atm. the temperature of equilibrium is 17.1°. Increase of
+pressure, therefore, raises the temperature at which the three phases can
+coexist.
+
+Again, addition of heat to the invariant system at F will cause the
+disappearance of the solid phase, and there will be formed the univariant
+system solution I.--solution II.--vapour. In the case of this system the
+vapour pressure increases as the temperature rises, as represented by the
+curve FG. Such a system is analogous to the case of ether and water, or
+other two partially miscible liquids (p. 103). As the temperature changes,
+the composition of the two liquid phases will undergo change; but this
+system has not been studied fully.
+
+The fourth curve, which ends at the quadruple point F, is {173} that
+representing the vapour pressure of the system hydrate--solution I.--vapour
+(FH). This curve has been followed to a temperature of 0°, the pressure at
+this point being 113 cm. The metastable prolongation of GF has also been
+determined. Although, theoretically, this curve must lie below FH, it was
+found that the difference in the pressure for the two curves was within the
+error of experiment.
+
+Bivariant Systems.--The different bivariant systems, consisting of two
+phases, which can exist within the range of temperature and pressure
+included in Fig. 45, were given on p. 170. The conditions under which these
+systems can exist are represented by the areas in the diagram, and the
+fields of the different bivariant systems are indicated by letters,
+corresponding to the letters on p. 170. Just as in the case of
+one-component systems (p. 29), we found that the field lying between any
+two curves gave the conditions of existence of that phase which was common
+to the two curves, so also in the case of two-component systems, a
+bivariant two-phase system occurs in the field enclosed[252] by the two
+curves to which the two phases are common. As can be seen, the same
+bivariant system can occur in more than one field.
+
+As is evident from Fig. 45, three different bivariant systems are capable
+of existing in the area HFI; which of these will be obtained will depend on
+the relative masses of the different phases in the univariant or invariant
+system. Thus, starting with a system represented by a point on the curve
+HF, diminution of volume at constant temperature will cause the
+condensation of a portion of the vapour, which is rich in sulphur dioxide;
+since this would increase the concentration of sulphur dioxide in the
+solution, it must be counteracted by the passage of a portion of the
+hydrate (which is relatively poor in sulphur dioxide) into the solution.
+If, therefore, the amount of hydrate present is relatively very small, the
+final result of the compression will be the production of the system _f_,
+solution I.--vapour. On the other hand, if the vapour is present in
+relatively small amount, it will be the first phase to disappear, {174} and
+the bivariant system _a_, hydrate--solution I., will be obtained. Finally,
+if we start with the invariant system at F, compression will cause the
+condensation of vapour, while the composition of the two solutions will
+remain unchanged. When all the vapour has disappeared, the univariant
+system hydrate--solution I.--solution II. will be left. If, now, the
+pressure is still further increased, while the temperature is kept below
+12°, more and more hydrate must be formed at the expense of the two liquid
+phases (because 12° is the lower limit for the coexistence of the two
+liquid phases), and if the amount of the solution I. (containing excess of
+sulphur dioxide) is relatively small, it will disappear before solution
+II., and there will be obtained the bivariant system hydrate--solution II.
+(bivariant system _b_).
+
+In a similar manner, account can be taken of the formation of the other
+bivariant systems.
+
+A behaviour similar to that of sulphur dioxide and water is shown by
+chlorine and water and by bromine and water, although these have not been
+so fully studied.[253] In the case of hydrogen bromide and water, and of
+hydrogen chloride and water, a hydrate, viz. HBr,2H_{2}O and HCl,2H_{2}O,
+is formed which possesses a definite melting point, as in the case of
+iodine trichloride. In these cases, therefore, a retroflex curve is
+obtained. Further, just as in the case of the chlorides of iodine the upper
+branch of the retroflex curve ended in a eutectic point, so also in the
+case of the hydrate HBr,2H_{2}O the upper branch of the curve ends in a
+eutectic point at which the system dihydrate--monohydrate--solution--vapour
+can exist. Before the melting point of the monohydrate is reached, two
+liquid phases are formed, as in the case of sulphur dioxide and water.
+
+       *       *       *       *       *
+
+
+{175}
+
+CHAPTER X
+
+SOLID SOLUTIONS. MIXED CRYSTALS
+
+General.--With the conception of gaseous and liquid solutions, every one is
+familiar. Gases can mix in all proportions to form homogeneous solutions.
+Gases can dissolve in or be "absorbed" by liquids; and solids, also, when
+brought in contact with liquids, "pass into solution" and yield a
+homogeneous liquid phase. On the other hand, the conception of a _solid
+solution_ is one which in many cases is found more difficult to appreciate;
+and the existence and behaviour of solid solutions, in spite of their not
+uncommon occurrence and importance, are in general comparatively little
+known.
+
+The reason of this is to be found, to some extent, no doubt, in the fact
+that the term "solid solution" was introduced at a comparatively recent
+date,[254] but it is probably also due in some measure to a somewhat hazy
+comprehension of the definition of the term "solution" itself. As has
+already been said (p. 92), a solution is a homogeneous phase, the
+composition of which can vary continuously within certain limits; the
+definition involves, therefore, no condition as to the physical state of
+the substances. Accordingly, solid solutions are homogeneous solid phases,
+the composition of which can undergo continuous variation within certain
+limits. Just as we saw that the range of variation of composition is more
+limited in the case of liquids than in the case of gases, so also we find
+that the limits of miscibility are in general still more restricted in the
+case of solids. Examples of complete miscibility are, however, not unknown
+even in the case of solid substances.
+
+Solid solutions have long been known, although, of course, {176} they were
+not defined as such. Thus, the phenomena of "occlusion" of gases by metals
+and other substances (occlusion of hydrogen by palladium; occlusion of
+hydrogen by iron) are due to the formation of solid solutions. The same is
+probably also true of the phenomena of "adsorption," as in the removal of
+organic colouring matter by charcoal, although, in this case, surface
+tension no doubt plays a considerable part.[255]
+
+As examples of the solution of gases in solids there may be cited (in
+addition to the phenomena of occlusion already mentioned), the hydrated
+silicates and the zeolites. During dehydration these crystalline substances
+remain clear and transparent, and the pressure of the water vapour which
+they emit varies with the degree of hydration or the concentration of water
+in the mineral.[256] As examples of the solution of solids in solids we
+have the cementation of iron by charcoal, the formation of glass, and the
+crystallization together of isomorphous substances.
+
+Although we have here spoken of the glasses as "solid solutions," it should
+be mentioned that the term "solid" is used in its popular sense. Strictly
+speaking, the glasses are to be regarded as supercooled liquids (see also
+p. 53, footnote).
+
+In discussing the equilibria in systems containing a solid solution, it is
+of essential importance to remember that a solid solution constitutes only
+_one_ phase, a phase of varying composition, as in the case of liquid
+solutions.
+
+Solution of Gases in Solids.--Comparatively little work has been done in
+this connection, the investigations being limited chiefly to the phenomena
+of occlusion or adsorption of gases by charcoal.[257] We shall, therefore,
+indicate only briefly {177} and in a general manner, the behaviour which
+the Phase Rule enables us to foresee.[258]
+
+In dealing with the systems formed by the two phases gas--solid, three
+chief cases call for mention:--
+
+I. _The gas is not absorbed by the solid, but when the pressure reaches a
+certain value, combination of the two components can result._
+
+[Illustration: Fig. 46.]
+
+The graphic representation of such a system is shown in Fig. 46, the
+ordinates being the pressures of the gas, and the abscissæ the
+concentrations of the gaseous component in the solid phase. Since there is
+no formation of a solid solution, the concentration of gas in the solid
+phase remains zero until the pressure has increased to the point A. At this
+point combination can take place. There will now be three phases present,
+viz. solid component, compound, and vapour. The system is therefore
+univariant, and if the temperature is maintained constant, the vapour
+pressure will be constant, irrespective of the amount of compound formed,
+_i.e._ irrespective of the relative amounts of gas and solid. This is
+indicated by the line AB. When the solid component has entirely
+disappeared, the system ceases to be univariant, and if no absorption
+occurs, the pressure will increase again, as shown by BC. If a second
+compound can be formed, then a second _pc_-line will be obtained, similar
+to the preceding. To this group belong the salt hydrates (Chap. VII.).
+
+II. _The gas may be absorbed and may also form a compound._
+
+If absorption of gas occurs with formation of a solid solution, then, as
+the system consists of two phases, solution--vapour, it is bivariant. At
+constant temperature, therefore, the pressure will still vary with the
+concentration of the gaseous component in the solid phase. This is
+represented by the curve AB in Fig. 47. When, however, the pressure has
+reached a certain value, combination can take place; and since there are
+now three phases present, the system is {178} univariant, and at constant
+temperature the pressure is constant, as shown by the line BC.
+
+III. _Absorption of gas occurs, but at a certain concentration the solid
+solution can separate into two immiscible solid solutions._
+
+We have seen, in Chapter VI., that two liquids can form two immiscible
+solutions, and the same has also been found true of solid solutions, as we
+shall presently learn more fully. If, now, two immiscible solutions are
+formed, then the system will become univariant, and at constant temperature
+the _pc_-curve will be a straight line, as in the case of the formation of
+a compound (cf. p. 86). The behaviour of this system will, therefore, also
+be represented diagrammatically by Fig. 47.
+
+[Illustration: FIG. 47.]
+
+_Palladium and Hydrogen._--The phenomenon of the absorption of hydrogen by
+palladium, to which Graham gave the name "occlusion," is one that has
+claimed the attention of several investigators. Although Graham was not of
+opinion that a compound is formed, but rather that the gas undergoes very
+great condensation, acts as a quasi-metal (to which he gave the name
+hydrogenium), and forms a homogeneous alloy with the palladium, later
+investigations, especially those of Troost and Hautefeuille,[259] pointed
+to the formation of a definite chemical compound, having the formula
+Pd_{2}H. This conclusion has, however, not been confirmed by subsequent
+investigation.[260]
+
+Roozeboom and Hoitsema[261] sought to arrive at a final decision as to the
+nature of the phenomenon by an investigation of the equilibrium between
+hydrogen and palladium on the basis of the Phase Rule classification given
+above. If a compound is formed, diminution of volume would cause no
+increase of pressure, but only an increase in the amount of the compound.
+
+As this is the only case of gas absorption which has been {179} accurately
+studied from this point of view, a brief account of the results obtained
+will be given here, although these are not so clear and free from ambiguity
+as one would desire.
+
+The scientists just mentioned investigated the variation of the pressure of
+hydrogen with the amount absorbed by the metal at different temperatures,
+and a few of their results, typical of all, are represented graphically in
+Fig. 48; the curves indicating the variation of the gas pressure with the
+concentration of the hydrogen in the palladium at the temperatures 120°,
+170°, and 200°. As can be seen, the curve consists of three parts, an
+ascending portion which passes gradually and continuously into an almost
+horizontal but slightly ascending middle part, which in turn passes without
+break into a second rapidly ascending curve. This, as Fig. 48 indicates, is
+the general form of the curve; but the length of the middle portion varies
+with the temperature, being shorter at higher than at lower temperatures.
+
+[Illustration: FIG. 48.]
+
+What is the interpretation to be put on these curves? With regard to the
+two end portions, these represent bivariant, two-phase systems, consisting
+of a solid solution and gas. They correspond, therefore, to curve AB in
+Fig. 47. If the middle portion were horizontal, it would indicate either
+the formation of a compound or of two immiscible solid solutions. If a
+compound Pd_{2}H were formed, then the middle portion would at all
+temperatures end at the same value of the concentration, viz. that
+corresponding to 0.5 atoms of hydrogen to 1 atom of palladium. As the
+figure shows, however, this is not the case; the higher the temperature,
+the lower is the concentration at which the middle passes into the terminal
+portion of the curve. {180} Such a behaviour would, however, agree with the
+assumption of the formation of two solid solutions, the "miscibility" of
+which increases with the temperature, as in the case of the liquid
+solutions of phenol and water (p. 97). Nevertheless, although the
+assumption of the formation of two solid solutions is more satisfactory
+than that of the formation of a compound, it does not entirely explain the
+facts. If two solid solutions are formed, the pressure curve should be
+horizontal, but this is not the case; and the deviation from the horizontal
+does not appear to be due to impurities either in the gas or in the metal,
+but is apparently a peculiarity of the system. Further, the gradual instead
+of abrupt passage of the three portions of the curve into one another
+remains unexplained. Hoitsema has expressed the opinion that the occlusion
+of hydrogen by palladium is a process of continuous absorption, the
+peculiar form of the curve--the flat middle portion--being possibly due to
+a condensation of the gas, even at temperatures far above the critical
+temperature of liquid hydrogen.
+
+While, therefore, the occlusion of hydrogen by palladium still presents
+some unexplained phenomena, the behaviour found by Hoitsema would appear to
+disprove conclusively the formation of a definite chemical compound.[262]
+
+SOLUTION OF SOLIDS IN SOLIDS. MIXED CRYSTALS.
+
+The introduction by van't Hoff of the term "solid solution" resulted from
+the discovery of a number of deviations from the Raoult-van't Hoff law for
+the depression of the freezing point by dissolved substances. In all cases,
+the depression was too small; in some instances, indeed, the freezing point
+may be raised. To explain these irregularities, van't Hoff assumed that the
+dissolved substance crystallized out along with the solid solvent; and he
+showed how this would account for the {181} deviations from the law of the
+depression of the freezing point, which had been developed on the
+assumption that only the pure solvent crystallized out from the
+solution.[263]
+
+The "mixed crystals" which were thus obtained, and which van't Hoff called
+dilute solid solutions, showed great resemblance in their behaviour to
+ordinary liquid solutions, and obeyed the laws applicable to these. These
+laws, however, can no longer be applied in the case of the concentrated
+solid solutions formed by the crystallization together of isomorphous
+substances, and known as isomorphous mixtures. Indeed, it has been
+contended[264] that these isomorphous mixtures should not be considered as
+solid solutions at all, although no sharp line of demarcation can be drawn
+between the two classes. The differences, however, in the behaviour of the
+two groups are of a quantitative rather than a qualitative nature; and
+since we are concerned at present only with the qualitative behaviour, we
+shall make no distinction between the crystalline solid solutions and the
+isomorphous mixtures, but shall study the behaviour of the two classes
+under the head of "mixed crystals."
+
+Mixed crystals can be formed either by sublimation[265] or from a liquid
+phase; and in the latter case the mixed crystals can be deposited either
+from solution in a common solvent or from a mixture of the fused
+components. In this method of formation, which alone will be discussed in
+the present chapter, we are dealing with the fusion curves of two
+substances, where, however, the liquid solution is in equilibrium not with
+one of the pure components, but with a solid solution or mixed crystal. The
+simple scheme (Fig. 29, p. 117) which was obtained in the case of two
+components which crystallize out in the pure state, is no longer sufficient
+in the case of the formation of mixed crystals. With the help of the Phase
+Rule, however, the different possible systems can be classified; and
+examples of the different cases predicted by the Phase Rule have also been
+obtained by experiment.
+
+{182}
+
+We shall now consider briefly the formation of mixed crystals by
+isomorphous substances; the consideration of the formation of mixed
+crystals of isodimorphous substances will, on account of the complexity of
+the relationships, not be undertaken here.[266]
+
+_Formation of Mixed Crystals of Isomorphous Substances._
+
+For the purpose of representing the relationships found here we shall
+employ a temperature-concentration diagram,[267] in which the ordinates
+represent the temperature and the abscissæ the concentration of the
+components. Since there are two solutions, the liquid and the solid, and
+since the concentration of the components in these two phases is not, in
+general, the same, two curves will be required for each system, one
+relating to the liquid phase, the other relating to the solid. The
+temperature at which solid begins to be deposited from the liquid solution
+will be called the _freezing point_ of the mixture, and the temperature at
+which the solid solution just begins to liquefy will be called the _melting
+point_ of the solid solution. The temperature-concentration curve for the
+liquid phase will therefore be the freezing-point curve; that for the solid
+solution, the melting-point curve. The latter will be represented by a
+dotted line.[268]
+
+{183}
+
+I.--THE TWO COMPONENTS CAN FORM AN UNBROKEN SERIES OF MIXED CRYSTALS.
+
+Since, as has already been pointed out (p. 176), a mixed crystal (solid
+solution) constitutes only one phase, it is evident that if the two
+components are miscible with one another in all proportions in the solid
+state, there can never be more than one solid phase present, viz. the solid
+solution or mixed crystal. If the components are completely miscible in the
+solid state, they will also be completely miscible in the liquid state, and
+there can therefore be only one liquid phase. The system can at no point
+become invariant, because there can never be more than three phases
+present. When, therefore, the two components form a continuous series of
+mixed crystals, the equilibrium curve must also be continuous. Of these
+systems three types are found.
+
+[Illustration: FIG. 49.]
+
+(_a_) _The freezing points of all mixtures lie between the freezing points
+of the pure components_ (Curve I., Fig. 49).
+
+Examples.--This type of curve is represented by the mixed crystals of
+naphthalene and [beta]-naphthol.[269] The addition of [beta]-naphthol to
+naphthalene raises the freezing point of the latter, and the rise is
+directly proportional to the amount of naphthol added. The freezing point
+curve is therefore a straight line joining the melting points of the two
+components. This behaviour, however, is rather exceptional, the
+freezing-point curve lying generally above, sometimes also below, the
+straight line joining the melting points of the pure components. Thus the
+freezing-point curve of mixtures of [alpha]-monochlorocinnamic aldehyde and
+[alpha]-monobromocinnamic aldehyde[270] lies above the {184} straight line
+joining the melting points of the pure components (31.22° and 69.56°), as
+is evident from the following table:--
+
+  ----------------------------------------------------------------------
+  Molecules of bromo-        |                 |
+  cinnamic aldehyde in       | Freezing point. | Deviation from straight
+  100 mols. of mixture.      |                 | line.
+  ----------------------------------------------------------------------
+          0.00               |   31.22°        |       --
+         10.48               |   37.28°        |      2.04°
+         21.91               |   43.12°        |      3.50°
+         30.07               |   46.80°        |      4.05°
+         45.04               |   52.94°        |      4.45°
+         62.16               |   58.82°        |      3.77°
+         82.98               |   65.07°        |      2.03°
+         93.50               |   67.91°        |      0.84°
+        100.00               |   69.56°        |       --
+  ----------------------------------------------------------------------
+
+Melting-point Curve.--This curve, like the freezing-point curve, must also
+be continuous, and the melting points of the different solid solutions will
+lie between the melting points of the pure components. This is represented
+by the dotted line in Fig. 49, I. The relative position of the two curves,
+which can be deduced with the help of thermodynamics and also by
+experimental determination, is found in all cases to be in accordance with
+the following rule: At any given temperature, _the concentration of that
+component by the addition of which the freezing point is depressed, is
+greater in the liquid than in the solid phase_; or, conversely, _the
+concentration of that component by the addition of which the freezing point
+is raised, is greater in the solid than in the liquid phase_. An
+illustration of this rule is afforded by the two substances chloro- and
+bromo-cinnamic aldehyde already mentioned. As can be seen from the above
+table, the addition of chlorocinnamic aldehyde lowers the melting point of
+the bromo-compound. In accordance with the rule, therefore, the
+concentration of the chloro-compound in the liquid phase must be greater
+than in the solid phase; and this was found experimentally. At a
+temperature of 49.44°, the liquid contained 58.52 per cent., the solid only
+52.57 per cent. of the chlorocinnamic aldehyde.
+
+From this it will also be clear that on cooling a fused mixture of two
+substances capable of forming mixed crystals, {185} the temperature of
+solidification will not remain constant during the separation of the solid;
+nor, on the other hand, will the temperature of liquefaction of the solid
+solution be constant. Thus, for example, if a liquid solution of two
+components, A and B, having the composition represented by the point _x_
+(Fig. 50), is allowed to cool, the system will pass along the line _xx'_.
+At the temperature of the point _a_, mixed crystals will be deposited, the
+composition of which will be that represented by b. As the temperature
+continues to fall, more and more solid will be deposited; and since the
+solid phase is relatively rich in the component B, the liquid will become
+relatively poorer in this. The composition of the liquid solution will
+therefore pass along the curve _ad_, the composition of the solid solution
+at the same time passing along the curve _bc_; at the point _c_ the liquid
+will solidify completely.[271]
+
+[Illustration: FIG. 50.]
+
+Conversely, if mixed crystals of the composition and at the temperature
+_x'_ are heated, liquefaction will begin at the temperature _c_, yielding a
+liquid of the composition d. On continuing to add heat, the temperature of
+the mass will rise, more of the solid will melt, and the composition of the
+two phases will change as represented by the curves _da_ and _cb_. When the
+temperature has risen to _a_, complete liquefaction will have occurred. The
+process of solidification or of liquefaction is therefore extended over a
+temperature interval _ac_.
+
+Even when the freezing-point curve is a straight line joining {186} the
+melting points of the pure components, the melting-point curve will not
+necessarily coincide with the freezing-point curve, although it may
+approach very near to it; complete coincidence can take place only when the
+melting points of the two components are identical. An example of this will
+be given later (Chap. XII.).
+
+(_b_) _The freezing-point curve passes through a maximum_ (Curve II., Fig.
+49).
+
+[Illustration: FIG. 51.]
+
+This curve exhibits the greatest degree of contrast to the freezing-point
+curve which is obtained when the pure components crystallize out. For,
+since the curve passes through a maximum, it is evident that the freezing
+point of each of the components must be _raised_ by the addition of the
+other component.
+
+Example.--Very few cases belonging to this type are known. The best example
+is found in the freezing-point curve of mixtures of _d_- and
+_l_-carvoxime[272] (C_{10}H_{14}N.OH). The freezing points and melting
+points of the different mixtures of _d_- and _l_-carvoxime are given in the
+following table, and represented graphically in Fig. 51:--
+
+  ---------------+----------------+-----------------+-----------------
+  Per cent. of   | Per cent. of   | Freezing point. | Melting point.
+  _d_-carvoxime. | _l_-carvoxime. |                 |
+  ---------------+----------------+-----------------+-----------------
+     100         |      0         |      72.0°      |     72.0°
+      99         |      1         |      72.4°      |      --
+      98         |      2         |      73.0°      |      --
+      95         |      5         |      75.4°      |     73.0°
+      90         |     10         |      79.0°      |     75.0°
+      80         |     20         |      84.6°      |     80.0°
+      70         |     30         |      88.2°      |     85.0°
+      60         |     40         |      90.4°      |      --
+      50         |     50         |      91.4°      |     91.4°
+      25         |     75         |      86.4°      |     82.0°
+       8         |     92         |      77.4°      |      --
+       1         |     99         |      72.4°      |      --
+       0         |    100         |      72.0°      |     72.0°
+  ---------------+----------------+-----------------+-----------------
+
+{187}
+
+In this figure, the melting-point curve, _i.e._ the
+temperature-concentration curve for the mixed crystals, is represented by
+the lower curve. Since the addition of the lævo-form to the dextro-form
+raises the melting point of the latter, the concentration of the lævo-form
+(on the right-hand branch of the curve) must, in accordance with the rule
+given, be greater in the solid phase than in the liquid. Similarly, since
+addition of the dextro-form raises the melting point of the lævo-form, the
+solid phase (on the left-hand branch of the curve) must be richer in
+dextro- than in lævo-carvoxime. At the maximum point, the melting-point and
+freezing-point curves touch; at this point, therefore, the composition of
+the solid and liquid phases must be identical. It is evident, therefore,
+that at the maximum point the liquid will solidify, or the solid will
+liquefy completely without change of temperature; and, accordingly, mixed
+crystals of the composition represented by the maximum point will exhibit a
+definite melting point, and will in this respect behave like a simple
+substance.
+
+(_c_) _The freezing-point curve passes through a minimum_ (Curve III., Fig.
+49).
+
+In this case, as in the case of those systems where the pure components are
+deposited, a minimum freezing point is obtained. In the latter case,
+however, there are two freezing-point curves which intersect at a eutectic
+point; in the case where mixed crystals are formed there is only one
+continuous curve. On one side of the minimum point the liquid phase
+contains relatively more, on the other side relatively less, of the one
+component than does the solid phase; while at the minimum point the
+composition of the two phases is the same. At this point, therefore,
+complete solidification or complete liquefaction will occur without change
+of temperature, and the mixed crystals will accordingly exhibit a definite
+melting point.
+
+[Illustration: FIG. 52.]
+
+{188}
+
+Example.--As an example of this there may be taken the mixed crystals of
+mercuric bromide and iodide.[273] Mercuric bromide melts at 236.5°, and
+mercuric iodide at 255.4°. The mixed crystal of definite constant melting
+point (minimum point) contains 59 mols. per cent. of mercuric bromide, the
+melting point being 216.1°.
+
+The numerical data are contained in the following table, and represented
+graphically in Fig. 52:--
+
+  -----------------------------------------------------
+  Mols. per cent. of |                 |
+  HgBr_{2}.          | Freezing point. | Melting point.
+  -----------------------------------------------------
+        100          |     236.5°      |      236°
+         90          |     228.8°      |      226°
+         80          |     222.2°      |      219°
+         70          |     217.8°      |      217°
+         65          |     216.6°      |      216°
+         60          |     216.1°      |      215.5°
+         55          |     216.3°      |      216°
+         50          |     217.3°      |      216°
+         40          |     221.1°      |      218°
+         30          |     227.8°      |      223°
+         20          |     236.2°      |      231°
+         10          |     245.5°      |      242°
+          0          |     255.4°      |      254°
+  -----------------------------------------------------
+
+[Illustration: FIG. 53.]
+
+Fractional Crystallization of Mixed Crystals.--With the help of the
+diagrams already given it will be possible to predict what will be the
+result of the fractional crystallization of a fused mixture of two
+substances which can form mixed crystals. Suppose, for example, a fused
+mixture of the composition _x_ (Fig. 53) is cooled down; then, as we have
+already seen, when the temperature has fallen to _a_, mixed crystals of
+composition, _b_, are deposited. If the temperature is allowed to fall
+{189} to _x'_, and the solid then separated from the liquid, the mixed
+crystals so obtained will have the composition represented by e. If, now,
+the mixed crystals _e_ are completely fused and the fused mass allowed to
+cool, separation of solid will occur when the temperature has fallen to the
+point _f_. The mixed crystals which are deposited have now the composition
+represented by _g_, i.e. _they are richer in B than the original mixed
+crystals_. By repeating this process, the composition of the successive
+crops of mixed crystals which are obtained approximates more and more to
+that of the pure component B, while, on the other hand, the composition of
+the liquid phase produced tends to that of pure A. By a systematic and
+methodical repetition of the process of fractional crystallization,
+therefore, a _practically_ complete separation of the components can be
+effected; a perfect separation is theoretically impossible.
+
+From this it will be readily understood that in the case of substances the
+freezing point of which passes through a maximum, fractional
+crystallization will ultimately lead to mixed crystals having the
+composition of the maximum point, while the liquid phase will more and more
+assume the composition of either pure A or pure B, according as the initial
+composition was on the A side or the B side of the maximum point. In those
+cases, however, where the curves exhibit a minimum, the solid phase which
+separates out will ultimately be one of the pure components, while a liquid
+phase will finally be obtained which has the composition of the minimum
+point.
+
+II.--THE TWO COMPONENTS DO NOT FORM A CONTINUOUS SERIES OF MIXED CRYSTALS.
+
+This case corresponds to that of the partial miscibility of liquids. The
+solid component A can "dissolve" the component B until the concentration of
+the latter in the mixed crystal has reached a certain value. Addition of a
+further amount of B will not alter the composition of the mixed crystal,
+but there will be formed a second solid phase consisting {190} of a
+solution of A in B. At this point the four phases, mixed crystals
+containing excess of A, mixed crystals containing excess of B, liquid
+solution, vapour, can coexist; this will therefore be an invariant point.
+The temperature-concentration curves will therefore no longer be
+continuous, but will exhibit a break or discontinuity at the point at which
+the invariant system is formed.
+
+(_a_) _The freezing-point curve exhibits a transition point_ (Curve I.,
+Fig. 54).
+
+As is evident from the figure, addition of B raises the melting point of A,
+and, in accordance with the rule previously given, the concentration of B
+in the mixed crystals will be greater than in the solution. This is
+represented in the figure by the dotted curve AD. On the other hand,
+addition of A lowers the melting point of B, and the two curves BC and BE
+are obtained for the liquid and solid phases respectively. At the
+temperature of the line CDE the liquid solution of the composition
+represented by C is in equilibrium with the two different mixed crystals
+represented by D and E. At this temperature, therefore, the _tc_-curve for
+the solid phase exhibits a discontinuity; and, since the solid phase
+undergoes change at this point, the freezing-point curve must show a break
+(p. 111).
+
+[Illustration: FIG. 54.]
+
+Example.--Curves of the form given in Fig. 54 I. have been found
+experimentally in the case of silver nitrate and sodium nitrate.[274] The
+following table contains the numerical data, which are also represented
+graphically in Fig. 55:--
+
+{191}
+
+  -----------------------------------------------------
+  Molecules NaNO_{3} | Freezing point. | Melting point.
+      per cent.      |                 |
+  -----------------------------------------------------
+          0          |      208.6°     |     208.6°
+          8          |      211.4°     |     210°
+         15.06       |      215°       |     212°
+         19.46       |      217.2°     |     214.8°
+         21.9        |      222°       |     215°
+         26          |      228.4°     |     216.5°
+         29.7        |      234.8°     |     217.5°
+         36.2        |      244.4°     |     217.5°
+         47.3        |      259.4°     |     237.6°
+         58.9        |      272°       |     257°
+         72          |      284°       |     274°
+        100          |      308°       |     308°
+  -----------------------------------------------------
+
+The temperature of the transition point is 217.5°; at this point the liquid
+contains 19.5, and the two conjugate solid solutions 26 and 38 molecules of
+sodium nitrate per cent. respectively.
+
+[Illustration: FIG. 55.]
+
+[Illustration: FIG. 56.]
+
+(_b_) _The freezing-point curve exhibits a eutectic point_ (Curve II., Fig.
+54). {192}
+
+In this case the freezing point of each of the components is lowered by the
+addition of the other, until at last a point is reached at which the liquid
+solution solidifies to a mixture or conglomerate of two mixed crystals.
+
+Examples.--Curves belonging to this class have been obtained in the case of
+potassium and thallium nitrates[275] and of naphthalene and monochloracetic
+acid.[276] The data for the latter are given in the following table and
+represented in Fig. 56:--
+
+  -------------------------------------------------------------------------
+               |         Liquid solution.     |        Solid solution.
+               ------------------------------------------------------------
+  Temperature. |              |               |              |
+               |  Per cent.   |   Per cent.   |  Per cent.   |   Per cent.
+               | naphthalene. |     acid.     | naphthalene. |     acid.
+  -------------------------------------------------------------------------
+      62°      |     --       |     100       |     --       |     100
+      60°      |      4.0     |      96.0     |      1.7     |      98.3
+      55°      |     21.0     |      79.0     |      2.1     |      97.9
+      53.5°    |     29.4     |      70.0     |     --       |      --
+      55°      |     31.3     |      68.7     |     59.6     |      40.4
+      60°      |     42.4     |      57.6     |     80.3     |      19.7
+      65°      |     53.3     |      46.7     |     89.2     |      10.8
+      70°      |     69.7     |       2.3     |     95.4     |       4.6
+      75°      |     84.4     |      15.6     |     96.6     |       3.4
+      79.9°    |    100       |      --       |    100       |      --
+  -------------------------------------------------------------------------
+
+At the eutectic point the liquid solution is in equilibrium with two
+different mixed crystals the composition of which is represented by D and E
+respectively. If, therefore, a fused mixture containing the two components
+A and B in the proportions represented by C is cooled down, it will, when
+the temperature has reached the point C, solidify completely to a
+_conglomerate_ of mixed crystals, D and E.
+
+[Illustration: FIG. 57.]
+
+[Illustration: FIG. 58.]
+
+Changes in Mixed Crystals with the Temperature.--In the case of the
+different types of systems represented in Fig. 49, a homogeneous liquid
+solution of the two components will exist at temperatures above the
+freezing-point curve, a homogeneous mixed crystal at temperatures below the
+melting-point curve, while at any point between the freezing-point and
+melting-point {193} curves the mixture will separate into a solid phase and
+a liquid phase. In the case, however, of the two types shown in Fig. 54 the
+relationships are somewhat more complicated. As before, the area above the
+freezing-point curve gives the conditions under which homogeneous liquid
+solutions can exist; but below the melting-point curve two different mixed
+crystals can coexist. This will be best understood from Figs. 57 and 58. D
+and E represent, as we have seen, the composition of two mixed crystals
+which are in equilibrium with the liquid solution at the temperature of the
+point C. These two mixed crystals represent, in the one case, a saturated
+solution of B in A (point D), and the other a saturated solution of A in B
+(point E). Just as we saw that the mutual solubility of two liquids varied
+with the temperature, so also in the case of two solids; as the temperature
+alters, the solubility of the two solid components in one another will
+change. This alteration is indicated diagrammatically in Figs. 57 and 58 by
+the dotted curve similar to the solubility curves for two mutually soluble
+liquids (p. 101).
+
+Suppose, now, that a mixed crystal of the composition _x_ is cooled down,
+it will remain unchanged until, when the temperature has fallen to _t'_,
+the homogeneous mixed crystal breaks up into a conglomerate of two mixed
+crystals the composition of {194} which is represented by _x'_ and _x"_
+respectively. From this, then, it can be seen that in the case of
+substances which form two solid solutions, the mixed crystals which are
+desposited from the liquid fused mass need not remain unchanged in the
+solid state, but may at some lower temperature lose their homogeneity. This
+fact is of considerable importance for the formation of alloys.[277]
+
+A good example of this will soon be met with in the case of the iron and
+carbon alloys. The alloys of copper and tin also furnish examples of the
+great changes which may take place in the alloy between the temperature at
+which it separates out from the fused mass and the ordinary temperature.
+Thus, for example, one of the alloys of copper and tin which separates out
+from the liquid as a solid solution breaks up, on cooling, into the
+compound Cu_{3}Sn and liquid:[278] a striking example of a solid substance
+partially liquefying on being cooled.
+
+       *       *       *       *       *
+
+
+{195}
+
+CHAPTER XI
+
+EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES
+
+It has long been known that certain substances, _e.g._ acetoacetic ester,
+are capable when in solution or in the fused state, of reacting as if they
+possessed two different constitutions; and in order to explain this
+behaviour the view was advanced (by Laar) that in such cases a hydrogen
+atom oscillated between two positions in the molecule, being at one time
+attached to oxygen, at another time to carbon, as represented by the
+formula--
+
+  CH_{3}.C--CH.CO_{2}C_{2}H_{5}
+         .  ^
+         .  |
+         O<-H
+
+When the hydrogen is in one position, the substance will act as an
+hydroxy-compound; with hydrogen in the other position, as a ketone.
+Substances possessing this double function are called _tautomeric_.
+
+Doubt, however, arose as to the validity of the above explanation, and this
+doubt was confirmed by the isolation of the two isomerides in the solid
+state, and also by the fact that the velocity of change of the one
+isomeride into the other could in some cases be quantitatively measured.
+These and other observations then led to the view, in harmony with the laws
+of chemical dynamics, that tautomeric substances in the dissolved or fused
+state represent a _mixture_ of two isomeric forms, and that equilibrium is
+established not by _intra_- but by _inter_-molecular change, as expressed
+by the equation--
+
+  CH_{3}.CO.CH_{2}.CO_{2}C_{2}H_{5} <--> CH_{3}.C(OH):CH.CO_{2}C_{2}H_{5}
+
+{196} In the solid state, the one or other of the isomerides represents the
+stable form; but in the liquid state (solution or fusion) the stable
+condition is an equilibrium between the two forms.
+
+A similar behaviour is also found in the case of other isomeric substances
+where the isomerism is due to difference of structure, _i.e._ structure
+isomerism (_e.g._ in the case of the oximes
+
+  C_{6}H_{5}.C.H      C_{6}H_{5}.C.H
+             ||   and            ||  ),
+             N.OH             HO.N
+
+or to difference in configuration, _i.e._ stereoisomerism (_e.g._ optically
+active substances), or to polymerism (_e.g._ acetaldehyde and paraldehyde).
+In all such cases, although the different solid forms correspond to a
+single definite constitution, in the liquid state a condition of
+equilibrium between the two modifications is established. As a general name
+for these different classes of substances, the term "dynamic isomerides"
+has been introduced; and the different kinds of isomerism are classed
+together under the title "dynamic isomerism."[279]
+
+By reason of the importance of these phenomena in the study more especially
+of Organic Chemistry, a brief account of the equilibrium relations
+exhibited by systems composed of dynamic isomerides may be given here.[280]
+
+In studying the fusion and solidification of those substances which exhibit
+the relationships of dynamic isomerism, the phenomena observed will vary
+somewhat according as the reversible transformation of the one form into
+the other takes place with measurable velocity at temperatures in the
+neighbourhood of the melting points, or only at some higher temperature. If
+the transformation is very rapid, the system will behave like a
+one-component system, but if the isomeric change is comparatively slow, the
+behaviour will be that of a two-component system.
+
+Temperature-Concentration Diagram.--The relationships which are met with
+here will be most readily understood with {197} the help of Fig. 59.
+Suppose, in the first instance, that isomeric transformation does not take
+place at the temperature of the melting point, then the freezing point
+curve will have the simple form ACB; the formation of compounds being for
+the present excluded. This is the simplest type of curve, and gives the
+composition of the solutions in equilibrium with the one modification
+([alpha] modification) at different temperatures (curve AC); and of the
+solutions in equilibrium with the other modification ([beta] modification)
+at different temperatures (curve BC). C is the eutectic point at which the
+two solid isomerides can exist side by side in contact with the solution.
+
+[Illustration: FIG. 59.]
+
+Now, suppose that isomeric transformation takes place with measurable
+velocity. If the pure [alpha]-modification is heated to a temperature _t'_
+above its melting point, and the liquid maintained at that temperature
+until equilibrium has been established, a certain amount of the [beta]-form
+will be present in the liquid, the composition of which will be represented
+by the point _x'_. The same condition of equilibrium will also be reached
+by starting with pure [beta]. Similarly, if the temperature of the liquid
+is maintained at the temperature _t"_, equilibrium will be reached, we
+shall suppose, when the solution has the composition _x"_. The curve DE,
+therefore, which passes through all the different values of _x_
+corresponding to different values of _t_, will represent the change of
+equilibrium with the temperature. It will slope to the right (as in the
+figure) if the transformation of [alpha] into [beta] is accompanied by
+absorption of heat; to the left if the transformation is accompanied by
+evolution of heat, in accordance with van't Hoff's Law of movable
+equilibrium. If transformation occurs without heat effect, the equilibrium
+will be independent of the {198} temperature, and the equilibrium curve DE
+will therefore be perpendicular and parallel to the temperature axis.
+
+We must now find the meaning of the point D. Suppose the pure [alpha]- or
+pure [beta]-form heated to the temperature _t'_, and the temperature
+maintained constant until the liquid has the composition _x'_ corresponding
+to the equilibrium at that temperature. If the temperature is now allowed
+to fall sufficiently slowly so that the condition of equilibrium is
+continually readjusted as the temperature changes, the composition of the
+solution will gradually alter as represented by the curve _x'_D. Since D is
+on the freezing point curve of pure [alpha], this form will be deposited on
+cooling; and since D is also on the equilibrium curve of the liquid, D is
+the only point at which solid can exist in stable equilibrium with the
+liquid phase. (The vapour phase may be omitted from consideration, as we
+shall suppose the experiments carried out in open vessels.) All systems
+consisting of the two hylotropic[281] isomeric substances [alpha] and
+[beta] will, therefore, ultimately freeze at the point D, which is called
+the "natural" freezing point[282] of the system; provided, of course, that
+sufficient time is allowed for equilibrium to be established. From this it
+is apparent that _the stable modification at temperatures in the
+neighbourhood of the melting point is that which is in equilibrium with the
+liquid phase at the natural freezing point_.
+
+From what has been said, it will be easy to predict what will be the
+behaviour of the system under different conditions. If pure [alpha] is
+heated, a temperature will be reached at which it will melt, but this
+melting point will be sharp only if the velocity of isomeric transformation
+is comparatively slow; _i.e._ slow in comparison with the determination of
+the melting point. If the substance be maintained in the fused condition
+for some time, a certain amount of the [beta] modification will be formed,
+and on lowering the temperature the pure [alpha] form will be deposited,
+not at the temperature of the melting point, but at some lower temperature
+depending on the concentration of the [beta] modification in the liquid
+phase. If isomeric transformation {199} takes place slowly in comparison
+with the rate at which deposition of the solid occurs, the liquid will
+become increasingly rich in the [beta] modification, and the freezing point
+will, therefore, sink continuously. At the eutectic point, however, the
+[beta] modification will also be deposited, and the temperature will remain
+constant until all has become solid. If, on the other hand, the velocity of
+transformation is sufficiently rapid, then as quickly as the [alpha]
+modification is deposited, the equilibrium between the two isomeric forms
+in the liquid phase will continuously readjust itself, and the end-point of
+solidification will be the natural freezing point.
+
+Similarly, starting with the pure [beta] modification, the freezing point
+after fusion will gradually fall owing to the formation of the [alpha]
+modification; and the composition of the liquid phase will pass along the
+curve BC. If, now, the rate of cooling is not too great, or if the velocity
+of isomeric transformation is sufficiently rapid, complete solidification
+will not occur at the eutectic point; for at this temperature solid and
+liquid are not in stable equilibrium with one another. On the contrary, a
+further quantity of the [beta] modification will undergo isomeric change,
+the liquid phase will become richer in the [alpha] form, and the freezing
+point will _rise_; the solid phase in contact with the liquid being now the
+[alpha] modification. The freezing point will continue to rise until the
+point D is reached, at which complete solidification will take place
+without further change of temperature.
+
+The diagram also allows us to predict what will be the result of rapidly
+cooling a fused mixture of the two isomerides. Suppose that either the
+[alpha] or the [beta] modification has been maintained in the fused state
+at the temperature _t'_ sufficiently long for equilibrium to be
+established. The composition of the liquid phase will be represented by
+_x'_. If the liquid is now _rapidly_ cooled, the composition will remain
+unchanged as represented by the dotted line _x'_G. At the temperature of
+the point G solid [alpha] modification will be deposited. If the cooling is
+not carried below the point G, so as to cause complete solidification, the
+freezing point will be found to rise with time, owing to the conversion of
+some of the [beta] form into the [alpha] form {200} in the liquid phase;
+and this will continue until the composition of the liquid has reached the
+point D. From what has just been said, it can also be seen that if the
+freezing point curves can be obtained by actual determination of the
+freezing points of different synthetic mixtures of the two isomerides, it
+will be possible to determine the condition of equilibrium in the fused
+state at any given temperature without having recourse to analysis. All
+that is necessary is to rapidly cool the fused mass, after equilibrium has
+been established, and find the freezing point at which solid is deposited;
+that is, find the point at which the line of constant temperature cuts the
+freezing point curve. The composition corresponding to this temperature
+gives the composition of the equilibrium mixture at the given temperature.
+
+It will be evident, from what has gone before, that the degree of
+completeness with which the different curves can be realised will depend on
+the velocity with which isomeric change takes place, and on the rapidity
+with which the determinations of the freezing point can be carried out. As
+the two extremes we have, on the one hand, practically instantaneous
+transformation, and on the other, practically infinite slowness of
+transformation. In the former case, only one melting and freezing point
+will be found, viz. the natural freezing point; in the latter case, the two
+isomerides will behave as two perfectly independent components, and the
+equilibrium curve DE will not be realised.
+
+The diagram which is obtained when isomeric transformation does not occur
+within measurable time at the temperature of the melting point is somewhat
+different from that already given in Fig. 59. In this case, the two
+freezing point curves AC and BC (Fig. 60) can be readily realized, as no
+isomeric change occurs in the liquid phase. Suppose, however, that at a
+higher temperature, _t'_, reversible isomeric transformation can take
+place, the composition of the liquid phase will alter until at the point
+_x'_ a condition of equilibrium is reached; and the composition of the
+liquid at higher temperatures will be represented by the curve _x'_F. Below
+the temperature _t'_ the position of the equilibrium curve is hypothetical;
+but as the temperature {201} falls the velocity of transformation
+diminishes, and at last becomes _practically_ zero. The equilibrium curve
+can therefore be regarded as dividing into two branches _x'_G and _x'_H. At
+temperatures between G and _t'_ the [alpha] modification can undergo
+isomeric change leading to a point on the curve G_x'_; and the [beta]
+modification can undergo change leading to a point on the curve H_x'_. The
+same condition of equilibrium is therefore not reached from each side, and
+we are therefore dealing not with true but with false equilibrium (p. 5).
+Below the temperatures G and H, isomeric transformation does not occur in
+measurable time. We shall not, however, enter into a detailed discussion of
+the equilibria in such systems, more especially as they are not systems in
+true equilibrium, and as the temperature at which true equilibrium can be
+established with appreciable velocity alters under the influence of
+catalytic agents.[283] Examples of such systems will no doubt be found in
+the case of optically active substances, where both isomerides are
+apparently quite stable at the melting point. In the case of such
+substances, also, the action of catalytic agents in producing isomeric
+transformation (racemisation) is well known.
+
+[Illustration: FIG. 60.]
+
+Transformation of the Unstable into the Stable Form.--As has already been
+stated, the stable modification in the neighbourhood of the melting point
+is that one which is in equilibrium with the liquid phase at the natural
+freezing point. In the case of polymorphic substances, we have seen (p. 39)
+that that form which is stable in the neighbourhood of the melting point
+melts at the higher temperature. That was a {202} consequence of the fact
+that the two polymorphic forms on melting gave identical liquid phases. In
+the present case, however, the above rule does not apply, for the simple
+reason that the liquid phase obtained by the fusion of the one modification
+is not identical with that obtained by the fusion of the other. In the case
+of isomeric substances, therefore, the form of lower melting point _may_ be
+the more stable; and where this behaviour is found it is a sign that the
+two forms are isomeric (or polymeric) and not polymorphic.[284] An example
+of this is found in the case of the isomeric benzaldoximes (p. 203).
+
+Since in Fig. 59 the [alpha] modification has been represented as the
+stable form, the transformation of the [beta] into the [alpha] form will be
+possible at all temperatures down to the transition point. At temperatures
+below the eutectic point, transformation will occur without formation of a
+liquid phase; but at temperatures above the eutectic point liquefaction can
+take place. This will be more readily understood by drawing a line of
+constant temperature, HK, at some point between C and B. Then if the [beta]
+modification is maintained for a sufficiently long time at that
+temperature, a certain amount of the [alpha] modification will be formed;
+and when the composition of the mixture has reached the point H, fusion
+will occur. If the temperature is maintained constant, isomeric
+transformation will continue to take place in the liquid phase until the
+equilibrium point for that temperature is reached. If this temperature is
+higher than the natural melting point, the mixture will remain liquid all
+the time; but if it is below the natural melting point, then the [alpha]
+modification will be deposited when the system reaches the condition
+represented by the point on the curve AC corresponding to the particular
+temperature. As isomeric transformation continues, the freezing point of
+the system will rise until it reaches the natural freezing point D.
+Similarly, if the [alpha] modification is maintained at a temperature above
+that of the point D, liquefaction will ultimately occur, and the system
+will again reach the final state represented by D.[285]
+
+{203}
+
+Examples.--_Benzaldoximes._ The relationships which have just been
+discussed from the theoretical point of view will be rendered clearer by a
+brief description of cases which have been experimentally investigated. The
+first we shall consider is that of the two isomeric benzaldoximes:[286]--
+
+  C_{6}H_{5}.C.H            C_{6}H_{5}.C.H
+             ||                        ||
+          HO.N                         N.OH
+
+  Benzantialdoxime          Benzsynaldoxime
+  ([alpha]-modification).    ([beta]-modification).
+
+Fig. 61 gives a graphic representation of the results obtained.
+
+The melting point of the [alpha] modification is 34-35°; the melting point
+of the unstable [beta]-modification being 130°. The freezing curves AC and
+BC were obtained by determining the freezing points of different mixtures
+of known composition, and the numbers so obtained are given in the
+following table.
+
+{204}
+
+  ----------------------------------------------------
+  Grams of the [alpha] modification |
+    in 100 gm. of mixture.          | Freezing point.
+  ----------------------------------+-----------------
+               26.2                 |     101°
+               49.2                 |      79°
+               73.7                 |      46°
+               91.7                 |      26.2°
+               95.0                 |      28.6°
+               96.0                 |      30.0°
+  ----------------------------------------------------
+
+[Illustration: FIG. 61.]
+
+The eutectic point C was found to lie at 25-26°, and the natural freezing
+point D was found to be 27.7°. The equilibrium curve DE was determined by
+heating the liquid mixtures at different temperatures until equilibrium was
+attained, and then rapidly cooling the liquid. In all cases the freezing
+point was practically that of the point D. From this it is seen that the
+equilibrium curve must be a straight line parallel to the temperature axis;
+and, therefore, isomeric transformation in the case of the two
+benzaldoximes is not accompanied by any heat effect (p. 197). This
+behaviour has also been found in the case of acetaldoxime.[287]
+
+The isomeric benzaldoximes are also of interest from the fact that the
+stable modification has the _lower_ melting point (_v._ p. 202).
+
+_Acetaldehyde and Paraldehyde._--As a second example of the equilibria
+between two isomerides, we shall take the two isomeric (polymeric) forms of
+acetaldehyde, which have recently been exhaustively studied.[288]
+
+{205}
+
+In the case of these two substances the reaction
+
+  3CH_{3}.CHO <--> (CH_{3}.CHO)_{3}
+
+takes place at the ordinary temperature with very great slowness. For this
+reason it is possible to determine the freezing point curves of
+acetaldehyde and paraldehyde. The three chief points on these curves,
+represented graphically in Fig. 62, are:--
+
+  m.p. of acetaldehyde - 118.45°
+  m.p. of paraldehyde  +  12.55°
+  eutectic point       - 119.9°
+
+[Illustration: FIG. 62.]
+
+In order to determine the position of the natural melting point, it was
+necessary, on account of the slowness of transformation, to employ a
+catalytic agent in order to increase the velocity with which the
+equilibrium was established. A drop of concentrated sulphuric acid served
+the purpose. In presence of a trace of this substance, isomeric
+transformation very speedily occurs, and leads to the condition of
+equilibrium. Starting in the one case with fused paraldehyde, and in the
+other case with acetaldehyde, the same freezing point, viz. 6.75°, was
+obtained, the solid phase being paraldehyde. This temperature, 6.75°, is
+therefore the natural freezing point, and paraldehyde, the solid in
+equilibrium with the liquid phase at this point, is the stable form.
+
+With regard to the change of equilibrium with the temperature, it was found
+that whereas the liquid phase contained 11.7 molecules per cent. of
+acetaldehyde at the natural freezing point, the liquid at the temperature
+of 41.6° contains 46.6 molecules per cent. of acetaldehyde. As the
+temperature {206} rises, therefore, there is increased formation of
+acetaldehyde, or a decreasing amount of polymerisation. This is in harmony
+with the fact that the polymerisation of acetaldehyde is accompanied by
+evolution of heat.
+
+While speaking of these isomerides, it may be mentioned that at the
+temperature 41.6° the equilibrium mixture has a vapour pressure equal to
+the atmospheric pressure. At this temperature, therefore, the equilibrium
+mixture (obtained quickly with the help of a trace of sulphuric acid)
+boils.[289]
+
+       *       *       *       *       *
+
+
+{207}
+
+CHAPTER XII
+
+SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF TWO
+COMPONENTS
+
+In this concluding chapter on two-component systems, it is proposed to
+indicate briefly how the Phase Rule has been applied to the elucidation of
+a number of problems connected with the equilibria between two components,
+and how it has been employed for the interpretation of the data obtained by
+experiment. It is hoped that the practical value of the Phase Rule may
+thereby become more apparent, and its application to other cases be
+rendered easier.
+
+The interest and importance of investigations into the conditions of
+equilibrium between two substances, lie in the determination not only of
+the conditions for the stable existence of the participating substances,
+but also of whether or not chemical action takes place between these two
+components; and if combination occurs, in the determination of the nature
+of the compounds formed and the range of their existence. In all such
+investigations, the Phase Rule becomes of conspicuous value on account of
+the fact that its principles afford, as it were, a touchstone by which the
+character of the system can be determined, and that from the form of the
+equilibrium curves obtained, conclusions can be drawn as to the nature of
+the interaction between the two substances. In order to exemplify the
+application of the principles of the Phase Rule more fully than has already
+been done, illustrations will be drawn from investigations on the
+interaction of organic compounds; on the equilibria between optically
+active compounds; and on alloys. {208}
+
+Summary of the Different Systems of Two Components.--Before passing to the
+consideration of the application of the Phase Rule to the investigation of
+particular problems, it will be well to collect together the different
+types of equilibrium curves with which we are already acquainted; to
+compare them with one another, in order that we may then employ these
+characteristic curves for the interpretation of the curves obtained as the
+result of experiment.
+
+In investigating the equilibria between two components, three chief classes
+of curves will be obtained according as--
+
+I. No combination takes place between the two components.
+
+II. The components can form definite compounds.
+
+III. The components separate out in the form of mixed crystals.
+
+The different types of curves which are obtained in these three cases are
+represented in Figs. 63, 64, 65. These different diagrams represent the
+whole series of equilibria, from the melting point of the one component (A)
+to that of the other component (B). The curves represent, in all cases, the
+composition of the solution, or phase of variable composition; the
+temperature being measured along one axis, and the composition along the
+other.
+
+We shall now recapitulate very briefly the characteristics of the different
+curves.
+
+[Illustration: FIG. 63.]
+
+If no compound is formed between the two components, {209} the general form
+of the equilibrium curve will be that of curve I. or II., Fig. 63. Type I.
+is the simplest form of curve found, and consists, as the diagram shows, of
+only two branches, AC and BC, meeting at the point C, _which lies below the
+melting point of either component_. The solid phase which is in equilibrium
+with the solutions AC is pure A; that in equilibrium with BC, pure B. C is
+the eutectic point. Although at the eutectic point the solution solidifies
+entirely without change of temperature, the solid which is deposited is not
+a homogeneous solid phase, but a mixture, or conglomerate of the two
+components. _The eutectic point, therefore, represents the melting or
+freezing point, not of a compound, but of a mixture_ (p. 119).
+
+Curve II., Fig. 63, is obtained when two liquid phases are formed. C is an
+eutectic point, D and F are transition points at which there can co-exist
+the four phases--solid, two liquid phases, vapour. DEF represents the
+change in the composition of the two liquid phases with rise of
+temperature; the curve might also have the reversed form with the critical
+solution point below the transition points D and F.
+
+[Illustration: FIG. 64.]
+
+In the second class of systems (Fig. 64), that in which combination between
+the components occurs, there are again two types according as the compound
+formed has a definite melting point (_i.e._ can exist in equilibrium with a
+solution of the same composition), or undergoes only partial fusion; that
+is, exhibits a transition point.
+
+If a compound possessing a definite melting point is formed, the
+equilibrium curve will have the general form shown by curve I., Fig. 64. A,
+B, and D are the melting points of pure A, pure B, and of the compound
+A_{x}B_{y} respectively. AC {210} is the freezing point curve of A in
+presence of B; BE that of B in presence of A; and DC and DE the freezing
+point curves of the compound in presence of a solution containing excess of
+one of the components. C and E are eutectic points at which mixtures of A
+and A_{x}B_{y}, or B and A_{x}B_{y} can co-exist in contact with solution.
+The curve CDE may be large or small, and the melting point of the compound,
+D, may lie above or below that of each of the components, or may have an
+intermediate position. If more than one compound can be formed, a series of
+curves similar to CDE will be obtained (_cf._ p. 152).
+
+On the other hand, if the compound undergoes transition to another solid
+phase at a temperature below its melting point, a curve of the form II.,
+Fig. 64, will be found. This corresponds to the case where a compound can
+exist only in contact with solutions containing excess of one of the
+components. The metastable continuation of the equilibrium curve for the
+compound is indicated by the dotted line, the summit of which would be the
+melting point of the compound. Before this temperature is reached, however,
+the solid compound ceases to be able to exist in contact with solution, and
+transition to a different solid phase occurs at the point E (_cf._ p. 134).
+This point, therefore, represents the limit of the existence of the
+compound AB. If a series of compounds can be formed none of which possess a
+definite melting point, then a series of curves will be obtained which do
+not exhibit a temperature-maximum, and there will be only one eutectic
+point. The limits of existence of each compound will be marked by a break
+in the curve (_cf._ p. 143).
+
+[Illustration: FIG. 65.]
+
+Turning, lastly, to the third class of systems, in which formation of mixed
+crystals can occur, five different types of curves can be obtained, as
+shown in Fig. 65. With regard to the first three types, curves I., II., and
+III., {211} these differ entirely from those of the previous classes, in
+that they are continuous; they exhibit no eutectic point, and no transition
+point. Curve II. bears some resemblance to the melting-point curve of a
+compound (_e.g._ CDE, Fig. 64, I.), but differs markedly from it in not
+ending in eutectic points.
+
+Further, in the case of the formation of a compound, the composition of the
+solid phase remains unchanged throughout the whole curve between the
+eutectic points; whereas, when mixed crystals are produced, the composition
+of the solid phase varies with the composition of the liquid solution. On
+passing through the maximum, the relative proportions of A and B in the
+solid and the liquid phase undergo change; on the one side of the maximum,
+the solid phase contains relatively more A, and on the other side of the
+maximum, relatively more B than the liquid phase. Lastly, when mixed
+crystals are formed, the temperature at which complete solidification
+occurs changes as the composition of the solution changes, whereas in the
+case of the formation of compounds, the temperature of complete
+solidification for all solutions is a eutectic point.
+
+The third type of curve, Fig. 65, can be distinguished in a similar manner
+from the ordinary eutectic curve, Fig. 63, I., to which it bears a certain
+resemblance. Whereas in the case of the latter, the eutectic point is the
+temperature of complete solidification of all solutions, the point of
+minimum temperature in the case of the formation of mixed crystals, is the
+solidification point only of solutions having one particular composition;
+that, namely, of the minimum point. For all other solutions, the
+temperature of complete solidification is different. Whereas, also, in the
+case of the simple eutectic curve, the solid which separates out from the
+solutions represented by either curve remains the same throughout the whole
+extent of that curve, the composition of the mixed crystal varies with
+variation of the composition of the liquid phase, and the relative
+proportions of the two components in the solid and the liquid phase are
+reversed on passing through the minimum.[290]
+
+In a similar manner, type IV., Fig. 65, can be distinguished from type II.,
+Fig. 64, by the fact that it does not exhibit a {212} eutectic point, and
+that the composition of the solid phase undergoes continuous variation with
+variation of the liquid phase on either side of the transition point.
+Lastly, type V., which does exhibit a eutectic point, differs from the
+eutectic curve of Fig. 63, in that the eutectic point does not constitute
+the point of complete solidification for all solutions, and that the
+composition of the solid phase varies with the composition of the liquid
+phase.
+
+Such, then, are the chief general types of equilibrium curves for
+two-components; they are the pattern curves with which other curves,
+experimentally determined, can be compared; and from the comparison it will
+be possible to draw conclusions as to the nature of the equilibria between
+the two components under investigation.
+
+1. _Organic Compounds._
+
+[Illustration: FIG. 66.]
+
+The principles of the Phase Rule have been applied to the investigation of
+the equilibria between organic compounds, and Figs. 66-69 reproduce some of
+the results which have been obtained.[291]
+
+{213}
+
+Fig. 66, the freezing point curve (curve of equilibrium) for
+_o_-nitrophenol and _p_-toluidine, shows a curve of the simplest type[292]
+(type I., Fig. 63), in which two branches meet at an eutectic point. The
+solid phase in equilibrium with solutions represented by the left-hand
+branch of the curve was _o_-nitrophenol (m.p. 44.1°); that in equilibrium
+with the solutions represented by the right-hand branch, was _p_-toluidine
+(m.p. 43.3°). At the eutectic point (15.6°), these two solid phases could
+co-exist with the liquid phase. This equilibrium curve, therefore, shows
+that _o_-nitrophenol and _p_-toluidine do not combine with one another.
+
+In connection with this curve, attention may be called to the interesting
+fact that although the solid produced by cooling the liquid phase at the
+eutectic point has a composition approximating to that of a compound of
+equimolecular proportions of the phenol and toluidine, and a constant
+melting point, it is nevertheless a _mixture_. Although, as a rule, the
+constituents of the eutectic mixture are not present in simple molecular
+proportions, there is no reason why they should not be so; and it is
+therefore necessary to beware of assuming the formation of compounds in
+such cases.[293]
+
+Fig. 67, on the other hand, indicates with perfect certainty the formation
+of a compound between phenol and [alpha]-naphthylamine.[294] (_Cf._ curve
+I., Fig. 64.)
+
+Phenol freezes at 40.4°, but the addition of [alpha]-naphthylamine lowers
+the freezing point as represented by the curve AC. At C (16.0°) the
+compound C_{6}H_{5}OH,C_{10}H_{7}NH_{2} is formed, and the system becomes
+invariant. On increasing the amount of the amine, the temperature of
+equilibrium rises, the solid phase now being the compound. At D, the curve
+passes through a maximum (28.8°), at which the solid and liquid phases have
+the same composition. This is the melting point of the compound. Further
+addition of the amine lowers the temperature of equilibrium, until at E
+solid [alpha]-naphthylamine separates out, and a second eutectic point
+(24.0°) is obtained. BE is the {214} freezing-point curve of
+[alpha]-naphthylamine in presence of phenol, the freezing point of the pure
+amine being 48.3°.
+
+On account of the great sluggishness with which the compound of phenol and
+[alpha]-naphthylamine crystallizes, it was found possible to follow the
+freezing point curves of phenol and the amine to temperatures considerably
+below the eutectic points, as shown by the curves CF and EG.
+
+[Illustration: FIG. 67.]
+
+Phenol can also combine with _p_-toluidine in equimolecular proportions;
+and this compound is of interest, from the fact that it exists in two
+crystalline forms melting at 28.5° and 30°. Each of these forms now must
+have its own equilibrium curve, and it was found that the intermediate
+portion of the freezing point curve was duplicated, as shown in Fig.
+68.[295]
+
+{215}
+
+[Illustration: FIG. 68.]
+
+[Illustration: FIG. 69.]
+
+{216}
+
+Lastly, a curve is given, Fig. 69,[296] which corresponds with curve II.,
+Fig. 64. Picric acid and benzene can form a compound, which, however, can
+exist only in contact with solutions _containing excess of benzene_. When
+the temperature is raised, a point (K) is reached at which the compound
+melts with separation of solid picric acid. The point, K, is, therefore, a
+_transition point_; analysis, however, showed that the composition of the
+solution at this point is very nearly that of the compound
+C_{6}H_{2}(NO_{2})_{3}OH,C_{6}H_{6}, so that the melting point of the
+compound can almost be reached. The fusion of the compound of benzene and
+picric acid with separation of the latter is analogous to the (partial)
+fusion of Glauber's salt with separation of anhydrous sodium sulphate.
+
+2. _Optically Active Substances._
+
+The question as to whether a resolvable inactive body is a mixture of the
+two oppositely active constituents (a _dl_-mixture), or a racemic compound,
+is one which has given rise to considerable discussion during the past
+decade; and several investigators have endeavoured to establish general
+rules by which the question could be decided. In the case of inactive
+liquids it is a matter of great difficulty to arrive at a certain
+conclusion as to whether one is dealing with a mixture or a compound, for
+in this case the usual physical methods give but a dubious answer; and
+although the existence of a racemate in the liquid state (in the case of
+conine) has been asserted,[297] most chemists incline to the belief that
+such a thing is improbable.
+
+Even in the case of crystalline substances, where the differences between
+the various forms is greater, it was not always easy to discriminate
+between the _dl_-mixture and the racemic compound. The occurrence of
+hemihedral faces was considered by Pasteur to be a sufficient criterion for
+an optically active substance. It has, however, been found that hemihedry
+in crystals, although a frequent accompaniment of {217} optical activity,
+is by no means a necessary or constant expression of this property. Other
+rules, also, which were given, although in some cases reliable, were in
+other cases insufficient; and all were in so far unsatisfactory that they
+lacked a theoretical basis.
+
+With the help of the Phase Rule, however, it is possible from a study of
+the solubility or fusion curves of the optically active and inactive
+substances, to decide the nature of the inactive substance, at least under
+certain conditions. On account of the interest and importance which these
+compounds possess, a brief description of the application of the Phase Rule
+to the study of such substances will be given here;[298] the two optical
+antipodes being regarded as the two components.
+
+In the present chapter we shall consider only the fusion curves, the
+solubility curves being discussed in the next section on three-component
+systems. The rules which are hereby obtained, have reference only to the
+nature of the inactive substance in the neighbourhood of the melting
+points.
+
+I. _The inactive substance is a _dl_-mixture._
+
+In this case the fusion curves will have the simple form shown in type I,
+Fig. 63. A and B are the melting points of the two optical isomerides, and
+C the eutectic point at which the inactive mixture consisting of equal
+amounts of d- and l-form melts. Owing to the similar effect of the one form
+on the freezing point of the other, the figure is symmetrical. No example
+of this simple case has been investigated.
+
+II. _The two components form a racemic compound._
+
+In this case there will be three melting point curves as in Fig. 64, type
+I. In this case also the figure must be symmetrical.
+
+Examples.--As examples of this, may be taken dimethyl tartrate and mandelic
+acid, the freezing point curves of which are given in Figs. 70 and 71.[299]
+As can be seen, the curve for the racemic tartrate occupies a large part of
+the diagram, {218} while that for racemic mandelic acid is much smaller. In
+the case of dimethyldiacetyl tartrate, this middle portion is still less.
+
+[Illustration: FIG. 70.]
+
+[Illustration: FIG. 71.]
+
+[Illustration: FIG. 72.]
+
+Active dimethyl tartrate melts at 43.3°; racemic dimethyl tartrate at
+89.4°. Active mandelic acid melts at 132.8°; the racemic acid at 118.0°. In
+the one case, therefore, the racemic compound has a higher, in the other a
+lower melting point than the active forms. {219}
+
+In the case of partially racemic compounds (_i.e._ the compound of a
+racemate with an optically active substance) the type of curve will be the
+same, but the figure will no longer be symmetrical. Such a curve has been
+found in the case of the l-menthyl esters of d- and l-mandelic acid (Fig.
+72).[300] The freezing point of l-menthyl d-mandelate is 97.2°, of
+l-menthyl l-mandelate 77.6°, and of l-menthyl r-mandelate 83.7.° It will be
+observed that the summit of the curve for the partially racemic mandelate
+is very flat, indicating that the compound is largely dissociated into its
+components at the temperature of fusion.
+
+III. _The inactive substance is a pseudo-racemic mixed crystal._
+
+In cases where the active components can form mixed crystals, the
+freezing-point curve will exhibit one of the forms given in Fig. 65. The
+inactive mixed crystal containing 50 per cent. of the dextro and laevo
+compound, is known as a pseudo-racemic mixed crystal.[301] So far, only
+curves of the types I. and II. have been obtained.
+
+Examples.--The two active camphor oximes are of interest from the fact that
+they form a continuous series of mixed crystals, _all of which have the
+same melting point_. The curve which is obtained in this case is,
+therefore, a straight line joining the melting points of the pure active
+components; the melting point of the active isomerides and of the whole
+series of mixed crystals being 118.8°.
+
+[Illustration: FIG. 73.]
+
+In the case of the carvoximes mixed crystals are also formed, but the
+equilibrium curve in this case exhibits a maximum (Fig. 73). At this
+maximum point the composition of the solid and of the liquid solution is
+the same. Since the curve must be symmetrical, this maximum point must
+occur in the case of the solution containing 50 per cent. {220} of each
+component, which will therefore be inactive. Further, this inactive mixed
+crystal will melt and solidify at the same temperature, and behave,
+therefore, like a chemical compound (p. 187). The melting point of the
+active compounds is 72°; that of the inactive pseudo-racemic mixed crystal
+is 91.4°·
+
+Transformations.--As has already been remarked, the conclusions which can
+be drawn from the fusion curves regarding the nature of the inactive
+substances formed hold only for temperatures in the neighbourhood of the
+melting points. At temperatures below the melting point transformation may
+occur; _e.g._ a racemate may break up into a _dl_-mixture, or a
+pseudo-racemic mixed crystal may form a racemic compound. We shall at a
+later point meet with examples of a racemic compound changing into a
+_dl_-mixture at a definite transition point; and the pseudo-racemic mixed
+crystal of camphoroxime is an example of the second transformation.
+Although at temperatures in the neighbourhood of the melting point the two
+active camphoroximes form only mixed crystals but no compound, a racemic
+compound is formed at temperatures below 103°. At this temperature the
+inactive pseudo-racemic mixed crystal changes into a racemic compound; and
+in the case of the other mixed crystals transformation to racemate and
+(excess of) active component also occurs, although at a lower temperature
+than in the case of the inactive mixed crystal. Although this behaviour is
+one of considerable importance, this brief reference to it must suffice
+here.[302]
+
+3. _Alloys._
+
+One of the most important classes of substances in the study of which the
+Phase Rule has been of very considerable importance, is that formed by the
+mixtures or compounds of metals with one another known as alloys. Although
+in the investigation of the nature of these bodies various methods are
+employed, one of the most important is the determination of the character
+of the freezing-point curve; for from the form of this, valuable
+information can, as we have already learned, be {221} obtained regarding
+the nature of the solid substances which separate out from the molten
+mixture.
+
+Although it is impossible here to discuss fully the experimental results
+and the oftentimes very complicated relationships which the study of the
+alloys has brought to light, a brief reference to these bodies will be
+advisable on account both of the scientific interest and of the industrial
+importance attaching to them.[303]
+
+We have already seen that there are three chief types of freezing-point
+curves in systems of two components, viz. those obtained when (1) the pure
+components crystallize out from the molten mass; (2) the components form
+one or more compounds; (3) the components form mixed crystals. In the case
+of the metals, representatives of these three classes are also found.
+
+1. _The components separate out in the pure state._
+
+In this case the freezing-point curve is of the simple type, Fig. 63, I.
+Such curves have been obtained in the case of a number of pairs of metals,
+_e.g._ zinc--cadmium, zinc--aluminium, copper--silver (Heycock and
+Neville), tin--zinc, bismuth--lead (Gautier), and in other cases. From
+molten mixtures represented by one branch of the freezing-point curve one
+of the metals will be deposited; while from mixtures represented by the
+other branch, the other metal will separate out. At the eutectic point the
+molten mass will solidify to a _heterogeneous mixture_ of the two metals,
+forming what is known as the _eutectic alloy_. Such an alloy, therefore,
+will melt at a definite temperature lower than the melting point of either
+of the pure metals.
+
+{222}
+
+In the following table are given the temperature and the composition of the
+liquid at the eutectic point, for three pairs of metals:--
+
+  -------------------------------------------------------------------
+                  | Temperature. |      Composition of liquid.
+  -------------------------------------------------------------------
+  Zinc--cadmium   |    264.5°    | 73.5 atoms per cent. of cadmium.
+  Zinc--aluminium |    380.5°    | 11     "      "         aluminium.
+  Copper--silver  |    778°      | 40     "      "         copper.
+  -------------------------------------------------------------------
+
+The melting points of the pure metals are, zinc, 419°; cadmium, 322°;
+silver, 960°; copper, 1081°; aluminium, 650°.
+
+2. _The two metals can form one or more compounds._
+
+In this case there will be obtained not only the freezing-point curves of
+the pure metals, but each compound formed will have its own freezing-point
+curve, exhibiting a point of maximum temperature, and ending on either side
+in an eutectic point. The simplest curve of this type will be obtained when
+only one compound is formed, as is the case with mercury and thallium.[304]
+This curve is represented in Fig. 74, where the summit of the intermediate
+curve corresponds with a composition TlHg_{2}. Similar curves are also
+given by nickel and tin, by aluminium and silver, and by other metals, the
+formation of definite compounds between these pairs of metals being thereby
+indicated.[305]
+
+[Illustration: FIG. 74.]
+
+{223}
+
+A curve belonging to the same type, but more complicated, is obtained with
+gold and aluminium;[306] in this case, several compounds are formed, some
+of which have a definite melting point, while others exhibit only a
+transition point. The chief compound is AuAl_{2}, which has practically the
+same melting point as pure gold.
+
+3. _The two metals form mixed crystals (solid solutions)._
+
+The simplest case in which the metals crystallize out together is found in
+silver and gold.[307] The freezing-point curve in this case is an almost
+straight line joining the freezing points of the pure metals (_cf._ curve
+I., Fig. 65, p. 210). These two metals, therefore, can form an unbroken
+series of mixed crystals.
+
+In some cases, however, the two metals do not form an unbroken series of
+mixed crystals. In the case of zinc and silver,[308] for example, the
+addition of silver _raises_ the freezing point of the mixture, until a
+transition point is reached. This corresponds with curve IV., Fig. 65.
+Silver and copper, and gold and copper, on the other hand, do not form
+unbroken series of mixed crystals, but the freezing-point curve exhibits an
+eutectic point, as in curve V., Fig. 65.
+
+Not only may there be these three different types of curves, but there may
+also be combinations of these. Thus the two metals may not only form
+compounds, but one of the metals may not separate out in the pure state at
+all, but form mixed crystals. In this case the freezing point may rise (as
+in the case of silver and zinc), and one of the eutectic points will be
+absent.
+
+Iron-Carbon Alloys.--Of all the different binary alloys, probably the most
+important are those formed by iron and carbon: alloys consisting not of two
+metals, but of a metal and a non-metal. On account of the importance of
+these alloys, an attempt will be made to describe in brief some of the most
+important relationships met with.
+
+Before proceeding to discuss the applications of the Phase Rule to the
+study of the iron-carbon alloys, however, the main {224} facts with which
+we have to deal may be stated very briefly. With regard to the metal
+itself, it is known to exist in three different allotropic modifications,
+called [alpha]-, [beta]-, and [gamma]-ferrite respectively. Like the two
+modifications of sulphur and of tin, these different forms exhibit
+transition points at which the relative stability of the forms changes.
+Thus the transition point for [alpha]- and [beta]-ferrite is about 780°;
+and below this temperature the [alpha]- form, above it the [beta]- form is
+stable. For [beta]- and [gamma]-ferrite, the transition point is about
+870°, the [gamma]- form being the stable modification above this
+temperature.
+
+The different modifications of iron also possess different properties.
+Thus, [alpha]-ferrite is magnetic, but does not possess the power of
+dissolving carbon; [beta]-ferrite is non-magnetic, and likewise does not
+dissolve carbon; [gamma]-ferrite is also non-magnetic, but possesses the
+power of dissolving carbon, and of thus giving rise to solid solutions of
+carbon in iron.
+
+Various alloys of iron and carbon, also, have to be distinguished. First of
+all, there is _hard steel_, which contains varying amounts of carbon up to
+2 per cent. Microscopic examination shows that these mixtures are all
+homogeneous; and they are therefore to be regarded as solid solutions of
+carbon in iron ([gamma]-ferrite). To these solutions the name _martensite_
+has been given. _Pearlite_ contains about 0.8 per cent. of carbon, and, on
+microscopic examination, is found to be a heterogeneous mixture. If heated
+above 670°, pearlite becomes homogeneous, and forms martensite. Lastly,
+there is a definite compound of iron and carbon, iron carbide or
+_cementite_, having the formula Fe_{3}C.
+
+A short description may now be given of the application of the Phase Rule
+to the two-component system iron--carbon; and of the diagram showing how
+the different systems are related, and with the help of which the behaviour
+of the different mixtures under given conditions can be predicted.
+Although, with regard to the main features of this diagram, the different
+areas to be mapped and the position of the frontier lines, there is general
+agreement; a final decision has not yet been reached with regard to the
+interpretation to be put on all the curves.
+
+[Illustration: FIG. 75.]
+
+The chief relationships met with in the case of the {225} iron-carbon
+alloys are represented graphically in Fig. 75.[309] The curve AC is the
+freezing-point curve for iron,[310] BC the unknown freezing-point curve for
+graphite. C is an eutectic point. Suppose, now, that we start with a
+mixture of iron and carbon, represented by the point _x_. On lowering the
+temperature, a point, _y_, will be reached at which solid begins to
+separate out. This solid phase, however, is not pure iron, but a solid
+solution of carbon in iron, having the composition represented by _y'_ (cf.
+p. 185). As the temperature continues to fall, the {226} composition of the
+liquid phase changes in the direction of _y_C, while the composition of the
+solid which separates out changes in the direction _y'_D; and, finally,
+when the composition of the molten mass is that of the point C (4.3 per
+cent. of carbon), the whole mass solidifies to a heterogeneous mixture of
+two solid solutions, one of which is represented by D (containing 2 per
+cent. of carbon), while the other will consist practically of pure
+graphite, and is not shown in the figure. The temperature of the eutectic
+point is 1130°.
+
+Even below the solidification point, however, changes can take place. As
+has been said, the solid phase which finally separates out from the molten
+mass is a solid solution represented by the point D; and the curve DE
+represents the change in the composition of this solid solution with the
+temperature. As indicated in the figure, DE forms a part of a curve
+representing the mutual solubility of graphite in iron and iron in
+graphite; the latter solutions, however, not being shown, as they would lie
+far outside the diagram. As the temperature falls below 1130°, more and
+more graphite separates out, until at E, when the temperature is 1000°, the
+solid solution contains only 1.8 per cent. of carbon. At this temperature
+cementite also begins to be formed, so that as the temperature continues to
+fall, separation of cementite (represented by the line E'F') occurs, and
+the composition of the solid solution undergoes alteration, as represented
+by the curve EF. Below the temperature of the point F (670°) the martensite
+becomes heterogeneous, and forms pearlite.
+
+From the above description, therefore, it follows that if we start with a
+molten mixture of iron and carbon, the composition of which is represented
+by any point between D and C (from 2 to 4.3 per cent. of carbon), we shall
+obtain, on cooling the mass, first of all solid solutions, the composition
+of which will be represented by points on the line AD; that then, after the
+mass has completely solidified at 1130°, further cooling will lead to a
+separation of graphite and a change in the composition of the martensite
+(from 2 to 1.8 per cent. of carbon). On cooling below 1000°, however, the
+martensite and graphite will give rise to cementite and solid solutions
+{227} containing less carbon than before, until, at temperatures below
+670°, we are left with a mixture of pearlite and cementite.
+
+We have already said that iron consists in three allotropic modifications,
+the regions of stability of which are separated by definite transition
+points. The transition point for [alpha]- and [beta]-ferrite (780°) is
+represented in Fig. 75 by the point H; and the transition point for [beta]-
+and [gamma]-ferrite (870°) by the point I. Since neither the [alpha]- nor
+the [beta]-ferrite dissolves carbon, the transition point will be
+unaffected by addition of carbon, and we therefore obtain the horizontal
+transition curve HG. In the case of the [beta]- and [gamma]-ferrite,
+however, the latter dissolves carbon, and the transition point is
+consequently affected by the amount of carbon present. This is shown by the
+line IG.
+
+If a martensite containing less carbon than that represented by the point G
+is cooled down from a temperature of, say, 900°, then when the temperature
+has fallen to that, represented by a point on the curve IG, [beta]-ferrite
+will separate out, and, as the temperature falls, the composition of the
+solid solution will alter as represented by IG. On passing below the
+temperature of HG, the [beta]-ferrite will be converted into
+[alpha]-ferrite, and, as the temperature falls, the latter will separate
+out more and more, while the composition of the solid solution alters in
+the direction GF. On passing to still lower temperatures, the solid
+solution at F (0.8 per cent. of carbon) breaks up into pearlite. If the
+percentage of carbon in the original solid solution was between that
+represented by the points G and F, then, on cooling down, no
+[beta]-ferrite, but only [alpha]-ferrite would separate out.
+
+We see, therefore, that when martensite is allowed to cool _slowly_, it
+yields a heterogeneous mixture either of ferrite and pearlite (when the
+original mixture contained up to 0.8 per cent. of carbon), or pearlite and
+cementite (when the original mixture contained between 0.8 and 2 per cent.
+of carbon). These heterogeneous mixtures constitute soft steels, or, when
+the carbon content is low, wrought iron.
+
+The case, however, is different if the solid solution of carbon in iron is
+_rapidly_ cooled (quenched) from a temperature above the curve IGFE to a
+temperature below this {228} curve. In this case, the rapid cooling does
+not allow time for the various changes which have been described to take
+place; so that the homogeneous solid solution, on being rapidly cooled,
+remains homogeneous. In this way hard steel is obtained. By varying the
+rapidity of cooling, as is done in the tempering of steel, varying degrees
+of hardness can be obtained.
+
+The interpretation of the curves given above is that due essentially to
+Roozeboom, who concluded from the experimental data that at temperatures
+below 1000° the stable systems are martensite and cementite, or ferrite and
+cementite, graphite being labile. It has, however, been pointed out, more
+especially by E. Heyn,[311] that this is not in harmony with the facts of
+metallurgy, which show that graphite is undoubtedly formed on slow cooling,
+and more especially when small quantities of silicon are present in the
+iron.[312] While, therefore, the relationships represented by Fig. 75 are
+obtained under certain conditions (especially when manganese is present),
+Heyn considers that all the curves in that figure, except ACB, represent
+_metastable_ systems--systems, therefore, akin to supercooled liquids.
+Rapid cooling will favour the production of the metastable systems
+containing cementite, and therefore give rise to relationships represented
+by Fig. 75; whereas slow cooling will lead to the stable system ferrite and
+graphite. Presence of silicon tends to prevent, presence of manganese tends
+to assist, the production of the metastable systems.
+
+Although this view put forward by Heyn has not been conclusively proved, it
+must be said that there is much evidence in its favour. Further
+investigation is, however, required before a final decision as to the
+interpretation of the curves can be reached.
+
+Determination of the Composition of Compounds, without Analysis.--Since the
+equilibrium between a solid and a liquid phase depends not only on the
+composition of the liquid (solution) but also on that of the solid, it is
+necessary {229} to determine the composition of the latter. In some cases
+this is easily effected by separating the solid from the liquid phase and
+analyzing it. In other cases, however, this method is inapplicable, or is
+accompanied by difficulties, due either to the fact that the solid phase
+undergoes decomposition (_e.g._ when it contains a volatile constituent),
+or to the difficulty of completely separating the mother liquor; as, for
+example, in the case of alloys. In all such cases, therefore, recourse must
+be had to other methods.
+
+In the first place, synthetic methods may be employed.[313] In this case we
+start with a solution of the two components, to which a third substance is
+added, which, however, does not enter into the solid phase.[314] We will
+assume that the initial solution contains _x_ gm. of A and _y_ gm. of B to
+1 gm. of C. After the solution has been cooled down to such a temperature
+that solid substance separates out, a portion of the liquid phase is
+removed with a pipette and analyzed. If, now, the composition of the
+solution is such that there are _x'_ gm. of A and _y'_ gm. of B to 1 gm. of
+C., then the composition of the solid phase is _x_ - _x'_ gm. of A and _y_
+- _y'_ gm. of B. When _x_ = _x'_, the solid phase is pure B; when _y_ =
+_y'_, the solid phase is pure A.
+
+We have assumed here that there is only one solid phase present, containing
+A and B. To make sure that the solid phase is not a solid solution in which
+A and B are present in the same ratio as in the liquid solution, a second
+determination of the composition must be made, with different initial and
+end concentrations. If the solid phase is a solid solution, the composition
+will now be found different from that found previously.
+
+The composition of the solid phase can, however, be determined in another
+manner, viz. by studying the fusion curve and the curve of cooling. From
+the form of the fusion curve alone, it is possible to decide whether the
+two components {230} form a compound or not; and if the compounds which may
+be formed have a definite melting point, the position of the latter gives
+at once the composition of the compounds (cf. p. 231).
+
+This method, however, cannot be applied when the compounds undergo
+decomposition before the melting point is reached. In such cases, however,
+the form of the cooling curve enables one to decide the composition of the
+solid phase.[315] If a solution is allowed to cool slowly, and the
+temperature noted at definite times, the graphic representation of the rate
+of cooling will give a continuous curve; _e.g._ _ab_ in Fig. 76. So soon,
+however, as a solid phase begins to be formed, the rate of cooling alters
+abruptly, and the cooling curve then exhibits a break, or change in
+direction (point _b_). When the eutectic point is reached, the temperature
+remains constant, until all the liquid has solidified. This is represented
+by the line _cd_. When complete solidification has occurred, the fall of
+temperature again becomes uniform (_de_).
+
+[Illustration: FIG. 76.]
+
+[Illustration: FIG. 77.]
+
+[Illustration: FIG. 78.]
+
+The length of time during which the temperature remains constant at the
+point _c_, depends, of course, on the eutectic solution. If, therefore, we
+take equal amounts of solution having a different initial composition, the
+period of constant temperature in the cooling curve will evidently be
+greatest in the case of the solution having the composition of the eutectic
+point; and the period will become less and less as we increase the amount
+of one of the components. The relationship between initial composition of
+solution and the duration of constant temperature at the eutectic point is
+represented by the curve _a'c'b'_ (Fig. 77). When a compound possessing a
+definite melting point is formed, it behaves as a pure substance. If,
+therefore, the initial composition of the {231} solution is the same as
+that of the compound, no eutectic solution will be obtained; and therefore
+no line of constant temperature, such as _cd_ (Fig. 76). In such a case, if
+we represent graphically the relation between the initial composition of
+the solution and the duration of constant temperature, a diagram is
+obtained such as shown in Fig. 78. The two maxima on the time-composition
+curve represent eutectic points, and the minima, _a'_, _b'_, _e'_, pure
+substances. The position of _e'_ gives the composition of the compound.
+When a series of compounds is formed, then for each compound a minimum is
+found on the time-composition curve.
+
+[Illustration: FIG. 79.]
+
+If the compound formed has no definite melting point, the diagram obtained
+is like that shown in Fig. 79. If we start with a solution, the composition
+of which is represented by a point between _d_ and _b_, then, on cooling,
+_b_ will separate out first, and the temperature will fall until the point
+_d_ is reached. The temperature then remains constant until the component
+_b_, which has separated out, is converted into the compound. After this
+the temperature again falls, until it again remains constant at the
+eutectic point c. In the case of the first halt, the period of constant
+temperature is greatest when the initial composition of the solution is the
+same as that of the compound; and it becomes shorter and shorter with {232}
+increase in the amount of either component. In this way we obtain the
+time-composition curve _b'e"d'_, of which the maximum point _e"_ gives the
+composition of the compound.
+
+On the other hand, the period of constant temperature for the eutectic
+point _c_ is greatest in the case of solutions having the same initial
+_composition_ as that corresponding with the eutectic point; and it
+decreases the more the initial composition approaches that of the pure
+component _a_ or the component e. In this way we obtain the
+time-composition curve _a'c'e'_. Here also the point _e'_ represents the
+composition of the compound. We see, therefore, that from the graphic
+representation of the freezing-point curve, and from the duration of the
+temperature-arrests on the cooling curve, for solutions of different
+initial composition, it is possible, without having recourse to analysis,
+to decide what solid phases are formed, and what is their composition.
+
+Formation of Minerals.--Important and interesting as is the application of
+the Phase Rule to the study of alloys, its application to the study of the
+conditions regulating the formation of minerals is no less so; and although
+we do not propose to consider different cases in detail here, still
+attention must be drawn to certain points connected with this interesting
+subject.
+
+In the first place, it will be evident from what has already been said,
+that that mineral which first crystallizes out from a molten magma is not
+necessarily the one with the highest melting point. The _composition_ of
+the fused mass must be taken into account. When the system consists of two
+components which do not form a compound, one or other of these will
+separate out in a pure state, according as the composition of the molten
+mass lies on one or other side of the eutectic composition; and the
+separation of the one component will continue until the composition of the
+eutectic point is reached. Further cooling will then lead to the
+simultaneous separation of the two components.
+
+If, however, the two components form a stable compound (_e.g._ orthoclase,
+from a fused mixture of silica and potassium aluminate), then the
+freezing-point curve will resemble that {233} shown in Fig. 64; _i.e._
+there will be a middle curve possessing a dystectic point, and ending on
+either side at a eutectic point. This curve would represent the conditions
+under which orthoclase is in equilibrium with the molten magma. If the
+initial composition of the magma is represented by a point between the two
+eutectic points, orthoclase will separate first. The composition of the
+magma will thereby change, and the mass will finally solidify to a mixture
+of orthoclase and silica, or orthoclase and potassium aluminate, according
+to the initial composition.
+
+What has just been said holds, however, only for stable equilibria, and it
+must not be forgotten that complications can arise owing to suspended
+transformation (when, for example, the magma is rapidly cooled) and the
+production of metastable equilibria. These conditions occur very frequently
+in nature.
+
+The study of the formation of minerals from the point of view of the Phase
+Rule is still in its initial stages, but the results which have already
+been obtained give promise of a rich harvest in the future.[316]
+
+       *       *       *       *       *
+
+
+{234}
+
+CHAPTER XIII
+
+SYSTEMS OF THREE COMPONENTS
+
+General.--It has already been made evident that an increase in the number
+of the components from one to two gives rise to a considerable increase in
+the possible number of systems, and introduces not a few complications into
+the equilibrium relations of these. No less is this the case when the
+number of components increases from two to three; and although examples of
+all the possible types of systems of three components have not been
+investigated, nor, indeed, any one type fully, nevertheless, among the
+systems which have been studied experimentally, cases occur which not only
+possess a high scientific interest, but are also of great industrial
+importance. On account not only of the number, but more especially of the
+complexity of the systems constituted of three components, no attempt will
+be made to give a full account, or, indeed, even a survey of all the cases
+which have been subjected to a more or less complete experimental
+investigation; on the contrary, only a few of the more important classes
+will be selected, and the most important points in connection with the
+behaviour of these described.
+
+On applying the Phase Rule
+
+  P + F = C + 2
+
+to the systems of three components, we see that in order that the system
+shall be invariant, no fewer than five phases must be present together, and
+an invariant system will therefore exist at a _quintuple_ point. Since the
+number of liquid phases can never exceed the number of the components, and
+since there can be only one vapour phase, it is evident that in this case,
+{235} as in others, there must always be at least one solid phase present
+at the quintuple point. As the number of phases diminishes, the variability
+of the system can increase from one to four, so that in the last case the
+condition of the system will not be completely defined until not only the
+temperature and the total pressure of the system, but also the
+concentrations of two of the components have been fixed. Or, instead of the
+concentrations, the partial pressures of the components may also be taken
+as independent variables.
+
+Graphic Representation.--Hitherto the concentrations of the components have
+been represented by means of rectangular co-ordinates, although the
+numerical relationships have been expressed in two different ways. In the
+one case, the concentration of the one component was expressed in terms of
+a fixed amount of the other component. Thus, the solubility of a salt was
+expressed by the number of grams of salt dissolved by 100 grams of water or
+other solvent; and the numbers so obtained were measured along one of the
+co-ordinates. The second co-ordinate was then employed to indicate the
+change of another independent variable, _e.g._ temperature. In the other
+case, the combined weights of the two components A and B were put equal to
+unity, and the concentration of the one expressed as a fraction of the
+whole amount. This method allows of the representation of the complete
+series of concentrations, from pure A to pure B, and was employed, for
+example, in the graphic representation of the freezing point curves.
+
+Even in the case of three components rectangular co-ordinates can also be
+employed, and, indeed, are the most convenient in those cases where the
+behaviour of two of the components to one another is very different from
+their behaviour to the third component; as, for example, in the case of two
+salts and water. In these cases, the composition of the system can be
+represented by measuring the amounts of each of the two components in a
+given weight of the third, along two co-ordinates at right angles to one
+another; and the change of the system with the temperature can then be
+represented by a third axis at right angles to the first two. In those
+cases, {236} however, where the three components behave in much the same
+manner towards one another, the rectangular co-ordinates are not at all
+suitable, and instead of these a _triangular diagram_ is employed. Various
+methods have been proposed for the graphic representation of systems of
+three components by means of a triangle, but only two of these have been
+employed to any considerable extent; and a short description of these two
+methods will therefore suffice.[317]
+
+[Illustration: FIG. 80.]
+
+In the method proposed by Gibbs an equilateral triangle of unit height is
+used (Fig 80).[318] The quantities of the different components are
+expressed as fractional parts of the whole, and the sum of their
+concentrations is therefore equal to unity, and can be represented by the
+height of the triangle. The corners {237} of the triangle represent the
+pure substances A, B, and C respectively. A point on one of the sides of
+the triangle will give the composition of a mixture in which only two
+components are present, while a point within the triangle will represent
+the composition of a ternary mixture. Since every point within the triangle
+has the property that the sum of the perpendiculars from that point on the
+sides of the triangle is equal to unity (the height of the triangle), it is
+evident that the composition of a ternary mixture can be represented by
+fixing a point within the triangle such that the lengths of the
+_perpendiculars_ from the point to the sides of the triangle are equal
+respectively to the fractional amounts of the three components present; the
+fractional amount of A, B, or C being represented by the perpendicular
+distance from the side of the triangle _opposite_ the corners A, B, and C
+respectively.
+
+The location of this point is simplified by dividing the normals from each
+of the corners on the opposite side into ten or one hundred parts, and
+drawing through these divisions lines at right angles to the normal and
+parallel to the side of the triangle. A network of rhombohedra is thus
+obtained, and the position of any point can be read off in practically the
+same manner as in the case of rectangular co-ordinates. Thus the point P in
+Fig. 80 represents a ternary mixture of the composition A = 0.5, B = 0.3, C
+= 0.2; the perpendiculars P_a_, P_b_, and P_c_ being equal respectively to
+0.5, 0.2, and 0.3 of the height of the triangle.
+
+Another method of representation, due to Roozeboom, consists in employing
+an equilateral triangle, the length of whose _side_ is made equal to unity,
+or one hundred; the sum of the fractional or percentage amounts of the
+three components being represented therefore by a side of the triangle. In
+this case the composition of a ternary mixture is obtained by determining,
+not the _perpendicular_ distance of a point P from the three sides of the
+triangle, but the distance in a direction _parallel_ to the sides of the
+triangle (Fig. 81). Conversely, in order to represent a mixture consisting
+of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one
+side of the triangle, say AB, is first of all divided into ten or one {238}
+hundred parts; a portion, B_x_ = _a_, is then measured off, and represents
+the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off
+and represents the fractional amount of B, while the remainder, _xx'_ =
+_c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel
+to the sides of the triangle, and the point of intersection, P, represents
+the composition of the ternary mixture of given composition; for, as is
+evident from the figure, the distance of the point P from the three sides
+of the triangle, when measured in directions _parallel_ to the sides, is
+equal to _a_, _b_, and _c_ respectively. From the division marks on the
+side AB, it is seen that the point P in this figure also represents a
+mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives
+exactly the same result as the previous method. The employment of a
+right-angled isosceles triangle has also been suggested,[319] but is not in
+general use.
+
+[Illustration: FIG. 81.]
+
+In employing the triangular diagram, it will be of use to note a property
+of the equilateral triangle. A line drawn from one corner of the triangle
+to the opposite side, represents the composition of all mixtures in which
+the _relative_ amounts of two of the components remain unchanged. Thus, as
+Fig. 82 shows, if the component C is added to a mixture x, in which A and B
+are present in the proportions of _a_ : _b_, a mixture _x'_, which is
+thereby obtained, also contains A and B in the ratio _a_ : b. For the two
+triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and
+KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_
+and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ :
+E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore,
+the ratio of A to B is the same.
+
+[Illustration: FIG. 82.]
+
+[Illustration: FIG. 83.]
+
+If it is desired to represent at the same time the change of another
+independent variable, _e.g._ temperature, this can be done by measuring the
+latter along axes drawn perpendicular to the corners of the triangle. In
+this way a right prism (Fig. 83) is obtained, and each section of this cut
+parallel to the base represents therefore an _isothermal surface_.
+
+       *       *       *       *       *
+
+
+{240}
+
+CHAPTER XIV
+
+SOLUTIONS OF LIQUIDS IN LIQUIDS
+
+We have already seen (p. 95) that when two liquids are brought together,
+they may mix in all proportions and form one homogeneous liquid phase; or,
+only partial miscibility may occur, and two phases be formed consisting of
+two mutually saturated solutions. In the latter case, the concentration of
+the components in either phase and also the vapour pressure of the system
+had, at a given temperature, perfectly definite values. In the case of
+three liquid components, a similar behaviour may be found, although
+complete miscibility of three components with the formation of only one
+liquid phase is of much rarer occurrence than in the case of two
+components. When only partial miscibility occurs, various cases are met
+with according as the three components form one, two, or three pairs of
+partially miscible liquids. Further, when two of the components are only
+partially miscible, the addition of the third may cause either an increase
+or a diminution in the mutual solubility of these. An increase in the
+mutual solubility is generally found when the third component dissolves
+readily in each of the other two; but when the third component dissolves
+only sparingly in the other two, its addition diminishes the mutual
+solubility of the latter.
+
+We shall consider here only a few examples illustrating the three chief
+cases which can occur, viz. (1) A and B, and also B and C are miscible in
+all proportions, while A and C are only partially miscible. (2) A and B are
+miscible in all proportions, but A and C and B and C are only partially
+miscible. (3) A and B, B and C, and A and C are only partially miscible. A,
+B, and C here represent the three components.
+
+1.--_The three components form only one pair of partially miscible
+liquids._ {241}
+
+An example of this is found in the three substances: chloroform, water, and
+acetic acid.[320] Chloroform and acetic acid, and water and acetic acid,
+are miscible with one another in all proportions, but chloroform and water
+are only partially miscible with one another. If, therefore, chloroform is
+shaken with a larger quantity of water than it can dissolve, two layers
+will be formed consisting one of a saturated solution of water in
+chloroform, the other of a saturated solution of chloroform in water. The
+composition of these two solutions at a temperature of about 18°, will be
+represented by the points _a_ and _b_ in Fig. 84; _a_ representing a
+solution of the composition: chloroform, 99 per cent.; water, 1 per cent.;
+and _b_ a solution of the composition: chloroform, 0.8 per cent.; water,
+99.2 per cent. When acetic acid is added, it distributes itself between the
+two liquid layers, and two conjugate _ternary_ solutions, consisting of
+chloroform, water, and acetic acid are thereby produced which are in
+equilibrium with one another, and the composition of which will be
+represented by two points inside the triangle. In this way a series of
+pairs of ternary solutions will be obtained by the addition of acetic acid
+to the mixture of chloroform and water. By this addition, also, not only do
+the two liquid phases become increasingly rich in acetic acid, but the
+mutual solubility of the chloroform and water increases; so that the layer
+_a_ becomes relatively richer in water, and layer _b_ relatively richer in
+chloroform. This is seen from the following table, which gives the
+percentage composition of different conjugate ternary solutions at 18°.
+
+  -------------------------------------------------------------------------
+             Heavier layer.           |            Lighter layer.
+  -------------------------------------------------------------------------
+  Chloroform. | Water. | Acetic acid. | Chloroform. | Water. | Acetic acid.
+  -------------------------------------------------------------------------
+     99.01    |  0.99  |     0        |     0.84    | 99.16  |     0
+     91.85    |  1.38  |     6.77     |     1.21    | 73.69  |    25.10
+     80.00    |  2.28  |    17.72     |     7.30    | 48.58  |    44.12
+     70.13    |  4.12  |    25.75     |    15.11    | 34.71  |    50.18
+     67.15    |  5.20  |    27.65     |    18.33    | 31.11  |    50.56
+     59.99    |  7.93  |    32.08     |    25.20    | 25.39  |    49.41
+     55.81    |  9.58  |    34.61     |    28.85    | 23.28  |    47.87
+  -------------------------------------------------------------------------
+
+{242}
+
+By the continued addition of acetic acid, the composition of the successive
+conjugate solutions in equilibrium with one another becomes, as the table
+shows, more nearly the same, and a point is at length reached at which the
+two solutions become identical. This will therefore be a _critical point_
+(p. 98). Increased addition of acetic acid beyond this point will lead to a
+single homogeneous solution.
+
+These relationships are represented graphically by the curve _a_K_b_, Fig.
+84. The points on the branch _a_K represent the composition of the
+solutions relatively rich in chloroform (heavier layer), those on the curve
+_b_K the composition of solutions relatively rich in water (lighter layer);
+and the points on these two branches representing conjugate solutions are
+joined together by "tie-lines." Thus, the points _a'b'_ represent conjugate
+solutions, and the line _a'b'_ is a tie-line.
+
+[Illustration: FIG. 84.]
+
+Since, now, acetic acid when added to a heterogeneous mixture of chloroform
+and water does not enter in equal amounts into the two layers, but in
+amounts depending on its coefficient of distribution between chloroform and
+water,[321] the {243} tie-lines will not be parallel to AB, but will be
+inclined at an angle. As the solutions become more nearly the same, the
+tie-lines diminish in length, and at last, when the conjugate solutions
+become identical, shrink to a point. For the reason that the tie-lines are,
+in general, not parallel to the side of the triangle, the critical point at
+which the tie-line vanishes will not be at the summit of the curve, but
+somewhere below this, as represented by the point K.
+
+The curve _a_K_b_, further, forms the boundary between the heterogeneous
+and homogeneous systems. A mixture of chloroform, water, and acetic acid
+represented by any point outside the curve _a_K_b_, will form only one
+homogeneous phase; while any mixture represented by a point within the
+curve, will separate into two layers having the composition represented by
+the ends of the tie-line passing through that point. Thus, a mixture of the
+total composition _x_, will separate into two layers having the composition
+_a'_ and _b'_ respectively.
+
+Since three components existing in three phases (two liquid and a vapour
+phase) constitute a bivariant system, the final result, _i.e._ the
+composition of the two layers and the total vapour pressure, will not
+depend merely on the temperature, as in the case of two-component systems
+(p. 102), but also on the composition of the mixture with which we start.
+At constant temperature, however, all mixtures, the composition of which is
+represented by a point on one and the same tie-line, will separate into the
+same two liquid phases, although the relative _amounts_ of the two phases
+will vary. If we omit the vapour phase, the condition of the system will
+depend on the pressure as well as on the temperature and composition of the
+initial mixture. By keeping the pressure constant, _e.g._ at atmospheric
+pressure (by working with open vessels), the system again becomes
+bivariant. We see, therefore, that the position of the curve _a_K_b_, or,
+in other words, the composition of the different conjugate ternary
+solutions, will vary with the temperature, and only with the temperature,
+if we assume either constancy of pressure or the presence of the vapour
+phase. Since at the critical point the condition is imposed that the two
+liquid phases become identical, one degree of freedom is thereby {244}
+lost, and therefore only one degree of freedom remains. The critical point,
+therefore, depends on the temperature, and only on the temperature; always
+on the assumption, of course, that the pressure is constant, or that a
+vapour phase is present. Fig. 84, therefore, represents an isothermal
+(p. 239).
+
+It is of importance to note that the composition of the different ternary
+solutions obtained by the addition of acetic acid to a heterogeneous
+mixture of chloroform and water, will depend not only on the amount of
+acetic acid added, but also on the relative amounts of chloroform and water
+at the commencement. Suppose, for example, that we start with chloroform
+and water in the proportions represented by the point _c'_ (Fig. 84). On
+mixing these, two liquid layers having the composition _a_ and _b_
+respectively will be formed. Since by the addition of acetic acid the
+relative amounts of these two substances in the system as a whole cannot
+undergo alteration, the total composition of the different ternary systems
+which will be obtained must be represented by a point on the line C_c'_
+(p. 238). Thus, for example, by the addition of acetic acid a system may be
+obtained, the total composition of which is represented by the point _c"_.
+Such a system, however, will separate into two conjugate ternary solutions,
+the composition of which will be represented by the ends of the tie-line
+passing through the point _c"_. So long as the total composition of the
+system lies below the point S, _i.e._ the point of intersection of the line
+C_c'_ with the boundary curve, two liquid layers will be formed; while all
+systems having a total composition represented by a point on the line
+C_c'_, above S, will form only one homogeneous solution.
+
+From the figure, also, it is evident that as the amount of acetic acid is
+increased, the relative amounts of the two liquid layers formed differ more
+and more until at S a limiting position is reached, when the amount of the
+one liquid layer dwindles to nought, and only one solution remains.
+
+The same reasoning can be carried through for different initial amounts of
+chloroform and water, but it would be fruitless to discuss all the
+different systems which can be obtained. The reason for the preceding
+discussion was to show that {245} although the addition of acetic acid to a
+mixture of chloroform and water will, in all cases, lead ultimately to a
+limiting system, beyond which homogeneity occurs, that point is not
+necessarily the critical point. On the contrary, in order that addition of
+acetic acid shall lead to the critical mixture, it is necessary to start
+with a binary mixture of chloroform and water in the proportions
+represented by the point _c'_. In this case, addition of acetic acid will
+give rise to a series of conjugate ternary solutions, the composition of
+which will gradually approach to one another, and at last become identical.
+
+From the foregoing it will be evident that the amount of acetic acid
+required to produce a homogenous solution, will depend on the relative
+amounts of chloroform and water from which we start, and can be ascertained
+by joining the corner C with the point on the line AB representing the
+total composition of the initial binary system. The point where this line
+intersects the boundary curve _a_K_b_ will indicate the minimum amount of
+acetic acid which, under these particular conditions, is necessary to give
+one homogeneous solution.
+
+Retrograde Solubility.--As a consequence of the fact that acetic acid
+distributes itself unequally between chloroform and water, and the critical
+point K, therefore, does not lie at the summit of the curve, it is possible
+to start with a homogeneous solution in which the percentage amount of
+acetic acid is greater than at the critical point, and to pass from this
+first to a heterogenous and then again to a homogenous system merely by
+altering the relative amounts of chloroform and water. This phenomenon, to
+which the term _retrograde solubility_ is applied, will be observed not
+only in the case of chloroform, water, and acetic acid, but in all other
+systems in which the critical point lies below the highest point of the
+boundary curve for heterogeneous systems. This will be seen from the
+diagram, Fig. 85. Starting with the homogeneous system represented by _x_,
+in which, therefore, the concentration of C is greater than in the critical
+mixture (K), if the relative amounts of A and B are altered in the
+direction _xx'_, while the amount of C is maintained constant, the system
+will become heterogeneous when the composition reaches the point _y_, and
+will remain {246} heterogeneous with changing composition until the point
+_y'_ is passed, when it will again become homogeneous. If the relative
+concentration of C is increased above that represented by the line SS, this
+phenomenon will, of course, no longer be observed.
+
+[Illustration: FIG. 85.]
+
+Relationships similar to those described for chloroform, water, and acetic
+acid are also found in the case of a number of other trios, _e.g._ ether,
+water, and alcohol; chloroform, water, and alcohol.[322] They have also
+been observed in the case of a considerable number of molten metals.[323]
+Thus, molten lead and silver, as well as molten zinc and silver, mix in all
+proportions; but molten lead and zinc are only partially miscible with one
+another. When melted together, therefore, the last two metals will separate
+into two liquid layers, one rich in lead, the other rich in zinc. If silver
+is now added, and the temperature maintained above the freezing point of
+the mixture, the silver passes for the most part, in accordance with the
+law of distribution, into the upper layer, which is rich in zinc; silver
+being more soluble in molten zinc than in molten lead. This is clearly
+shown by the following figures:--[324]
+
+{247}
+
+  --------------------------------------------------
+       Heavier alloy.     |     Lighter alloy.
+  --------------------------------------------------
+    Percentage amount of  |  Percentage amount of
+  Silver. | Lead. | Zinc. | Silver. | Lead. | Zinc.
+  --------------------------------------------------
+    1.25  | 96.69 |  2.06 |  38.91  |  3.12 | 57.97
+    1.71  | 96.43 |  1.86 |  45.01  |  3.37 | 51.62
+    5.55  | 93.16 |  1.29 |  54.93  |  4.21 | 40.86
+  --------------------------------------------------
+
+The numbers in the same horizontal row give the composition of the
+conjugate alloys, and it is evident that the upper layer consists almost
+entirely of silver and zinc. On allowing the mixture to cool slightly, the
+upper layer solidifies first, and can be separated from the still molten
+lead layer. It is on this behaviour of silver towards a mixture of molten
+lead and zinc that the Parkes's method for the desilverization of lead
+depends.[325] If aluminium is also added, a still larger proportion of
+silver passes into the lighter layer, and the desilverization of the lead
+is more complete.[326]
+
+[Illustration: FIG. 86.]
+
+[Illustration: FIG. 87.]
+
+The Influence of Temperature.--As has already been said, a ternary system
+existing in three phases possesses two degrees of freedom; and the state of
+the system is therefore dependent not only on the relative concentration of
+the components, but also on the temperature. As the temperature changes,
+therefore, the boundary curve of the heterogeneous system will also alter;
+and in order to represent this alteration we shall make use of the right
+prism, in which the temperature is measured upwards. In this way the
+boundary curve passes into a boundary surface (called a dineric surface),
+as shown in Fig. 86. In this figure the curve _akb_ is the isothermal for
+the ternary system; the curve _a_K_b_ shows the change in the _binary_
+system AB with the temperature, with {248} a critical point at K. This
+curve has the same meaning as those given in Chapter VI. The curve _k_K is
+a critical curve joining together the critical points of the different
+isothermals. In such a case as is shown in Fig. 86, there does not exist
+any real critical temperature for the ternary system, for as the
+temperature is raised, the amount of C in the "critical" solution becomes
+less and less, and at K only two components, A and B, are present. In the
+case, however, represented in Fig. 87, a real ternary critical point is
+found. In this figure _ak'b_ is an isothermal, _ak"_ is the curve for the
+binary system, and K is the ternary critical point. All points outside the
+helmet-shaped boundary surface represent homogeneous ternary solutions,
+while all points within the surface belong to heterogeneous systems. Above
+the temperature of the point K, the three components are miscible in all
+proportions. An example of a ternary system yielding such a boundary
+surface is that consisting of phenol, water, and acetone.[327] In this case
+the critical temperature K is 92°, and the composition at this ternary
+critical point is--
+
+  Water   59 per cent.
+  Acetone 12    "
+  Phenol  29    "
+
+[Illustration: FIG. 88.]
+
+The difference between the two classes of systems just mentioned, is seen
+very clearly by a glance at the Figs. 88 and 89, which show the projection
+of the isothermals on the base of the prism. In Fig. 88, the projections
+yield paraboloid curves, the two branches of which are cut by one side of
+the triangle; and the critical point is represented by a point on {249}
+this side. In the second case (Fig. 89), however, the projections of the
+isothermals form ellipsoidal curves surrounding the supreme critical point,
+which now lies _inside the triangle_. At lower temperatures, these
+isothermal boundary curves are cut by a side of the triangle; at the
+critical temperature, _k"_, of the binary system AB, the boundary curve
+_touches_ the side AB, while at still higher temperatures the boundary
+curve comes to lie entirely within the triangle. At any given temperature,
+therefore, between the critical point of the binary system (_k"_), and the
+supreme critical point of the ternary system (K), each pair of the three
+components are miscible with one another in all proportions; for the region
+of heterogeneous systems is now bounded by a closed curve lying entirely
+within the triangle. Outside this curve only homogeneous systems are found.
+Binary mixtures, therefore, represented by any point on one of the sides of
+the triangle must be homogeneous, for they all lie outside the boundary
+curve for heterogeneous states.
+
+[Illustration: FIG. 89.]
+
+2. _The three components can form two pairs of partially miscible liquids._
+
+In the case of the three components water, alcohol, and succinic nitrile,
+water and alcohol are miscible in all proportions, but not so water and
+succinic nitrile, or alcohol and succinic nitrile.
+
+[Illustration: FIG. 90.]
+
+[Illustration: FIG. 91.]
+
+As we have already seen (p. 122), water and succinic nitrile can form two
+liquid layers between the temperatures 18.5° and 55.5°; while alcohol and
+nitrile can form two liquid layers between 13° and 31°. If, then, between
+these two temperature limits, alcohol is added to a heterogeneous mixture
+of water and nitrile, or water is added to a mixture of alcohol and
+nitrile, two heterogeneous ternary systems will be formed, {250} and two
+boundary curves will be obtained in the triangular diagram, as shown in
+Fig. 90.[328] On changing the temperature, the boundary curves will also
+undergo alteration, in a manner similar to that just discussed. As the
+temperature falls, the two curves will spread out more and more into the
+centre of the triangle, and might at last meet one another; while at still
+lower temperatures we may imagine the curves still further expanding so
+that the two heterogeneous regions flow into one another and form a _band_
+on the triangular diagram (Fig. 91). This, certainly, has not been realized
+in the case of the three components mentioned, because at a temperature
+higher than that at which the two heterogeneous regions could fuse
+together, solid separates out.
+
+[Illustration: FIG. 92.]
+
+The gradual expansion of a paraboloid into a band-like area of
+heterogeneous ternary systems, has, however, been observed in the case of
+water, phenol, and aniline.[329] In Fig. 92 are shown three isothermals,
+viz. those for 148°, 95°, and 50°. At 148°, water and aniline form two
+layers having the composition--
+
+  Water,   83.5 per cent. }     { water,   20 per cent.
+                          } and {
+  Aniline, 16.5    "      }     { aniline, 80    "
+
+{251}
+
+and the critical point _k'_ has the composition--
+
+  Water, 65; phenol, 13.2; aniline, 21.8 per cent.
+
+At 95°, the composition of the two binary solutions is--
+
+  Water,  93 per cent. }     { water     8 per cent.
+                       } and {
+  Aniline, 7    "      }     { aniline, 92    "
+
+while the point _k"_ has the composition
+
+  Water, 69.9; phenol, 26.6; aniline, 3.5 per cent.
+
+At 50°, the region of heterogeneous states now forms a band, and the two
+layers formed by water and aniline have the composition--
+
+  Water,  96.5 per cent. }     { water,    5.5 per cent.
+                         } and {
+  Aniline, 3.5    "      }     { aniline, 94.5    "
+
+while the two layers formed by water and phenol have the composition--
+
+  Water,  89 per cent.}     { water,  38 per cent.
+                      } and {
+  Phenol, 11    "     }     { phenol, 62    "
+
+All mixtures of water, phenol, and aniline, therefore, the composition of
+which is represented by any point within the band _abcd_, will form two
+ternary solutions; while if the composition is represented by a point
+outside the band, only one homogeneous solution will be produced.
+
+3. _The three components form three pairs of partially miscible liquids._
+
+[Illustration: FIG. 93.]
+
+The third chief case which can occur is that no two of the components are
+completely miscible with one another. In this case, therefore, we shall
+obtain three paraboloid boundary curves, as shown in Fig. 93. If, now, we
+imagine these three curves to expand in towards the centre of the triangle,
+as might happen, for example, by lowering the temperature, a point will
+{252} be reached at which the curves partly overlap, and we shall get the
+appearance shown in Fig. 94.
+
+The points _a_, _b_, and _c_ represent the points where the three curves
+cut, and the triangle _abc_ is a region where the curves overlap. From this
+diagram we can see that any mixture having a composition represented by a
+point in one of the clear spaces at the corners of the larger triangle,
+will form a homogeneous solution; if the composition corresponds to any
+point lying in one of the quadrilateral regions _x__{1}, _x__{2} or
+_x__{3}, two ternary solutions will be formed; while, if the composition is
+represented by any point in the inner triangle, separation into three
+layers will occur.
+
+[Illustration: FIG. 94.]
+
+Since in the clear regions at the corners of the triangle we have three
+components in two phases, liquid and vapour, the systems have three degrees
+of freedom. At constant temperature, therefore, the condition of the system
+is not defined until the concentrations of two of the components are fixed.
+A system belonging to one of the quadrilateral spaces has, as we have seen,
+two degrees of freedom; besides the temperature, one concentration must be
+fixed. Lastly, a system the composition of which falls within the inner
+triangle _abc_, will form three layers, and will therefore possess only one
+degree of freedom. If the temperature is fixed, the composition of the
+three layers is also determined, viz. that of the points _a_, _b_, and _c_
+respectively; and a change in the composition of the original mixture can
+lead only to a difference in the relative amounts of the three layers, not
+to a difference in their composition.
+
+An example of a system which can form three liquid phases is found in
+water, ether, and succinic nitrile.[330]
+
+       *       *       *       *       *
+
+
+{253}
+
+CHAPTER XV
+
+PRESENCE OF SOLID PHASES
+
+A. The Ternary Eutectic Point.--In passing to the consideration of those
+ternary systems in which one or more solid phases can exist together with
+one liquid phase, we shall first discuss not the solubility curves, as in
+the case of two-component systems, but the simpler relationships met with
+at the freezing point. That is, we shall first of all examine the freezing
+point curves of ternary systems.
+
+[Illustration: FIG. 95.]
+
+Since it is necessary to take into account not only the changing
+composition of the liquid phase, but also the variation of the temperature,
+we shall employ the right prism for the graphic representation of the
+systems, as shown in Fig. 95. A, B, and C in this figure, therefore, denote
+the melting points of the pure components. If we start with the component A
+at its melting point, and add B, which is capable of dissolving in liquid
+A, the freezing point of A will be lowered; and, similarly, the freezing
+point of B by addition of A. In this way we get the freezing point curve
+A_k__{1}B for the binary system; _k__{1}; being an eutectic point. This
+curve will of course lie in the plane formed by one face of the prism. In a
+similar manner we obtain the freezing point curves A_k__{2}C and B_k__{3}C.
+These curves give the composition of the binary liquid phases in
+equilibrium {254} with one of the pure components, or at the eutectic
+points, with a mixture of two solid components. If, now, to the system
+represented say by the point _k__{1}, a small quantity of the third
+component, C, is added, the temperature at which the two solid phases A and
+B can exist in equilibrium with the liquid phase is lowered; and this
+depression of the eutectic point is all the greater the larger the addition
+of C. In this way we obtain the curve _k__{1}K, which slopes inwards and
+downwards, and indicates the varying composition of the ternary liquid
+phase with which a mixture of solid A and B are in equilibrium. Similarly,
+the curves _k__{2}K and _k__{3}K are the corresponding eutectic curves for
+A and C, and B and C in equilibrium with ternary solutions. At the point K,
+the three solid components are in equilibrium with the liquid phase; and
+this point, therefore, represents _the lowest temperature attainable with
+the three components given_. Each of the ternary eutectic curves, as they
+may be called, is produced by the intersection of two surfaces, while at
+the ternary eutectic point, three surfaces, viz. A_k__{1}K_k__{2},
+B_k__{1}K_k__{3}, and C_k__{1}K_k__{3} intersect. Any point on one of these
+surfaces represents a ternary solution in equilibrium with only one
+component in the solid state; the lines or curves of intersection of these
+represent equilibria with two solid phases, while at the point K, the
+ternary eutectic point, there are three solid phases in equilibrium with a
+liquid and a vapour phase. The surfaces just mentioned represent bivariant
+systems. One component in the solid state can exist in equilibrium with a
+ternary liquid phase under varying conditions of temperature and
+concentration of the components in the solution; and before the state of
+the system is defined, these two variables, temperature and composition of
+the liquid phase, must be fixed. On the other hand, the curves formed by
+the intersection of these planes represent univariant systems; at a given
+temperature two solid phases can exist in equilibrium with a ternary
+solution, only when the latter has a definite composition. Lastly, the
+ternary eutectic point, K, represents an invariant system; three solid
+phases can exist in equilibrium with a ternary solution, only when the
+latter has one fixed composition and when the temperature has a definite
+value. This eutectic point, therefore, {255} has a perfectly definite
+position, depending only on the nature of the three components.
+
+Instead of employing the prism, the change in the composition of the
+ternary solutions can also be indicated by means of the _projections_ of
+the curves _k__{1}K, _k__{2}K, and _k__{3}K on the base of the prism, the
+particular temperature being written beside the different eutectic points
+and curves. This is shown in Fig. 96.
+
+[Illustration: FIG. 96.]
+
+The numbers which are given in this diagram refer to the eutectic points
+for the system bismuth--lead--tin, the data for which are as
+follows:--[331]
+
+  --------------------------------------------------------------------
+  Melting point of | Percentage composition of | Temperature of binary
+     pure  metal.  |  binary eutectic mixture. |    eutectic point.
+  --------------------------------------------------------------------
+                   |      Bi    Pb    Sn       |
+    Bismuth, 268°  |      55    45    --       |      Bi--Pb, 127°
+    Lead, 325°     |      58    --    42       |      Bi--Sn, 133°
+    Tin, 232°      |      --    37    63       |      Pb--Sn, 182°
+  --------------------------------------------------------------------
+
+  --------------------------------------------------
+  Percentage composition of | Temperature of ternary
+  ternary eutectic mixture. |    eutectic point.
+  --------------------------------------------------
+       Bi    Pb    Sn       |
+       52    32    16       |          96°
+  --------------------------------------------------
+
+Formation of Compounds.--In the case just discussed, the components
+crystallized out from solution in the pure state. If, however, combination
+can take place between two of the components, the relationships will be
+somewhat different; the curves which are obtained in such a case being
+represented in Fig. 97. From the figure, we see that the two components B
+{256} and C form a compound, and the freezing point curve of the binary
+system has therefore the form shown in Fig. 64 (p. 209). Further, there are
+two _ternary_ eutectic points, K_{1} and K_{2}, the solid phases present
+being A, B, and compound, and A, C, and compound respectively.
+
+[Illustration: FIG. 97.]
+
+The particular point, now, to which it is desired to draw attention is
+this. Suppose the ternary eutectic curves projected on a plane parallel to
+the face of the prism containing B and C, _i.e._ suppose the concentrations
+of the two components B and C, between which interaction can occur,
+expressed in terms of a constant amount of the third component A,[332]
+curves will then be obtained which are in every respect analogous to the
+freezing point curves of binary systems. Thus, suppose the eutectic curves
+_k__{1}K and _k__{2}K in Fig. 95 projected on the face BC of the prism,
+then evidently a curve will be obtained consisting of two branches
+meeting in an eutectic point. On the other hand, the projection of the
+ternary eutectic curves in Fig. 97 on the face BC of the prism, will
+give a curve consisting of three portions, as shown by the outline
+_k__{1}K_{1}K_{2}_k__{2} in Fig. 97.
+
+Various examples of this have been studied, and the following table
+contains some of the data for the system ethylene bromide (A), picric acid
+(B), and [beta]-naphthol (C), obtained by Bruni.[333]
+
+{257}
+
+  -------------------------------------------------------------------------
+                     | Temperature |            Solid phases present.
+  -------------------------------------------------------------------------
+  Point _k__{1}      |    9.41°    | Ethylene bromide, picric acid.
+  Curve _k__{1}K_{1} |     --      |         "               "
+  Point K_{1}        |    9.32°    | Ethylene bromide, picric acid, and
+                     |             |  [beta]-naphthol picrate.
+  Curve K_{1}D'K_{2} |     --      | Ethylene bromide,
+                     |             |               [beta]-naphthol picrate.
+  Point D'           |    9.75°    |     "       "         "        "
+  Point K_{2}        |    8.89°    |     "       "     [beta]-naphthol,
+                     |             |                           and picrate.
+  Curve K_{2}_k__{2} |     --      |     "       "     [beta]-naphthol.
+  Point _k__{2}      |    9.04°    |     "       "               "
+  -------------------------------------------------------------------------
+
+From what has been said, it will be apparent that if the ternary eutectic
+curve of a three-component system (in which one of the components is
+present in constant amount) is determined, it will be possible to state,
+from the form of curve obtained, whether or not the two components present
+in varying amount crystallize out pure or combine with one another to form
+a compound. It may be left to the reader to work out the curves for the
+other possible systems; but it will be apparent, that the projections of
+the ternary eutectic curves in the manner given will yield a series of
+curves alike in all points to the binary curves given in Figs. 63-65,
+pp. 208-210.
+
+Since, from the method of investigation, the temperatures of the eutectic
+curves will depend on the melting point of the third component (A), it is
+possible, by employing substances with widely differing melting points, to
+investigate the interaction of the two components (_e.g._ two optical
+antipodes) B and C over a range of temperature; and thus determine the
+range of stability of the compound, if one is formed. Since, in some cases,
+two substances which at one temperature form mixed crystals combine at
+another temperature to form a definite compound, the relationships which
+have just been described can be employed, and indeed, have been employed,
+to determine the temperature at which this change occurs.[334] By means of
+this method, Adriani found that below 103° _i_-camphoroxime exists as a
+racemic compound, while above {258} that temperature it occurs as a racemic
+mixed crystal[335] (_cf._ p. 219).
+
+B. Equilibria at Higher Temperatures. Formation of Double Salts.--After
+having studied the relationships which are found in the neighbourhood of
+the freezing points of the components, we now pass to the discussion of the
+equilibria which are met with at higher temperatures. In this connection we
+shall confine the discussion entirely to the systems formed of two salts
+and water, dealing more particularly with those cases in which the water is
+present in relatively large amount and acts as solvent. Further, in
+studying these systems, one restriction must be made, viz. that the single
+salts are salts either of the same base or of the same acid; or are, in
+other words, capable of yielding a common ion in solution. Such a
+restriction is necessary, because otherwise the system would be one not of
+three but of four components.[336]
+
+Transition Point.--As is very well known, there exist a number of hydrated
+salts which, on being heated, undergo apparent partial fusion; and in
+Chapter V. the behaviour of such hydrates was more fully studied in the
+light of the Phase Rule. Glauber's salt, or sodium sulphate decahydrate,
+for example, on being heated to a temperature of about 32.5°, partially
+liquefies, owing to the fact that the water of crystallization is split off
+and anhydrous sodium sulphate formed, as shown by the equation--
+
+  Na_{2}SO_{4},10H_{2}O = Na_{2}SO_{4} + 10H_{2}O
+
+The temperature of 32.5°, it was learned, constituted a _transition point_
+for the decahydrate and anhydrous salt plus water; decomposition of the
+hydrated salt occurring above this temperature, combination of the
+anhydrous salt and water below it.
+
+Analogous phenomena are met with in systems constituted of two salts and
+water in which the formation of double salts can take place. Thus, for
+example, if _d_-sodium potassium {259} tartrate is heated to above 55°,
+apparent partial fusion occurs, and the two single salts, _d_-sodium
+tartrate and _d_-potassium tartrate, are deposited, the change which occurs
+being represented by the equation--
+
+  4NaKC_{4}O_{6}H_{4},4H_{2}O = 2Na_{2}C_{4}O_{6}H_{4},2H_{2}O
+                              + 2K_{2}C_{4}O_{6}H_{4},½H_{2}O + 11H_{2}O
+
+On the other hand, if sodium and potassium tartrates are mixed with water
+in the proportions shown on the right side of the equation, the system will
+remain partially liquid so long as the temperature is maintained above 55°
+(in a closed vessel to prevent loss of water), but on allowing the
+temperature to fall below this point, complete solidification will ensue,
+owing to the formation of the hydrated double salt. Below 55°, therefore,
+the hydrated double salt is the stable system, while above this temperature
+the two single salts plus saturated solution are stable.[337]
+
+A similar behaviour is found in the case of the double salt copper
+dipotassium chloride (CuCl_{2},2KCl,2H_{2}O or CuK_{2}Cl_{4},2H_{2}O).[338]
+When this salt is heated to 92°, partial liquefaction occurs, and the
+original blue plate-shaped crystals give place to brown crystalline needles
+and white cubes; while on allowing the temperature to fall, re-formation of
+the blue double salt ensues. The temperature 92° is, therefore, a
+transition point at which the reversible reaction--
+
+  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O
+
+takes place.
+
+The decomposition of sodium potassium tartrate, or of copper dipotassium
+chloride, differs in so far from that of Glauber's salt that _two_ new
+solid phases are formed; and in the case of copper dipotassium chloride,
+one of the decomposition products is itself a double salt.
+
+In the two examples of double salt decomposition which have just been
+mentioned, sufficient water was yielded to cause a partial liquefaction;
+but other cases are known where this is not so. Thus, when copper calcium
+acetate is heated to a {260} temperature of 75°, although decomposition of
+the double salt into the two single salts occurs as represented by the
+equation[339]--
+
+  CuCa(C_{2}H_{3}O_{2})_{4},8H_{2}O = Cu(C_{2}H_{3}O_{2})_{2},H_{2}O
+                                    + Ca(C_{2}H_{3}O_{2})_{2},H_{2}O
+                                    + 6H_{2}O
+
+the amount of water split off is insufficient to give the appearance of
+partial fusion, and, therefore, only a change in the crystals is observed.
+
+The preceding examples, in which decomposition of the double salt was
+effected by a rise of temperature, were chosen for first consideration as
+being more analogous to the case of Glauber's salt; but not a few examples
+are known where the reverse change takes place, formation of the double
+salt occurring _above_ the transition point, and decomposition into the
+constituent salts below it. Instances of this behaviour are found in the
+case of the formation of astracanite from sodium and magnesium sulphates,
+and of sodium ammonium racemate from the two sodium ammonium tartrates, to
+which reference will be made later. Between these various systems, however,
+there is no essential difference; and whether decomposition or formation of
+the double salt occurs at temperatures above the transition point, will of
+course depend on the heat of change at that point. For, in accordance with
+van't Hoff's law of movable equilibrium (p. 58), that change will take
+place at the higher temperature which is accompanied by an absorption of
+heat. If, therefore, the formation of the double salt from the single salts
+is accompanied by an absorption of heat, the double salt will be formed
+from the single salts on raising the temperature; but if the reverse is the
+case, then the double salt on being heated will decompose into the
+constituent salts.[340]
+
+In those cases, now, which have so far been studied, the change at the
+transition point is accompanied by a taking up or a splitting off of water;
+and _in such cases the general rule can be given, that if the water of
+crystallization of the two constituent {261} salts together is greater than
+that of the double salt, the latter will be produced from the former on
+raising the temperature_ (_e.g._ astracanite from sodium and magnesium
+sulphates); _but if the double salt contains more water of crystallization
+than the two single salts, increase of temperature will effect the
+decomposition of the double salt_. When we seek for the connection between
+this rule and the law of van't Hoff, it is found in the fact that the heat
+effect involved in the hydration or dehydration of the salts is much
+greater than that of the other changes which occur, and determines,
+therefore, the sign of the total heat effect.[341]
+
+Vapour Pressure. Quintuple Point.--In the case of Glauber's salt, we saw
+that at a certain temperature the vapour pressure curve of the hydrated
+salt cut that of the saturated solution of anhydrous sodium sulphate. That
+point, it will be remembered, was a quadruple point at which the four
+phases sodium sulphate decahydrate, anhydrous sodium sulphate, solution,
+and vapour, could co-exist; and was also the point of intersection of the
+curves for four univariant systems. In the case of the formation of double
+salts, similar relationships are met with; and also certain differences,
+due to the fact that we are now dealing with systems of three components.
+Two cases will be chosen here for brief description, one in which
+formation, the other in which decomposition of the double salt occurs with
+rise of temperature.
+
+On heating a mixture of sodium sulphate decahydrate and magnesium sulphate
+heptahydrate, it is found that at 22° partial liquefaction occurs with
+formation of astracanite. At this temperature, therefore, there can coexist
+the five phases--
+
+  Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O;
+  solution; vapour.
+
+This constitutes, therefore, a _quintuple point_; and since there are three
+components present in five phases, the system is invariant. This point,
+also, will be the point of intersection of curves for five univariant
+systems, which, in this case, must each be composed of four phases. These
+systems are--
+
+{262}
+
+    I. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
+    Na_{2}Mg(SO_{4})_{2},4H_{2}O; vapour.
+
+    II. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; solution; vapour.
+
+    III. MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution; vapour.
+
+    IV. Na_{2}SO_{4},10H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution;
+    vapour.
+
+    V. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
+    Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution.
+
+[Illustration: FIG. 98.]
+
+On representing the vapour pressures of these different systems
+graphically, a diagram is obtained such as is shown in Fig. 98,[342] the
+curves being numbered in accordance with the above list. When the system I.
+is heated, the vapour pressure increases until at the quintuple point the
+liquid phase (solution) is formed, and it will then depend on the relative
+amounts of the different phases whether on further heating there is formed
+system III., IV., or V. If either of the first two is produced, we shall
+obtain the vapour pressure of the solutions saturated with respect to both
+double salt and one of the single salts; while if the vapour phase
+disappears, there will be obtained the pressure of the condensed systems
+formed of double salt, two single salts and solution. This curve,
+therefore, indicates the _change of the transition point with pressure_;
+and since in the ordinary determinations of the transition point in open
+vessels, we are in reality dealing with condensed systems under the
+pressure of 1 atm., it will be evident that the transition point does not
+accurately coincide with the quintuple point (at which the system is under
+the pressure of its own vapour). As in the case of other condensed systems,
+however, pressure has only a slight influence on the temperature of the
+transition point. Whether or not pressure raises or lowers the transition
+point will depend on whether transformation is accompanied by an increase
+or {263} diminution of volume (theorem of Le Chatelier, p. 58). In the case
+of the formation of astracanite, expansion occurs, and the transition point
+will therefore be raised by increase of pressure. Although measurements
+have not been made in the case of this system, the existence of such a
+curve has been experimentally verified in the case of copper and calcium
+acetates and water (v. _infra_).[343]
+
+[Illustration: FIG. 99.]
+
+The vapour pressure diagram in the case of copper calcium acetate and water
+(Fig. 99), is almost the reverse of that already discussed. In this case,
+the double salt decomposes on heating, and the decomposition is accompanied
+by a contraction. Curve I. is the vapour pressure curve for double salt,
+two single salts (p. 260), and vapour; curves II. and III. give the vapour
+pressures of solutions saturated with respect to double salt and one of the
+single salts; curve IV. is the curve of pressures for the solutions
+saturated with respect to the two single salts; while curve V. again
+represents the change of the transition point with pressure. On examining
+this diagram, it is seen that whereas {264} astracanite could exist both
+above and below the quintuple point, copper calcium acetate can exist only
+_below_ the quintuple point. This behaviour is found only in those cases in
+which the double salt is decomposed by rise of temperature, and where the
+decomposition is accompanied by a diminution of volume.[344]
+
+As already mentioned, the decomposition of copper calcium acetate into the
+single salts and saturated solution is accompanied by a contraction, and it
+was therefore to be expected that increase of pressure would _lower_ the
+transition point. This expectation of theory was confirmed by experiment,
+for van't Hoff and Spring found that although the transition point under
+atmospheric pressure is about 75°, decomposition of the double salt took
+place even at the ordinary temperature when the pressure was increased to
+6000 atm.[345]
+
+Solubility Curves at the Transition Point.--At the transition point, as has
+already been shown, the double salt and the two constituent salts can exist
+in equilibrium with the same solution. The transition point, therefore,
+must be the point of intersection of two solubility curves; the solubility
+curve of the double salt and the solubility curve of the mixtures of the
+two constituent salts. It should be noted here that we are not dealing with
+the solubility curves of the single salts separately, for since the systems
+are composed of three components, a single solid phase can, at a given
+temperature, be in equilibrium with solutions of different composition, and
+two solid phases in contact with solution (and vapour) are therefore
+necessary to give an univariant system. The same applies, of course, to the
+solubility of the double salt; for a double salt also constitutes a single
+phase, and can therefore exist in equilibrium with solutions of varying
+composition. If, however, we make the restriction (which we do for the
+present) that the double salt is not decomposed by water, then the solution
+will contain the constituent salts in the same relative proportions as they
+are contained in the double salt, and the system may therefore be regarded
+as one of _two_ components, viz. double salt and water. In this case one
+solid phase is sufficient, with solution and {265} vapour, to give an
+univariant system; and at a given temperature, therefore, the solubility
+will have a perfectly definite value.
+
+Since in almost all cases the solubility is determined in open vessels, we
+shall in the following discussion consider that the vapour phase is absent,
+and that the system is under a constant pressure, that of the atmosphere.
+With this restriction, therefore, four phases will constitute an invariant
+system, three phases an univariant, and two phases a bivariant system.
+
+It has already been learned that in the case of sodium sulphate and water,
+the solubility curve of the salt undergoes a sudden change in direction at
+the transition point, and that this is accompanied by a change in the solid
+phase in equilibrium with the solution. The same behaviour is also found in
+the case of double salts. To illustrate this, we shall briefly discuss the
+solubility relations of a few double salts, beginning with one of the
+simplest cases, that of the formation of rubidium racemate from rubidium
+_d_- and _l_-tartrates. The solubilities are represented diagrammatically
+in Fig. 100, the numerical data being contained in the following table, in
+which the solubility is expressed as the number of gram-molecules
+Rb_{2}C_{4}H_{4}O_{6} in 100 gm.-molecules of water.[346]
+
+  ---------------------------------------------------------------
+  Temperature. | Solubility of tartrate | Solubility of racemate.
+               |        mixture.        |
+  ---------------------------------------------------------------
+      25°      |         13.03          |          10.91
+      35°      |          --            |          12.63
+      40.4°    |          --            |          13.48
+      40.7°    |         13.46          |           --
+      54°      |         13.83          |           --
+  ---------------------------------------------------------------
+
+In Fig. 100 the curve AB represents the solubility of the racemate, while
+A'BC represents the solubility of the mixed tartrates. Below the transition
+point, therefore, the solubility of the racemate is less than that of the
+mixed tartrates. The solution, saturated with respect to the latter, will
+be supersaturated with respect to the racemate; and if a nucleus of this is
+present, racemate will be deposited, and the mixed tartrates, if present in
+equimolecular amounts, will ultimately {266} entirely disappear, and only
+racemate will be left as solid phase. The solution will then have the
+composition represented by a point on the curve AB. Conversely, above the
+transition point, the saturated solution of the racemate would be
+supersaturated with respect to the two tartrates, and transformation into
+the latter would ensue. If, therefore, a solution of equimolecular
+proportions of rubidium _d_- and _l_-tartrates is allowed to evaporate at a
+temperature above 40°, a mixture of the two tartrates will be deposited;
+while at temperatures below 40° the racemate will separate out.
+
+[Illustration: FIG. 100.]
+
+Similar relationships are met with in the case of sodium ammonium _d_- and
+_l_-tartrate and sodium ammonium racemate; but in this case the racemate is
+the stable form in contact with solution above the transition point
+(27°).[347] Below the transition point, therefore, the solubility curve of
+the mixed tartrates will lie below the solubility curve of the racemate.
+Below the transition point, therefore, sodium ammonium racemate will break
+up in contact with solution into a mixture of sodium ammonium _d_- and
+_l_-tartrates. At a higher temperature, 35°, sodium ammonium racemate
+undergoes decomposition into sodium racemate and ammonium racemate.[348]
+
+The behaviour of sodium ammonium racemate is of interest from the fact that
+it was the first racemic substance to be resolved into its optically active
+forms by a process of crystallization. On neutralizing a solution of
+racemic tartaric acid, half with soda and half with ammonia, and allowing
+the solution to evaporate, Pasteur[349] obtained a mixture of sodium
+ammonium {267} _d_- and _l_-tartrates. Since Pasteur was unaware of the
+existence of a transition point, the success of his experiment was due to
+the happy chance that he allowed the solution to evaporate at a temperature
+below 27°; for had he employed a temperature above this, separation of the
+racemate into the two enantiomorphous forms would not have occurred. For
+this reason the attempt of Staedel to perform the same resolution met only
+with failure.[350]
+
+Decomposition of the Double Salt by Water.--In the two cases just
+described, the solubility relationships at the transition point are of a
+simpler character than in the case of most double salts. If, at a
+temperature above the transition point, a mixture of rubidium _d_- and
+_l_-tartrates in equimolecular proportions is brought in contact with water
+a solution will be obtained, which is saturated with respect to both
+enantiomorphous forms; and since the solubility of the two optical
+antipodes is identical, and the effect of one on the solubility of the
+other also the same, the solution will contain equimolecular amounts of the
+_d_- and _l_-salt. If, now, the solution is cooled down in contact with the
+solid salts to just below the transition point, it becomes supersaturated
+with respect to the racemate, and this will be deposited. The solution
+thereby becomes unsaturated with respect to the mixture of the active
+salts, and these must therefore pass into solution. As the latter are
+equally soluble, equal amounts of each will dissolve, and a further
+quantity of the racemate will be deposited. These processes of solution and
+deposition will continue until the single tartrates have completely
+disappeared, and only racemate is left as solid phase. As a consequence of
+the identical solubility of the two tartrates, therefore, no excess of
+either form will be left on passing through the transition point. From this
+it will be evident that the racemate can exist as single solid phase in
+contact with its saturated solution at the transition point; or, in other
+words, the racemate is not decomposed by water at the transition point. The
+same behaviour will evidently be exhibited by sodium ammonium racemate at
+27°, for the two enantiomorphous sodium ammonium tartrates have also
+identical solubility.
+
+{268}
+
+Very different, however, is the behaviour of, say, astracanite, or of the
+majority of double salts; for the solubility of the constituent salts is
+now no longer the same. If, for example, excess of a mixture of sodium
+sulphate and magnesium sulphate, in equimolecular proportions, is brought
+in contact with water below the transition point (22°), more magnesium
+sulphate than sodium sulphate will dissolve, the solubility of these two
+salts in a common solution being given by the following figures, which
+express number of molecules of the salt in 100 molecules of water.[351]
+
+  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
+  Na_{2}SO_{4},10H_{2}O AND MgSO_{4},7H_{2}O.
+
+  ----------------------------------------
+  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
+  ----------------------------------------
+      18.5°    |      2.16     |   4.57
+      24.5°    |      3.43     |   4.68
+  ----------------------------------------
+
+At the transition point, then, it is evident that the solution contains
+more magnesium sulphate than sodium sulphate: and this must still be the
+case when astracanite, which contains sodium sulphate and magnesium
+sulphate in equimolecular proportions, separates out. If, therefore, the
+temperature is raised slightly above the transition point, magnesium
+sulphate and sodium sulphate will pass into solution, the former, however,
+in larger quantities than the latter, and astracanite will be deposited;
+and this will go on until all the magnesium sulphate has disappeared, and a
+mixture of astracanite and sodium sulphate decahydrate is left as solid
+phases. Since there are now three phases present, the system is univariant
+(by reason of the restriction previously made that the vapour phase is
+absent), and at a given temperature the solution will have a definite
+composition; as given in the following table:--
+
+  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
+  Na_{2}Mg(SO_{4})_{2},4H_{2}O AND Na_{2}SO_{4},10H_{2}O.
+
+  ----------------------------------------
+  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
+  ----------------------------------------
+      22°      |      2.95     |   4.70
+      24.5°    |      3.45     |   3.62
+  ----------------------------------------
+
+{269}
+
+From the above figures, therefore, it will be seen that at a temperature
+just above the transition point a solution in contact with the two solid
+phases, astracanite and Glauber's salt, contains a relatively smaller
+amount of sodium sulphate than a pure solution of astracanite would; for in
+this case there would be equal molecular amounts of Na_{2}SO_{4} and
+MgSO_{4}. A solution which is saturated with respect to astracanite alone,
+will contain more sodium sulphate than the solution saturated with respect
+to astracanite plus Glauber's salt, and the latter will therefore be
+deposited. From this, therefore, it is clear that if astracanite is brought
+in contact with water at about the transition point, it will undergo
+decomposition with separation of Glauber's salt (supersaturation being
+excluded).
+
+[Illustration: FIG. 101.]
+
+This will perhaps be made clearer by considering Fig. 101. In this diagram
+the ordinates represent the ratio of sodium sulphate to magnesium sulphate
+in the solutions, and the abscissæ represent the temperatures. The line AB
+represents solutions saturated with respect to a mixture of the single
+salts (p. 268); BC refers to solutions in equilibrium with astracanite and
+magnesium sulphate; while BX represents the composition of solutions in
+contact with the solid phases astracanite and Glauber's salt. The values of
+the solubility are contained in the following table, and in that on p. 268,
+and are, as before, expressed in gm.-molecules of salt in 100 gm.-molecules
+of water.[352]
+
+{270}
+
+  -------------------------------------------------------------------------
+               | Astracanite                | Astracanite
+  Temperature. |        + sodium sulphate.  |        + magnesium sulphate.
+               |----------------------------|------------------------------
+               | Na_{2}SO_{4}. |  MgSO_{4}. | Na_{2}SO_{4}. |  MgSO_{4}.
+  -------------------------------------------------------------------------
+      18.5°    |       --      |      --    |      3.41     |    4.27
+      22°      |      2.95     |     4.70   |      2.85     |    4.63
+      24.5°    |      3.45     |     3.62   |      2.68     |    4.76
+      30°      |      4.58     |     2.91   |      2.30     |    5.31
+      35°      |      4.30     |     2.76   |      1.73     |    5.88
+  -------------------------------------------------------------------------
+
+At the transition point the ratio of sodium sulphate to magnesium sulphate
+is approximately 1 : 1.6. In the case of solutions saturated with respect
+to both astracanite and Glauber's salt, the relative amount of sodium
+sulphate increases as the temperature rises, while in the solutions
+saturated for astracanite and magnesium sulphate, the ratio of sodium
+sulphate to magnesium sulphate decreases.
+
+If, now, we consider only the temperatures above the transition point, we
+see from the figure that solutions represented by points above the line BX
+contain relatively more sodium sulphate than solutions in contact with
+astracanite and Glauber's salt; and solutions lying below the line BC
+contain relatively more magnesium sulphate than solutions saturated with
+this salt and astracanite. These solutions will therefore not be stable,
+but will deposit in the one case, astracanite and Glauber's salt, and in
+the other case, astracanite and magnesium sulphate, until a point on BX or
+BC is reached. All solutions, however, lying to the right of CBX, will be
+_unsaturated_ with respect to these two pairs of salts, and only the
+solutions represented by the line XY (and which contain equimolecular
+amounts of sodium and magnesium sulphates) will be saturated with respect
+to the pure double salt.
+
+Transition Interval.--Fig. 101 will also render intelligible a point of
+great importance in connection with astracanite, and of double salts
+generally. At temperatures between those represented by the points B and X,
+the double salt when brought in contact with water will be decomposed with
+separation of sodium sulphate. Above the temperature of the point {271} X,
+however, the solution of the pure double salt is stable, because it can
+still take up a little of either of the components. At temperatures, then,
+above that at which the solution in contact with the double salt and the
+less soluble single salt, contains the single salts in the ratio in which
+they are present in the double salt, solution of the latter will take place
+without decomposition. _The range of temperature between that at which
+double salt can begin to be formed (the transition point) and that at which
+it ceases to be decomposed by water is called the transition
+interval._[353] If the two single salts have identical solubility at the
+transition point, the transition interval diminishes to nought.
+
+In those cases where the double salt is the stable form below the
+transition point, the transition interval will extend downwards to a lower
+temperature. Fig. 101 will then have the reverse form.
+
+Summary.--With regard to double salts we have learned that their formation
+from and their decomposition into the single salts, is connected with a
+definite temperature, the _transition temperature_. At this transition
+temperature two vapour pressure curves cut, viz. a curve of dehydration of
+a mixture of the single salts and the solubility curve of the double salt;
+or the dehydration curve of the double salt and the solubility curve of the
+mixed single salts. The solubility curves, also, of these two systems
+intersect at the transition point, but although the formation of the double
+salt commences at the transition point, complete stability in contact with
+water may not be attained till some temperature above (or below) that
+point. _Only when the temperature is beyond the transition interval, will a
+double salt dissolve in water without decomposition (_e.g._ the alums)._
+
+       *       *       *       *       *
+
+
+{272}
+
+CHAPTER XVI
+
+ISOTHERMAL CURVES AND THE SPACE MODEL
+
+In the preceding chapter we considered the changes in the solubility of
+double salts and of mixtures of their constituent salts with the
+temperature; noting, more especially, the relationships between the two
+systems at the transition point. It is now proposed to conclude the study
+of the three-component systems by discussing very briefly the solubility
+relations at constant temperature, or the isothermal solubility curves. In
+this way fresh light will be thrown on the change in the solubility of one
+component by the addition of another component, and also on the conditions
+of formation and stable existence of double salts in solution. With the
+help of these isothermal curves, also, the phenomena of crystallization at
+constant temperature--phenomena which have not only a scientific interest
+but also an important bearing on the industrial preparation of double
+salts--will be more clearly understood.[354]
+
+A brief description will also be given of the method of representing the
+variation of the concentration of the two salts in the solution with the
+temperature.
+
+Non-formation of Double Salts.--In Fig. 102 are shown the solubility curves
+of two salts, A and B, which at the given temperature do not form a double
+salt.[355] The ordinates represent the amount of A, the abscissæ the amount
+of B in a _constant amount_ of the third component, the solvent. The {273}
+point A, therefore, represents the solubility of the salt A at the given
+temperature; and similarly, point B represents the solubility of B. Since
+we are dealing with a three-component system, one solid phase in contact
+with solution will constitute a bivariant system (in the absence of the
+vapour phase and under a constant pressure). At any given temperature,
+therefore, the concentration of the solution in equilibrium with the solid
+can undergo change. If, now, to a pure solution of A a small quantity of B
+is added, the solubility of A will in general be altered; as a rule it is
+diminished, but sometimes it is increased.[356] The curve AC represents the
+varying composition of the solution in equilibrium with the solid component
+A. Similarly, the curve BC represents the composition of the solutions in
+contact with pure B as solid phase. At the point, C, where these two curves
+intersect, there are two solid phases, viz. pure A and pure B, in
+equilibrium with solution, and the system becomes invariant. At this point
+the solution is saturated with respect to both A and B, and at a given
+temperature must have a perfectly definite composition. To take an example,
+if we suppose A to represent sodium sulphate decahydrate, and B, magnesium
+sulphate heptahydrate, and the temperature to be 18.5° (_i.e._ below the
+transition point), the point C would represent a solution containing 2.16
+gm.-molecules Na_{2}SO_{4} and 4.57 gm.-molecules MgSO_{4} per 100
+gm.-molecules of water (p. 268). The curve ACB is the boundary curve for
+saturated solutions; solutions lying outside this curve are supersaturated,
+those lying within the area ACBO, are unsaturated.
+
+[Illustration: FIG. 102.]
+
+[Illustration: FIG. 103.]
+
+[Illustration: FIG. 104.]
+
+Formation of Double Salt.--We have already learned in the preceding chapter
+that if the temperature is outside[357] the {274} transition interval, it
+is possible to prepare a pure saturated solution of the double salt. If,
+now, we suppose the double salt to contain the two constituent salts in
+equimolecular proportions, its saturated solution must be represented by a
+point lying on the line which bisects the angle AOB; _e.g._ point D, Fig.
+103. But a double salt constitutes only a single phase, and can exist,
+therefore, in contact with solutions of varying concentration, as
+represented by EDF.
+
+Let us compare, now, the relations between the solubility curve for the
+double salt, and those for the two constituent salts. We shall suppose that
+the double salt is formed from the single salts when the temperature is
+raised above a certain point (as in the formation of astracanite). At a
+temperature below the transition point, as we have already seen, the
+solubility of the double salt is greater than that of a mixture of the
+single salts. The curve EDF, therefore, must lie above the point C, in the
+region representing solutions supersaturated with respect to the single
+salts (Fig. 104). Such a solution, however, would be metastable, and on
+being brought in contact with the single salts would deposit these and
+yield a solution represented by the point C. At this particular
+temperature, therefore, the isothermal solubility curve will consist of
+only two branches.
+
+[Illustration: FIG. 105.]
+
+Suppose, now, that the temperature is that of the transition point. At this
+point, the double salt can exist together with the single salts in contact
+with solution. The solubility curve {275} of the double salt must,
+therefore, pass through the point C, as shown in Fig. 105.
+
+From this figure, now, it is seen that a solution saturated with respect to
+double salt alone (point D), is supersaturated with respect to the
+component A. If, then, at the temperature of the transition point, excess
+of the double salt is brought in contact with water,[358] and if
+supersaturation is excluded, _the double salt will undergo decomposition
+and the component A will be deposited_. The relative concentration of the
+component B in the solution will, therefore, increase, and the composition
+of the solution will be thereby altered in the direction DC. When the
+solution has the composition of C, the single salt ceases to be deposited,
+for at this point the solution is saturated for both double and single
+salt; and the system becomes invariant.
+
+This diagram explains very clearly the phenomenon of the decomposition of a
+double salt at the transition point. As is evident, this decomposition will
+occur when the solution which is saturated at the temperature of the
+transition point, with respect to the two single salts (point C), does not
+contain these salts in the same ratio in which they are present in the
+double salt. If point C lay on the dotted line bisecting the right angle,
+then the pure saturated solution of the double salt would not be
+supersaturated with respect to either of the single salts, and the double
+salt would, therefore, not be decomposed by water. As has already been
+mentioned, this behaviour is found in the case of optically active
+isomerides, the solubilities of which are identical.
+
+At the transition point, therefore, the isothermal curve also consists of
+two branches; but the point of intersection of the two branches now
+represents a solution which is saturated not {276} only with respect to the
+single salts, but also for the double salt in presence of the single salts.
+
+We have just seen that by a change of temperature the two solubility
+curves, that for the two single salts and that for the double salt, were
+made to approach one another (_cf._ Figs. 104 and 105). In the previous
+chapter, however, we found that on passing the transition point to the
+region of stability for the double salt, the solution which is saturated
+for a mixture of the two constituent salts, is supersaturated for the
+double salt. In this case, therefore, point C must lie above the solubility
+curve of the pure double salt (Fig. 106), and a solution of the composition
+C, if brought in contact with double salt, will deposit the latter. If the
+single salts were also present, then as the double salt separated out, the
+single salts would pass into solution, because so long as the two single
+salts are present, the composition of the solution must remain unaltered.
+If one of the single salts disappear before the other, there will be left
+double salt plus A or double salt plus B, according to which was in excess;
+and the composition of the solution will be either that represented by D
+(saturated for double salt plus A), or that of the point F (saturated for
+double salt plus B).
+
+[Illustration: FIG. 106.]
+
+In connection with the isothermal represented in Fig. 106, it should be
+noted that at this particular temperature a solution saturated with respect
+to the pure double salt is no longer supersaturated for one of the single
+salts (point D); so that at the temperature of this isothermal the double
+salt is not decomposed by water. At this temperature, further, the boundary
+curve consists of three branches AD, DF, and FB, which give the composition
+of the solutions in equilibrium with pure A, double salt, and pure B
+respectively; while the points D and F represent solutions saturated for
+double salt plus A and double salt plus B.
+
+On continuing to alter the temperature in the same direction {277} as
+before, the relative shifting of the solubility curves becomes more marked,
+as shown in Fig. 107. At the temperature of this isothermal, the solution
+saturated for the double salt now lies in a region of distinct unsaturation
+with respect to the single salts; and the double salt can now exist as
+solid phase in contact with solutions containing both relatively more of A
+(curve ED), and relatively more of B (curve DF), than is contained in the
+double salt itself.
+
+[Illustration: FIG. 107.]
+
+Transition Interval.--From what has been said, and from an examination of
+the isothermal diagrams, Figs. 104-107, it will be seen that by a variation
+of the temperature we can pass from a condition where the double salt is
+quite incapable of existing in contact with solution (supersaturation being
+excluded), to a condition where the existence of the double salt in
+presence of solution becomes possible; only in the presence, however, of
+one of the single salts (_transition point_, Fig. 105). A further change of
+temperature leads to a condition where the stable existence of the pure
+double salt in contact with solution just becomes possible (Fig. 106); and
+from this point onwards, pure saturated solutions of the double salt can be
+obtained (Fig. 107). _At any temperature, therefore, between that
+represented by Fig. 105, and that represented by Fig. 106, the double salt
+undergoes partial decomposition, with deposition of one of the constituent
+salts._ The temperature range between the transition point and the
+temperature at which a stable saturated solution of the pure double salt
+just begins to be possible, is known as the _transition interval_ (p. 270).
+As the figures show, the transition interval is limited on the one side by
+the transition temperature, and on the other by the temperature at which
+the solution saturated for double salt and the less soluble of the single
+salts, contains the component salts in the same ratio as they are present
+in the double salt. The greater the difference in the solubility of the
+single salts, the larger will be the transition interval. {278}
+
+Isothermal Evaporation.--The isothermal solubility curves are of great
+importance for obtaining an insight into the behaviour of a solution when
+subjected to isothermal evaporation. To simplify the discussion of the
+relationships found here, we shall still suppose that the double salt
+contains the single salts in equimolecular proportions; and we shall, in
+the first instance, suppose that the unsaturated solution with which we
+commence, also contains the single salts in the same ratio. The composition
+of the solution must, therefore, be represented by some point lying on the
+line OD, the bisectrix of the right angle.
+
+From what has been said, it is evident that when the formation of a double
+salt can occur, three temperature intervals can be distinguished, viz. the
+single-salt interval, the transition interval, and the double-salt
+interval.[359] When the temperature lies in the first interval, evaporation
+leads first of all to the crystallization of one of the single salts, and
+then to the separation of both the single salts together. In the second
+temperature interval, evaporation again leads, in the first place, to the
+deposition of one of the single salts, and afterwards to the
+crystallization of the double salt. In the third temperature interval, only
+the double salt crystallizes out. This will become clearer from what
+follows.
+
+[Illustration: FIG. 108.]
+
+[Illustration: FIG. 109.]
+
+If an unsaturated solution of the two single salts in equimolecular
+proportion (_e.g._ point _x_, Fig. 108) is evaporated at a temperature at
+which the formation of double salt is impossible, the component A, the
+solubility curve of which is {279} cut by the line OD, will first separate
+out; the solution will thereby become richer in B. On continued
+evaporation, more A will be deposited, and the composition of the solution
+will change until it attains the composition represented by the point C,
+when both A and B will be deposited, and the composition of the solution
+will remain unchanged. The result of evaporation will therefore be a
+mixture of the two components.
+
+If the formation of double salt is possible, but if the temperature lies
+within the transition interval, the relations will be represented by a
+diagram like Fig. 109. Isothermal evaporation of the solution X will lead
+to the deposition of the component A, and the composition of the solution
+will alter in the direction DE; at the latter point the double salt will be
+formed, and the composition of the solution will remain unchanged so long
+as the two solid phases are present. As can be seen from the diagram,
+however, the solution in E contains less of component A than is contained
+in the double salt. Deposition of the double salt at E, therefore, would
+lead to a relative decrease in the concentration of A in the solution, and
+to counterbalance this, _the salt which separated out at the commencement
+must redissolve_.
+
+Since the salts were originally present in equimolecular proportions, the
+final result of evaporation will be the pure double salt. If when the
+solution has reached the point E the salt A which had separated out is
+removed, double salt only will be left as solid phase. At a given
+temperature, however, a single solid phase can exist in equilibrium with
+solutions of different composition. If, therefore, isothermal evaporation
+is continued after the removal of the salt A, double salt will be
+deposited, and the composition of the solution will change in the direction
+EF. At the point F the salt B will separate out, and on evaporation both
+double salt and the salt B will be deposited. In the former case (when the
+salt A disappears on evaporation) we are dealing with an _incongruently
+saturated solution_; but in the latter case, where both solid phases
+continue to be deposited, the solution is said to be _congruently
+saturated_.[360]
+
+A "congruently saturated solution" is one from which the {280} solid phases
+are continuously deposited during isothermal evaporation to dryness,
+whereas in the case of "incongruently saturated solutions," at least one of
+the solid phases disappears during the process of evaporation.
+
+[Illustration: FIG. 110.]
+
+Lastly, if the temperature lies outside the transition interval, isothermal
+evaporation of an unsaturated solution of the composition X (Fig. 110) will
+lead to the deposition of pure double salt from beginning to end. If a
+solution of the composition Y is evaporated, the component A will first be
+deposited and the composition of the solution will alter in the direction
+of E, at which point double salt will separate out. Since the solution at
+this point contains relatively more of A than is present in the double
+salt, both the double salt and the single salt A will be deposited on
+continued evaporation, in order that the composition of the solution shall
+remain unchanged. In the case of solution Z, first component B and
+afterwards the double salt will be deposited. The result will, therefore,
+be a mixture of double salt and the salt B (congruently saturated
+solutions),
+
+It may be stated here that the same relationships as have been explained
+above for double salts are also found in the resolution of racemic
+compounds by means of optically active substances (third method of
+Pasteur). In this case the single salts are doubly active substances
+(_e.g._ strychnine-_d_-tartrate and strychnine-_l_-tartrate), and the
+double salt is a partially racemic compound.[361]
+
+Crystallization of Double Salt from Solutions containing Excess of One
+Component.--One more case of isothermal crystallization may be discussed.
+It is well known that a double salt which is decomposed by pure water can
+nevertheless be obtained pure by crystallization from a solution containing
+excess of one of the single salts (_e.g._ in the case of carnallite). Since
+the double salt is partially decomposed by water, the temperature of the
+experiment must be within the transition {281} interval, and the relations
+will, therefore, be represented by a diagram like Fig. 109. If, now,
+instead of starting with an unsaturated solution containing the single
+salts in equimolecular proportions, we commence with one in which excess of
+one of the salts is present, as represented by the point Y, isothermal
+evaporation will cause the composition to alter in the direction YD', the
+relative amounts of the single salts remaining the same throughout. When
+the composition of the solution reaches the point D', pure double salt will
+be deposited. The separation of double salt will, however, cause a relative
+decrease in the concentration of the salt A, and the composition of the
+solution will, therefore, alter in the direction D'F. If the evaporation is
+discontinued before the solution has attained the composition F, only
+double salt will have separated out. Even within the transition interval,
+therefore, pure double salt can be obtained by crystallization, provided
+the original solution has a composition represented by a point lying
+between the two lines OE and OF. Since, as already shown, the composition
+of the solution alters on evaporation in the direction EF, it will be best
+to employ a solution having a composition near to the line OE.
+
+Formation of Mixed Crystals.--If the two single salts A and B do not
+crystallize out pure from solution, but form an unbroken series of mixed
+crystals, it is evident that an invariant system cannot be produced. The
+solubility curve will therefore be continuous from A to B; the liquid
+solutions of varying composition being in equilibrium with solid solutions
+also of varying composition. If, however, the series of mixed crystals is
+not continuous, there will be a break in the solubility curve at which two
+solid solutions of different composition will be in equilibrium with liquid
+solution. This, of course, will constitute an invariant system, and the
+point will correspond to the point C in Fig. 108. A full discussion of
+these systems would, however, lead us too far, and the above indication of
+the behaviour must suffice.[362]
+
+{282}
+
+Application to the Characterization of Racemates.--The form of the
+isothermal solubility curves is also of great value for determining whether
+an inactive substance is a racemic compound or a conglomerate of equal
+proportions of the optical antipodes.[363]
+
+As has already been pointed out, the formation of racemic compounds from
+the two enantiomorphous isomerides, is analogous to the formation of double
+salts. The isothermal solubility curves, also, have a similar form. In the
+case of the latter, indeed, the relationships are simplified by the fact
+that the two enantiomorphous forms have identical solubility, and the
+solubility curves are therefore symmetrical to the line bisecting the angle
+of the co-ordinates. Further, with the exception of the partially racemic
+compounds to be mentioned later, there is no transition interval.
+
+In Fig. 111, are given diagrammatically two isothermal solubility curves
+for optically active substances. From what has been said in the immediately
+preceding pages, the figure ought really to explain itself. The upper
+isothermal _acb_ represents the solubility relations when the formation of
+a racemic compound is excluded, as, _e.g._ in the case of rubidium _d_- and
+_l_-tartrates above the transition point (p. 265). The solution at the
+point _c_ is, of course, inactive, and _is unaffected by addition of either
+the _d_- or _l_- form_. The lower isothermal, on the other hand, would be
+obtained at a temperature at which the racemic compound could be formed.
+The curve _a'e_ is the solubility curve for the _l_- form; _b'f_, that for
+the _d_- form; and _edf_, that for the racemic compound in presence of
+solutions of varying concentration. The point _d_ corresponds to saturation
+for the pure racemic compound.
+
+[Illustration: FIG. 111.]
+
+From these curves now, it will be evident that it will be possible, in any
+given case, to decide whether or not an inactive body is a mixture or a
+racemic compound. For this purpose, {283} two solubility determinations are
+made, first with the inactive material alone (in excess), and then with the
+inactive material plus excess of one of the optically active forms. If we
+are dealing with a mixture, the two solutions thus obtained will be
+identical; both will have the composition corresponding to the point _c_,
+and will be inactive. If, however, the inactive material is a racemic
+compound, then two different solutions will be obtained; namely, an
+inactive solution corresponding to the point _d_ (Fig. 111), and an
+_active_ solution corresponding either to _e_ or to _f_, according to which
+enantiomorphous form was added.
+
+_Partially racemic compounds._[364] In this case we are no longer dealing
+with enantiomorphous forms, and the solubility of the two oppositely active
+isomerides is no longer the same. The symmetry of the solubility curves
+therefore disappears, and a figure is obtained which is identical in its
+general form with that found in the case of ordinary double salts (Fig.
+112). In this case there is a transition interval.
+
+[Illustration: FIG. 112.]
+
+The curves _acb_ belong to a temperature at which the partially racemic
+compound cannot be formed; _a'dfb'_, to the temperature at which the
+compound just begins to be stable in contact with water, and _a"ed'f'b"_
+belongs to a temperature at which the partially racemic compound is quite
+stable in contact with water. Suppose now solubility determinations, made
+in the first case with the original material alone, and then with the
+original body plus each of the two compounds, formed from the
+enantiomorphous substances separately, then if the original body was a
+mixture, identical solutions will be obtained in all three cases (point
+_c_); if it was a partially racemic compound, three different solutions
+(_e_, _d'_, and _f'_) will be obtained if the temperature was outside the
+transition interval, and two solutions, _d_ and _f_, if the temperature
+belonged to the transition interval.
+
+{284}
+
+_Representation in Space._
+
+Space Model for Carnallite.--Interesting and important as the isothermal
+solubility curves are, they are insufficient for the purpose of obtaining a
+clear insight into the complete behaviour of the systems of two salts and
+water. A short description will, therefore, be given here of the
+representation in space of the solubility relations of potassium and
+magnesium chlorides, and of the double salt which they form,
+carnallite.[365]
+
+[Illustration: FIG. 113.]
+
+Fig. 113 is a diagrammatic sketch of the model for carnallite looked at
+sideways from above. Along the X-axis is measured the concentration of
+magnesium chloride in the {285} solution; along the Y-axis, the
+concentration of potassium chloride; while along the T-axis is measured the
+temperature. The three axes are at right angles to one another. The
+XT-plane, therefore, contains the solubility curve of magnesium chloride;
+the YT-plane, the solubility curve of potassium chloride, and in the space
+between the two planes, there are represented the composition of solutions
+containing both magnesium and potassium chlorides. Any _surface_ between
+the two planes will represent the various solutions in equilibrium with
+only one solid phase, and will therefore indicate the area or field of
+existence of bivariant ternary systems. A _line_ or _curve_ formed by the
+intersection of two surfaces will represent solutions in equilibrium with
+two solid phases (viz. those belonging to the intersecting surfaces), and
+will show the conditions for the existence of univariant systems. Lastly,
+_points_ formed by the intersection of three surfaces will represent
+invariant systems, in which a solution can exist in equilibrium with three
+solid phases (viz. those belonging to the three surfaces).
+
+We shall first consider the solubility relations of the single salts. The
+complete equilibrium curve for magnesium chloride and water is represented
+in Fig. 113 by the series of curves ABF_{1} G_{1} H_{1} J_{1} L_{1} N_{1}.
+AB is the freezing-point curve of ice in contact with solutions containing
+magnesium chloride, and B is the cryohydric point at which the solid phases
+ice and MgCl_{2},12H_{2}O can co-exist with solution. BFG is the solubility
+curve of magnesium chloride dodecahydrate. This curve shows a point of
+maximum temperature at F_{1}, and a retroflex portion F_{1}G_{1}. The curve
+is therefore of the form exhibited by calcium chloride hexahydrate, or the
+hydrates of ferric chloride (Chapter VIII.). G_{1} is a transition point at
+which the solid phase changes from dodecahydrate to octahydrate, the
+solubility of which is represented by the curve G_{1}H_{1}. At H_{1} the
+octahydrate gives place to the hexahydrate, which is the solid phase in
+equilibrium with the solutions represented by the curve H_{1}J_{1}. J_{1}
+and L_{1} are also transition points at which the solid phase undergoes
+change, in the former case from hexahydrate to tetrahydrate; and in the
+latter case, {286} from tetrahydrate to dihydrate. The complete curve of
+equilibrium for magnesium chloride and water is, therefore, somewhat
+complicated, and is a good example of the solubility curves obtained with
+salts capable of forming several hydrates.
+
+The solubility curve of potassium chloride is of the simplest form,
+consisting only of the two branches AC, the freezing-point curve of ice,
+and CO, the solubility curve of the salt. C is the cryohydric point. This
+point and the two curves lie in the YT-plane.
+
+On passing to the ternary systems, the composition of the solutions must be
+represented by points or curves situated _between_ the two planes. We shall
+now turn to the consideration of these. BD and CD are ternary eutectic
+curves (p. 284). They give the composition of solutions in equilibrium with
+ice and magnesium chloride dodecahydrate (BD), and with ice and potassium
+chloride (CD). D is a _ternary cryohydric point_. If the temperature is
+raised and the ice allowed to disappear, we shall pass to the solubility
+curve for MgCl_{2},12H_{2}O + KCl (curve DE). At E carnallite is formed and
+the potassium chloride disappears; EFG is then the solubility curve for
+MgCl_{2},12H_{2}O + carnallite (KMgCl_{3},6H_{2}O). This curve also shows a
+point of maximum temperature (F) and a retroflex portion. GH and HJ
+represent the solubility curves of carnallite + MgCl_{2},8H_{2}O and
+carnallite + MgCl_{2},6H_{2}O, G and H being transition points. JK is the
+solubility curve for carnallite + MgCl_{2},4H_{2}O. At the point K we have
+the _highest temperature at which carnallite can exist with magnesium
+chloride in contact with solution_. Above this temperature decomposition
+takes place and potassium chloride separates out.
+
+If at the point E, at which the two single salts and the double salt are
+present, excess of potassium chloride is added, the magnesium chloride will
+all disappear owing to the formation of carnallite, and there will be left
+carnallite and potassium chloride. The solubility curve for a mixture of
+these two salts is represented by EMK; a simple curve exhibiting, however,
+a temperature maximum at M. This maximum point corresponds with the fact
+that dry carnallite melts at this temperature with separation of potassium
+chloride. _At all temperatures {287} above this point, the formation of
+double salt is impossible_. The retroflex portion of the curve represents
+solutions in equilibrium with carnallite and potassium chloride, but in
+which the ratio MgCl_{2} : KCl is greater than in the double salt.
+
+Throughout its whole course, _the curve EMK represents solutions in which
+the ratio of MgCl_{2} : KCl is greater than in the double salt_. As this is
+a point of some importance, it will be well, perhaps, to make it clearer by
+giving one of the isothermal curves, _e.g._ the curve for 10°, which is
+represented diagrammatically in Fig. 114. E and F here represent solutions
+saturated for carnallite plus magnesium chloride hydrate, and for
+carnallite plus potassium chloride. As is evident, the point F lies above
+the line representing equimolecular proportions of the salts (OD).
+
+[Illustration: FIG. 114.]
+
+Summary and Numerical Data.--We may now sum up the different systems which
+can be formed, and give the numerical data from which the model is
+constructed.[366]
+
+  I. _Bivariant Systems._
+
+  --------------------------------------
+     Solid phase.   | Area of existence.
+  --------------------------------------
+  Ice               | ABDC
+  KCl               | CDEMKLNO
+  Carnallite        | EFGHJKM
+  MgCl_{2},12H_{2}O | BF_{1}G_{1}GFED
+  MgCl_{2},8H_{2}O  | G_{1}H_{1}HG
+  MgCl_{2},6H_{2}O  | H_{1}I_{1}IH
+  MgCl_{2},4H_{2}O  | I_{1}L_{1}LKI
+  MgCl_{2},2H_{2}O  | L_{1}N_{1}NL
+  --------------------------------------
+
+II. _Univariant Systems._--The different univariant systems have already
+been described. The course of the curves will be sufficiently indicated if
+the temperature and composition of the solutions for the different
+invariant systems are given.
+
+{288}
+
+  III.--_Invariant Systems--Binary and Ternary._
+
+  -------------------------------------------------------------------------
+         |                           |          | Composition of solution.
+  Point. |      Solid Phases.        | Temper-  | Gram-molecules of salt
+         |                           |  ature.  | per 1000 gram-mol. water.
+  -------------------------------------------------------------------------
+  A      | Ice                       |    0°    | --
+         |                           |          |
+  B      | Ice; MgCl_{2},12H_{2}O    |  -33.6°  | 49.2 MgCl_{2}
+         |                           |          |
+  C      | Ice; KCl                  |  -11.1°  | 59.4 KCl
+         |                           |          |
+  D      |{ Ice; MgCl_{2},12H_{2}O; }|  -34.3°  | 43 MgCl_{2}; 3 KCl
+         |{ KCl                     }|          |
+         |                           |          |
+  E      |{ MgCl_{2},12H_{2}O; KCl; }|  -21°    | 66.1 MgCl_{2}; 4.9 KCl
+         |{ carnallite              }|          |
+         |                           |          |
+  F_{1}  | MgCl_{2},12H_{2}O         |  -16.4°  | 83.33 MgCl_{2}
+         |                           |          |
+  F      |{ MgCl_{2},12H_{2}O;      }|  -16.6°  |{ Almost same as F_{1};
+         |{ carnallite              }|          |{ contains small amount
+         |                           |          |{ of KCl
+         |                           |          |
+  G_{1}  |{ MgCl_{2},12H_{2}O;      }|  -16.8°  | 87.5 MgCl_{2}
+         |{ MgCl_{2},8H_{2}O        }|          |
+         |                           |          |
+  G      |{ MgCl_{2},12H_{2}O;      }|  -16.9°  |{ Almost same as G_{1},
+         |{ MgCl_{2},8H_{2}O;       }|          |{ but contains small
+         |{ carnallite              }|          |{ quantity of KCl
+         |                           |          |
+  H_{1}  |{ MgCl_{2},8H_{2}O;       }|   -3.4°  | 99 MgCl_{2}
+         |{ MgCl_{2},6H_{2}O        }|          |
+         |                           |          |
+  H      |{ MgCl_{2},8H_{2}O;       }|ca. -3.4° |{ Almost same as H_{1},
+         |{ MgCl_{2},6H_{2}O;       }|          |{ but contains small
+         |{ carnallite              }|          |{ amount of KCl
+         |                           |          |
+  J_{1}  |{ MgCl_{2},6H_{2}O;       }|  116.67° | 161.8 MgCl_{2}
+         |{ MgCl_{2},4H_{2}O        }|          |
+         |                           |          |
+  J      |{ MgCl_{2},6H_{2}O;       }|  115.7°  | 162 MgCl_{2}; 4 KCl
+         |{ MgCl_{2},4H_{2}O;       }|          |
+         |{ carnallite              }|          |
+         |                           |          |
+  K      |{ MgCl_{2},4H_{2}O; KCl;  }|  152.5°  | 200 MgCl_{2}; 24 KCl
+         |{ carnallite              }|          |
+         |                           |          |
+  L_{1}  |{ MgCl_{2},4H_{2}O;       }|  181°    | 238.1 MgCl_{2}
+         |{ MgCl_{2},2H_{2}O        }|          |
+         |                           |          |
+  L      |{ MgCl_{2},4H_{2}O;       }|  176°    | 240 MgCl_{2}; 41 KCl
+         |{ MgCl_{2},2H_{2}O; KCl   }|          |
+         |                           |          |
+  M      | Carnallite; KCl           |  167.5°  | 166.7 MgCl_{2}; 41.7 KCl
+         |                           |          |
+  [N_{1} | MgCl_{2},2H_{2}O          |  186°    | ca. 241 MgCl_{2}]
+         |                           |          |
+  N      | MgCl_{2},2H_{2}O; KCl     |  186°    | 240 MgCl_{2}; 63 KCl
+         |                           |          |
+  [O     | KCl                       |  186°    | 195.6 KCl]
+  -------------------------------------------------------------------------
+
+With the help of the data in the preceding table and of the solid model it
+will be possible to state in any given case what will be the behaviour of a
+system composed of magnesium chloride, potassium chloride and water. One or
+two different cases will be very briefly described; and the reader should
+have no difficulty in working out the behaviour under other conditions with
+the help of the model and the numerical data just given. {289}
+
+In the first place it may be again noted that at a temperature above 167.5°
+(point M) carnallite cannot exist. If, therefore, a solution of magnesium
+and potassium chlorides is evaporated at a temperature above this point,
+the result will be a mixture of potassium chloride and either magnesium
+chloride tetrahydrate or magnesium chloride dihydrate, according as the
+temperature is below or above 176°. The isothermal curve here consists of
+only two branches.
+
+Further, reference has already been made to the fact that all points of the
+carnallite area correspond to solutions in equilibrium with carnallite, but
+in which the ratio of MgCl_{2} to KCl is greater than in the double salt. A
+solution which is saturated with respect to double salt alone will be
+supersaturated with respect to potassium chloride. At all temperatures,
+therefore, carnallite is decomposed by water with separation of potassium
+chloride; hence all solutions obtained by adding excess of carnallite to
+water will lie on the curve EM. _A pure saturated solution of carnallite
+cannot be obtained._
+
+If an unsaturated solution of the two salts in equimolecular amounts is
+evaporated, potassium chloride will first be deposited, because the plane
+bisecting the right angle formed by the X and Y axes cuts the area for that
+salt. Deposition of potassium chloride will lead to a relative increase in
+the concentration of magnesium chloride in the solution; and on continued
+evaporation a point (on the curve EM) will be reached at which carnallite
+will separate out. So long as the two solid phases are present, the
+composition of the solution must remain unchanged. Since the separation of
+carnallite causes a decrease in the relative concentration of the potassium
+chloride in the solution, the portion of this salt which was deposited at
+the commencement must _redissolve_, and carnallite will be left on
+evaporating to dryness. (_Incongruently saturated solution._)
+
+Although carnallite is decomposed by pure water, it will be possible to
+crystallize it from a solution having a composition represented by any
+point in the carnallite area. Since during the separation of the double
+salt the relative amount of magnesium chloride increases, it is most
+advantageous to {290} commence with a solution the composition of which is
+represented by a point lying just above the curve EM (cf. p. 281).
+
+From the above description of the behaviour of carnallite in solution, the
+processes usually employed for obtaining potassium chloride will be readily
+intelligible.[367]
+
+Ferric Chloride--Hydrogen Chloride--Water.--In the case of another system
+of three components which we shall now describe, the relationships are
+considerably more complicated than in those already discussed. They deserve
+discussion, however, on account of the fact that they exhibit a number of
+new phenomena.
+
+In the system formed by the three components, ferric chloride, hydrogen
+chloride, and water, not only can various compounds of ferric chloride and
+water (p. 152), and of hydrogen chloride and water be formed, each of which
+possesses a definite melting point, but various ternary compounds are also
+known. Thus we have the following solid phases:--
+
+  2FeCl_{3},12H_{2}O   HCl,3H_{2}O   2FeCl_{3},2HCl,12H_{2}O
+  2FeCl_{3},7H_{2}O    HCl,2H_{2}O   2FeCl_{3},2HCl,8H_{2}O
+  2FeCl_{3},5H_{2}O    HCl,H_{2}O    2FeCl_{3},2HCl,4H_{2}O
+  2FeCl_{3},4H_{2}O
+  FeCl_{3}
+
+From this it will be readily understood that the complete study of the
+conditions of temperature and concentration under which solutions can
+exist, either with one solid phase or with two or three solid phases, are
+exceedingly complicated; and, as a matter of fact, only a few of the
+possible equilibria have been investigated. We shall attempt here only a
+brief description of the most important of these.[368]
+
+If we again employ rectangular co-ordinates for the graphic {291}
+representation of the results, we have the two planes XOT and YOT (Fig.
+115): the concentration of ferric chloride being measured along the X-axis,
+the concentration of hydrogen chloride along the Y-axis, and the
+temperature along the T-axis. The curve ABCDEFGHJK is, therefore, the
+solubility curve of ferric chloride in water (p. 152), and the curve
+A'B'C'D'E'F' the solubility curve of hydrogen chloride and its hydrates. B'
+and D' are the melting points of the hydrates HCl,3H_{2}O and HCl,2H_{2}O.
+In the space between these two planes are represented those systems in
+which all three components are present. As already stated, only a few of
+the possible ternary systems have been investigated, and these are
+represented in Fig. 116. The figure shows the model resting on the
+XOT-plane, so that the lower edge represents the solubility curve of ferric
+chloride, the concentration increasing from right to left. The
+concentration of hydrogen chloride is measured upwards, and the temperature
+forwards. The further end of the model represents the isothermal surface
+for -30°. The surface of the model on the left does not correspond with the
+plane YOT in Fig. 115, but with a parallel plane which cuts the
+concentration axis for ferric chloride at a point representing 65
+gm.-molecules FeCl_{3} in 100 gm.-molecules of water. The upper surface
+corresponds with a plane parallel to the axis XOT, at a distance
+corresponding with the concentration of 50 gm.-molecules HCl in 100
+gm.-molecules of water.
+
+[Illustration: FIG. 115.]
+
+Ternary Systems.--We pass over the binary system FeCl_{3}--H_{2}O, which
+has already been discussed (p. 152), and the similar system HCl--H_{2}O
+(see Fig. 115), and turn to the discussion of some of the ternary systems
+represented by {292} points on the surface of the model between the planes
+XOT and YOT. As in the case of carnallite, a plane represents the
+conditions of concentration of solution and temperature under which a
+ternary solution can be in equilibrium with a _single_ solid phase
+(bivariant systems), a line represents the conditions for the coexistence
+of a solution with two solid phases (univariant systems), and a point the
+conditions for equilibrium with three solid phases (invariant systems).
+
+[Illustration: FIG. 116.]
+
+In the case of a binary system, in which 2FeCl_{3},12H_{2}O is in
+equilibrium with a solution of the same composition, addition of hydrogen
+chloride must evidently lower the temperature at which equilibrium can
+exist; and the same holds, of course, {293} for all other binary solutions
+in equilibrium with this solid phase. In this way we obtain the surface I.,
+which represents the temperatures and concentrations of solutions in which
+2FeCl_{3},12H_{2}O can be in equilibrium with a ternary solution containing
+ferric chloride, hydrogen chloride, and water. This surface is analogous to
+the curved surface K_{1}K_{2}_k__{4}_k__{3} in Fig. 97 (p. 256). Similarly,
+the surfaces II., III., IV., and V. represent the conditions for
+equilibrium between the solid phases 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O;
+2FeCl_{3},4H_{2}O; FeCl_{3} and ternary solutions respectively. The lines
+CL, EM, GN, and IO on the model represent univariant systems in which a
+ternary solution is in equilibrium with two solid phases, viz. with those
+represented by the adjoining fields. These lines correspond with the
+ternary eutectic curves _k__{3}K_{1} and _k__{4}K_{2} in Fig. 97. Besides
+the surfaces already mentioned, there are still three others, VI., VII.,
+and VIII., which also represent the conditions for equilibrium between one
+solid phase and a ternary solution; but in these cases, the solid phase is
+not a binary compound or an anhydrous salt, but a ternary compound
+containing all three components. The solid phases which are in equilibrium
+with the ternary solutions represented by the surfaces VI., VII., and
+VIII., are 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
+2FeCl_{3},2HCl,12H_{2}O respectively.
+
+The model for FeCl_{3}--HCl--H_{2}O exhibits certain other peculiarities
+not found in the case of MgCl_{2}--KCl--H_{2}O. On examining the model more
+closely, it is found that the field of the ternary compound
+2FeCl_{3},2HCl,8H_{2}O (VII.) resembles the surface of a sugar cone, and
+has a projecting point, the end of which corresponds with a higher
+temperature than does any other point of the surface. At the point of
+maximum temperature the composition of the liquid phase is the same as that
+of the solid. This point, therefore, represents the melting point of the
+double salt of the above composition.
+
+The curves representing univariant systems are of two kinds. In the one
+case, the two solid phases present are both binary compounds; or one is a
+binary compound and the other is one of the components. In the other case,
+either one or both solid phases are ternary compounds. Curves belonging
+{294} to the former class (so-called _border curves_) start from binary
+eutectic points, and their course is always towards lower temperatures,
+_e.g._ CL, EM, GN, IO. Curves belonging to the latter class (so-called
+_medial curves_) would, in a triangular diagram, lie entirely within the
+triangle. Such curves are YV, WV, VL, LM, MV, NS, ST, SO, OZ. These curves
+do not always run from higher to lower temperatures, but may even exhibit a
+point of maximum temperature. Such maxima are found, for example, at U
+(Fig. 116), and also on the curves ST and LV.
+
+Finally, whereas all the other ternary univariant curves run in valleys
+between the adjoining surfaces, we find at the point X a similar appearance
+to that found in the case of carnallite, as the univariant curve here rises
+above the surrounding surface. The point X, therefore, does not correspond
+with a eutectic point, but with a transition point. At this point the
+ternary compound 2FeCl_{3},2HCl,12H_{2}O melts with separation of
+2FeCl_{3},12H_{2}O, just as carnallite melts at 168° with separation of
+potassium chloride.
+
+The Isothermal Curves.--A deeper insight into the behaviour of the system
+FeCl_{3}--HCl--H_{2}O is obtained from a study of the isothermal curves,
+the complete series of which, so far as they have been studied, is given in
+Fig. 117.[369] In this figure the lightly drawn curves represent isothermal
+solubility curves, the particular temperature being printed beside the
+curve.[370] The dark lines give the composition of the univariant systems
+at different temperatures. The point of intersection of a dark with a light
+curve gives the composition of the univariant solution at the temperature
+represented by the light curve; and the point of intersection of two dark
+lines gives the composition of the invariant solution in equilibrium with
+three solid phases. The dotted lines represent metastable systems, and the
+points P, Q, and R represent solutions of {295} the composition of the
+ternary salts, 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
+2FeCl_{3},2HCl,12H_{2}O.
+
+[Illustration: FIG. 117.]
+
+The farther end of the model (Fig. 116) corresponds, as already mentioned,
+to the temperature -30°, so that the outline evidently represents the
+isothermal curve for that temperature. Fig. 117 does not show this. We can,
+however, follow the isothermal for -20°, which is the extreme curve on the
+right in Fig. 117. Point A represents the solubility of 2FeCl_{3},12H_{2}O
+in water. If hydrogen chloride is added, the concentration of ferric
+chloride in the solution first decreases and then increases, until at point
+34 the ternary double salt 2FeCl_{3},2HCl,12H_{2}O is formed. If the
+addition of hydrogen chloride is continued, the ferric chloride disappears
+ultimately, and only the ternary double salt remains. This salt can coexist
+with solutions of the composition represented by the curve which passes
+through the points 173, 174, 175. At the last-mentioned point, the ternary
+salt with 8H_{2}O is formed. The composition of the solutions with which
+this salt is in equilibrium at -20° is represented by the curve which
+passes through a point of maximal concentration with respect to HCl, and
+cuts the curve SN at the point 112, at which the solution is in equilibrium
+with the two solid phases 2FeCl_{3},4H_{2}O and 2FeCl_{3},2HCl,8H_{2}O. The
+succeeding portion of the isotherm represents the solubility curve at -20°
+of 2FeCl_{3},4H_{2}O, which cuts the dark line OS at point 113, at which
+the solution is in equilibrium with the two solid phases 2FeCl_{3},4H_{2}O
+and 2FeCl_{3},2HCl,4H_{2}O. Thereafter comes the solubility curve of the
+latter compound.
+
+The other isothermal curves can be followed in a similar manner. If the
+temperature is raised, the region of existence of the ternary double salts
+becomes smaller and smaller, and at temperatures above 30° the ternary
+salts with 12H_{2}O and 8H_{2}O are no longer capable of existing. If the
+temperature is raised above 46°, only the binary compounds of ferric
+chloride and water and the anhydrous salt can exist as solid phases.
+The isothermal curve for 0° represents the solubility curve for
+2FeCl_{3},12H_{2}O; 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O; and
+2FeCl_{3},4H_{2}O. {296}
+
+Finally, in the case of the system FeCl_{3}--HCl--H_{2}O, we find
+_closed_ isothermal curves. Since, as already stated, the salt
+2FeCl_{3},2HCl,8H_{2}O has a definite melting point, the temperature of
+which is therefore higher than that at which this compound is in
+equilibrium with solutions of other composition, it follows that the line
+of intersection of an isothermal plane corresponding with a temperature
+immediately below the melting point of the salt with the cone-shaped
+surface of its region of existence, will form a closed curve. This is shown
+by the isotherm for -4.5°, which surrounds the point Q, the melting point
+of the ternary salt.
+
+The following table gives some of the numerical data from which the curves
+and the model have been constructed:--
+
+  -------------------------------------------------------------------------
+         |                             |          | Composition of the sol-
+         |                             |          | ution in gm.-mols. salt
+  Point. |         Solid phases.       | Temper-  | to 100 gm.-mols. water.
+         |                             |  ature.  |------------------------
+         |                             |          |   HCl   |  FeCl_{3}
+  -------------------------------------------------------------------------
+  A      | 2FeCl_{3},12H_{2}O          |  -20°    |   --    |   6.56
+         |                             |          |         |
+  C      |{ 2FeCl_{3},12H_{2}O;       }|   27.4°  |   --    |  24.30
+         |{ 2FeCl_{3},7H_{2}O         }|          |         |
+         |                             |          |         |
+  E      |{ 2FeCl_{3},7H_{2}O;        }|   30°    |   --    |  30.24
+         |{ 2FeCl_{3},5H_{2}O         }|          |         |
+         |                             |          |         |
+  G      |{ 2FeCl_{3},5H_{2}O;        }|   55°    |   --    |  40.64
+         |{ 2FeCl_{3},4H_{2}O         }|          |         |
+         |                             |          |         |
+  J      | 2FeCl_{3},4H_{2}O; FeCl_{3} |   66°    |   --    |  58.40
+         |                             |          |         |
+         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
+  L      |{ 2FeCl_{3},7H_{2}O;        }|   -7.5°  |  19.22  |  23.72
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},7H_{2}O;        }|          |         |
+  M      |{ 2FeCl_{3},5H_{2}O;        }|   -7.3°  |  23.08  |  28.55
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},5H_{2}O;        }|          |         |
+  N      |{ 2FeCl_{3},4H_{2}O;        }|  -16°    |  28.40  |  31.89
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
+  S      |{ 2FeCl_{3},2HCl,8H_{2}O;   }|  -27.5°  |  32.33  |  34.21
+         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
+  O      |{ FeCl_{3};                 }|   29°    |  33.71  |  49.84
+         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
+         |                             |          |         |
+  U      |{ 2FeCl_{3},7H_{2}O;        }|   -4.5°  |  20.66  |  25.74
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
+  V      |{ 2FeCl_{3},2HCl,12H_{2}O;  }|  -13°    |  22.40  |  18.00
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+  X      |{ 2FeCl_{3},12H_{2}O;       }|  -12.5°  |  22.14  |  16.69
+         |{ 2FeCl_{3},2HCl,12H_{2}O   }|          |         |
+         |                             |          |         |
+  Q      | 2FeCl_{3},2HCl,8H_{2}O      |   -3° (melting point)
+  -------------------------------------------------------------------------
+
+Basic Salts.--Another class of systems in the study of {297} which the
+Phase Rule has performed exceptional service, is that of the basic salts.
+In many cases it is impossible, by the ordinary methods of analysis, to
+decide whether one is dealing with a definite chemical individual or with a
+mixture. The question whether a solid phase is a chemical individual can,
+however, be answered, in most cases, with the help of the principles which
+we have already learnt. Let us consider, for example, the formation of
+basic salts from bismuth nitrate, and water. In this case we can choose as
+components Bi_{2}O_{3}, N_{2}O_{5}, and H_{2}O; since all the systems
+consist of these in varying amounts. If we are dealing with a condition of
+equilibrium at constant temperature between liquid and solid phases, three
+cases can be distinguished,[371] viz.--
+
+1. The solutions in different experiments have the same composition, but
+the composition of the precipitate alters. In this case there must be two
+solid phases.
+
+2. The solutions in different experiments can have varying composition,
+while the composition of the precipitate remains unchanged. In this case
+only one solid phase exists, a definite compound.
+
+3. The composition both of the solution and of the precipitate varies. In
+this case the solid phase is a solid solution or a mixed crystal.
+
+In order, therefore, to decide what is the nature of a precipitate produced
+by the hydrolysis of a normal salt, it is only necessary to ascertain
+whether and how the composition of the precipitate alters with alteration
+in the composition of the solution. If the composition of the solution is
+represented by abscissæ, and the composition of the precipitate by
+ordinates, the form of the curves obtained would enable us to answer our
+question; for vertical lines would indicate the presence of two solid
+phases (1st case), horizontal lines the presence of only one solid phase
+(2nd case), and slanting lines the presence of mixed crystals (3rd case).
+This method of representation cannot, however, be carried out in most
+cases. It is, however, {298} generally possible to find one pair or several
+pairs of components, the _relative amounts_ of which in the solution or in
+the precipitate undergo change when, and only when, the composition of the
+solution or of the precipitate changes. Thus, in the case of bismuth,
+nitrate, and water, we can represent the ratio of Bi_{2}O_{3} : N_{2}O_{5}
+in the precipitate as ordinates, and N_{2}O_{5} : H_{2}O in the solution as
+abscissæ. A horizontal line then indicates a single solid phase, and a
+vertical line two solid phases. An example of this is given in Fig.
+118.[372]
+
+[Illustration: FIG. 118.]
+
+Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O.--Although various systems have been
+studied in which there is formation of basic salts,[373] we shall content
+ourselves here with the description of some of the conditions for the
+formation of basic salts of bismuth nitrate, and for their equilibrium in
+contact with solutions.[374]
+
+Three normal salts of bismuth oxide and nitric acid are known, viz.
+Bi_{2}O_{3},3N_{2}O_{5},10H_{2}O(S_{10});
+Bi_{2}O_{3},3N_{2}O_{5},4H_{2}O(S_{4}); and
+Bi_{2}O_{3},3N_{2}O_{5},3H_{2}O(S_{3}). Besides these normal salts, there
+are the following basic salts:--
+
+{299}
+
+  Bi_{2}O_{3},N_{2}O_{5},2H_{2}O   (represented by B_{1-1-2})
+  Bi_{2}O_{3},N_{2}O_{5},H_{2}O    (     "      "  B_{1-1-1})
+  6Bi_{2}O_{3},5N_{2}O_{5},9H_{2}O (     "      "  B_{6-5-9})
+  2Bi_{2}O_{3},N_{2}O_{5},H_{2}O   (     "      "  B_{2-1-1})
+
+Probably some others also exist. The problem now is to find the conditions
+under which these different normal and basic salts can be in equilibrium
+with solutions of varying concentration of the three components. Having
+determined the equilibrium conditions for the different salts, it is then
+possible to construct a model similar to that for MgCl_{2}--KCl--H_{2}O or
+for FeCl_{3}--HCl--H_{2}O, from which it will be possible to determine the
+limits of stability of the different salts, and to predict what will occur
+when we bring the salts in contact with solutions of nitric acid of
+different concentrations and at different temperatures.
+
+For our present purpose it is sufficient to pick out only some of the
+equilibria which have been studied, and which are represented in the model
+(Fig. 119). In this case use has been made of the triangular method of
+representation, so that the surface of the model lies within the prism.
+
+[Illustration: FIG. 119.]
+
+This model shows the three surfaces, A, B, and C, which represent the
+conditions for the stable existence of the salts B_{1-1-1}, S_{10}, and
+S_{3} in contact with solution at different {300} temperatures. The front
+surface of the model represents the temperature 9°, and the farther end the
+temperature 75.5°. The dotted curve represents the isotherm for 20°. The
+prominences between the surfaces represent, of course, solutions which are
+saturated in respect of two solid phases. Thus, for example, _pabc_
+represents solutions in equilibrium with B_{1-1-1} and S_{10}; and the
+ridge _qdc_, solutions in equilibrium with S_{10} and S_{3}. The point _b_,
+which lies at 75.5°, is the point of maximum temperature for S_{10}. If the
+temperature is raised above this point, S_{10} decomposes into the basic
+salt B_{1-1-1} and solution. This point is therefore analogous to the point
+M in the carnallite model, at which this salt decomposes into potassium
+chloride and solution (p. 284); or to the point at which the salt
+2FeCl_{3},2HCl,12H_{2}O decomposes into 2FeCl_{3},12H_{2}O and solution
+(p. 294). The curve _pab_ has been followed to the temperature of 72°
+(point _c_). The end of the model is incomplete, but it is probable that in
+the neighbourhood of the point _c_ there exists a quintuple point at which
+the basic salt B_{1-2-2} appears. In the neighbourhood of _e_ also there
+probably exists another quintuple point at which S_{4} is formed. These
+systems have, however, not been studied.
+
+The following tables give some of the numerical data:--
+
+  ISOTHERM FOR 20°.
+
+  ----------------------------------------------------------------------
+                               | Composition of the solution. Gram-mols.
+                               | in 1000 gm.-mols. of water.
+  Solid phase.                 |----------------------------------------
+                               |    Bi_{2}O_{3}  |      N_{2}O_{5}
+  -----------------------------|-----------------|----------------------
+  B_{1-1-1}                    |      10.50      |       38.65
+  --                           |      27.20      |       83.84
+  B_{1-1-1}; S_{10}            |      30.15      |       97.97
+  S_{10}                       |      29.70      |       96.57
+  --                           |      19.65      |       98.76
+  --                           |      10.51      |      162.58
+  --                           |      33.51      |      355.87
+  S_{10}; S_{3}                |      51.00      |      403.0
+  S_{3}                        |      14.35      |      492.0
+  --                           |       7.45      |      592.9
+  ----------------------------------------------------------------------
+
+SYSTEMS IN EQUILIBRIUM WITH B_{1-1-1} AND S_{10} (CURVE _pabc_).
+
+  ------------------------------------------------------------
+                     | Composition of the solution. Gram-mols.
+                     |  in 1000 gm.-mols. of water.
+  Temperature.       |----------------------------------------
+                     |  Bi_{2}O_{3}  |  N_{2}O_{5}
+  -------------------|---------------|------------------------
+   9°                |    26.7       |     88.2
+   20° (point _a_)   |    30.15      |     97.97
+   30°               |    33.6       |    112.3
+   50°               |    41.8       |    148.4
+   65°               |    57.21      |    190.8
+   75.5° (point _b_) |    87.9       |    288.4
+   72° (point _c_)   |    96.0       |    327.0
+  ------------------------------------------------------------
+
+  SYSTEMS IN EQUILIBRIUM WITH S_{10} AND S_{3} (CURVE _qde_).
+
+  --------------------------------------------------------
+                | Composition of the solution. Gram-mols.
+                |  in 1000 gm.-mols. of water.
+  Temperature.  |-----------------------------------------
+                |  Bi_{2}O_{3}  |  N_{2}O_{5}
+  --------------|---------------|-------------------------
+   11.5°        |    44.5       |    396
+   20°          |    51.0       |    405.4
+   50°          |    66.5       |    444.2
+   65°          |    80.0       |    454.4
+  --------------------------------------------------------
+
+Basic Mercury Salts.--The Phase Rule has also been applied by A. J.
+Cox[375] in an investigation of the basic salts of mercury, the result of
+which has been to show that, of the salts mentioned in text-books, quite a
+number are incorrectly stated to be chemical compounds or chemical
+individuals (p. 92). The investigation, which was carried out essentially
+in the manner described above, included the salts mentioned in the
+following table; and of the basic salts said to be derived from them, only
+those mentioned really exist. In the following table, the numbers in the
+second column give the minimum values of the concentration of the acid,
+expressed in equivalent normality, necessary for the existence of the {301}
+corresponding salts in contact with solution at the temperature given in
+the third column:--
+
+  -------------------------------------------------------------
+               Salt.             | Normality of  | Temperature.
+                                 |     acid.     |
+  -------------------------------------------------------------
+  HgCrO_{4}                      |     1.41      |     50°
+  3HgO.CrO_{3}                   | 2.6 × 10^{-4} |     50°
+                                 |               |
+  Hg(NO_{3})_{2}.H_{2}O          |    18.72      |     25°
+  3HgO.N_{2}O_{5}                |     0.159     |     25°
+                                 |               |
+  HgSO_{4}                       |     6.87      |     25°
+  3HgO.SO_{3}                    | 1.3 × 10^{-3} |     25°
+                                 |               |
+  HgF_{2}                        |     1.14      |     25°
+                                 |               |
+  HgNO_{3}.H_{2}O                |     2.95      |     25°
+  5Hg_{2}O.3N_{2}O_{5}.2H_{2}O   | ca. 0.293     |     25°
+  2Hg_{2}O.N_{2}O_{5}(?)         |     0.110     |     25°
+  3Hg_{2}O.N_{2}O_{5}.2H_{2}O(?) | 1.7 × 10^{-3} |     25°
+                                 |               |
+  Hg_{2}SO_{4}                   | 4.2 × 10^{-3} |     25°
+  2Hg_{2}O.SO_{3}.H_{2}O         | 5.6 × 10^{-4} |     25°
+  -------------------------------------------------------------
+
+Mercuric fluoride does not form any basic salt.
+
+Since two succeeding members of a series can coexist only in contact with a
+solution of definite concentration, we can prepare acid solutions of
+definite concentration by bringing an excess of two such salts in contact
+with water.
+
+Indirect Determination of the Composition of the Solid Phase.--It has
+already been shown (p. 228) how the composition of the solid phase in a
+system of two components can be determined without analysis, and we shall
+now describe how this can be done in a system of three components.[376]
+
+We shall assume that we are dealing with the aqueous solution of two salts
+which can give rise to a double salt, in which case we can represent the
+solubility relations in a system of rectangular co-ordinates. In this case
+we should obtain, as before (Fig. 120), the isotherm _adcb_, if we express
+the {302} composition of the solution in gram-molecules of A or of B to 100
+gram-molecules of water.
+
+[Illustration: FIG. 120.]
+
+Let us suppose, now, that the double salt is in equilibrium with the
+solution at a definite temperature, and that the composition of the
+solution is represented by the point e. The greater part of the solution is
+now separated from the solid phase, and the latter, _together with the
+adhering mother liquor_, is analyzed. The composition (expressed, as
+before, in gram-molecules of A and B to 100 gram molecules of water) will
+be represented by a point (_e.g._ _f_) on the line _e_S, where S represents
+the composition of the double salt. That this is so will be evident when
+one considers that the composition of the whole mass must lie between the
+composition of the solution and that of the double salt, no matter what the
+relative amounts of the solid phase and the mother liquor.
+
+If, in a similar manner, we analyze a solution of a different composition
+in equilibrium with the same double salt (not necessarily at the same
+temperature as before), and also the mixture of solid phase and solution,
+we shall obtain two other points, as, for example, _g_ and _h_, and the
+line joining these must likewise pass through S. The method of finding the
+{303} composition of an unknown double salt consists, therefore, in
+finding, in the manner just described, the position of two lines such as
+_ef_ and _gh_. The point of intersection of these lines then gives the
+composition of the double salt.
+
+If the double salt is anhydrous, the point S lies at infinity, and the
+lines _ef_ and _gh_ are parallel to each other.
+
+The same result is arrived at by means of the triangular method of
+representation.[377] If we start with the three components in known
+amounts, and represent the initial composition of the whole by a point in
+the triangle, and then ascertain the final composition of the solution in
+equilibrium with the solid phase at a definite temperature, the line
+joining the points representing the initial and end concentration passes
+through the point representing the composition of the solid phase. If two
+determinations are made with solutions having different initial and final
+concentrations in equilibrium with the same solid phase, then the point of
+intersection of the two lines so obtained gives the composition of the
+solid phase.
+
+       *       *       *       *       *
+
+
+{304}
+
+CHAPTER XVII
+
+ABSENCE OF A LIQUID PHASE
+
+In the preceding chapters dealing with equilibria in three-component
+systems, our attention was directed only to those cases in which liquid
+solutions formed one or more phases. Mention must, however, be made of
+certain systems which contain no liquid phase, and in which only solids and
+gases are in equilibrium. Since, in all cases, there can be but one gas
+phase, four solid phases will be necessary in order to form an invariant
+system. When only three solid phases are present, the system is univariant;
+and when only two solid phases coexist with gas, it is bivariant. If,
+however, we make the restriction that the gas pressure is constant, we
+diminish the variability by one.
+
+On account of their great industrial importance, we shall describe briefly
+some of the systems belonging to this class.
+
+Iron, Carbon Monoxide, Carbon Dioxide.--Some of the most important systems
+of three components in which equilibrium exists between solid and gas
+phases are those formed by the three components--iron, carbon monoxide, and
+carbon dioxide--and they are of importance especially for the study of the
+processes occurring in the blast furnace.
+
+If carbon monoxide is passed over reduced iron powder at a temperature of
+about 600°, the iron is oxidized and the carbon monoxide reduced with
+separation of carbon in accordance with the equation
+
+  Fe + CO = FeO + C
+
+This reaction is succeeded by the two reactions
+
+  FeO + CO = Fe + CO_{2}
+    CO_{2} + C = 2CO
+
+{305}
+
+[Illustration: FIG. 121.]
+
+The former of these reactions is not complete, but leads to a definite
+equilibrium. The result of the different reactions is therefore an
+equilibrium between the three solid phases, carbon, iron, and ferrous
+oxide, and the gas phase consisting of carbon monoxide and dioxide. We have
+here four phases; and if the total pressure is maintained constant,
+equilibrium can occur only at a definite temperature.
+
+Since, under certain conditions, we can also have the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+{306} a second series of equilibria can be obtained of a character similar
+to the former. These various equilibria have been investigated by Baur and
+Glaessner,[378] and the following is a short account of the results of
+their work.
+
+Mixtures of the solid phases in equilibrium with carbon monoxide and
+dioxide were heated in a porcelain tube at a definite temperature until
+equilibrium was produced, and the gas was then pumped off and analyzed. The
+results which were obtained are given in the following tables, and
+represented graphically in Fig. 121.
+
+  SOLID PHASES: Fe_{3}O_{4}; FeO.
+
+  -----------------------------------------------------------------
+      |             | Duration of    |              | Percentage of
+  No. | Tube filled | the experiment | Temperature. |
+      |    with     | in hours.      |              | CO_{2} | CO
+  -----------------------------------------------------------------
+    1 |    CO       |       14       |     600°     |  59.3  | 40.7
+    2 |    CO       |       15       |     590°     |  54.7  | 45.3
+    3 |    CO_{2}   |       16       |     590°     |  64.6  | 35.4
+    4 |    CO       |       24       |     590°     |  58.4  | 41.6
+    5 |    CO       |       22       |     730°     |  67.7  | 32.3
+    6 |    CO_{2}   |       22       |     730°     |  86.1  | 31.9
+    7 |    CO       |       22       |     750°     |  68.4  | 31.6
+    8 |    CO_{2}   |       22       |     610°     |  64.9  | 35.1
+    9 |    CO       |       23       |     420°     |  56.0  | 44.0
+   10 |    CO       |       47       |     350°     |  65.6  | 34.4
+   11 |    CO_{2}   |       46       |     350°     |  72.8  | 27.2
+   12 |    CO       |       53       |     350°     |  64.0  | 36.0
+   13 |    CO       |       18       |     570°     |  53.4  | 46.6
+   14 |    CO       |       19       |     680°     |  60.5  | 39.5
+   15 |    CO_{2}   |       24       |     540°     |  55.5  | 44.5
+   16 |    CO       |       21       |     630°     |  57.5  | 42.5
+   17 |    CO_{2}   |       17       |     690°     |  65.5  | 34.5
+   18 |    CO_{2}   |       17       |     670°     |  67.0  | 33.0
+   19 |    CO_{2}   |       24       |     410°     |  58.5  | 41.5
+   20 |    CO       |       24       |     490°     |  51.7  | 48.8
+   21 |    CO_{2}   |       23       |     590°     |  54.4  | 45.6
+   22 |    CO_{2}   |        4       |     950°     |  77.0  | 23.0
+   23 |    CO_{2}   |       15       |     850°     |  73.4  | 26.6
+   24 |    CO       |        8       |     800°     |  71.2  | 28.8
+   25 |    CO_{2}   |       24       |     540°     |  56.7  | 43.3
+  -----------------------------------------------------------------
+
+{307}
+
+  SOLID PHASES: FeO; Fe.
+
+  ------+-------------+-------------+--------------+---------------
+        |             | Duration of |              | Percentage of
+   No.  | Tube filled | experiment  | Temperature. |
+        | with        | in hours.   |              | CO_{2} |  CO
+  ------+-------------+-------------+--------------+--------+------
+  I.    |    CO       |      15     |     800°     |  35.2  | 64.8
+  II.   |    CO       |      18     |     530°     |  29.1  | 70.9
+  III.  |    CO       |      13     |     880°     |  30.2  | 69.6
+  IV.   |    CO_{2}   |      24     |     870°     |  32.3  | 67.7
+  V.    |    CO       |      18     |     760°     |  36.9  | 63.1
+  VI.   |    CO_{2}   |      16     |     820°     |  34.7  | 65.3
+  VII.  |    CO_{2}   |      18     |     730°     |  41.1  | 58.9
+  VIII. |    CO       |      18     |     630°     |  34.9  | 65.1
+  IX.   |    CO_{2}   |      17     |     630°     |  61.6  | 58.4
+  X.    |    CO       |      18     |     540°     |  25.0  | 75.0
+  XI.   |    CO_{2}   |      25     |     540°     |  36.5  | 63.5
+  ------+-------------+-------------+--------------+--------+------
+
+As is evident from the above tables and from the curves in Fig. 121, the
+curve of equilibrium in the case of the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+exhibits a maximum for the ratio CO : CO_{2}, at 490°, while, for the
+reaction
+
+  FeO + CO = Fe + CO_{2}
+
+this ratio has a minimum value at 680°. From these curves can be derived
+the conditions under which the different solid phases can exist in contact
+with gas. Thus, for example, at a temperature of 690°, FeO and Fe_{3}O_{4}
+can coexist with a mixture of 65.5 per cent. of CO_{2} and 34.5 per cent.
+of CO. If the partial pressure of CO_{2} is increased, there occurs the
+reaction
+
+  3FeO + CO_{2} = Fe_{3}O_{4} + CO
+
+and if carbon dioxide is added in sufficient amount, the ferrous oxide
+finally disappears completely. If, on the other hand, the partial pressure
+of CO is increased, there occurs the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+and all the ferric oxide can be made to disappear. We see, therefore, that
+Fe_{3}O_{4} can only exist at temperatures and in {308} contact with
+mixtures of carbon monoxide and dioxide, represented by the area which lies
+below the under curve in Fig. 121. Similarly, the region of existence of
+FeO is that represented by the area between the two curves; while metallic
+iron can exist under the conditions of temperature and composition of gas
+phase represented by the area above the upper curve in Fig. 121. If,
+therefore, ferric oxide or metallic iron is heated for a sufficiently long
+time at temperatures above 700° (to the right of the dotted line; _vide
+infra_), complete transformation to ferrous oxide finally occurs.
+
+In another series of equilibria which can be obtained, carbon is one of the
+solid phases. In Fig. 121 the equilibria between carbon, carbon monoxide,
+and carbon dioxide under pressures of one and of a quarter atmosphere, are
+represented by dotted lines.[379]
+
+If we consider only the dotted line on the right, representing the
+equilibria under atmospheric pressure, we see that the points in which the
+dotted line cuts the other two curves must represent systems in which
+carbon monoxide and carbon dioxide are in equilibrium with FeO +
+Fe_{3}O_{4} + C, on the one hand, and with Fe + FeO + C on the other. These
+systems can only exist at one definite temperature, if we make the
+restriction that the pressure is maintained constant (atmospheric
+pressure). Starting, therefore, with the equilibrium FeO + Fe_{3}O_{4} + CO
++ CO_{2} at a temperature of about 670°, and then add carbon to the system,
+the reaction
+
+  C + CO_{2} = 2CO
+
+will occur, because the concentration of CO_{2} is greater than what
+corresponds with the system FeO + Fe_{3}O_{4} + C in equilibrium with
+carbon monoxide and dioxide. In consequence of this reaction, the
+equilibrium between FeO + Fe_{3}O_{4} and the gas phase is disturbed, and
+the change in the composition of the gas phase is opposed by the reaction
+Fe_{3}O_{4} + CO = 3FeO + CO_{2}, which continues until either all the
+carbon {309} or all the ferric oxide is used up. If the ferric oxide first
+disappears, the equilibrium corresponds with a point on the dotted line in
+the middle area of Fig. 121, which represents equilibria between FeO + C as
+solid phases, and a mixture of carbon monoxide and dioxide as gas phase. If
+the temperature is higher than 685°, at which temperature the curve for
+C--CO--CO_{2} cuts that for Fe--FeO--CO--CO_{2}; then, when all the ferric
+oxide has disappeared, the concentration of CO_{2} is still too great for
+the coexistence of FeO and C. Consequently, there occurs the reaction C +
+CO_{2} = 2CO, and the composition of the gas phase alters until a point on
+the upper curve is reached. A further increase in the concentration of CO
+is opposed by the reaction FeO + CO = Fe + CO_{2}, and the pressure remains
+constant until all the ferrous oxide is reduced and only iron and carbon
+remain in equilibrium with gas. If the quantities of the substances have
+been rightly chosen, we ultimately reach a point on the dotted curve in the
+upper part of Fig. 121.
+
+Fig. 121 shows us, also, what are the conditions under which the reduction
+of ferric to ferrous oxide by carbon can occur. Let us suppose, for
+example, that we start with a mixture of carbon monoxide and dioxide at
+about 600° (the lowest point on the dotted line), and maintain the total
+pressure constant and equal to one atmosphere. If the temperature is
+increased, the concentration of the carbon dioxide will diminish, owing to
+the reaction C + CO_{2} = 2CO, but the ferric oxide will undergo no change
+until the temperature reaches 647°, the point of intersection of the dotted
+curve with the curve for FeO and Fe_{3}O_{4}. At this point further
+increase in the concentration of carbon monoxide is opposed by the
+reduction of ferric oxide in accordance with the equation Fe_{3}O_{4} + CO
+= 3FeO + CO_{2}. The pressure, therefore, remains constant until all the
+ferric oxide has disappeared. If the temperature is still further raised,
+we again obtain a univariant system, FeO + C, in equilibrium with gas
+(univariant because the total pressure is constant); and if the temperature
+is raised the composition of the gas must undergo change. This is effected
+by the reaction C + CO_{2} = 2CO. When the {310} temperature rises to 685°,
+at which the dotted curve cuts the curve for Fe--FeO, further change is
+prevented by the reaction FeO + CO = Fe + CO_{2}. When all the ferrous
+oxide is used up, we obtain the system Fe + C in equilibrium with gas. If
+the temperature is now raised, the composition of the gas undergoes change,
+as shown by the dotted line. The two temperatures, 647° and 685°, give,
+evidently, the limits within which ferric or ferrous oxide can be reduced
+directly by carbon.
+
+It is further evident that at any temperature to the right of the dotted
+line, carbon is unstable in presence of iron or its oxides; while at
+temperatures lower than those represented by the dotted line, it is stable.
+In the blast furnace, therefore, separation of carbon can occur only at
+lower temperatures, and the carbon must disappear on raising the
+temperature.
+
+Finally, it may be remarked that the equilibrium curves show that ferrous
+oxide is most easily reduced at 680°, since the concentration of the carbon
+monoxide required at this temperature is a minimum. On the other hand,
+ferric oxide is reduced with greatest difficulty at 490°, since at this
+temperature the requisite concentration of carbon monoxide is a maximum.
+
+Other equilibria between solid and gas phases are: Equilibrium between
+iron, ferric oxide, water vapour, and hydrogen,[380] and the equilibria
+between carbon, carbon monoxide, carbon dioxide, water vapour, and
+hydrogen,[381] which is of importance for the manufacture of water gas.
+
+       *       *       *       *       *
+
+
+{311}
+
+CHAPTER XVIII
+
+SYSTEMS OF FOUR COMPONENTS
+
+In the systems which have so far been studied, we have met with cases where
+two or three components could enter into combination; but in no case did we
+find double decomposition occurring. The reason of this is that in the
+systems previously studied, in which double decomposition might have been
+possible, namely in those systems in which two salts acted as components,
+the restriction was imposed that either the basic or the acid constituent
+of these salts must be the same; a restriction imposed, indeed, for the
+very purpose of excluding double decomposition. Now, however, we shall
+allow this restriction to fall, thereby extending the range of study.
+
+Hitherto, in connection with four-component systems, the attention has been
+directed solely to the study of aqueous solutions of salts, and more
+especially of the salts which occur in sea-water, _i.e._ chiefly, the
+sulphates and chlorides of magnesium, potassium, and sodium. The importance
+of these investigations will be recognized when one recollects that by the
+evaporation of sea-water there have been formed the enormous salt-beds at
+Stassfurt, which constitute at present the chief source of the sulphates
+and chlorides of magnesium and potassium. The investigations, therefore,
+are not only of great geological interest as tending to elucidate the
+conditions under which these salt-beds have been formed, but are of no less
+importance for the industrial working of the deposits.
+
+It is, however, not the intention to enter here into any detailed
+description of the different systems which have so far been studied, and of
+the sometimes very complex relationships {312} met with, but merely to
+refer briefly to some points of more general import in connection with
+these systems.[382]
+
+Reciprocal Salt-Pairs. Choice of Components.--When two salts undergo double
+decomposition, the interaction can be expressed by an equation such as
+
+  NH_{4}Cl + NaNO_{3} = NaCl + NH_{4}NO_{3}
+
+Since one pair of salts--NaCl + NH_{4}NO_{3}--is formed from the other
+pair--NH_{4}Cl + NaNO_{3}--by double decomposition, the two pairs of salts
+are known as _reciprocal salt-pairs_.[383] It is with systems in which the
+component salts form reciprocal salt-pairs that we have to deal here.
+
+It must be noted, however, that the four salts formed by two reciprocal
+salt-pairs do not constitute a system of four, but only of _three_
+components. This will be understood if it is recalled that only so many
+constituents are taken as components as are necessary to _express_ the
+composition of all the phases present (p. 12). It will be seen, now, that
+the composition of each of the four salts which can be present together can
+be expressed in terms of three of them. Thus, for example, in the case of
+NH_{4}Cl, NaNO_{3}, NH_{4}NO_{3}, NaCl, we can express the composition of
+NH_{4}Cl by NH_{4}NO_{3} + NaCl - NaNO_{3}; or of NaNO_{3} by NH_{4}NO_{3}
++ NaCl - NH_{4}Cl. In all these cases it will be seen that negative
+quantities of one of the components must be employed; but that we have seen
+to be quite permissible (p. 12). The number of components is, therefore,
+three; but any three of the four salts can be chosen.
+
+Since, then, two reciprocal salt-pairs constitute only three {313}
+components or independently variable constituents, another component is
+necessary in order to obtain a four-component system. As such, we shall
+choose water.
+
+Transition Point.--In the case of the formation of double salts from two
+single salts, we saw that there was a point--the _quintuple point_--at
+which five phases could coexist. This point we also saw to be a transition
+point, on one side of which the double salt, on the other side the two
+single salts in contact with solution, were found to be the stable system.
+A similar behaviour is found in the case of reciprocal salt-pairs. The
+four-component system, two reciprocal salt-pairs and water, can give rise
+to an invariant system in which the six phases, four salts, solution,
+vapour, can coexist; the temperature at which this is possible constitutes
+a _sextuple point_. Now, this sextuple point is also a transition point, on
+the one side of which the one salt-pair, on the other side the reciprocal
+salt-pair, is stable in contact with solution.
+
+The sextuple point is the point of intersection of the curves of six
+univariant systems, viz. four solubility curves with three solid phases
+each, a vapour-pressure curve for the system: two reciprocal
+salt-pairs--vapour; and a transition curve for the condensed system: two
+reciprocal salt-pairs--solution. If we omit the vapour phase and work under
+atmospheric pressure (in open vessels), we find that the transition point
+is the point of intersection of four solubility curves.
+
+Just as in the case of three-component systems we saw that the presence of
+one of the single salts along with the double salt was necessary in order
+to give a univariant system, so in the four-component systems the presence
+of a third salt is necessary as solid phase along with one of the
+salt-pairs. In the case of the reciprocal salt-pairs mentioned above, the
+transition point would be the point of intersection of the solubility
+curves of the systems with the following groups of salts as solid phases:
+Below the transition point: NH_{4}Cl + NaNO_{3} + NaCl; NH_{4}Cl + NaNO_{3}
++ NH_{4}NO_{3}; above the transition point: NaCl + NH_{4}NO_{3} + NaNO_{3};
+NaCl + NH_{4}NO_{3} + NH_{4}Cl. From this we see that the two salts
+NH_{4}Cl and NaNO_{3} would be able to exist together with solution below
+the transition point, but not above it. This transition point has not been
+determined. {314}
+
+Formation of Double Salts.--In all cases of four-component systems so far
+studied, the transition points have not been points at which one salt-pair
+passed into its reciprocal, but at which a double salt was formed. Thus, at
+4.4° Glauber's salt and potassium chloride form glaserite and sodium
+chloride, according to the equation
+
+  2Na_{2}SO_{4},10H_{2}O + 3KCl = K_{3}Na(SO_{4})_{2} + 3NaCl + 20H_{2}O
+
+Above the transition point, therefore, there would be K_{3}Na(SO_{4})_{2},
+NaCl and KCl; and it may be considered that at a higher temperature the
+double salt would interact with the potassium chloride according to the
+equation
+
+  K_{3}Na(SO_{4})_{2} + KCl = 2K_{2}SO_{4} + NaCl
+
+thus giving the reciprocal of the original salt-pair. This point has,
+however, not been experimentally realized.[384]
+
+Transition Interval.--A double salt, we learned (p. 277), when brought in
+contact with water at the transition point undergoes partial decomposition
+with separation of one of the constituent salts; and only after a certain
+range of temperature (transition interval) has been passed, can a pure
+saturated solution be obtained. A similar behaviour is also found in the
+case of reciprocal salt-pairs. If one of the salt-pairs is brought in
+contact with water at the transition point, interaction will occur and one
+of the salts of the reciprocal salt-pair will be deposited; and this will
+be the case throughout a certain range of temperature, after which it will
+be possible to prepare a solution saturated only for the one salt-pair. In
+the case of ammonium chloride and sodium nitrate the lower limit of the
+transition interval is 5.5°, so that above this temperature and up to that
+of the transition point (unknown), ammonium chloride and sodium nitrate in
+contact with water would give rise to a third salt by double decomposition,
+in this case to sodium chloride.[385]
+
+{315}
+
+Graphic Representation.--For the graphic representation of systems of four
+components, four axes may be chosen intersecting at a point like the edges
+of a regular octahedron (Fig. 122).[386] Along these different axes the
+equivalent molecular amounts of the different salts are measured.
+
+[Illustration: FIG. 122.]
+
+[Illustration: FIG. 123.]
+
+To represent a given system consisting of _x_B, _y_C, and _z_D in a given
+amount of water (where B, C, and D represent equivalent molecular amounts
+of the salts), measure off on OB and OC lengths equal to _x_ and _y_
+respectively. The point of intersection _a_ (Fig. 122) represents a
+solution containing _x_B and _y_C (_ab_ = _x_; _ac_ = _y_). From _a_ a line
+_a_P is drawn parallel to OD and equal to _z_. P then represents the
+solution of the above composition.
+
+It is usual, however, not to employ the three-dimensional figure, but its
+horizontal and vertical projections. Fig. 122, if projected on the base of
+the octahedron, would yield a diagram such as is shown in Fig. 123. The
+projection of the edges of the octahedron form two axes at right angles and
+give rise to four quadrants similar to those employed for the
+representation of ternary solutions (p. 273). Here, the point _a_
+represents a ternary solution saturated with respect to B and C; and _a_P,
+quaternary solutions in equilibrium with the same two salts as solid
+phases. Such a diagram represents the conditions of equilibrium only for
+one definite temperature, and corresponds, therefore, to the isothermal
+diagrams for ternary systems (p. 273). In such a diagram, since the
+temperature and {316} pressure are constant (vessels open to the air), a
+surface will represent a solution in equilibrium with only one solid phase;
+a line, a solution with two solid phases, and a point, one in equilibrium
+with three solid phases.
+
+[Illustration: FIG. 124.]
+
+Example.--As an example of the complete isothermal diagram, there may be
+given one representing the equilibria in the system composed of water and
+the reciprocal salt-pair sodium sulphate--potassium chloride for the
+temperature 0° (Fig. 124).[387] The amounts of the different salts are
+measured along the four axes, and the composition of the solution is {317}
+expressed in equivalent gram-molecules per 1000 gram-molecules of
+water.[388]
+
+The outline of this figure represents four ternary solutions in which the
+component salts have a common acid or basic constituent; viz. sodium
+chloride--sodium sulphate, sodium sulphate--potassium sulphate, potassium
+sulphate--potassium chloride, potassium chloride--sodium chloride. These
+four sets of curves are therefore similar to those discussed in the
+previous chapter. In the case of sodium and potassium sulphate, a double
+salt, _glaserite_ [K_{3}Na(SO_{4})_{2}] is formed. Whether glaserite is
+really a definite compound or not is still a matter of doubt, since
+isomorphic mixtures of Na_{2}SO_{4} and K_{2}SO_{4} have been obtained.
+According to van't Hoff and Barscholl,[389] glaserite is an isomorphous
+mixture; but Gossner[390] considers it to be a definite compound having the
+formula K_{3}Na(SO_{4})_{2}. Points VIII. and IX. represent solutions
+saturated with respect to glaserite and sodium sulphate, and glaserite and
+potassium sulphate respectively.
+
+The lines which pass inwards from these boundary curves represent solutions
+containing three salts, but in contact with only two solid phases; and the
+points where three lines meet, or where three fields meet, represent
+solutions in equilibrium with three solid phases; with the phases, namely,
+belonging to the three concurrent fields.
+
+If it is desired to represent a solution containing the salts say in the
+proportions, 51Na_{2}Cl_{2}, 9.5K_{2}Cl_{2}, 3.5K_{2}SO_{4}, the difficulty
+is met with that two of the salts, sodium chloride and potassium sulphate,
+lie on opposite axes. To overcome this difficulty the difference 51 - 3.5 =
+47.5 is taken and measured off along the sodium chloride axis; and the
+solution is therefore represented by the point 47.5Na_{2}Cl_{2},
+9.5K_{2}Cl_{2}. In order, therefore, to find the amount of potassium
+sulphate present {318} from such a diagram, it is necessary to know the
+total number of salt molecules in the solution. When this is known, it is
+only necessary to subtract from it the sum of the molecules of sodium and
+potassium chloride, and the result is equal to twice the number of
+potassium sulphate molecules. Thus, in the above example, the total number
+of salt molecules is 64. The number of molecules of sodium and potassium
+chloride is 57; 64 - 57 = 7, and therefore the number of potassium sulphate
+molecules is 3.5.
+
+Another method of representation employed is to indicate the amounts of
+only two of the salts in a plane diagram, and to measure off the total
+number of molecules along a vertical axis. In this way a solid model is
+obtained.
+
+The numerical data from which Fig. 124 was constructed are contained in the
+following table, which gives the composition of the different solutions at
+0°:--[391]
+
+  ----------------------------------------
+         |                               |
+         |                               |
+  Point. |       Solid phases.           |
+         |                               |
+  ----------------------------------------
+      I. | NaCl                          |
+         |                               |
+     II. | KCl                           |
+         |                               |
+    III. | Na_{2}SO_{4},10H_{2}O         |
+         |                               |
+     IV. | K_{2}SO_{4}                   |
+         |                               |
+      V. | NaCl; KCl                     |
+         |                               |
+     VI. | NaCl; Na_{2}SO_{4},10H_{2}O   |
+         |                               |
+    VII. | KCl; K_{2}SO_{4}              |
+         |                               |
+   VIII. |{ Glaserite;                  }|
+         |{ Na_{2}SO_{4},10H_{2}O       }|
+         |                               |
+     IX. |  Glaserite; K_{2}SO_{4}       |
+         |                               |
+      X. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
+         |{ NaCl                        }|
+         |                               |
+     XI. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
+         |{ glaserite                   }|
+         |                               |
+    XII. |  K_{2}SO_{4}; KCl; glaserite  |
+  ----------------------------------------
+  [Transcriber's note: table continued below...]
+  -------------------------------------------------------------------------
+   Composition of solution in gram-mols.                       | Total
+   per 1000 gram-mols. water.                                  | number
+  -------------------------------------------------------------| of salt
+   Na_{2}Cl_{2}. | K_{2}Cl_{2}. | Na_{2}SO_{4}. | K_{2}SO_{4}. | molecules.
+  -------------------------------------------------------------------------
+        55       |      --      |      --       |      --      |   55
+                 |              |               |              |
+         --      |     34.5     |      --       |      --      |   34.5
+                 |              |               |              |
+         --      |      --      |      6        |      --      |    6
+                 |              |               |              |
+         --      |      --      |      --       |      9       |    9
+                 |              |               |              |
+        46.5     |     12.5     |      --       |      --      |   59
+                 |              |               |              |
+        47.5     |      --      |      8        |      --      |   55.5
+                 |              |               |              |
+         --      |     34.5     |      --       |      1       |   35.5
+                 |              |               |              |
+         --      |      --      |     10        |     10       |   20
+                 |              |               |              |
+                 |              |               |              |
+         --      |      --      |      7.5      |     10       |   17.5
+                 |              |               |              |
+        51       |      9.5     |      --       |      3.5     |   64
+                 |              |               |              |
+                 |              |               |              |
+        40.5     |     13       |      --       |      3.5     |   57
+                 |              |               |              |
+                 |              |               |              |
+        18       |     23       |      --       |      3       |   44
+  -------------------------------------------------------------------------
+
+From the aspect of these diagrams the conditions under which the salts can
+coexist can be read at a glance. Thus, {319} for example, Fig. 124 shows
+that at 0° Glauber's salt and potassium chloride can exist together with
+solution; namely, in contact with solutions having the composition X--XI.
+This temperature must therefore be below the transition point of this
+salt-pair (p. 314). On raising the temperature to 4.4°, it is found that
+the curve VIII.--XI. moves so that the point XI. coincides with point X. At
+this point, therefore, there will be _four_ concurrent fields, viz.
+Glauber's salt, potassium chloride, glaserite, and sodium chloride. But
+these four salts can coexist with solution only at the transition point; so
+that 4.4° is the transition temperature of the salt-pair: Glauber's
+salt--potassium chloride. At higher temperatures the line VIII.--XI. moves
+still further to the left, so that the field for Glauber's salt becomes
+entirely separated from the field for potassium chloride. This shows that
+at temperatures above the transition point the salt-pair Glauber's
+salt--potassium chloride cannot coexist in presence of solution.
+
+[Illustration: FIG. 125.]
+
+If it is only desired to indicate the mutual relationships of the different
+components and the conditions for their coexistence (_paragenesis_), a
+simpler diagram than Fig. 124 can be employed. Thus if the boundary curves
+of Fig. 124 are so drawn that they cut one another at right angles, a
+figure such as Fig. 125 is obtained, the Roman numerals here corresponding
+with those in Fig. 124.
+
+Ammonia-Soda Process.--One of the most important applications of the Phase
+Rule to systems of four components with reciprocal salt-pairs has recently
+been made by Fedotieff[392] in his investigations of the conditions for the
+formation of sodium carbonate by the so-called ammonia-soda (Solvay) {320}
+process.[393] This process consists, as is well known, in passing carbon
+dioxide through a solution of common salt saturated with ammonia.
+
+Whatever differences of detail there may be in the process as carried out
+in different manufactories, the reaction which forms the basis of the
+process is that represented by the equation
+
+  NaCl + NH_{4}HCO_{3} = NaHCO_{3} + NH_{4}Cl
+
+We are dealing here, therefore, with reciprocal salt-pairs, the behaviour
+of which has just been discussed in the preceding pages. The present case
+is, however, simpler than that of the salt-pair Na_{2}SO_{4}.10H_{2}O +
+KCl, inasmuch as under the conditions of experiment neither hydrates nor
+double salts are formed. Since the study of the reaction is rendered more
+difficult on account of the fact that ammonium bicarbonate in solution,
+when under atmospheric pressure, undergoes decomposition at temperatures
+above 15°, this temperature was the one chosen for the detailed
+investigation of the conditions of equilibrium. Since, further, it has been
+shown by Bodländer[394] that the bicarbonates possess a definite solubility
+only when the pressure of carbon dioxide in the solution has a definite
+value, the measurements were carried out in solutions saturated with this
+gas. This, however, does not constitute another component, because we have
+made the restriction that the sum of the partial pressures of carbon
+dioxide and water vapour is equal to 1 atmosphere. The concentration of the
+carbon dioxide is, therefore, not independently variable (p. 10).
+
+[Illustration: FIG. 126.]
+
+In order to obtain the data necessary for a discussion of the conditions of
+soda formation by the ammonia-soda process, solubility determinations with
+the four salts, NaCl, NH_{4}Cl, NH_{4}HCO_{3}, and NaHCO_{3} were made,
+first with the single salts and then {321} with the salts in pairs. The
+results obtained are represented graphically in Fig. 126, which is an
+isothermal diagram similar to that given by Fig. 124. The points I., II.,
+III., IV., represent the composition of solutions in equilibrium with two
+solid salts. We have, however, seen (p. 314) that the transition point,
+when the experiment is carried out under constant pressure (atmospheric
+pressure), is the point of intersection of four solubility curves, each of
+which represents the composition of solutions in equilibrium with three
+salts, viz. one of the reciprocal salt-pairs along with a third salt.
+Since, now, it was found that the stable salt-pair at temperatures between
+0° and 30° is sodium bicarbonate and ammonium chloride, determinations were
+made of the composition of solutions in equilibrium with NaHCO_{3} +
+NH_{4}Cl + NH_{4}HCO_{3} and with NaHCO_{3} + NH_{4}Cl + NaCl as solid
+phases. Under the {322} conditions of experiment (temperature = 15°) sodium
+chloride and ammonium bicarbonate cannot coexist in contact with solution.
+These determinations gave the data necessary for the construction of the
+complete isothermal diagram (Fig. 127). The most important of these data
+are given in the following table (temperature, 15°):--
+
+  -------------------------------------------------------------------------
+         |                  | Composition of the solution in gram-molecules
+         |                  | to 1000 gram-molecules
+  Point. |  Solid phases.   | of water.
+         |                  |----------------------------------------------
+         |                  | NaHCO_{3} | NaCl | NH_{4}HCO_{3} | NH_{4}Cl
+  -------------------------------------------------------------------------
+   --    | NaHCO_{3}        |    1.08   |  --  |       --      |     --
+   --    | NaCl             |     --    | 6.12 |       --      |     --
+   --    | NH_{4}HCO_{3}    |     --    |  --  |      2.36     |     --
+   --    | NH_{4}Cl         |     --    |  --  |       --      |    6.64
+   I.    | NaHCO_{3}; NaCl  |    0.12   | 6.06 |       --      |     --
+   II.   | NaCl; NH_{4}Cl   |     --    | 4.55 |       --      |    3.72
+   III.  | NH_{4}Cl;        |     --    |  --  |      0.81     |    6.40
+         | NH_{4}HCO_{3}    |           |      |               |
+   IV.   | NaHCO_{3};       |    0.71   |  --  |      2.16     |     --
+         | NH_{4}HCO_{3}    |           |      |               |
+   P_{1} | NaHCO_{3};       |    0.93   | 0.51 |       --      |    6.28
+         | NH_{4}HCO_{3};   |           |      |               |
+         | NH_{4}Cl         |           |      |               |
+   P_{2} | NaHCO_{3};       |    0.18   | 4.44 |       --      |    3.73
+         | NaCl; NH_{4}Cl   |           |      |               |
+  -------------------------------------------------------------------------
+
+With reference to the solution represented by the point P_{1}, it may be
+remarked that it is an incongruently saturated solution (p. 279). If sodium
+chloride is added to this solution, the composition of the latter undergoes
+change; and if a sufficient amount of the salt is added, the solution P_{2}
+is obtained.
+
+Turning now to the practical application of the data so obtained, consider
+first what is the influence of concentration on the yield of soda. Since
+the reaction consists essentially in a double decomposition between sodium
+chloride and ammonium bicarbonate, then, after the deposition of the sodium
+bicarbonate, we obtain a solution containing sodium chloride, ammonium
+chloride, and sodium bicarbonate. In order to ascertain to what extent the
+sodium chloride has been converted into solid sodium bicarbonate, it is
+necessary to examine the composition of the solution which is obtained
+{323} with definite amounts of sodium chloride and ammonium bicarbonate.
+
+[Illustration: FIG. 127.]
+
+Consider, in the first place, the solutions represented by the curve
+P_{2}P_{1}. With the help of this curve we can state the conditions under
+which a solution, saturated for ammonium chloride, is obtained, after
+deposition of sodium bicarbonate. In the following table the composition of
+the solutions is given which are obtained with different initial amounts of
+sodium chloride and ammonium bicarbonate. The last two columns give the
+percentage amount of the sodium used, which is deposited as solid sodium
+bicarbonate (U_{Na}); and likewise the percentage amount of ammonium
+bicarbonate which is usefully converted into sodium bicarbonate, that is to
+say, the amount of the radical HCO_{3} deposited (U_{NH_{4}}):-- {324}
+
+  ------+---------------------+
+        |Initial composition  |
+        |of the solutions:    |
+        |grams of salt to 1000|
+  Point.|grams of water.      |
+        +------+--------------+
+        | NaCl | NH_{4}HCO_{3}|
+  ------+------+--------------+
+  P_{2} | 479  |     295      |
+  --    | 448  |     360      |
+  --    | 417  |     431      |
+  P_{1} | 397  |     496      |
+  ------+------+--------------+
+  [Transcriber's note: table continued below...]
+  +----------------------------------+---------+----------
+  |                                  |         |
+  |Composition of solutions obtained:|         |
+  |gram-equivalents per 1000 grams   |U_{Na}   |U_{NH_{4}}
+  |of water.                         |per cent.|per cent.
+  +----------+------+------+---------+         |
+  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |         |
+  +----------+------+------+---------+---------+----------
+  |   0.18   | 8.17 | 4.62 |  3.73   |  43.4   |   95.1
+  |   0.31   | 7.65 | 3.39 |  4.56   |  55.7   |   93.4
+  |   0.51   | 7.13 | 2.19 |  5.45   |  69.2   |   90.5
+  |   0.92   | 6.79 | 1.44 |  6.28   |  78.8   |   85.1
+  +----------+------+------+---------+---------+----------
+
+This table shows that the greater the excess of sodium chloride, the
+greater is the percentage utilization of ammonia (Point P_{2}); and the
+more the amount of sodium chloride decreases, the greater is the percentage
+amount of sodium chloride converted into bicarbonate. In the latter case,
+however, the percentage utilization of the ammonium bicarbonate decreases;
+that is to say, less sodium bicarbonate is deposited, or more of it remains
+in solution.
+
+Consider, in the same manner, the relations for solutions represented by
+the curve P_{2}IV, which gives the composition of solutions saturated with
+respect to sodium bicarbonate and ammonium bicarbonate. In this case we
+obtain the following results:--
+
+  ------+---------------------+
+        |Initial composition  |
+        |of the solutions:    |
+        |grams of salt to 1000|
+  Point.|grams of water.      |
+        +------+--------------+
+        | NaCl | NH_{4}HCO_{3}|
+  ------+------+--------------+
+  P_{1} | 397  |     496      |
+  --    | 351  |     446      |
+  --    | 316  |     412      |
+  --    | 294  |     389      |
+  --    | 234  |     327      |
+  ------+------+--------------+
+  [Transcriber's note: table continued below...]
+  +----------------------------------+------+----------
+  |                                  |      |
+  |Composition of solutions obtained:|      |
+  |in gram-equivalents per 1000 grams|U_{Na}|U_{NH_{4}}
+  |of water.                         |      |
+  +----------+------+------+---------+      |
+  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |      |
+  +----------+------+------+---------+------+----------
+  |   0.92   | 6.79 | 1.44 |  6.28   | 78.8 |   85.1
+  |   0.99   | 6.00 | 1.34 |  5.65   | 77.7 |   82.5
+  |   1.07   | 5.41 | 1.27 |  5.21   | 76.4 |   79.5
+  |   1.12   | 5.03 | 1.23 |  4.92   | 75.5 |   75.1
+  |   1.30   | 4.00 | 1.16 |  4.14   | 71.0 |   68.6
+  +----------+------+------+---------+------+----------
+
+As is evident from this table, diminution in the relative amount of sodium
+chloride exercises only a slight influence {325} on the utilization of this
+salt, but is accompanied by a rapid diminution of the effective
+transformation of the ammonium bicarbonate. So far as the efficient
+conversion of the sodium is concerned, we see that it reaches its maximum
+at the point P_{1}, and that it decreases both with increase and with
+decrease of the relative amount of sodium chloride employed; and faster,
+indeed, in the former than in the latter case. On the other hand, the
+effective transformation of the ammonium bicarbonate reaches its maximum at
+the point P_{2}, and diminishes with increase in the relative amount of
+ammonium bicarbonate employed. Since sodium chloride is, in comparison with
+ammonia--even when this is regenerated--a cheap material, it is evidently
+more advantageous to work with solutions which are relatively rich in
+sodium chloride (solutions represented by the curve P_{1}P_{2}). This fact
+has also been established empirically.
+
+When, as is the case in industrial practice, we are dealing with solutions
+which are saturated not for two salts but only for sodium bicarbonate, it
+is evident that we have then to do with solutions the composition of which
+is represented by points in the area P_{1}P_{2}I,IV. Since in the
+commercial manufacture, the aim must be to obtain as complete a utilization
+of the materials as possible, the solutions employed industrially must lie
+in the neighbourhood of the curves P_{2}P_{1}IV, as is indicated by the
+shaded portion in Fig. 127. The best results, from the manufacturer's
+standpoint, will be obtained, as already stated, when the composition of
+the solutions approaches that given by a point on the curve P_{2}P_{1}.
+Considered from the chemical standpoint, the results of the experiments
+lead to the conclusion that the Solvay process, _i.e._ passage of carbon
+dioxide through a solution of sodium chloride saturated with ammonia, is
+not so good as the newer method of Schlösing, which consists in bringing
+together sodium chloride and ammonium bicarbonate with water.[395]
+
+{326}
+
+Preparation of Barium Nitrite.--Mention may also be made here of the
+preparation of barium nitrite by double decomposition of barium chloride
+and sodium nitrite.[396]
+
+The reaction with which we are dealing here is represented by the equation
+
+  BaCl_{2} + 2NaNO_{2} = 2NaCl + Ba(NO_{2})_{2}
+
+It was found that at the ordinary temperature NaCl and Ba(NO_{2})_{2} form
+the stable salt-pair. If, therefore, barium chloride and sodium nitrite are
+brought together with an amount of water insufficient for complete
+solution, transformation to the stable salt-pair occurs, and sodium
+chloride and barium nitrite are deposited. When, however, a stable
+salt-pair is in its transition interval (p. 315), a third salt--in this
+case barium chloride--will be deposited, as we have already learned. On
+bringing barium chloride and sodium nitrite together with water, therefore,
+three solid phases are obtained, viz. BaCl_{2}, NaCl, Ba(NO_{2})_{2}. These
+three phases, together with solution and vapour, constitute a univariant
+system, so that at each temperature the composition of the solution must be
+constant.
+
+Witt and Ludwig found that the presence of solid barium chloride can be
+prevented by adding an excess of sodium nitrite, as can be readily foreseen
+from what has been said. Since the solution in presence of the three solid
+phases must have a definite composition at a definite temperature, the
+addition of sodium nitrite to the solution must have, as its consequence,
+the solution of an equivalent amount of barium chloride, and the deposition
+of an equivalent amount of sodium chloride and barium nitrite. By
+sufficient addition of sodium nitrite, the complete disappearance of the
+solid barium chloride can be effected, and there will remain only the
+stable salt-pair sodium chloride and barium nitrite. As was pointed out by
+Meyerhoffer, however, the disappearance of the barium chloride is effected,
+not by a change in the {327} composition of the solution, but by the
+necessity for the composition of the solution remaining constant.
+
+[Illustration: FIG. 128.]
+
+Barium Carbonate and Potassium Sulphate.--As has been found by
+Meyerhoffer,[397] these two salts form the stable pair, not only at the
+ordinary temperature, but also at the melting point. For the ordinary
+temperatures this was proved in the following manner: A solution with the
+solid phases K_{2}SO_{4} and K_{2}CO_{3}.2H_{2}O in excess can only coexist
+in contact either with BaCO_{3} or with BaSO_{4}, since, evidently, in one
+of the two groups the stable system must be present. Two solutions were
+prepared, each with excess of K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O, {328} and
+to one was added BaCO_{3} and to the other BaSO_{4}. After stirring for a
+few days, the barium sulphate was completely transformed to BaCO_{3},
+whereas the barium carbonate remained unchanged. Consequently, BaCO_{3} +
+K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O is stable, and, therefore, so also is
+BaCO_{3} + K_{2}SO_{4}. That BaCO_{3} + K_{2}SO_{4} is the stable pair also
+at the melting point was proved by a special analytical method which allows
+of the detection of K_{2}CO_{3} in a mixture of the four solid salts. This
+analysis showed that a mixture of BaCO_{3} + K_{2}SO_{4}, after being fused
+and allowed to solidify, contains only small amounts of K_{2}CO_{3}; and
+this is due entirely to the fact that BaCO_{3} + K_{2}SO_{4} on fusion
+deposits a little BaSO_{4}, thereby giving rise at the same time to the
+separation of an equivalent amount of K_{2}CO_{3}.
+
+The different solubilities are shown in Fig. 128. In this diagram the
+solubility of the two barium salts has been neglected. A is the solubility
+of K_{2}CO_{3}.2H_{2}O; addition of BaCO_{3} does not alter this. B is the
+solubility of K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} + BaCO_{3}. A and B almost
+coincide, since the potassium sulphate is very slightly soluble in the
+concentrated solution of potassium carbonate. D gives the concentration of
+the solution in equilibrium with K_{2}SO_{4} + BaSO_{4}. The most
+interesting point is C. This solution is obtained by adding a small
+quantity of water to BaCO_{3} + K_{2}SO_{4}, whereupon, being in the
+transition interval, BaSO_{4} separates out and an equivalent amount of
+K_{2}CO_{3} goes into solution. C is the end point of the curve CO, which
+is called the Guldberg-Waage curve, because these investigators determined
+several points on it.
+
+In their experiments, Guldberg and Waage found the ratio K_{2}CO_{3} :
+K_{2}SO_{4} in solution to be constant and equal to 4. This result is,
+however, not exact, for the curve CO is not a straight line, as it should
+be if the above ratio were constant; but it is concave to the abscissa
+axis, and more so at lower than at higher temperatures.
+
+The following table refers to the temperature of 25°. The Roman numbers in
+the first column refer to the points in Fig. 128. The numbers in the column
+[Sigma]_k__{2} give the amount, {329} in gram-molecules, of K_{2}CO_{3} +
+K_{2}SO_{4} contained in 1000 gram-molecules of water:--
+
+  SOLUBILITY DETERMINATIONS AT 25°.
+
+  -----+-------------------------------------+-----------------------+
+       |                                     | 100 gms. of the       |
+       |                                     | solution contain,     |
+   No. |           Solid phases.             | in grams,             |
+       |                                     |           |           |
+       |                                     |K_{2}CO_{3}|K_{2}SO_{4}|
+  -----+-------------------------------------+-----------+-----------+
+  I.   | K_{2}CO_{3}.2H_{2}O + BaCO_{3}      |    53.2   |     --    |
+       |                                     |           |           |
+  II.  |{ K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} }|    53.0   |    0.023  |
+       |{ + BaCO_{3}                        }|           |           |
+       |                                     |           |           |
+  III.}| K_{2}SO_{4} + BaCO_{3}              |  { 28.5   |    0.886  |
+  IV. }|                                     |  { 22.1   |    1.72   |
+       |                                     |           |           |
+  V.   | BaCO_{3} + K_{2}SO_{4} + BaSO_{4}   |    17.81  |    2.485  |
+       |                                     |           |           |
+  VI. }| K_{2}SO_{4} + BaSO_{4}              |  { 12.6   |    3.92   |
+  VII.}|                                     |  {  5.85  |    6.76   |
+       |                                     |           |           |
+  VIII.| K_{2}SO_{4}                         |     --    |   10.76   |
+       |                                     |           |           |
+  IX. }| BaCO_{3} + BaSO_{4}                 |  {  7.35  |    0.602  |
+  X.  }|                                     |  {  2.85  |    0.173  |
+  -----+-------------------------------------+-----------+-----------+
+  [Transcriber's note: table continued below...]
+  -----+-----------------------+-----------------+-----------
+       | 1000 moles            |                 |
+       | of water contain,     |                 | K_{2}CO_{3}
+   No. | in moles,             |[Sigma]_k__{2}   | -----------
+       |           |           |                 | K_{2}SO_{4}
+       |K_{2}CO_{3}|K_{2}SO_{4}|                 |
+  -----+-----------+-----------+-----------------+-----------
+  I.   |   147.9   |     --    |        --       |    --
+       |           |           |                 |
+  II.  |   147.8   |    0.051  |        --       |    --
+       |           |           |                 |
+       |           |           |                 |
+  III.}|    52.58  |    1.296  |        --       |    --
+  IV. }|    37.79  |    2.333  |        --       |    --
+       |           |           |                 |
+  V.   |    29.11  |    3.220  |       32.32     |   9.03
+       |           |           |                 |
+  VI. }|    19.66  |    4.853  |        --       |    --
+  VII.}|     8.724 |    7.995  |        --       |    --
+       |           |           |                 |
+  VIII.|     --    |   12.47   |        --       |    --
+       |           |           |                 |
+  IX. }|    10.43  |    0.676  |       11.11     |  15.0
+  X.  }|     3.828 |    0.184  |        4.0      |  21.0
+  -----+-----------+-----------+-----------------+-----------
+
+The Guldberg-Waage curve at 100° was also determined, and it was found that
+the ratio K_{2}CO_{3}: K_{2}SO_{4} is also not constant, although the
+variations are not so great as at 25°.
+
+  GULDBERG-WAAGE CURVE AT 100°.
+
+  ----------------------+-----------------------+-----------------+-------
+                        |100 moles of water     |                 | K2CO3
+       Solid phases.    |contain, in moles,     | [Sigma]_k__{2}  | -----
+                        |           |           |                 | K2SO4
+                        |K_{2}CO_{3}|K_{2}SO_{4}|                 |
+  ----------------------+-----------+-----------+-----------------+-------
+  BaCO_{3} + K_{2}SO_{4}|   23.9    |   12.65   |      35.65      |  1.82
+         + BaSO_{4}     |           |           |                 |
+  BaCO_{3} + BaSO_{4}   |    6.28   |    2.02   |       8.3       |  3.1
+     "           "      |    3.17   |    0.851  |       4.025     |  3.7
+  ----------------------+-----------+-----------+-----------------+-------
+
+       *       *       *       *       *
+
+
+{330}
+
+APPENDIX
+
+EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT
+
+For the purpose of determining the transition temperature, a number of
+methods have been employed, and the most important of these will be briefly
+described here. In any given case it is sometimes possible to employ more
+than one method, but all are not equally suitable, and the values of the
+transition point obtained by the different methods are not always
+identical. Indeed, a difference of several degrees in the value found may
+quite well occur.[398] In each case, therefore, some care must be taken to
+select the method most suitable for the purpose.
+
+I. The Dilatometric Method.--Since, in the majority of cases,
+transformation at the transition point is accompanied by an appreciable
+change of volume, it is only necessary to ascertain the temperature at
+which this change of volume occurs, in order to determine the transition
+point. For this purpose the _dilatometer_ is employed, an apparatus which
+consists of a bulb with capillary tube attached, and which constitutes a
+sort of large thermometer (Fig. 129). Some of the substance to be examined
+is passed into the bulb A through the tube B, which is then sealed off. The
+rest of the bulb and a small portion of the capillary tube is then filled
+with some liquid, which, of course, must be without chemical action on the
+substance under investigation. A liquid, however, may be employed which
+dissolves the substance, for, as we have seen (p. 70), the transformation
+at the transition point is, as a rule, accelerated by the presence of a
+solvent. On the other hand, the liquid must not dissolve in the substance
+under examination, for the temperature of transformation would be thereby
+altered.
+
+{331}
+
+In using the dilatometer, two methods of procedure may be followed.
+According to the first method, the dilatometer containing the form stable
+at lower temperatures is placed in a thermostat, maintained at a constant
+temperature, until it has taken the temperature of the bath. The height of
+the meniscus is then read on a millimetre scale attached to the capillary.
+The temperature of the thermostat is then raised degree by degree, and the
+height of the meniscus at each point ascertained. If, now, no change takes
+place in the solid, the expansion will be practically uniform, or the rise
+in the level of the meniscus per degree of temperature will be practically
+the same at the different temperatures, as represented diagrammatically by
+the line AB in Fig. 130. On passing through the transition point, however,
+there will be a more or less sudden increase in the rise of the meniscus
+per degree (line BC) if the specific volume of the form stable at higher
+temperatures is greater than that of the original modification; thereafter,
+the expansion will again be uniform (line CD). Similarly, on cooling,
+contraction will at first be uniform and then at the transition point there
+will be a relatively large diminution of volume.
+
+[Illustration: FIG. 129.]
+
+[Illustration: FIG. 130.]
+
+If, now, transformation occurred immediately the transition point was
+reached, the sudden expansion and contraction would take place at the same
+temperature. It is, however, generally found that there is a lag, and that
+with rising temperature the relatively large expansion does not take place
+until a temperature somewhat higher than the transition point; and with
+falling temperature the contraction occurs at a temperature somewhat below
+the transition point. This is represented in Fig. 130 by the lines BC and
+EF. The amount of lag will vary from case to case, and will {332} also
+depend on the length of time during which the dilatometer is maintained at
+constant temperature.
+
+As an example, there may be given the results obtained in the determination
+of the transition point at which sodium sulphate and magnesium sulphate
+form astracanite (p. 268).[399] The dilatometer was charged with a mixture
+of the two sulphates.
+
+  --------------------------------------------------------
+  Temperature. | Level of oil in capillary. | Rise per 1°.
+  --------------------------------------------------------
+     15.6°     |            134             |
+     16.6°     |            141             |       7
+     17.6°     |            148             |       7
+     18.6°     |            154             |       6
+     19.6°     |            161             |       7
+     20.6°     |            168             |       7
+     21.6°     |            241             |      73
+     22.6°     |            243             |       2
+     23.6°     |            251             |       8
+     24.6°     |            259             |       8
+  --------------------------------------------------------
+
+The transition point, therefore, lies about 21.6° (p. 268).
+
+The second method of manipulation depends on the fact that, while above or
+below the transition point transformation of one form into the other can
+take place, at the transition point the two forms undergo no change. The
+bulb of the dilatometer is, therefore, charged with a mixture of the stable
+and metastable forms and a suitable liquid, and is then immersed in a bath
+at constant temperature. After the temperature of the bath has been
+acquired, readings of the height of the meniscus are made from time to time
+to ascertain whether expansion or contraction occurs. If expansion is
+found, the temperature of the thermostat is altered until a temperature is
+obtained at which a gradual contraction takes place. The transition point
+must then lie between these two temperatures; and by repeating the
+determinations it will be possible to reduce the difference between the
+temperatures at which expansion and contraction take place to, say, 1°, and
+to fix the temperature of the transition point, therefore, to within half a
+degree. By this method the transition point, for example, of sulphur was
+found to be 95.6° under a pressure of 4 atm.[400] The following are the
+figures obtained by Reicher, who used a mixture {333} of 1 part of carbon
+disulphide (solvent for sulphur) and 5 parts of turpentine as the measuring
+liquid.
+
+  TEMPERATURE 95.1°.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      343.5
+         30        |      340.5
+         55        |      335.75
+         65        |      333
+  -----------------------------------
+
+  TEMPERATURE 96.1°.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      342.75
+         30        |      354.75
+         55        |      360.5
+         60        |      361.5
+  -----------------------------------
+
+  TEMPERATURE 95.6°.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      368.75
+        100        |      368
+        110        |      368.75
+  -----------------------------------
+
+At a temperature of 95.1° there is a contraction, _i.e._ monoclinic sulphur
+passes into the rhombic, the specific volume of the former being greater
+than that of the latter. At 96.1°, however, there is expansion, showing
+that at this temperature rhombic sulphur passes into monoclinic; while at
+95.6° there is neither expansion nor contraction. This is, therefore, the
+transition temperature; and since the dilatometer was sealed up to prevent
+evaporation of the liquid, the pressure within it was 4 atm.
+
+II. Measurement of the Vapour Pressure.--In the preceding pages it has been
+seen repeatedly that the vapour pressures of the two systems undergoing
+reciprocal transformation become identical at the transition point (more
+strictly, at the triple or {334} multiple point), and the latter can
+therefore be determined by ascertaining the temperature at which this
+identity of vapour pressure is established. The apparatus usually employed
+for this purpose is the Bremer-Frowein tensimeter (p. 91).
+
+Although this method has not as yet been applied to systems of one
+component, it has been used to a considerable extent in the case of systems
+containing water or other volatile component. An example of this has
+already been given in Glauber's salt (p. 139).
+
+III. Solubility Measurements.--The temperature of the transition point can
+also be fixed by means of solubility measurements, for at that point the
+solubility of the two systems becomes identical. Reference has already been
+made to several cases in which this method was employed, _e.g._ ammonium
+nitrate (p. 112), Glauber's salt (p. 134), astracanite and sodium and
+magnesium sulphates (p. 268).
+
+The determinations of the solubility can be carried out in various ways.
+One of the simplest methods, which also gives sufficiently accurate results
+when the temperature is not high or when the solvent is not very volatile,
+can be carried out in the following manner. The solid substance is finely
+powdered (in order to accelerate the process of solution), and placed in
+sufficient quantity along with the solvent in a tube carefully closed by a
+glass stopper; the latter is protected by a rubber cap, such as a rubber
+finger-stall. The tube is then rotated in a thermostat, the temperature of
+which does not vary more than one or two tenths of a degree, until
+saturation is produced. The solution is withdrawn by means of a pipette to
+which a small glass tube, filled with cotton wool to act as a filter, is
+attached. The solution is then run into a weighing bottle, and weighed;
+after which the amount of solid in solution is determined in a suitable
+manner.
+
+For more accurate determinations of the solubility, especially when the
+solvent is appreciably volatile at the temperature of experiment, other
+methods are preferable. In Fig. 131 is shown the apparatus employed by H.
+Goldschmidt,[401] and used to a considerable extent in the laboratory of
+van't Hoff. This consists essentially of three parts: _a_, a tube in which
+the solvent and salt are placed; this is closed at the foot by an
+india-rubber stopper. Through this stopper there passes the bent tube _cb_,
+which connects the tube _a_ with the weighing-tube d. At _c_ there is a
+plug of cotton wool. Tube _e_ is open to the air. The wider portion of the
+tube _cb_, which passes through the rubber stopper in _a_, can be closed by
+a plug {335} attached to a glass rod _ff_, which passes up through a hollow
+Witt stirrer, _g_. After being fitted together, the whole apparatus is
+immersed in the thermostat. After the solution has become saturated, the
+stopper of the bent tube is raised by means of the rod _ff_ and a
+suction-pump attached to the end of e. The solution is thereby drawn into
+the weighing-tube _d_, the undissolved salt being retained by the plug at
+c. The apparatus is then removed from the thermostat, tube _d_ detached and
+immediately closed by a ground stopper. It is then carefully dried and
+weighed.
+
+[Illustration: FIG. 131.]
+
+Another form of solubility vessel, due to Meyerhoffer and Saunders, is
+shown in Fig. 132.[402] This consists of a single tube, and the stirring is
+effected by means of a glass screw.
+
+[Illustration: FIG. 132.]
+
+The progress of the solution towards saturation can be very well tested by
+determining the density of the solution from time to {336} time. This is
+most conveniently carried out by means of the pipette shown in Fig.
+133.[403] With this pipette the solution can not only be removed for
+weighing, but the volume can be determined at the same time. It consists of
+the wide tube _a_, to which the graduated capillary _b_, furnished with a
+cap _c_, is attached. To the lower end of the pipette the tube _e_, with
+plug of cotton wool, can be fixed. After the pipette has been filled by
+sucking at the end of _b_, the stop-cock _d_ is closed and the cap _c_
+placed on the capillary. The apparatus can then be weighed, and the volume
+of the solution be ascertained by means of the graduations.
+
+As has already been insisted, particular care must be paid to the
+characterization of the solid in contact with the solution.
+
+[Illustration: FIG. 133.]
+
+IV. Thermometric Method.--If a substance is heated, its temperature will
+gradually rise until the melting point is reached, and the temperature will
+then remain constant until all the solid has passed into liquid. Similarly,
+if a substance which can undergo transformation is heated, the temperature
+will rise until the transition point is reached, and will then remain
+constant until complete transformation has taken place.
+
+This method, it will be remembered, was employed by Richards for the
+determination of the transition point of sodium sulphate decahydrate
+(p. 136). The following figures give the results obtained by Meyerhoffer in
+the case of the transformation:--
+
+  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O
+
+the temperature being noted from minute to minute: 95°, 93°, 91.8°, 91.7°,
+92°, 92.3°, 92.4°, 92.2°, 92.2°, 92°, 90.5°, 89°, and then a rapid fall in
+the temperature. From this we see that the transition point is about 92.2°.
+It is also evident that a slight supercooling took place (91.7°), owing to
+a delay in the transformation, but that then the temperature rose to the
+transition point. This is analogous to the supercooling of a liquid.
+
+A similar halt in the temperature would be observed on passing from lower
+to higher temperatures; but owing to a lag in the transformation, the same
+temperature is not always obtained.
+
+{337}
+
+V. Optical Method.--The transition point can sometimes be determined by
+noting the temperature at which some alteration in the appearance of the
+substance occurs, such as a change of colour or of the crystalline form.
+Thus mercuric iodide changes colour from red to yellow, and the blue
+quadratic crystals of copper calcium acetate change, on passing the
+transition point, into green rhombs of copper acetate and white needles of
+calcium acetate (p. 260). Or again, changes in the double refraction of the
+crystals may be also employed to ascertain the temperature of the
+transition point. These changes are best observed by means of a microscope.
+
+For the purpose of regulating the temperature of the substance a small
+copper air-bath is employed.[404]
+
+VI. Electrical Methods.--Electrical methods for the determination of the
+transition point are of two kinds, based on measurements of conductivity or
+of electromotive force. Both methods are restricted in their application,
+but where applicable give very exact results.
+
+The former method, which has been employed in several cases, need not be
+described here. The second method, however, is of considerable interest and
+importance, and calls for special reference.[405]
+
+If two pieces, say, of zinc, connected together by a conducting wire, are
+placed in a solution of a zinc salt, _e.g._ zinc sulphate, the potential of
+the two electrodes will be the same, and no current will be produced in the
+connecting wire. If, however, the zinc electrodes are immersed in two
+solutions of _different_ concentration contained in separate vessels, but
+placed in connection with one another by means of a bent tube filled with a
+conducting solution, the potentials at the electrodes will no longer be the
+same, and a current will now flow through the connecting wire. The
+direction of this current _in the cell_ will be from the weaker to the more
+concentrated solution.
+
+The greater the difference in the concentration of the solutions with
+respect to zinc, the greater will be the difference of the potential at the
+two electrodes, or the greater will be the E.M.F. of the cell. When the
+concentration of the two solutions becomes the same, the E.M.F. will become
+zero, and no current will pass.
+
+It will be understood now how this method can be made use of {338} for
+determining the transition point of a salt, when we bear in mind that at
+the transition point the solubility of the two forms becomes identical.
+Thus, for example, the transition point of zinc sulphate heptahydrate into
+hexahydrate could be determined in the following manner. Tube A (Fig. 134)
+contains, say, a saturated solution of the heptahydrate along with some of
+the solid salt; tube B, a saturated solution of the hexahydrate along with
+the solid salt. The tube C is a connecting tube bent downwards so as to
+prevent the mixing of the solutions by convection currents. ZZ are two zinc
+electrodes immersed in the solution; the cell is placed in a thermostat and
+the zinc electrodes connected with a galvanometer. Since, now, at
+temperatures below the transition point the solubility of the hexahydrate
+(the metastable form) is greater than that of the heptahydrate, a current
+will be produced, flowing in the cell from heptahydrate to hexahydrate. As
+the temperature is raised towards the transition point, the solubilities of
+the two hydrates also approach, and the current produced will therefore
+become weaker, because the E.M.F. of the cell becomes less; and when the
+transition point is attained, the E.M.F. becomes zero, and the current
+ceases. If the temperature is raised above this, the solubility of the
+heptahydrate becomes greater than that of the hexahydrate, and a current
+will again be produced, but in the opposite direction. By noting the
+temperature, therefore, at which the current ceases, or the E.M.F. becomes
+zero, the transition temperature can be ascertained.[406]
+
+[Illustration: FIG. 134.]
+
+In the case just described, the electrodes consisted of the same metal as
+was contained in the salt. But in some cases, _e.g._ sodium sulphate,
+electrodes of the metal contained in the salt cannot be employed.
+Nevertheless, the above electrical method can be used {339} even in those
+cases, if a suitable non-polarizable mercury electrode is employed.[407]
+
+Although, as we saw, no current was produced when two pieces of zinc were
+immersed in the same solution of zinc salt, a current will be obtained if
+two different metals, or even two different modifications of the same
+metal, are employed. Thus an E.M.F. will be established when electrodes of
+grey and of white tin are immersed in the same solution of zinc salt, but
+at the transition point this E.M.F. will become zero. By this method Cohen
+determined the transition point of grey and white tin (p. 42).
+
+       *       *       *       *       *
+
+
+{340}
+
+NAME INDEX
+
+          A
+  Abegg, 52
+  Adriani, 186, 217, 220
+  Alexejeff, 97, 125
+  Allan, 298
+  Allen, L. E., 109
+  Allen, R. W., 63
+  Ampolla, 213
+  Andreä, 109
+  Aristotle, 41
+  Armstrong, E. F., 313
+  Armstrong, H. E., 196
+  Arzruni, 33
+  Aten, 147, 163
+  Auerbach, 326
+
+          B
+  Babo, 126
+  Bancroft, 102, 104, 161, 176, 196, 202, 229, 246, 260, 261, 272, 281, 302
+  Barnes, 331, 339
+  Barschall, 318
+  Barus, 67
+  Battelli, 23
+  Baur, 233, 307
+  Beckmann, 49
+  Bell, 229
+  Berthollet, 7
+  Bodländer, 181, 247, 311, 321
+  Bogojawlenski, 72
+  Boudouard, 309, 311
+  Braun, 107
+  Brauns, 40, 51, 74
+  Bredig, 52
+  Bremer, 91
+  Brodie, 34, 47
+  Bruner, 126
+  Bruni, 181, 182, 256, 257
+  Bunsen, 67
+
+          C
+  Cady, 192
+  Calvert, 130
+  Cameron, 203
+  Carnelley, 47
+  Carpenter, 225
+  Carveth, 204, 255
+  Centnerszwer, 158
+  Chapman, 47
+  Chappuis, 51, 176
+  Charpy, 255
+  Churchill, 140
+  Coehn, 52
+  Cohen, 41, 72, 136, 139, 140
+  Cooke, 331, 339
+  Cox, 301
+
+          D
+  Dawson, 263
+  Debray, 74, 81, 139
+  Deville, 49, 74
+  Dewar, 26, 51, 178
+  Dietz, 157
+  {341}
+  Doelter, 233
+  Donnan, 8, 18
+  Dreyer, 73
+  Duhem, 56, 151
+  Dutoit, 204
+
+          E
+  Etard, 115, 135
+
+          F
+  Fahrenheit, 30
+  Faraday, 82, 89
+  Fath, 204
+  Fedotieff, 315, 320
+  Findlay, 111, 204, 206, 219
+  Foote, 69
+  Friedländer, 72
+  Fritsche, 41
+  Frowein, 91
+  Füchtbauer, 75
+  Fyffe, 143
+
+          G
+  Gattermann, 51, 52
+  Gautier, 222, 223
+  Gay-Lussac, 135
+  Gernez, 72
+  Gibbs, 7, 8, 151, 236
+  Glaessner, 307
+  Goldschmidt, E., 41
+  Goldschmidt, H., 335
+  Goldschmidt, V., 32
+  Goossens, 26
+  Gossner, 318
+  Graham, 178
+  Guertler, 73
+  Guldberg, 7
+  Guthrie, 97, 104, 117, 118, 119, 233
+
+          H
+  Haber, 311
+  Hahn, 309, 311
+  Hallock, 35
+  Hammerl, 145
+  Hautefeuille, 46, 49, 50, 51, 178
+  Heller, 311
+  Henry, 94
+  Herold, 321
+  Hertz, 49
+  Heycock, 194, 221, 223
+  Heyn, 225, 228
+  Hickmans, 219
+  Hiorns, 228
+  Hissink, 115, 190
+  Hoitsema, 14, 90, 177, 178, 298
+  Hollmann, 204
+  Holsboer, 110
+  Horstmann, 8, 83, 89
+  Hudson, 102
+  Hulett, 10, 48, 52, 54, 67, 109
+
+          I
+  Isaac, 114
+  Isambert, 80, 82, 84
+
+          J
+  Jaffé, 74, 114
+  Joulin, 176
+  Juhlin, 23, 24, 30
+  von Jüptner, 225
+
+          K
+  Kastle, 71
+  Kaufler, 49
+  Kaufmann, 112
+  Kayser, 176
+  Keeling, 225
+  Kelvin, 25
+  Kenrick, 263, 297
+  Kipping, 219
+  Kirchhoff, 32
+  Knorr, 203
+  de Kock, 53, 182, 194
+  Konowaloff, 102, 103, 104
+  Krasnicki, 144
+  Kremann, 147, 212
+  Kuenen, 105
+  Kultascheff, 233
+  {342}
+  Kuriloff, 216
+  Kurnakoff, 221, 222, 230
+  Küster, 72, 181, 183
+
+          L
+  Laar, 195
+  Labenburg, 216
+  Lattey, 101
+  Le Chatelier, 58, 81, 233
+  Lehfeldt, 338, 340
+  Lehmann, 33, 52, 53
+  Lidbury, 147
+  Loewel, 134, 135
+  Loewenherz, 134, 316
+  Lowry, 196, 198
+  Ludwig, 327
+  Lumsden, 80, 109, 110
+  Lussana, 68
+  Luther, 22
+
+          M
+  Mack, 67
+  Magnus, 22
+  Mathews, 221
+  Mellor, 80
+  Meusser, 142
+  Meyer, J., 71
+  Meyer, V., 47
+  Meyerhoffer, 158, 233, 259, 268, 271, 278, 279, 280, 284, 313, 315, 317,
+      319, 327, 328, 336, 337
+  Middelberg, 116
+  Miers, 114
+  Miller, 297
+  Mitscherlich, 33, 49
+  Mond, 178
+  Moore, 72
+  Moss, 66
+  Müller, 112, 265
+  Mylius, 109, 142, 157
+
+          N
+  Naumann, 49
+  Neville, 194, 221, 223
+
+          O
+  Offer, 119
+  Ostwald, 8, 10, 13, 16, 22, 44, 58, 68, 70, 74, 85, 88, 92, 102, 110,
+      117, 125, 127, 130, 141, 198
+
+          P
+  Padoa, 73, 181
+  Parsons, 298
+  Pasteur, 266
+  Paternò, 213
+  Payen, 74
+  Pedler, 47
+  Pfaundler, 119
+  Philip, 213, 214
+  von Pickardt, 73
+  Planck, 68
+  Pope, 219
+  Poynting, 68
+  Preuner, 311
+  Puschin, 222
+
+          Q
+  Quincke, 52
+
+          R
+  Rabe, 113
+  Ramsay, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 90, 165, 178
+  Raoult, 180
+  Reed, 71
+  Regnault, 22
+  Reicher, 36, 37, 110, 260, 333
+  Reinders, 71, 185, 188
+  Reinitzer, 51, 52
+  Richards, 136, 140
+  Riddle, 47
+  Riecke, 48, 55
+  Roberts-Austen, 63, 194, 221, 223, 225
+  Roloff, 117
+  Roozeboom, 10, 38, 45, 47, 49, 50, 51, 54, 56, 57, 62, 63, 68, 88, 103,
+      126, 145, 147, 150, 151, 157, 162, 170, 174, 178, 182, 196, 201, 211,
+      217, 220, 225, 236, 238, 262, 264, 269, 272, 273, 281, 282, 290, 331
+  {343}
+  Rose, 223
+  Rotarski, 52
+  Rothmund, 97, 98, 100
+  Rutten, 298
+
+          S
+  Saposchnikoff, 212
+  Saunders, 313, 317, 319, 336, 337
+  Saurel, 151
+  Schaum, 49, 75
+  Scheel, 22, 23, 30
+  Schenck, 49, 52, 54, 311
+  Schneider, 52
+  Schönbeck, 75
+  Schreinemakers, 122, 126, 246, 248, 250, 252, 290, 302
+  Schrötter, 46
+  Schukowsky, 52
+  Schwarz, 331
+  Seitz, 52
+  Shenstone, 109, 115, 135
+  Shepherd, 221, 255
+  Shields, 178
+  Skirrow, 130
+  Spring, 63
+  von Stackelberg, 107, 110
+  Staedel, 267
+  Stansfield, 194, 221
+  Stokes, 236
+  Stortenbeker, 44, 147, 161, 164, 281
+
+          T
+  Taber, 229
+  Tammann, 26, 32, 33, 37, 38, 39, 48, 52, 65, 67, 68, 72, 73, 140, 151,
+      176, 221, 230
+  Thiesen, 22, 23, 30
+  Thomson, J., 25, 28, 32
+  Thomson, W., 25
+  Tilden, 109, 115, 135
+  Trevor, 16
+  Troost, 46, 49, 50, 51
+  Tumlirz, 72
+
+          V
+  Van Bemmelen, 180
+  Van Deventer, 110, 139, 266, 267, 333
+  Van Eyk, 41, 63, 192, 338
+  Van't Hoff, 36, 38, 58, 70, 90, 92, 108, 127, 139, 140, 165, 175, 225,
+      258, 260, 263, 265, 266, 267, 272, 284, 290, 313, 318, 333, 340
+  Van Leeuwen, 259
+  Van Wyk, 185
+  Vogt, 5, 233
+
+          W
+  Waage, 7
+  Wald, 92
+  Walden, 158
+  Walker, 80, 105, 122, 126, 143
+  Wegscheider, 10, 49, 202
+  Wells, 136
+  Wenzel, 7
+  Wiebe, 22
+  Witt, 327
+  Wright, 241, 246, 247
+  von Wrochem, 109, 142
+
+          Y
+  Young, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 105, 165
+
+          Z
+  Zacharias, 180
+  Zawidski, 63
+  Zenghelis, 35
+  Zimmermann, 311
+  Zincke, 44
+  Ziz, 141
+
+       *       *       *       *       *
+
+
+{344}
+
+SUBJECT INDEX
+
+          A
+  Acetaldehyde and paraldehyde, 204
+  Acetic acid, chloroform, water, 241
+  Acetone, phenol, water, 248
+  Adsorption, 176
+  Alcohol, chloroform, water, 246
+    ----, ether, water, 246
+  Alloys, equilibrium curves of, 221
+    ---- of copper and tin, liquefaction of, by cooling, 194
+    ---- of iron and carbon, 223
+    ---- of thallium and mercury, 222
+    ----, ternary, 246
+  Ammonia compounds of metal chlorides, 82
+  Ammonia silver chlorides, 82
+    ---- ---- ----, dissociation pressures of, 84
+  Ammonia-soda process, 320
+  Ammonium chloride, dissociation of, 3, 79
+    ---- cyanide, dissociation of, 80
+    ---- hydrosulphide, dissociation of, 80
+    ---- nitrate, solubility of, 113
+  Aniline, phenol, water, 250
+  Astracanite, 260, 261, 268, 274
+
+          B
+  Babo, law of, 126
+  Barium acetate, solubility of, 143
+  Barium carbonate and potassium sulphate, 328
+    ---- nitrite, preparation of, 327
+  Basic salts, 296
+  Benzaldoximes, 203
+  Benzene and picric acid, 216
+  Bismuth, effect of pressure on the melting point of, 67
+    ----, lead, tin, 255
+    ---- nitrates, basic, 298
+  Bivariant systems, 16
+  Bromocinnamic aldehyde and chlorocinnamic aldehyde, 183
+
+          C
+  Calcium carbonate, dissociation of, 3, 11, 81
+    ---- chloride hexahydrate, solubility of, 146
+    ---- ----, solubility of hydrates of, 148
+    ---- ----, vapour-pressure of hydrates of, 88
+  Camphor oximes, 219, 257
+  Carnallite, 284
+  Carvoximes, 186, 219
+  Cementite, 224
+  Chlorine and iodine, 161
+  Chlorocinnamic aldehyde and bromocinnamic aldehyde, 183
+  Chloroform, acetic acid, water, 241
+    ----, alcohol, water, 246
+  {345}
+  Classification of systems, 17
+  Component, 8, 10, 12
+    ----, systems of one, 21, 55
+  Components, choice of, 12, 13, 14, 76, 313
+    ----, determination of number of, 13
+    ----, systems of four, 312
+    ----, ---- of three, 234
+    ----, ---- of two, 76, 207
+    ----, variation in number of, 11, 14
+  Composition, determination of, without analysis, 228, 302
+  Concentration-temperature curve for two liquids, 101
+  Condensed systems, 36
+  Constituent, 10
+  Cooling curve, 230
+  Copper calcium acetate, 260
+    ---- chloride, heat of solution of, 110
+    ---- dipotassium chloride, 259
+    ---- sulphate, 85
+  Critical concentration, 98, 242
+    ---- pressure of water, 23
+    ---- solution temperature, 98
+    ---- temperature of water, 23
+  Cryohydrates, 117, 118
+  Cryohydric point, 117
+    ---- ----, changes at the, 119
+    ---- ---- for silver nitrate and ice, 116
+  Crystals, liquid, 51
+    ----, ----, equilibria of, 53
+    ----, ----, list of, 54
+    ----, ----, nature of, 52
+    ----, mixed, 180
+  Crystallization, velocity of, 72, 74
+    ----, spontaneous, 114
+
+          D
+  Deliquescence, 130
+  Devitrification, 73
+  Diethylamine and water, solubility of, 101
+  Dilatometer, determination of transition points by, 331
+  Dineric surface, 247
+  Dissociation equilibrium, effect of addition of dissociation products on,
+      4
+    ---- of ammonia compounds of metal chlorides, 82, 84
+    ---- of ammonium chloride, 3, 79
+    ---- ---- cyanide, 80
+    ---- ---- hydrosulphide, 80
+    ---- of calcium carbonate, 3, 81
+    ---- of compounds, degree of, 147
+    ---- of phosphonium bromide, 80
+    ---- of salt hydrates, 85
+    ----, phenomena of, 79
+  Dissociation pressure, 81
+  Distillation of supercooled liquid to solid, 32, 50
+  Double salt interval, 278
+    ---- salts, crystallization from solution, 280
+    ---- ----, decomposition by water, 267
+    ---- ----, formation of, 258, 273, 315
+
+          E
+  Efflorescence, 86
+  Electrical methods of determining transition points, 338
+  Enantiotropy, 44, 51
+  Equilibria, Gibbs's theory of, 8
+    ----, metastable, 69
+  Equilibrium apparent (false), 5, 6
+    ---- between ice and solution, 116
+    ---- between ice and water, 25
+    ---- between ice, water, vapour, 27
+    ---- between water and vapour, 21
+    ----, chemical, 3, 16
+    ----, heterogeneous, 5
+    ----, homogeneous, 5
+    ----, independence of, on amounts of phases, 9
+    ----, law of movable, 58
+  {346}
+    ----, physical, 3, 16
+    ---- real (true), 5, 6
+  Ether, alcohol, water, 246
+    ----, succinic nitrile, water, 252
+  Ethylene bromide, picric acid, [beta]-naphthol, 256
+  Eutectic mixtures, 117, 191, 209, 255, 257
+    ---- point, 117, 209, 213, 253
+
+          F
+  Ferric chloride, evaporation of solutions of, 155
+    ---- ----, hydrates of, 151, 153
+    ---- ----, hydrogen chloride and water, systems of, 290
+  Ferrite, modifications of, 224
+  Freedom, degree of, 14
+  Freezing mixtures, 120
+    ---- point, natural, 198
+  Fusion curve, 66
+    ---- ---- of ice, 25
+    ---- of ice, influence of pressure on, 26
+    ----, partial, 139
+
+          G
+  Glaserite, 315, 317
+  Glasses, 176
+  Glauber's salt, 13, 134
+    ---- ----, transition curve of, 68, 140
+  Graphic representation in space, 77, 284
+
+          H
+  Hydrates, range of existence of, 89
+    ---- chloride and water, 174
+  Hydrogen bromide and water, 174
+  Hylotropic substances, 198
+
+          I
+  Ice I., 32
+    ---- II., 32
+    ---- III., 32
+    ----, equilibrium between water and, 25
+    ----, influence of pressure on melting point of, 25, 26
+    ----, sublimation curve of, 24
+    ----, vapour pressure of, 25, 31
+  Indifferent point, 150
+  Individual, chemical, 92
+  Inversion temperature, 36
+  Iodine and chlorine, 161
+  Iron--carbon alloys, 223
+    ----, carbon monoxide and carbon dioxide, 305
+  Isomerides, dynamic, 195, 196
+    ----, ----, equilibrium between, 195, 196
+    ----, ----, equilibrium point of, 198
+    ----, transformation of unstable into stable, 201
+  Isomerism, dynamic, 196
+  Isothermal evaporation, 278
+    ---- solubility curves, 272
+
+          L
+  Lead, bismuth, tin, 255
+    ----, desilverization of, 247
+    ----, silver, zinc, 246
+  Le Chatelier, theorem of, 57
+  Lime, burning of, 3
+  Liquidus curve, 182
+
+          M
+  Mandelic acid, 217
+  Martensite, 224
+  Mass action, law of, 7
+  Melting point, influence of pressure on, 66
+  {347}
+    ---- ----, congruent, 146
+    ---- ----, incongruent, 139
+    ---- under the solvent, 122
+  Menthyl mandelates, 219
+  Mercuric bromide and iodide, 188
+  Mercury salts, basic, 301
+  Metastable equilibria, 69
+    ---- region, 30
+    ---- state, 30
+  Methylethyl ketone and water, 100
+  Minerals, formation of, 232
+  Miscibility of liquids, complete, 95, 104, 114
+    ---- ----, partial, 95, 96, 121
+  Mixed crystals, 180, 281
+    ---- ----, changes in, with temperature, 192
+    ---- ----, examples of, 183, 186, 187, 190, 192, 219, 223
+    ---- ----, formation of, 181, 182
+    ---- ----, fractional crystallization of, 188
+    ---- ----, freezing points of, 182
+    ---- ----, melting points of, 182, 184
+    ---- ----, pseudoracemic, 219
+  Mixtures, isomorphous, 181
+    ---- of constant boiling point, 105
+    ---- of constant melting point, 117, 186, 187, 192, 209, 255, 257
+  Monotropy, 44, 51
+  Multivariant systems, 16
+
+          N
+  Naphthalene and monochloracetic acid, 192
+    ---- and [beta]-naphthol, mixed crystals of, 183
+  [beta]-Naphthol, ethylene bromide, picric acid, 256
+  [alpha]-Naphthylamine and phenol, 213
+  Nickel iodate, solubility of, 142
+  _o_-Nitrophenol and _p_-toluidine, 213
+
+          O
+  Occlusion of gases, 176
+  Optical method of determining transition points, 338
+  Optically active substances, freezing-point curves of, 216
+  Order of a system, 13
+  Organic compounds, application of Phase Rule to, 212
+
+          P
+  Palladium and hydrogen, 90, 178
+  Paragenesis, 320
+  Paraldehyde and acetaldehyde, 204
+  Partial pressures of two components, 102
+  Pearlite, 224
+  Phase, 8
+    ---- Rule, 8, 16
+    ---- ----, deduction of, 18
+    ---- ----, scope of, 1
+  Phases, formation of new, 69
+    ----, number of, 9
+  Phenol, acetone, water, 248
+    ----, aniline, water, 250
+    ---- and [alpha]-naphthylamine, 213
+    ---- and _p_-toluidine, 214
+    ---- and water, solubility of, 97
+  Phosphonium bromide, dissociation of, 80
+    ---- chloride, 65
+  Phosphorus, 46
+    ----, distillation of white to red, 50
+    ----, melting point of red, 47
+    ----, ---- ---- of white, 48
+    ----, solubility of white and red, 47
+    ----, vapour pressure of white and red, 46
+  Picric acid and benzene, 216
+    ---- ----, ethylene bromide, and [beta]-naphthol, 256
+  Polymorphic forms, solubility of, 112
+  {348}
+    ---- substances, list of, 63
+  Polymorphism, 33
+  Potassium nitrate and thallium nitrate, 192
+  Potential, chemical, 19
+  Pressure-concentration diagram for two liquids, 102
+  Pressure-temperature diagram for solutions, 126
+  Pseudomonotropy, 45
+  Pseudo-racemic mixed crystals, 21
+  Pyridine and methyl iodide, 147
+  Pyrometer, registering, 230
+
+          Q
+  Quadruple point, 116
+  Quintuple point, 234, 261
+
+          R
+  Racemates, characterization of, 217, 282
+  Reactions, law of successive, 73
+  Reciprocal salt-pairs, 313
+    ---- ----, transition point of, 314
+  Rubidium tartrates, 265
+
+          S
+  Salt hydrates, 85
+    ---- ----, indefiniteness of vapour pressure of, 87
+    ---- ---- with definite melting point, 145
+  Separation of salt on evaporation, 130
+  Silicates, hydrated, 176
+  Silver, lead, zinc, 246
+  Silver nitrate, solubility of, 114
+    ---- ---- and sodium nitrate, 190
+  Single salt interval, 278
+  Sodium ammonium tartrates, 266
+    ---- nitrate and silver nitrate, 190
+    ---- sulphate and water, equilibria between, 134
+  Sodium sulphate and water, vapour pressures of, 138, 140
+    ---- ----, anhydrous, dehydration by, 138
+    ---- ----, solubility of, 135
+    ---- ---- decahydrate, solubility of, 134
+    ---- ---- ----, transition point of, 136, 139
+    ---- ---- heptahydrate, solubility of, 136
+    ---- ---- ----, transition point of, 137
+  Solidus curve, 182
+  Solubility curve at higher temperatures, 114
+    ---- ----, form of, 108
+    ---- ---- of anhydrous salts, 111
+    ---- ----, retroflex, 146, 151, 162
+    ---- curves, interpolation and extrapolation of, 111
+    ---- ---- of three component systems, 264
+    ----, determination of transition points by, 335
+    ----, influence of pressure on, 107
+    ----, ---- of subdivision on, 10
+    ----, ---- of temperature on, 109
+    ---- of metastable forms, 47, 112, 137
+  Solubility of polymorphic forms, 112
+    ---- of salt hydrates, 133, 145
+    ---- of supercooled liquids, 125
+    ----, retrograde, 245
+  Solute, 93
+  Solution, definition of, 92
+    ----, heat of, 109, 110
+    ----, saturated, 106, 108
+    ----, supersaturated, 108
+    ---- temperature, critical, 98
+    ----, unsaturated, 108
+  Solutions, bivariant systems, 129
+    ----, congruently saturated, 279
+    ---- conjugate, 97, 241
+  {349}
+    ----, incongruently saturated, 279, 289
+    ----, inevaporable, 157
+    ---- of gases in liquids, 93
+    ---- ---- in solids, 176
+    ---- of liquids in liquids (binary), 95
+    ---- ---- ---- (ternary), 240
+    ---- ----, influence of temperature on, 247
+    ---- of solids in liquids, 106
+    ---- ---- in solids, 180
+    ----, solid, 175, 180
+    ----, univariant systems, 127
+  Space model for carnallite, 284
+  Stability limit, 202
+  Steel, formation of, 223
+  Sublimation curve, 63
+    ---- ---- of ice, 24
+    ---- without fusion, 65
+  Succinic nitrile and water, 122
+    ---- ether, water, 252
+  Sulphur, 33, 34
+    ---- dioxide and water, 169
+    ---- ---- and potassium iodide, 158
+    ----, transition point of rhombic and monoclinic, 36
+  Supersaturation, 113, 114, 124
+    ----, limits of, 114
+  Systems, condensed, 36
+    ---- of one component, 21
+    ---- of two components, 76, 77, 207
+
+          T
+  Tachydrite, influence of pressure on the transition point of, 263
+  Tartrate, dimethyl, 217
+    ----, sodium potassium, 259
+  Tautomeric substances, 195
+  Tensimeter, 91
+  Thallium nitrate and potassium nitrate, 192
+  Theorem of van't Hoff and Le Chatelier, 57
+  Thermometric  determination of transition point, 337
+  Tin, 41
+    ----, lead, bismuth, 255
+    ---- plague, 43
+    ----, transition point of white and grey, 41
+  _p_-Toluidine and _o_-nitrophenol, 213
+    ---- and phenol, 214
+  Transformation of optically active substances, 220
+    ----, suspended, 37, 69, 89, 113, 137, 155
+    ----, velocity of, 70
+  Transition curve, 66
+    ---- ---- of Glauber's salt, 68, 140
+    ---- ---- of rhombic and monoclinic sulphur, 37
+    ---- interval, 270, 277, 315
+    ---- point, 34
+    ---- ---- for double salts, 258
+    ---- ----, influence of pressure on the, 68
+    ---- points, as fixed points in thermometry, 140
+    ---- ----, methods of determining, 331
+    ---- ---- of polymorphic substances, 63
+  Triangle, graphic representation by, 235
+  Triethylamine and water, 101
+  Triple point, 27, 55
+    ---- ----, arrangement of curves round, 56
+    ---- ----, changes at, 58
+    ---- ----, ice, water, vapour, 27
+    ---- ----, ice II., ice III., and water, 33
+    ---- ----, metastable, 38
+    ---- ----, monoclinic sulphur, liquid, vapour, 38
+    ---- ----, monoclinic and rhombic sulphur, liquid, 38
+    ---- ----, monoclinic and rhombic sulphur, vapour, 34
+  {350}
+    ---- ----, red phosphorus, liquid, vapour, 47
+    ---- ----, rhombic sulphur, liquid, vapour, 38
+    ---- ---- solid, solid, vapour, 62
+    ---- ----, white phosphorus, liquid, vapour, 48
+
+          U
+  Univariant systems, 16
+
+          V
+  Van't Hoff, theorem of, 57
+  Vaporization curve, 63
+    ---- ----, interpolation and extrapolation of, 66
+    ---- ---- of water, 21, 23
+  Vapour pressure, constancy of, and formation of compounds, 90
+    ---- ----, dependence of, on solid phase, 88
+    ---- ----, influence of surface tension on, 2
+    ---- ---- in three-component systems, 261
+    ---- ----, measurement of, 91, 334
+    ---- ---- of calcium chloride solutions, 150
+    ---- ---- of ice, 25, 31
+    ---- ---- of small drops, 10
+    ---- ---- of sodium sulphate and water, 138
+  Vapour pressure of solid, solution, vapour, 126
+    ---- ---- of water, 21, 31
+  Variability of a system, 14, 16
+  Variance of a system, 16
+  Volatile components, two, 161
+
+          W
+  Water, 21
+    ----, acetic acid, chloroform, 241
+    ----, acetone, phenol, 248
+    ----, alcohol, ether, 246
+    ----, ----, chloroform, 246
+    ----, aniline, phenol, 250
+    ----, bivariant systems of, 29
+    ----, critical pressure of, 23
+    ----, critical temperature of, 23
+    ----, equilibrium between ice and, 25
+    ----, ---- between vapour and, 21
+    ----, ether, succinic nitrile, 252
+    ----, supercooled, 30
+    ----, ----, vapour pressure of, 31
+    ----, vaporization curve of, 21
+    ----, vapour pressure of, 23
+
+          Z
+  Zeolites, 176
+  Zinc, lead, silver, 246
+  ---- chloride in water, solubility of, 157
+
+THE END
+
+PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.
+
+       *       *       *       *       *
+
+
+NOTES
+
+[1] Except when the volume of the liquid becomes exceedingly small, in
+which case the surface tension exerts an influence on the vapour pressure.
+
+[2] For reasons which will appear later (Chap. IV.), the volume of the
+vapour is supposed to be large in comparison with that of the solid and
+liquid.
+
+[3] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.
+
+[4] See, more especially, Vogt, _Die Silikatschmelzlösungen_. (Christiania,
+1903, 1904.)
+
+[5] _Trans. Connecticut Acad._, 1874-1878.
+
+[6] Lehre von der chemischen Verwandtschaft der Körper, 1777.
+
+[7] See Ostwald's _Klassiker_, No. 74.
+
+[8] Etudes sur les affinités chimiques, 1867; Ostwald's _Klassiker_, No.
+104.
+
+[9] Died April, 1903.
+
+[10] For a mathematical treatment of the Phase Rule the reader is referred
+to the volume in this series on Thermodynamics, by F. G. Donnan.
+
+[11] Liebig's _Annalen_, 1873, 170, 192; Ostwald, _Lehrbuch_, II. 2. 111.
+
+[12] The action of gravity and other forces being excluded (see p. 5).
+
+[13] It may seem as if this were a contradiction to what was said on p. 4
+as to the effect of the addition of ammonia or hydrogen chloride to the
+system constituted by solid ammonium chloride in contact with its products
+of dissociation. There is, however, no contradiction, because in the case
+of ammonium chloride the gaseous phase consists of ammonia and hydrogen
+chloride in equal proportions, and in adding ammonia or hydrogen chloride
+alone we are not adding the gaseous phase, but only a constituent of it.
+Addition of ammonia and hydrogen chloride together in the proportions in
+which they are combined to form ammonium chloride would cause no change in
+the equilibrium.
+
+[14] The vapour pressure of water in small drops is greater than that of
+water in mass, and the solubility of a solid is greater when in a state of
+fine subdivision than when in large pieces (_cf._ Hulett, _Zeitschr.
+physikal. Chem._, 1901, 37. 385).
+
+[15] See Ostwald, _Lehrbuch_, II. 2. 476, 934; Roozeboom, _Zeitschr.
+physikal. Chem._, 1894, 15. 150; _Heterogene Gleichgewichte_, I. p. 16;
+Wegscheider, _Zeitschr. physikal. Chem._, 1903, 43. 89.
+
+[16] Ostwald, _Lehrbuch_, II. 2. 478.
+
+[17] See also Hoitsema, _Zeitschr. physikal. Chem._ 1895, 17. 651.
+
+[18] The term "degree of freedom" employed here must not be confused with
+the same term used to denote the various movements of a gas molecule
+according to the kinetic theory.
+
+[19] Trevor, _Jour. Physical Chem._, 1902, 6. 136.
+
+[20] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
+Findlay, 2nd edit., p. 7. (Macmillan, 1904.)
+
+[21] See the volume in this series on _Thermodynamics_ by F. G. Donnan.
+
+[22] _Pogg. Annalen_, 1844, 61. 225.
+
+[23] _Mémoires de l'Acad._, 26. 751.
+
+[24] _Phil. Trans._ 1884, 175. 461; 1892, A, 183. 107.
+
+[25] _Bihang Svenska Akad. Handl._ 1891, 17. I. 1.
+
+[26] Abh_andl. physikal.-tech. Reichsanstalt_, 1900, 3. 71.
+
+[27] Ostwald-Luther, _Physiko-chemische Messungen_, 2nd edit., p. 156.
+
+[28] _Annales chim. et phys._, 1892 [6], 26. 425.
+
+[29] The vapour pressure of water at 0° has recently been very accurately
+determined by Thiesen and Scheel (_loc. cit._), and found to be 4.579 ±
+0.001 mm. of mercury (at 0°), or equal to 0.006025 atm.
+
+[30] Juhlin, _Bihang Svenska Akad. Handl._, 1891, 17. I. 58. See also
+Ramsay and Young, _loc. cit._
+
+[31] _Trans. Roy. Soc. Edin._, 1849, 16. 575.
+
+[32] _Proc. Roy. Soc. Edin._, 1850, 2, 267.
+
+[33] _Annalen der Physik_, 1899 [3], 68. 564; 1900 [4], 2. 1, 424. See
+also Dewar, _Proc. Roy. Soc._, 1880, 30. 533.
+
+[34] The pressure of 1 atmosphere is equal to 1.033 kilogm. per sq. cm.; or
+the pressure of 1 kilogm. per sq. cm. is equal to 0.968 atm.
+
+[35] Tammann, _loc. cit._, 1900, 2. 1, 424; cf. Goossens, _Arch. néerland_,
+1886, 20. 449.
+
+[36] J. Thomson, _Proc. Roy. Soc._, 1874, 22. 28.
+
+[37] A field is "enclosed" by two curves when these cut at an angle less
+than two right angles. It may be useful to remember that an invariant
+system is represented by a _point_, a univariant system by a _line_, and a
+bivariant system by an _area_.
+
+[38] _Phil. Trans._, 1724, 39. 78.
+
+[39] Juhlin, _loc. cit._, p. 61; cf. Ramsay and Young, _loc. cit._: Thiesen
+and Scheel, _loc. cit._
+
+[40] This small difference is due to experimental errors in the
+determination of the vapour pressures; a differential method betrayed no
+difference between the vapour pressure of ice and of water at 0°.
+
+[41] _Phil. Mag._, 1874 [4], 47. 447; _Proc. Roy. Soc._, 1873, 22. 27.
+
+[42] _Pogg. Annalen_, 1858, 103, 206.
+
+[43] See _Phil. Trans._, 1884, 175, 461.
+
+[44] This phenomenon of distillation from the supercooled liquid to the
+solid has been very clearly observed in the case of furfuraldoxime (V.
+Goldschmidt, _Zeitschr. f. Krystallographie_, 1897, 28. 169).
+
+[45] _Annalen der Physik_, 1900 [4], 2. 1, 424.
+
+[46] A similar triple point has been determined by Tammann in the case of
+phenol (_Annalen der Physik_, 1902 [4], 9. 249).
+
+[47] _Annales chim. et phys._, 1821, 19. 414.
+
+[48] Lehmann, _Molekularphysik_, I. 153.; Arzruni, _Physikalische Chemie
+der Krystalle_. (Graham-Otto, _Lehrbuch der Chemie_, I. 3.)
+
+[49] Brodie, _Proc. Roy. Soc._, 1855, 7. 24.
+
+[50] That solid sulphur does possess a certain vapour pressure has been
+shown by Hallock, who observed the formation at the ordinary temperature of
+copper sulphide in a tube containing copper and sulphur (_Amer. Jour.
+Sci._, 1889 [3], 37. 405). See also Zenghelis, _Zeitschr. physikal.
+Chem._, 1904, 50. 219.
+
+[51] _Zeitschr. für Krystallographie_, 1884, 8. 593.
+
+[52] Van't Hoff, _Studies on Chemical Dynamics_, p. 163.
+
+[53] Reicher, _loc. cit._ See also Tammann, _Annalen der Physik_, 1899
+[3], 68. 663.
+
+[54] Tammann, _Annalen der Physik_, 1899 [3], 68. 633.
+
+[55] Rec. Trav. _Chim. Pays-Bas_, 1887, 6. 314.
+
+[56] Cf. van't Hoff, _Lectures on Physical Chemistry_, I., p. 27 (Arnold).
+
+[57] _Annalen der Physik_, 1899 [3], 68. 663.
+
+[58] Brauns, _Jahrbuch für Mineralogie_, 1899-1901, 13. Beilage, p. 39.
+
+[59] Fritsche, _Ber._, 1869, 2. 112, 540.
+
+[60] _De mirabilibus Auscultationibus_, Cap. 51 (_v._ Cohen, _Zeitschr.
+physikal. Chem._, 1901, 36. 513).
+
+[61] E. Cohen and C. van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 601;
+Cohen, _ibid._, 1900, 33. 59; 35. 588; 1901, 36. 513; Cohen and E.
+Goldschmidt, _ibid._, 1904, 50. 225.
+
+[62] _Zeitschr. physikal. Chem._, 1900, 33, 58.
+
+[63] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. Trav.
+Chim. Pays-Bas_, 1888, 7. 152.
+
+[64] Zincke, _Ber._, 1871, 4. 576.
+
+[65] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 313.
+
+[66] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.
+
+[67] Roozeboom, _ibid._, p. 179.
+
+[68] Schrötter, _Pogg. Annalen_, 1850, 81. 276; Troost and Hautefeuille,
+_Annales de Chim. et Phys._ 1874 [5], 2. 153; _Ann. Scient. École Norm._
+1868 [2], II. 266.
+
+[69] Pedler, _Trans. Chem. Soc._, 1890, 57. 599.
+
+[70] Brodie, _Trans. Chem. Soc._, 1853, 5, 289.
+
+[71] This is a familiar fact in the case of the solubility in carbon
+disulphide.
+
+[72] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 170.
+
+[73] _Trans. Chem. Soc._, 1899, 57. 734.
+
+[74] Carnelley, _Trans. Chem. Soc._, 1876, 29. 489; 1878, 33. 275. V. Meyer
+and Riddle, _Ber._, 1893, 26. 2443.
+
+[75] Riecke, _Zeitschr. physikal. Chem._, 1890, 6. 411.
+
+[76] _Annalen der Physik._, 1898 [3], 66. 492.
+
+[77] _Zeitschr. physikal. Chem._, 1899, 28. 666.
+
+[78] See Naumann, _Ber._, 1872, 4. 646; Troost and Hautefeuille, _Compt.
+rend._, 1868, 66. 795; 1868, 67. 1345; Roozeboom, _Das Heterogene
+Gleichgewicht_, I. pp. 62, 171.
+
+[79] Mitscherlich, _Lieb. Annalen_, 1834, 12. 137; Deville and Troost,
+_Compt. rend._, 1863, 56. 891.
+
+[80] Beckmann, _Zeitschr. physikal. Chem._, 1890, 5. 79; Hertz, _ibid._, 6.
+358.
+
+[81] _Ber._, 1902, 35. 351. _Cf._ also, K. Schaum, _Annalen der Chem._,
+1898, 300. 221; R. Wegscheider and Kaufler, _Sitzungsber. kaiserl. Akad.
+Wissensch. in Wien_, 1901, 110, II. 606.
+
+[82] See also Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.
+
+[83] _Annales de Chim. et Phys._, 1874 [5], 2. 154.
+
+[84] _Compt. rend._, 1887, 104. 1505.
+
+[85] _Compt. rend._, 1868, 66. 795.
+
+[86] _Phil. Mag._, 1884 [5], 18. 210. See also Roozeboom, _Das Heterogene
+Gleichgewicht_, I. p. 177.
+
+[87] Brauns, _Neues Jahrbuch für Mineralogie_, 1900, 13. Beilage-Band, p.
+39; Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 181.
+
+[88] _Monatshefte_, 1888, 9. 435.
+
+[89] Gattermann, _Ber._, 1890, 53. 1738.
+
+[90] _Zeitschr. physikal. Chem._, 1889, 4. 468; _Annalen der Physik_, 1900
+[4], 2. 649.
+
+[91] Quincke, _Annalen der Physik_, 1894 [3], 53. 613; Tammann, _Annalen
+der Physik_, 1901 [4], 4. 524; 1902, 8. 103; Rotarski, _ibid._, 4. 528.
+
+[92] _Annalen der Physik_, 1900 [4], 2. 649.
+
+[93] _Annalen der Physik_, 1902 [4], 8. 911.
+
+[94] See, more especially, O. Lehmann, _Annalen der Physik_, 1900 [4], 2.
+649; Reinitzer, _Sitzungsber. kaiserl. Akad. zu Wien._, 1888, 94. (2), 719;
+97. (1), 167; Gattermann, _loc. cit._; Schenck, _Zeitschr. physikal.
+Chem._, 1897, 23. 703; 1898, 25. 337; 27. 170; 1899, 28. 280; Schenck and
+Schneider, _ibid._, 1899, 29. 546; Abegg and Seitz, _ibid._, 1899, 29. 491;
+Hulett, _ibid._, 1899, 28. 629; Coehn, _Zeitschr. Elektrochem._, 1904, 10.
+856: Bredig and Schukowsky, _ibid._, 3419. For a full account of the
+subject, the reader is referred to the work by Lehmann, _Flüssige
+Kristalle_ (Engelmann, 1904), or the smaller monograph by Schenck,
+_Kristallinische Flüssigkeiten und flüssige Kristalle_ (Engelmann, 1905).
+
+[95] A. C. de Kock, _Zeitschr. physikal. Chem._, 1904, 48. 129.
+
+[96] On account of the fact that all grades of rigidity have been realized
+between the ordinary solid and the liquid state, in the case both of
+crystalline and amorphous substances, it has been proposed to abandon the
+terms "solid" and "liquid," and to class bodies as "crystalline" or
+"amorphous," the passage from the one condition to the other being
+discontinuous; crystalline bodies possess a certain regular orientation of
+their molecules and a directive force, while in amorphous bodies these are
+wanting (see Lehmann, _Annalen der Physik_, 1900 [4], 2. 696).
+
+[97] Hulett, _loc. cit._
+
+[98] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 144. See also
+Schenck, _Kristallinische Flüssigkeiten und flüssige Kristalle_, p. 8
+(Engelmann, 1904).
+
+[99] The possible number of triple points in a one-component system is
+given by the expression (_n_(_n_ - 1)(_n_ - 2))/1.2.3, where _n_ is the
+number of phases (Riecke, _Zeitschr. physikal. Chem._, 1890, 6, 411). The
+number of triple points, therefore, increases very rapidly as the number of
+possible phases increases.
+
+[100] Duhem, _Zeitschr. physikal. Chem._, 1891, 8. 371. _Cf._ Roozeboom,
+_Das Heterogene Gleichgewicht_, p. 94 ff.
+
+[101] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 99.
+
+[102] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.
+
+[103] These changes can be predicted quantitatively by means of the
+thermodynamic equation, _dp_/_dt_ = Q/(T(_v_{2}_ - _v_{1}_)), provided the
+specific volumes of the phases are known, and the heat effect which
+accompanies the transformation of one phase into the other.
+
+[104] _Studies on Chemical Dynamics_, translated by Ewan, p. 218.
+
+[105] Le Chatelier, _Compt. rend._, 1884, 99. 786.
+
+[106] See _Principles of Inorganic Chemistry_, translated by Findlay, 2nd
+edit., p. 133. (Macmillan, 1904.)
+
+[107] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.
+
+[108] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 189.
+
+[109] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 125. See also
+Zawidski, _Zeitschr. physikal. Chem._, 1904, 47. 727; van Eyk, _ibid._,
+1905, 51. 720.
+
+[110] Roberts-Austen, _Proc. Roy. Soc._, 63. 454; Spring, _Zeitschr.
+physikal. Chem._, 1894, 15. 65. See also p. 35.
+
+[111] Ramsay and Young, _Phil. Trans._, 1884, 175. 461; Allen, _Trans.
+Chem. Soc._, 1900, 77. 413.
+
+[112] Ramsay and Young, _Phil. Trans._ 1886, 177. 87.
+
+[113] This is exemplified in the well-known experiment with the cryophorus.
+
+[114] Tammann has, however, found that the fusion curve (solid in contact
+with liquid) of phosphonium chloride can be followed up to temperatures
+above the critical point (_Arch. néer._, 1901 [2], 6. 244).
+
+[115] _Phil. Mag._, 1886, 21. 33. See also S. A. Moss, _Physical Review_,
+1903, 16. 356.
+
+[116] This is found also in the case of bismuth. See Tammann, _Zeitschr.
+anorgan. Chem._, 1904, 40. 54.
+
+[117] See p. 57, footnote.
+
+[118] _Pogg. Annalen_, 1850, 81. 562.
+
+[119] Barus, _Amer. Jour. Sci._, 1892, 42. 125; Mack, _Compt. rend._, 1898,
+127. 361; Hulett, _Zeitschr. physikal. Chem._, 1899, 38. 629.
+
+[120] _Annalen der Physik_, 1899 [3], 68. 553, 629; 1900 [4], 1. 275; 2.
+1; 3. 161. See also Tammann, _Kristallisieren und Schmelzen_ (Leipzig,
+1903).
+
+[121] Ostwald, _Lehrbuch_, II. 2. 373; Poynting, _Phil. Mag._, 1881 [5],
+12. 2; Planck, _Wied. Annalen_, 1882, 15. 446.
+
+[122] Bakhuis Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 91.
+
+[123] Lussana, _Il nuovo Cimento_, 1895 [4], 1. 105.
+
+[124] Tammann, _Zeitschr. physikal. Chem._, 1903, 46. 818.
+
+[125] Foote, _Zeitschr. physikal. Chem._, 1900, 33. 740.
+
+[126] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 289.
+
+[127] Van't Hoff, _Arch, néer._, 1901, 6. 471.
+
+[128] See, for example, the determinations of the solubility of rhombic and
+monoclinic sulphur, by J. Meyer, _Zeitschr. anorg. Chem._, 1902, 33. 140.
+
+[129] _Zeitschr. physikal. Chem._, 1899, 32. 506.
+
+[130] Kastle and Reed, _Amer. Chem. Jour._, 1902, 27. 209.
+
+[131] _Zeitschr. physikal. Chem._, 1900, 35. 581.
+
+[132] _Compt. rend._, 1882, 95. 1278; 1884, 97. 1298, 1366, 1433.
+
+[133] _Zeitschr. physikal. Chem._, 1893, 12. 545.
+
+[134] _Sitzungsber. Wiener Akad._, 1894, 103. IIa. 226.
+
+[135] _Zeitschr. physikal. Chem._, 23-29. See also Küster, _ibid._, 25-28.
+
+[136] _Zeitschr. physikal. Chem._, 1897, 24. 152.
+
+[137] _Ibid._, 1898, 27. 585.
+
+[138] See W. Guertler, _Zeitschr. anorgan. Chem._, 1904, 40. 268; Tammann,
+_Zeitschr. Elektrochem._, 1904, 10. 532.
+
+[139] E. von Pickardt, _Zeitschr. physikal. Chem._, 1902, 42. 17.
+
+[140] _Zeitschr. physikal. Chem._, 1904, 48. 467.
+
+[141] M. Padoa, _Accad. Lincei, Atti_, 1904, 13. 329.
+
+[142] Deville, _Compt. rend._, 1852, 34. 561; Payen, _ibid._, 1852, 34.
+508; Debray, _ibid._, 1858, 46. 576. It has also been found by Jaffé
+(_Zeitschr. physikal. Chem._, 1903, 43. 465) that when spontaneous
+crystallization from solution occurs, the less stable form always separates
+first when purification has been carried sufficiently far.
+
+[143] Brauns, _Neues Jahrbuch für Mineralogie_, 1899, 13. (Beilage Band)
+84.
+
+[144] _Lehrbuch_, II. 2. 445. See also _Principles of Inorganic Chemistry_,
+2nd edit., p. 210 ff.
+
+[145] Schaum and Schönbeck, _Annalen der Physik_, 1902 [4], 8. 652. See
+also Chr. Füchtbauer, _Zeitschr. physikal. Chem._, 1904, 48. 549.
+
+[146] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.
+
+[147] See volume in this series on _Chemical Dynamics_, by Dr. J. W.
+Mellor.
+
+[148] Isambert, _Compt. rend._, 1881, 92. 919; 1882, 94. 958; 1883, 96.
+643. Walker and Lumsden, _Jour. Chem. Soc._, 1897, 71. 428.
+
+[149] _Compt. rend._, 1867, 64. 603.
+
+[150] _Compt. rend._, 1883, 102. 1243.
+
+[151] _Compt. rend._, 1868, 66, 1259.
+
+[152] Horstmann, _Ber._, 1876, 9. 749.
+
+[153] _Loc. cit._
+
+[154] For the reasons for choosing anhydrous salt and water instead of salt
+hydrate and water as components, see p. 14.
+
+[155] See Ostwald, _Lehrbuch_, II. 2. 527.
+
+[156] Ostwald, _Lehrbuch_, II. 2. 538.
+
+[157] _Zeitschr. physikal. Chem._, 1889, 4. 43.
+
+[158] _Ber._, 1876, 9. 749.
+
+[159] See, for example, van't Hoff, _Lectures on Theoretical and Physical
+Chemistry_, I. p. 62 (Arnold).
+
+[160] _Jour. Chem. Soc._, 1877, 32. 395.
+
+[161] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.
+
+[162] _Zeitschr. physikal. Chem._, 1887, 1. 5; 1895, 17. 52.
+
+[163] It is important to powder the salt, since otherwise the dehydration
+of the hydrate and the production of equilibrium occurs with comparatively
+great tardiness.
+
+[164] A chemical individual is a substance which persists as a phase of
+constant composition when the conditions of temperature, pressure, and
+composition of the other phases present, undergo continuous alteration
+within certain limits--the limits of existence of the substance (Wald,
+_Zeitschr. physikal. Chem._, 1897, 24. 648).
+
+[165] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323; Ostwald,
+_Lehrbuch_, I. 606.
+
+[166] That mercury does dissolve in water can be argued from analogy, say,
+with mercury and bromonaphthalene. At the ordinary temperature these two
+liquids appear to be quite insoluble in one another, but at a temperature
+of 280° the mercury dissolves in appreciable quantity; for on heating a
+tube containing bromonaphthalene over mercury the latter sublimes _through_
+the liquid bromonaphthalene and condenses on the upper surface of the tube.
+
+[167] _Phil. Mag._, 1884, [5], 18. 22; 495.
+
+[168] _Wied. Annalen_, 1886, 28. 305.
+
+[169] _Zeitschr. physikal. Chem._, 1898, 26. 433.
+
+[170] Rothmund, _loc. cit._
+
+[171] Rothmund, _loc. cit._
+
+[172] A similar behaviour is found in the case of diethylamine and water
+(R. T. Lattey, _Phil. Mag._, 1905, [6], 10, 397).
+
+[173] C. S. Hudson, _Zeitschr. physikal. Chem._, 1904, 47. 113.
+
+[174] Konowaloff, _Wied. Annalen_, 1881, 14. 219. Ostwald, _Lehrbuch_, II.
+2. 687. Bancroft, _Phase Rule_, p. 96.
+
+[175] Konowaloff, _loc. cit._
+
+[176] Roozeboom, _Zeitschr. physikal. Chem._, 1891, 8. 526; _Rec. Trav.
+Chim. Pays-Bas_, 1884, 3. 38.
+
+[177] Konowaloff, _loc. cit._ Cf. Bancroft, _Phase Rule_, p. 100.
+
+[178] _Phil. Mag._, 1884 [5], 18. 503.
+
+[179] See, for example, Walker, _Introduction to Physical Chemistry_, 3rd
+edit., p. 86 (Macmillan, 1903). Consult also Young, _Fractional
+Distillation_ (Macmillan, 1903), or Kuenen, _Verdampfung und Verflüssigung
+von Gemischen_ (Barth, 1906), where the subject is fully treated.
+
+[180] Since this is the only phase of variable composition present.
+
+[181] E. von Stackelberg, _Zeitschr. physikal. Chem._, 1896, 20. 337. If
+the change of volume which accompanies solution, and the heat effect are
+known, the quantitative change of the solubility with the pressure can be
+calculated (Braun, _Zeitschr. physikal. Chem._, 1887, 1. 259).
+
+[182] Van't Hoff, _Arch. néerland._ 1901 [2], 6. 471.
+
+[183] Tilden and Shenstone, _Phil. Trans._ 1884, 175. 23; Hulett and Allen,
+_Jour. Amer. Chem. Soc._ 1902, 24. 667; Andreä, _Jour. prak. Chem._ 137.
+474; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Mylius and v. Wrochem,
+_Ber._ 1900, 33. 3689.
+
+[184] E. von Stackelberg, _Zeitschr. physikal. Chem._ 1896, 20. 159; 1898,
+26. 533; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Holsboer, _Zeitschr.
+physikal. Chem._, 1902, 39. 691.
+
+[185] Reicher and van Deventer, _Zeitschr. physikal. Chem._ 1890, 5. 559;
+cf. Ostwald, _Lehrbuch_, II. 2. 803.
+
+[186] It has been shown that the formula of Ramsay and Young (p. 66) can be
+applied (with certain restrictions) to the interpolation and extrapolation
+of the solubility curve of a substance provided two (or three) points on
+the curve are known. In this case T, T_{1}, etc., refer to the temperatures
+at which the two substances--one the solubility curve of which is known,
+the other the solubility curve of which is to be calculated--have equal
+solubilities, instead of, as in the previous case, equal vapour pressures.
+(Findlay, _Proc. Roy. Soc._, 1902, 69. 471; _Zeitschr. physikal. Chem._,
+1903, 42. 110.)
+
+[187] W. Müller and P. Kaufmann, _Zeitschr. physikal. Chem._ 1903, 42. 497.
+
+[188] W. O. Rabe, _Zeitschr. physikal. Chem._, 1901, 38. 175.
+
+[189] With regard to the limits of supersaturation and the spontaneous
+crystallization of the solute from supersaturated solutions, see Jaffé,
+_Zeitschr. physikal. Chem._, 1903, 43. 565, and the very interesting paper
+by Miers and Isaac, _Trans. Chem. Soc._, 1906, 89. 413.
+
+[190] _Annales chim. phys._, 1894 [7], 2. 524.
+
+[191] _Phil. Trans._, 1884, 175. 23.
+
+[192] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 543.
+
+[193] _Zeitschr. physikal. Chem._, 1903, 43. 313.
+
+[194] Guthrie, _Phil. Mag._, 1875, [4], 49. 1; 1884, [5], 17. 462.
+
+[195] See Roloff, _Zeitschr. physikal. Chem._, 1895, 17. 325; Guthrie,
+_loc. cit._
+
+[196] Guthrie, _Phil. Mag._, _loc. cit._ Cf. Ostwald, _Lehrbuch_, II. 2.
+843.
+
+[197] Guthrie, _Phil. Mag._, 1875 [4], 49. 269.
+
+[198] _Ber._, 1877, 20. 2223.
+
+[199] _Silz-Ber. Wien. Akad._, 1880, 81. II. 1058.
+
+[200] Guthrie, _Phil. Mag._, 1875 [4], 49. 206.
+
+[201] If in the neighbourhood of the cryohydric point solution should be
+accompanied by an evolution of heat, then as the solubility would in that
+case increase with fall of temperature, salt would pass into solution.
+
+[202] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193.
+
+[203] _Zeitschr. physikal. Chem._, 1897, 23. 418.
+
+[204] Provided the solid nitrile is not present in too great excess.
+
+[205] _Wied. Annalen_, 1886, 28. 328. Cf. Ostwald, _Lehrbuch_, II. 2. 872.
+
+[206] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193. Schreinemakers,
+_ibid._, 1897, 23. 417. Roozeboom, _Rec. trav. chim. Pays-Bays_, 1889, 8.
+257. Bruner, _Zeitschr. physikal. Chem._, 1897, 23. 542.
+
+[207] Van't Hoff, _Lectures on Theoretical Chemistry_, I. p. 42. Ostwald,
+_Lehrbuch_, II. 2. 824.
+
+[208] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
+Findlay, 2nd edit., p. 453 (Macmillan, 1904); Skirrow and Calvert,
+_Zeitschr. physikal. Chem._, 1901, 37. 217.
+
+[209] _Vide_ Loewel, _Annales chim. phys._, 1857 [3], 49. 32. Cf.
+Löwenherz, _Zeitschr. physikal. Chem._, 1895, 18. 82.
+
+[210] Loewel, _loc. cit._ Gay-Lussac, _Annales chim. phys._, 1819, 11. 296.
+For the solubility at higher temperatures, see Tilden and Shenstone, _Phil.
+Trans._, 1884, 175. 23. Étard, _Annales chim. phys._, 1894 [7], 2. 548.
+
+[211] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690; Richards and
+Wells, _ibid._, 1903, 43. 465. This temperature is not quite the same as
+that of the _quadruple point_ anhydrous salt--hydrated
+salt--solution--vapour, because the latter is the temperature at which the
+system is under the pressure of its own vapour. Since, however, the
+influence of pressure on the solubility is very slight (p. 107), the
+position of the two points will not be greatly different. The quadruple
+point was found by Cohen (_Zeitschr. physikal. Chem._, 1894, 14. 90) to be
+32.6° and 30.8 mm. of mercury.
+
+[212] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+185. Cf. Cohen, _ibid._, 1894, 14. 88.
+
+[213] Debray, _Compt. rend._, 1868, 66. 194.
+
+[214] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690. A number of
+other salt hydrates, having transition-points ranging from 20° to 78°,
+which might be used for the same purpose, have been given by Richards and
+Churchill, _ibid._, 1899, 28. 313.
+
+[215] _Zeitschr. physikal. Chem._, 1903, 46. 818.
+
+[216] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 67.
+
+[217] Cohen, _Zeitschr. physikal. Chem._, 1894, 14. 90.
+
+[218] Ziz, _Schweigger's Journal_, 1815, 15. 166. See Ostwald, _Lehrbuch_,
+II. 2. 717.
+
+[219] See, for example, the solubility determinations published in
+_Wissenschaftliche Abhandl. der physikalisch-technischen Reichsanstalt_,
+Vol. III., or in the _Berichte_, for the years 1897-1901.
+
+[220] Meusser, _Ber._, 1901, 34. 2440.
+
+[221] Mylius and von Wrochem, _Ber._, 1900, 33. 3693.
+
+[222] Walker and Fyffe, _Jour. Chem. Soc._, 1903, 83. 180.
+
+[223] _Monatshefte_, 1887, 8. 601.
+
+[224] The equilibria between calcium chloride and water have been most
+completely studied by Roozeboom (_Zeitschr. physikal. Chem._, 1889, 4. 31).
+
+[225] Hammerl, _Sitzungsber. Wien. Akad._, 2^{te} Abteil, 1878, 78. 59.
+Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.
+
+[226] Lidbury, _Zeitschr. physikal. Chem._, 1902, 39. 453. The curvature at
+the melting point is all the greater the more the compound is dissociated
+into its components in the liquid state. If the compound is _completely
+undissociated_, even in the vapour phase, the two branches of the curve
+will _intersect_, (_e.g._ pyridine and methyl iodide; Aten, _Versl. Konink.
+Akad. Wetensch. Amsterdam_, 1905, 13. 462). The smaller the degree of
+dissociation, therefore, the sharper will be the bend. (See Stortenbeker,
+_Zeitschr. physikal. Chem._, 1892, 10. 194.) From the extent of flattening
+of the curve, it is also possible, with some degree of approximation, to
+calculate the degree of dissociation of the substance in the fused state.
+(See Roozeboom and Aten, _Zeitschr. physikal. Chem._, 1905, 53. 463;
+Kremann, _Zeitschr. Elektrochem._, 1906, 12. 259.)
+
+[227] See Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.
+
+[228] Tammann, _Wied. Annalen_, 1899, 68. 577.
+
+[229] Duhem, _Journ. Physical Chem._, 1898, 2. 31.
+
+[230] Gibbs, _Trans. Conn. Acad._, 3. 155; Saurel, _Journ. Phys. Chem._,
+1901, 5. 35.
+
+[231] In the case of the fusion of a compound of two components with
+formation of a liquid phase of the same composition, the temperature is a
+maximum; in the case of liquid mixtures of constant boiling-point, the
+temperature may be a minimum (p. 105).
+
+[232] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477. The formula of
+ferric chloride has been doubled, in order to avoid fractions in the
+expression of the water of crystallization.
+
+[233] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477.
+
+[234] A similar series of hydrates is formed by zinc chloride and water
+(Dietz and Mylius, _Zeitschr. anorg. Chem._, 1905, 44. 209).
+
+[235] Meyerhoffer, _Ber._, 1897, 30. 1810.
+
+[236] Walden, _Ber._, 1899, 32. 2863.
+
+[237] _Zeitschr. physikal. Chem._, 1903, 42. 432.
+
+[238] This composition was also confirmed by measurements of the vapour
+pressure (cf. p. 90).
+
+[239] Since all substances are no doubt volatile to a certain extent at
+some temperature, it is to be understood here that the substances are
+appreciably volatile at the temperature of the experiment.
+
+[240] For a general discussion of the partial pressures in a system of two
+components, see Bancroft, _Journ. Physical Chem._, 1899, 3. 1.
+
+[241] _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. trav. chim.
+Pays-Bas_, 1888, 7. 152.
+
+[242] The composition of a solution is represented symbolically by placing
+a double wavy line between the symbols of the components, and indicating
+the number of atoms present in the ordinary manner: thus, I [wavy] Cl_{_x_}
+represents a solution containing _x_ atoms of chlorine to one atom of
+iodine (Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 450).
+
+[243] Since iodine monochloride in the liquid state is only very slightly
+dissociated, the bend at C is very sharp (see p. 147, footnote). See also
+the investigation of the system pyridine and methyl iodide (Aten, _Versl.
+Konink. Akad. Wetensch. Amsterdam_, 1905, 13. 462).
+
+[244] This upper branch of the curve is not shown in the figure, as the
+ordinate corresponding to 30° would be very great.
+
+[245] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 22.
+
+[246] Ramsay and Young, _Journ. Chem. Soc._, 1886, 49. 458.
+
+[247] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 77 (Arnold).
+
+[248] This is different from what we found in the case of non-volatile
+solutes (p. 126). In the present case, the _partial pressure_ of the iodine
+in the vapour will be lowered by addition of chlorine, but the _total
+pressure_ is increased.
+
+[249] The diminution of volume is supposed to be carried out at constant
+temperature. The pressure and the composition of the phases must,
+therefore, remain unchanged, and only the relative amounts of these can
+undergo alteration.
+
+[250] At point _b_ the ratio of chlorine to iodine in the solution is less
+than in the monochloride, so that by the separation of this the excess of
+chlorine yielded by the condensation of the vapour is removed.
+
+[251] Roozeboom, _Rec. trav. chim. Pays-Bas_, 1884, 3. 29; 1885, 4. 65;
+_Zeitschr. physikal. Chem._, 1888, 2. 450.
+
+[252] Two curves "enclose" a field when they form with one another an angle
+less than two right angles.
+
+[253] Roozeboom, _Zeitschr. physikal. Chem._, _loc. cit._
+
+[254] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323.
+
+[255] Bancroft has proposed to restrict the term "occlusion" to the
+formation of solid solutions, and to apply "adsorption" only to effects
+which are primarily due to surface tension. Such a distinction, however,
+would probably be very difficult to carry through, for although adsorption
+may, in large measure, be due to surface tension, the behaviour of adsorbed
+substances is similar to that of substances existing in solid solutions.
+
+[256] Tammann, _Wied. Annalen_, 1897, 63. 16; _Zeitschr. physikal. Chem._,
+1898, 27. 323.
+
+[257] See, for example, Chappuis, _Wied. Annalen_, 1881, 12. 161; Joulin,
+_Annal. chim. phys._, 1881, [5], 22. 398; Kayser, _Wied. Annalen_, 1881,
+12. 526.
+
+[258] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.
+
+[259] _Annales chim. phys._, 1874, [5], 2. 279.
+
+[260] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1; Dewar, _Phil.
+Mag._, 1874, [4], 47, 324, 342; Mond, Ramsay and Shields, _Proc. Royal
+Soc._, 1897, 62. 290.
+
+[261] _Loc. cit._
+
+[262] It is noteworthy that the form of curve obtained for hydrogen and
+palladium bears a striking resemblance to that for the dehydration of
+colloids containing absorbed water, _e.g._ silicic acid (_vide_ van
+Bemmelen, _Zeitschr. anorg. Chem._, 1897-1900. Cf. Zacharias, _Zeitschr.
+physikal. Chem._, 1902, 39. 480).
+
+[263] _Zeitschr. physikal. Chem._, 1890, 5. 322.
+
+[264] Küster, _Zeitschr. physikal. Chem._, 1895, 17. 367. Bodländer, _Neues
+Jahrbuch f. Mineralogie_, 1898-99, Beilage Band, 12. 92.
+
+[265] Bruni and Padoa, _Atti Accad. Lincei_, 1902 [5], 11. 1; 565.
+
+[266] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 385; Bruni, _Rend.
+Accad. Lincei_, 1898, 2. 138, 347. For a general account of "solid
+solutions" the reader is referred to Bruni, "_Ueber feste Lösungen_"
+(Ahrens'sche Sammlung), and to Bodländer, _loc. cit._ For the formation and
+transformation of liquid mixed crystals, see A. C. de Kock, _Zeitschr.
+physikal. Chem._, 1904, 48. 129.
+
+[267] In discussing the various systems which may be obtained here,
+Roozeboom (_loc. cit._) made use of the variation of the thermodynamic
+potential (p. 29) with the concentration. In spite of the advantages which
+such a treatment affords, the temperature-concentration diagram has been
+adopted as being more readily understood and as more suitable for an
+elementary discussion of the subject.
+
+[268] These curves are also called the "liquidus" and the "solidus" curve
+respectively.
+
+[269] Küster, _Zeitschr. physikal. Chem._, 1895, 17. 360.
+
+[270] Küster, _ibid._, 1891, 8. 589.
+
+[271] It should be remarked that the behaviour described here will hold
+strictly only when the solid mixed crystals undergo change sufficiently
+rapidly to be always in equilibrium with the liquid. This, however, is not
+always the case (see Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494;
+van Wyk, _Zeitschr. anorg. Chem._, 1905, 48. 25), and complete
+solidification will not in this case take place at the temperature
+corresponding with the line _dc_ in Fig. 50, but only at a lower
+temperature.
+
+[272] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 469.
+
+[273] Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494.
+
+[274] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 542.
+
+[275] Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 430.
+
+[276] Cady, _Journ. Physical. Chem._, 1899, 3. 127.
+
+[277] See Roberts-Austen and Stansfield, _Rapports du congrès international
+de physique_, 1900, I. 363.
+
+[278] Heycock and Neville, _Proc. Roy. Soc._, 1903, 71. 409. For the
+partial liquefaction of mixed crystals on cooling, see also A. C. de Kock
+(_Zeitschr. physikal. Chem._, 1904, 48. 129).
+
+[279] Armstrong, _Watt's Dictionary of Chemistry_ (Morley and Muir), III.,
+p. 88. See also Lowry, _Jour. Chem. Soc._, 1899, 75. 211.
+
+[280] See Bancroft, _Journ. Physical Chem._, 1898, 2. 143; Roozeboom,
+_Zeitschr. physikal. Chem._, 1899, 28. 288.
+
+[281] Hylotropic substances are such as can undergo transformation into
+other substances of the same composition (Ostwald, _Lehrbuch_, II. 2. 298).
+
+[282] Also called Equilibrium Point (Lowry).
+
+[283] For a discussion of these systems, see Roozeboom, _Zeitschr.
+physikal. Chem._, _loc. cit_.
+
+[284] See Bancroft, _loc. cit._, p. 147; Wegscheider, _Sitzungsber. Wiener
+Akad._, 1902, 110. 908.
+
+[285] Reference may be made here to the term "stability limit," introduced
+by Knorr (_Annalen_, 1896, 293. 88) to indicate that temperature above
+which liquefaction and isomeric change takes place. As employed by Knorr
+and others, the term does not appear to have a very precise meaning, since
+it is used to denote, not the temperature at which these changes can occur,
+but the temperature at which the change is rapid (vide _Annalen_, 1896,
+293. 91; 1899, 306. 334); and the introduction of an indefinite velocity of
+change renders the temperature of the stability limit also somewhat
+indefinite. The definiteness of the term is also not a little diminished by
+the fact that the "limit" can be altered by means of catalytic agents.
+Since, as we have seen, the stable modification can always undergo isomeric
+change and liquefy at temperatures above the natural freezing point, but
+not below that point; and, further, the less stable modification can
+undergo isomeric transformation and liquefy at temperatures above the
+eutectic point, but will not liquefy at temperatures below that; it seems
+to the author that it would be more precise to identify these two
+points--the natural freezing point and the eutectic point--which are not
+altered by catalytic agents, with the "stability limits" of the stable and
+unstable modification respectively. A perfectly definite meaning would
+thereby be given to the term. In the case of those substances which do not
+undergo appreciable isomeric change at the temperature of the melting
+point, the stability limits would be the points G and H, Fig. 60.
+
+[286] Cameron, _Journ. Physical Chem._, 1898, 2. 409.
+
+[287] Carveth, _Journ. Phys. Chem._, 1898, 2. 159. See also Dutoit and
+Fath, _Journ. chim. phys_., 1903, 1. 358; Findlay, _Trans. Chem. Soc._,
+1904, 85. 403.
+
+[288] Hollmann, _Zeitschr. physikal. Chem._, 1903, 43. 129.
+
+[289] For other examples of the application of the Phase Rule to isomeric
+substances, see _Journ. Physical Chem._, vols. 2. _et seq._; Findlay,
+_Trans. Chem. Soc._, 1904, 85. 403.
+
+[290] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 410.
+
+[291] See also Saposchnikoff, _Zeitschr. physikal. Chem._, 49. 688;
+Kremann, _Monatshefte_, 1904, 25. 1215, 1271, 1311.
+
+[292] J. C. Philip, _Journ. Chem. Soc._, 1903, 83. 821.
+
+[293] _Cf._ also Paterno and Ampolla, _Gazzetta chim. ital._, 1897, 27.
+481.
+
+[294] Philip, _loc. cit._, p. 826.
+
+[295] Philip, _loc. cit._, p. 829. Compare curves for iodine monochloride,
+Fig. 42, p. 162.
+
+[296] Kuriloff, _Zeitschr. physikal. Chem._, 1897, 23. 676.
+
+[297] Ladenburg, _Ber._, 1895, 28. 163; 1991.
+
+[298] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 494; Adriani,
+_ibid._, 1900, 33. 453.
+
+[299] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 453.
+
+[300] A. Findlay and Miss E. Hickmans.
+
+[301] Kipping and Pope, _Journ. Chem. Soc._, 1897, 71. 993.
+
+[302] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 512; Adriani,
+_ibid._, 1900, 33. 473; 1901, 36. 168.
+
+[303] In this connection reference should be made more especially to the
+paper by Roberts-Austen and Stansfield, "Sur la constitution des alliages
+métalliques," in the _Rapports du congrès international de physique_, 1900,
+I. 363; J. A. Mathews, _Journ. of the Franklin Inst._, 1902; Gautier,
+_Compt. rend._, 1896, 123. 109; Roberts-Austen, "Reports of the Alloys
+Research Committee," in _Journ. Inst. Mechan. Engineers_, from 1891 to
+1904; and the papers by Heycock and Neville, published in the _Journ. Chem.
+Soc._, and the _Trans. Roy. Soc._ since 1897; also Neville, _Reports of the
+British Association_, 1900, p. 131. Reference must also be made to the
+important metallographic investigations by Tammann and his pupils, and of
+Kurnakoff (_Zeitschr. anorgan. Chem._, vol. 40 and onwards), and also to
+those of Shepherd, _Journ. Physical Chem._, 8. A bibliography of the alloys
+is given in _Zeitschr. anorgan. Chem._, 1903, 35. 249.
+
+[304] Kurnakoff and Puschin, _Zeitschr. anorgan. Chem._, 1902, 30. 104.
+
+[305] Gautier, _Bull. Soc. d'Encouragement_, 1896 [5], 1. 1312.
+
+[306] Heycock and Neville, _Phil. Trans._, 1900, 194. 201.
+
+[307] Gautier, _loc. cit._ See also Roberts-Austen and Rose, _Proc. Roy.
+Soc._, 1903, 71. 161.
+
+[308] Heycock and Neville, _Journ. Chem. Soc._, 1897, 71. 414.
+
+[309] See Roberts-Austen, _Introduction to Metallurgy_, 5th edit., p. 102;
+Bakhuis Roozeboom, _Journ. Iron and Steel Inst._, 1900, II. 311; _Zeitschr.
+physikal. Chem._, 1900, 34. 437; von Jüptner, _Siderology_, p. 223
+(translation by C. Salter); van't Hoff, _Zinn, Gips, und Stahl_, p. 24, or
+_Acht Vorträge über physikalische Chemie_, p. 37. Further, Roozeboom,
+_Zeitschr. Elektrochem._, 1904, 10. 489; E. Heyn, _ibid._, p. 491;
+Carpenter and Keeling, _Journ. Iron and Steel Inst._, 1904, 65. 224.
+
+[310] The melting point of pure iron is given by Carpenter and Keeling
+(_Journ. Iron and Steel Inst._, 1904, 65. 224) as 1505°.
+
+[311] _Zeitschr. für Elektrochem._, 1904, 10. 491.
+
+[312] See also Hiorns, _Journ. Soc. Chem. Ind._, 1906, 25. 50.
+
+[313] Bancroft, _Jour. Physical Chem._, 1902, 6. 178; Bell and Taber,
+_ibid._, 1906, 10. 120.
+
+[314] The method to be followed when the third component enters into the
+solid phase will be explained later.
+
+[315] Tammann, _Zeitschr. anorg. Chem._, 1903, 37. 303; 1905, 45. 24.
+Reference may be made here to the registering pyrometer of Kurnakoff,
+_Zeitschr. anorg. Chem._, 1904, 42. 184.
+
+[316] In this connection, see Doelter, _Physikalisch-chemisch Mineralogie_
+(Barth, 1901); Meyerhoffer, _Zeitschr. f. Kristallographie_, 1902, 36. 593;
+Guthrie, _Phil. Mag._, 1884 [5], 17. 479; Le Chatelier, _Compt. rend._,
+1900, 130. 85; and especially E. Baur, _Zeitschr. physikal. Chem._, 1903,
+42. 567; J. H. L. Vogt, _Zeitschr. Elektrochem._, 1903, 9. 852, and _Die
+Silikatschmelzlösungen_, Parts I. and II. (Christiania, 1903, 1904). See
+also N. V. Kultascheff, _Zeitschr. anorg. Chem._, 1903, 35. 187.
+
+[317] G. G. Stokes, _Proc. Roy. Soc._, 1891, 49. 174; Gibbs, _Trans. Conn.
+Acad._, 1876, 3. 176; Roozeboom, _Zeitschr. physikal. Chem._, 1894, 15.
+147.
+
+[318] This figure has been taken from Ostwald's _Lehrbuch_, II. 2. 984.
+
+[319] Roozeboom, _Zeitschr. physikal. Chem._, 1893, 12. 369.
+
+[320] C. R. A. Wright, _Proc. Roy. Soc._, 1891, 49. 174; 1892, 50. 375.
+
+[321] The distribution coefficient will not remain constant because, apart
+from other reasons, the mutual solubility of chloroform and water is
+altered by the addition of the acid.
+
+[322] Bancroft, _Physical Review_, 1895, 3. 21; Schreinemakers, _Zeitschr.
+physikal. Chem._, 1897, 23. 652, and subsequent volumes.
+
+[323] C. R. A. Wright, _Proc. Roy. Soc._, 1889-1893.
+
+[324] C. R. A. Wright, _Proc. Roy. Soc._, 1892, 50. 390.
+
+[325] Bodländer, _Berg- und Hüttenmänn. Ztg._, 1897, 56. 331.
+
+[326] C. R. A. Wright, _Proc. Roy. Soc._, _loc. cit._
+
+[327] Schreinemakers, _Zeitschr. physikal. Chem._, 1900, 33. 78.
+
+[328] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 27. 95.
+
+[329] Schreinemakers, _Zeitschr. physikal. Chem._, 1899, 29. 577.
+
+[330] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 25. 543.
+
+[331] Charpy, _Compt. rend._, 1898, 126. 1569. Compare the curves for the
+system KNO_{3}--NaNO_{3}--LiNO_{3} (H. R. Carveth, _Journ. Physical Chem._,
+1898, 2. 209). Also alloys of Pb--Sn--Bi (E. S. Shepherd, _Journ. Physical
+Chem._, 1902, 6. 527).
+
+[332] It should be remembered that in the triangular diagram a _line_
+parallel to one of the sides indicates, at a given temperature, a constant
+amount of the component represented by the opposite corner of the triangle;
+and, hence, points in a _plane_, parallel to one face of a right prism,
+will indicate for different temperatures, variation in the amounts of two
+components, but constancy in the amount of the third.
+
+[333] _Gazzetta chim. ital._, 1898, 28. II. 520.
+
+[334] Bruni, _Gazzetta chim. ital._, 1898, 28. II. 508; 1900, 30. I. 35.
+
+[335] _Zeitschr. physikal. Chem._, 1900, 36. 168.
+
+[336] For a discussion of these systems, see van't Hoff, _Bildung und
+Spaltung von Doppelsalzen_ (Leipzig, 1897).
+
+[337] Van Leeuwen, _Zeitschr. physikal. Chem._, 1897, 23. 35.
+
+[338] Meyerhoffer, _Zeitschr. physikal. Chem._, 1889, 3. 336; 1890, 5. 97.
+
+[339] Reicher, _Zeitschr. physikal. Chem._, 1887, 1. 220.
+
+[340] For other examples of the formation and decomposition of double salts
+at a transition point, the reader is referred to the work by van't Hoff,
+already cited, on the _Bildung und Spaltung von Doppelsalzen_; or to
+Bancroft, _Phase Rule_, p. 180.
+
+[341] Bancroft, _Phase Rule_, p. 183.
+
+[342] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 514.
+
+[343] The influence of pressure on the transition point in the case of
+tachydrite has been determined by van't Hoff, Kenrick, and Dawson
+(_Zeitschr. physikal. Chem._, 1901, 39. 27, 34; van't Hoff, _Zur Bildung
+der ozeanischen Salzablagerungen_, I. p. 66--Brunswick, 1905). This salt is
+formed from magnesium chloride and calcium chloride at 22°, in accordance
+with the equation--
+
+  2MgCl_{2}.6H_{2}O + CaCl_{2}.6H_{2}O = Mg_{2}CaCl_{6}.12H_{2}O + 6H_{2}O
+
+Increase of pressure raises the transition point, because the formation of
+tachydrite is accompanied by increase of volume; the elevation being 0.016°
+for an increase of pressure of 1 atm. The number calculated from the
+theoretical formula (p. 57) is 0.013° for 1 atm.
+
+If one calculates the influence of the pressure of sea-water on the
+temperature of formation of tachydrite (which is of interest on account of
+the natural occurrence of this salt), it is found that a depth of water of
+1500 metres, exerting a pressure of 180 atm., would alter the temperature
+of formation of tachydrite by only 3°. The effect is, therefore,
+comparatively unimportant.
+
+[344] Roozeboom, _Zeitschr. physical. Chem._, 1887, 1. 227.
+
+[345] _Zeitschr. physical. Chem._, 1887, 1. 227.
+
+[346] Van't Hoff and Müller, _Ber._, 1898, 31. 2206.
+
+[347] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+165.
+
+[348] For a full discussion of the solubility relations of sodium ammonium
+racemate, see van't Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 81.
+
+[349] _Annales chim. phys._, 1848 [3], 24. 442.
+
+[350] See Van't Hoff and van Deventer, _Zeitschr. phys. Chem._, 1887, 1.
+165.
+
+[351] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 121.
+
+[352] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 518.
+
+[353] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 109. On the
+importance of the transition interval in the case of optically active
+substances, see Meyerhoffer, _Ber._, 1904, 37. 2604.
+
+[354] In connection with this chapter, see, more especially, van't Hoff,
+_Bildung und Spaltung von Doppelsalzen_, p. 3, _ff._; Roozeboom, _Zeitschr.
+physikal Chem._, 1892, 10. 158; Bancroft, _Phase Rule_, p. 201; 209.
+
+[355] The same restriction must be made here as was imposed in the
+preceding chapter, namely, that the two salts in solution give a common
+ion.
+
+[356] For example, addition of ammonium chloride to solutions of ferric
+chloride (Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 149).
+
+[357] It must, of course, be understood that the temperature is on that
+side of the transition point on which the double salt is stable.
+
+[358] Excess of the double salt must be taken, because otherwise an
+unsaturated solution might be formed, and this would, of course, not
+deposit any salt.
+
+[359] Meyerhoffer, _Ber._, 1904, 37. 2605.
+
+[360] Meyerhoffer, _Ber._, 1897, 30. 1809.
+
+[361] Meyerhoffer, _Ber._, 1904, 37. 2604.
+
+[362] Bancroft, _Phase Rule_, p. 203; Roozeboom, _Zeitschr. physikal.
+Chem._, 1891, 8. 504, 531; Stortenbeker, _ibid._, 1895, 17. 643; 1897, 22.
+60; 1900, 34. 108.
+
+[363] Roozeboom, _Zeitschr. phys. Chem._, 1899, 28. 494; _Ber._, 1899, 32.
+537.
+
+[364] As, for instance, strychnine racemate, a compound of racemic acid
+with the _optically active_ strychnine. This would be resolved into
+strychnine _d_-tartrate and strychnine _l_-tartrate, which are not
+enantiomorphous forms.
+
+[365] Van't Hoff and Meyerhoffer, _Zeitschr. physikal Chem._, 1898, 27. 75;
+1899, 30. 86. Fig. 113 is taken from the latter paper.
+
+[366] Solid models constructed of plaster of Paris can be obtained from Max
+Kaehler and Martini, Berlin.
+
+[367] Instead of the present method of obtaining potassium chloride by
+decomposing carnallite with water, advantage might be taken of the fact
+that carnallite when heated to 168° undergoes decomposition with separation
+of three-fourths of the potassium chloride (van't Hoff, _Acht Vorträge über
+physikalische Chemie_, 1902, p. 32).
+
+[368] Roozeboom and Schreinemakers, _Zeitschr. physikal. Chem._, 1894, 15.
+588.
+
+[369] These curves represent only portions of the isotherms, since the
+systems in which a ternary solution is in equilibrium with solid hydrogen
+chloride or a hydrate, have not been investigated.
+
+[370] The numbers printed beside the points on the curves refer to the
+number of the experiment in the original paper.
+
+[371] Lash, Miller and Kenrick, _Journ. Physical. Chem._, 1903, 7. 259;
+Allan, _Amer. Chem. Journ._, 1901, 25. 307.
+
+[372] Allan, _Amer. Chem. Journ._, 1901, 25. 307.
+
+[373] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 651; Allan, _loc.
+cit._
+
+[374] Rutten, _Zeitschr. anorgan. Chem._, 1902, 30. 342. Compare the system
+BeO--SO_{3}--H_{2}O; Parsons, _Zeitschr. anorgan. Chem._, 1904, 42. 250.
+
+[375] _Zeitschr. anorgan. Chem._, 1904, 40. 146.
+
+[376] Schreinemakers, _Zeitschr. physikal. Chem._, 1893, 11. 76; Bancroft,
+_Journ. Physical Chem._, 1902, 6. 179.
+
+[377] _Zeitschr. anorgan. Chem._, 1904, 40. 148.
+
+[378] _Zeitschr. physikal. Chem._, 1903, 43. 354.
+
+[379] These equilibria were obtained by Boudouard, _Annales chim. phys._,
+1901 [7], 24. 5. See also Hahn, _Zeitschr. physikal. Chem._, 1903, 42.
+705; 44. 513.
+
+[380] G. Preuner, _Zeitschr. physikal. Chem._, 1903, 47. 385.
+
+[381] See Hahn, _Zeitschr. physikal. Chem._, 1903, 42. 705; 44. 513;
+Boudouard, _Bull. Soc. chim._, [3], 25. 484; Bodländer, _Zeitschr. f.
+Elektrochem._, 1902, 8. 833; R. Schenck and Zimmermann, _Ber._, 1903, 36.
+1231, 3663; Schenck and Heller, _ibid._, 1905, 38. 2132; _Zeitschr. f.
+Elektrochem._, 1903, 9. 691; Haber, _Thermodynamik technischer
+Gasreaktionen_, p. 293 (Munich, 1903).
+
+[382] A very useful summary of the investigations carried out by van't Hoff
+and his pupils on the formation of the Stassfurt salt-beds is given by E.
+F. Armstrong, in the _Reports of the British Association for 1901_, p. 262.
+See also van't Hoff, _Zur Bildung der ozeanischen Salzablagerungen_
+(Brunswick, 1905).
+
+[383] See especially Meyerhoffer, _Silzungsber. Wien. Akad._, 1895, 104.
+II. _b_, 840; Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899,
+28. 453; 31. 370. The investigation of the equilibria between reciprocal
+salt-pairs alone (three-component systems) is of great importance for the
+artificial preparations of minerals, as also in analytical chemistry for
+the proper understanding of the methods of conversion of insoluble systems
+into soluble by fusion (see Meyerhoffer, _Zeitschr. physikal. Chem._, 1901,
+38. 307).
+
+[384] See Meyerhoffer, _Zeitschr. physikal. Chem._, 1899, 28. 459.
+
+[385] Compare the reciprocal salt-pair NaCl--NH_{4}HCO_{3} (p. 321). In
+this case the upper limit of the transition interval was found by
+extrapolation of the solubility curve for NaHCO_{3} + NH_{4}Cl +
+NH_{4}HCO_{3} and NaHCO_{3} + NH_{4}Cl + NaCl to be 32° (Fedotieff,
+_Zeitschr. phys. Chem._, 1904, 49. 179).
+
+[386] Löwenherz, _Zeitschr. physikal. Chem._, 1894, 13. 464.
+
+[387] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.
+
+[388] As the quantities of the salts are expressed in _equivalent_
+gram-molecules, the molecule of sodium and potassium chloride must be
+doubled in order to be equivalent to sodium sulphate and potassium
+sulphate.
+
+[389] _Sitz-Ber. der kgl. preuss. Akad. der Wiss._, 1903, p. 359. Van't
+Hoff, _Zur Bildung der ozeanischen Salzablagerungen_, I. p. 34 (Brunswick,
+1905).
+
+[390] _Zeitschr. für Kristallographie_, 1904, 39. 155.
+
+[391] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.
+
+[392] _Zeitschr. physikal. Chem._, 1904, 49. 162.
+
+[393] Another commercial process, in the study of which good service is
+done by the Phase Rule, is the caustification of the alkali salts (G.
+Bodländer, _Zeitschr. für Elektrochem._, 1905, 11. 186; J. Herold, _ibid._,
+418).
+
+[394] _Zeitschr. physikal. Chem._, 1900, 35. 32.
+
+[395] Mention may also be made here of the equilibria between magnesium
+carbonate and potassium carbonate, although these do not form a reciprocal
+salt-pair (Auerbach, _Zeitschr. für Elektrochem._, 1904, 10. 161).
+
+[396] O. N. Witt and K. Ludwig, _Ber._, 1903, 36. 4384; Meyerhoffer,
+_ibid._, 1904, 37. 261, 1116.
+
+[397] _Zeitschr. physikal. Chem._, 1905, 53. 513. Compare also, _ibid._,
+1903, 38. 307.
+
+[398] See Schwarz, _Beiträge zur Kenntnis der umkehrbaren Umwandlungen
+polymorpher Korper_ (Göttingen, 1892); or, Roozeboom, _Heterogen.
+Gleichgewicht_, I. p. 125. Also Barnes and Cooke, _Journ. Physical Chem._,
+1902, 6. 172.
+
+[399] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+173.
+
+[400] Reicher, _Zeitschr. für Krystallographie_, 1884, 8. 593.
+
+[401] _Zeitschr. physikal. Chem._, 1895, 17. 153.
+
+[402] _Zeitschr. physikal. Chem._, 1899, 28. 464.
+
+[403] Meyerhoffer and Saunders, _ibid._, p. 466.
+
+[404] See Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 446.
+
+[405] See in this connection the volume in this series on
+_Electro-chemistry_, by Dr. R. A. Lehfeldt.
+
+[406] Barnes and Cooke, _Journ. Physical Chem._, 1902, 6. 172.
+
+[407] For a description and explanation of these, the reader should consult
+the volume in this series by Dr. Lehfeldt on _Electro-chemistry_; and van't
+Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 48 _ff._
+
+
+
+
+       *       *       *       *       *
+
+
+
+
+Changes made to the printed original.
+
+Pages 30-31. "Fig. 3, p. 27.": 'p. 25." in original. So also page 33, "Fig.
+2, p. 27".
+
+Page 57. "pp. 29 and 35": 'pp. 25 and 38" in original.
+
+Page 65. "p. 57.": 'p. 60" in original (twice).
+
+Page 166. "there is the point C_{1}": C' in original.
+
+Page 225. "C is an eutectic point": 'eutetic' in original.
+
+Page 228. "Although this view put forward by Heyn": 'Athough' in original.
+
+Page 232. "the period of constant temperature for the eutectic point c":
+'the eutectic point e' in original.
+
+Page 249. "two liquid layers between 13° and 31°": 'betwen' in original.
+
+Page 257. Tables entries 4 and 7. "naphthol": 'napthol' in original.
+
+Page 287. "from which the model is constructed": 'he model' in original.
+
+
+
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+<body>
+<h1>The Project Gutenberg eBook, The Phase Rule and Its Applications, by
+Alexander Findlay</h1>
+<pre>
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at <a href = "http://www.gutenberg.org">www.gutenberg.org</a></pre>
+<p>Title: The Phase Rule and Its Applications</p>
+<p>Author: Alexander Findlay</p>
+<p>Release Date: November 27, 2010  [eBook #34457]</p>
+<p>Language: English</p>
+<p>Character set encoding: ISO-8859-1</p>
+<p>***START OF THE PROJECT GUTENBERG EBOOK THE PHASE RULE AND ITS APPLICATIONS***</p>
+<p>&nbsp;</p>
+<h3>E-text prepared by Juliet Sutherland, Keith Edkins,<br />
+    and the Online Distributed Proofreading Team<br />
+    (http://www.pgdp.net)</h3>
+<p>&nbsp;</p>
+<table border="0" cellpadding="10" style="background-color: #ccccff; ">
+<tr>
+<td style="width:25%; vertical-align:top">
+Transcriber's note:
+</td>
+<td>
+A few typographical errors have been corrected. They
+appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the
+explanation will appear when the mouse pointer is moved over the marked
+passage.<br /><br />
+</td>
+</tr>
+</table>
+<p>&nbsp;</p>
+<hr class="pg" />
+<p>&nbsp;</p>
+
+<h3><span class="sc">TEXT-BOOKS of PHYSICAL CHEMISTRY.</span></h3>
+
+<p class="cenhead"><span class="sc">Edited by SIR WILLIAM RAMSAY, K.C.B., F.R.S., D.Sc.</span></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="short" />
+
+<blockquote class="b1n">
+
+  <p>STOICHIOMETRY. By <span class="sc">Sydney Young</span>, D.Sc., F.R.S.,
+  Professor of Chemistry in the University of Dublin; together with an
+  INTRODUCTION TO THE STUDY OF PHYSICAL CHEMISTRY by Sir <span
+  class="sc">William Ramsay</span>, K.C.B., F.R.S., Editor of the Series.
+  Crown 8vo. 7<i>s.</i> 6<i>d.</i></p>
+
+  <p>AN INTRODUCTION TO THE STUDY OF PHYSICAL CHEMISTRY. Being a General
+  Introduction to the Series by Sir <span class="sc">William Ramsay</span>,
+  K.C.B., F.R.S., D.Sc. Crown 8vo. 1<i>s.</i> net.</p>
+
+  <p>CHEMICAL STATICS AND DYNAMICS, including <span class="sc">The Theories
+  of Chemical Change, Catalysis and Explosions</span>. BY <span
+  class="sc">J. W. Mellor</span>, D.Sc. (N.Z.), B.Sc. (Vict.) Crown 8vo.
+  7<i>s.</i> 6<i>d.</i></p>
+
+  <p>THE PHASE RULE AND ITS APPLICATIONS. By <span class="sc">Alex.
+  Findlay</span>, M.A., Ph.D., D.Sc., Lecturer and Demonstrator in
+  Chemistry, University of Birmingham. With 134 Figures in the Text. Crown
+  8vo. 5<i>s.</i></p>
+
+  <p>SPECTROSCOPY. By <span class="sc">E. C. C. Baly</span>, F.I.C.,
+  Lecturer on Spectroscopy and Assistant Professor of Chemistry, University
+  College, London. With 163 Illustrations. Crown 8vo. 10<i>s.</i>
+  6<i>d.</i></p>
+
+  <p>THERMOCHEMISTRY. By <span class="sc">Julius Thomsen</span>, Emeritus
+  Professor of Chemistry in the University of Copenhagen. Translated by
+  <span class="sc">Katharine A. Burke</span>, B.Sc. (Lond.), Assistant in
+  the Department of Chemistry, University College, London. Crown 8vo.
+  9<i>s.</i></p>
+
+  <p>ELECTRO-CHEMISTRY. <span class="sc">Part I.&mdash;General
+  Theory.</span> By <span class="sc">R. A. Lehfeldt</span>, D.Sc.,
+  Professor of Physics at the East London Technical College. Including a
+  Chapter on the Relation of Chemical Constitution to Conductivity, by
+  <span class="sc">T. S. Moore</span>, B.A., B.Sc., Lecturer in the
+  University of Birmingham. Crown 8vo. 5<i>s.</i></p>
+
+  <p><span class="sc">Part II.&mdash;Applications to Electrolysis, Primary
+  and Secondary Batteries</span>, etc. By <span class="sc">N. T. M.
+  Wilsmore</span>, D.Sc.</p>
+
+  <p class="author">[<i>In the press.</i></p>
+
+  <p>STEREOCHEMISTRY. By <span class="sc">A. W. Stewart</span>, D.Sc.,
+  Carnegie Research Fellow, Lecturer on Stereochemistry in University
+  College, London. With 87 Illustrations. Crown 8vo. 10<i>s.</i>
+  6<i>d.</i></p>
+
+  <p>RELATIONS BETWEEN CHEMICAL CONSTITUTION AND PHYSICAL PROPERTIES. By
+  <span class="sc">Samuel Smiles</span>, D.Sc.</p>
+
+  <p class="author">[<i>In preparation.</i></p>
+
+  <p>THERMODYNAMICS. By <span class="sc">F. G. Donnan</span>, M.A.,
+  Ph.D.</p>
+
+  <p class="author">[<i>In preparation.</i></p>
+
+  <p>ACTINOCHEMISTRY. By <span class="sc">C. E. K. Mees</span>, D.Sc., and
+  <span class="sc">S. E. Sheppard</span>, D.Sc.</p>
+
+  <p class="author">[<i>In preparation.</i></p>
+
+  <p>PRACTICAL SPECTROGRAPHIC ANALYSIS. By <span class="sc">J. H.
+  Pollok</span>, D.Sc.</p>
+
+  <p class="author">[<i>In preparation.</i></p>
+
+</blockquote>
+
+<p class="cenhead">LONGMANS, GREEN, AND CO.<br />
+39 PATERNOSTER ROW, LONDON<br />
+NEW YORK, BOMBAY, AND CALCUTTA</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<h1>THE PHASE RULE</h1>
+
+<h3>AND ITS APPLICATIONS</h3>
+
+<p class="cenhead">BY</p>
+
+<h3><span class="sc">ALEX. FINDLAY, M.A., Ph.D., D.Sc.</span></h3>
+
+<p class="cenhead">LECTURER ON PHYSICAL CHEMISTRY, UNIVERSITY OF BIRMINGHAM</p>
+
+<h3><i>WITH ONE HUNDRED AND THIRTY-FOUR FIGURES<br />
+IN THE TEXT</i></h3>
+
+<p class="cenhead">THIRD IMPRESSION</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<h3>LONGMANS, GREEN, AND CO.</h3>
+<p class="cenhead">39 PATERNOSTER ROW, LONDON<br />
+NEW YORK, BOMBAY, AND CALCUTTA<br />
+1908</p>
+
+<p class="cenhead"><i>All rights reserved</i></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p class="cenhead">DEDICATED</p>
+
+<p class="cenhead">TO</p>
+
+<h3>FRANCIS ROBERT JAPP, LL.D., F.R.S.</h3>
+
+<p class="cenhead">PROFESSOR OF CHEMISTRY, UNIVERSITY OF ABERDEEN,</p>
+
+<p class="cenhead">IN GRATITUDE FOR EARLY TRAINING</p>
+
+<p class="cenhead">AND ADVICE</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page vii --><span class="pagenum"><a name="pagevii"></a>{vii}</span></p>
+
+<h3>PREFACE TO THE SECOND
+EDITION.</h3>
+
+  <p>During the two years which have elapsed since the first edition of
+  this book appeared, the study of chemical equilibria has been prosecuted
+  with considerable activity, and valuable additions have been made to our
+  knowledge in several departments of this subject. In view of the scope of
+  the present work, it has been, of course, impossible to incorporate all
+  that has been done; but several new sections have been inserted, notably
+  those on the study of basic salts; the interpretation of cooling curves,
+  and the determination of the composition of solid phases without
+  analysis; the equilibria between iron, carbon monoxide, and carbon
+  dioxide, which are of importance in connection with the processes
+  occurring in the blast furnace; and the Phase Rule study of the
+  ammonia-soda process. I have also incorporated a short section on the
+  reciprocal salt-pair barium carbonate&mdash;potassium sulphate, which had
+  been written for the German edition of this book by the late Professor W.
+  Meyerhoffer. The section on the iron-carbon alloys, which in the first
+  edition was somewhat unsatisfactory, has been rewritten.</p>
+
+  <p class="author">A. F.</p>
+
+  <p class="address"><i>September, 1906.</i></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="short" />
+
+<p><!-- Page viii --><span class="pagenum"><a name="pageviii"></a>{viii}</span></p>
+
+<h3>PREFACE</h3>
+
+  <p>Although we are indebted to the late Professor Willard Gibbs for the
+  first enunciation of the Phase Rule, it was not till 1887 that its
+  practical applicability to the study of Chemical Equilibria was made
+  apparent. In that year Roozeboom disclosed the great generalization,
+  which for upwards of ten years had remained hidden and unknown save to a
+  very few, by stripping from it the garb of abstract Mathematics in which
+  it had been clothed by its first discoverer. The Phase Rule was thus made
+  generally accessible; and its adoption by Roozeboom as the basis of
+  classification of the different cases of chemical equilibrium then known
+  established its value, not only as a means of co-ordinating the large
+  number of isolated cases of equilibrium and of giving a deeper insight
+  into the relationships existing between the different systems, but also
+  as a guide in the investigation of unknown systems.</p>
+
+  <p>While the revelation of the principle embedded in the Phase Rule is
+  primarily due to Roozeboom, it should not be forgotten that, some years
+  previously, van't Hoff, in ignorance of the work of Willard Gibbs, had
+  enunciated his "law of the incompatibility of condensed systems," which
+  in some respects coincides with the Phase Rule; and it is only owing to
+  the more general applicability of the latter that the very <!-- Page ix
+  --><span class="pagenum"><a name="pageix"></a>{ix}</span>important
+  generalization of van't Hoff has been somewhat lost sight of.</p>
+
+  <p>The exposition of the Phase Rule and its applications given in the
+  following pages has been made entirely non-mathematical, the desire
+  having been to explain as clearly as possible the principles underlying
+  the Phase Rule, and to illustrate their application to the classification
+  and investigation of equilibria, by means of a number of cases actually
+  studied. While it has been sought to make the treatment sufficiently
+  elementary to be understood by the student just commencing the study of
+  chemical equilibria, an attempt has been made to advance his knowledge to
+  such a stage as to enable him to study with profit the larger works on
+  the subject, and to follow with intelligence the course of investigation
+  in this department of Physical Chemistry. It is also hoped that the
+  volume may be of use, not only to the student of Physical Chemistry, or
+  of the other branches of that science, but also to the student of
+  Metallurgy and of Geology, for whom an acquaintance with at least the
+  principles of the Phase Rule is becoming increasingly important.</p>
+
+  <p>In writing the following account of the Phase Rule, it is scarcely
+  necessary to say that I have been greatly indebted to the larger works on
+  Chemical Equilibria by Ostwald ("Lehrbuch"), Roozeboom ("Die Heterogenen
+  Gleichgewichte"), and Bancroft ("The Phase Rule"); and in the case of the
+  first-named, to the inspiration also of personal teaching. My
+  indebtedness to these and other authors I have indicated in the following
+  pages.</p>
+
+  <p>In conclusion, I would express my thanks to Sir William Ramsay, whose
+  guidance and counsel have been constantly <!-- Page x --><span
+  class="pagenum"><a name="pagex"></a>{x}</span>at my disposal; and to my
+  colleagues, Dr. T. Slater Price and Dr. A. McKenzie, for their friendly
+  criticism and advice. To Messrs. J. N. Friend, M.Sc., and W. E. S.
+  Turner, B.Sc., I am also indebted for their assistance in reading the
+  proof-sheets.</p>
+
+  <p class="author">A. F.</p>
+
+  <p class="address"><i>November, 1903.</i></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page xi --><span class="pagenum"><a name="pagexi"></a>{xi}</span></p>
+
+<h3>CONTENTS</h3>
+
+<table class="nobctr" summary="Contents" title="Contents">
+<tr><td class="spacsingle"> </td><td class="spacsingle" style="text-align:right"> PAGE</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER I</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Introduction</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page1">1</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> General, I. Homogeneous and heterogeneous equilibrium,
+<a href="#page5">5</a>. Real and apparent equilibrium, <a href="#page5">5</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER II</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">The Phase Rule</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page7">7</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Phases, <a href="#page8">8</a>. Components, <a href="#page10">10</a>. Degree of freedom. Variability
+of a system, <a href="#page14">14</a>. The Phase Rule, <a href="#page16">16</a>. Classification of systems
+according to the Phase Rule, <a href="#page17">17</a>. Deduction of the Phase
+Rule, <a href="#page18">18</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER III</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Typical Systems of One Component</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page21">21</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> A. <i>Water.</i> Equilibrium between liquid and vapour. Vaporization
+curve, <a href="#page21">21</a>. Upper limit of vaporization curve, <a href="#page23">23</a>.
+Sublimation curve of ice, <a href="#page24">24</a>. Equilibrium between ice and
+water. Curve of fusion, <a href="#page25">25</a>. Equilibrium between ice, water,
+and vapour. The triple point, <a href="#page27">27</a>. Bivariant systems of water,
+<a href="#page29">29</a>. Supercooled water. Metastable state, <a href="#page30">30</a>. Other systems
+of the substance water, <a href="#page32">32</a>. B. <i>Sulphur</i>, <a href="#page33">33</a>. Polymorphism, <a href="#page33">33</a>.
+Sulphur, <a href="#page34">34</a>. Triple point&mdash;Rhombic and monoclinic sulphur
+and vapour. Transition point, <a href="#page34">34</a>. Condensed systems, <a href="#page36">36</a>.
+Suspended transformation, <a href="#page37">37</a>. Transition curve&mdash;Rhombic
+and monoclinic sulphur, <a href="#page37">37</a>. Triple point&mdash;Monoclinic sulphur,
+liquid, and vapour. Melting point of monoclinic sulphur, <a href="#page38">38</a>.
+Triple point&mdash;Rhombic and monoclinic sulphur and liquid, <a href="#page38">38</a>.
+Triple point&mdash;Rhombic sulphur, liquid, and vapour. Metastable
+triple point, <a href="#page38">38</a>. Fusion curve of rhombic sulphur, <a href="#page39">39</a>.
+Bivariant systems, <a href="#page39">39</a>. C. <i>Tin</i>, <a href="#page41">41</a>. Transition point, <a href="#page41">41</a>.
+<!-- Page xii --><span class="pagenum"><a name="pagexii"></a>{xii}</span>
+Enantiotropy and monotropy, <a href="#page44">44</a>. D. <i>Phosphorus</i>, <a href="#page46">46</a>. Enantiotropy
+combined with monotropy, <a href="#page51">51</a>. E. <i>Liquid Crystals</i>, <a href="#page51">51</a>.
+Phenomena observed, <a href="#page51">51</a>. Nature of liquid crystals, <a href="#page52">52</a>. Equilibrium
+relations in the case of liquid crystals, <a href="#page53">53</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER IV</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">General Summary</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page55">55</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Triple point, <a href="#page55">55</a>. Theorems of van't Hoff and of Le Chatelier,
+<a href="#page57">57</a>. Changes at the triple point, <a href="#page58">58</a>. Triple point solid&mdash;solid&mdash;vapour,
+<a href="#page62">62</a>. Sublimation and vaporization curves,
+<a href="#page63">63</a>. Fusion curve&mdash;Transition curve, <a href="#page66">66</a>. Suspended transformation.
+Metastable equilibria, <a href="#page69">69</a>. Velocity of transformation,
+<a href="#page70">70</a>. Law of successive reactions, <a href="#page73">73</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER V</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Systems of Two Components&mdash;Phenomena of Dissociation</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page76">76</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Different systems of two components, <a href="#page77">77</a>. <span class="sc">Phenomena of Dissociation.</span> Bivariant systems, <a href="#page79">79</a>. Univariant systems,
+<a href="#page80">80</a>. Ammonia compounds of metal chlorides, <a href="#page82">82</a>. Salts with
+water of crystallization, <a href="#page85">85</a>. Efflorescence, <a href="#page86">86</a>. Indefiniteness
+of the vapour pressure of a hydrate, <a href="#page87">87</a>. Suspended transformation,
+<a href="#page89">89</a>. Range of existence of hydrates, <a href="#page90">90</a>. Constancy
+of vapour pressure and the formation of compounds, <a href="#page90">90</a>.
+Measurement of the vapour pressure of hydrates, <a href="#page91">91</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER VI</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Solutions</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page92">92</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Definition, <a href="#page92">92</a>. <span class="sc">Solutions of Gases in Liquids</span>, <a href="#page93">93</a>.
+<span class="sc">Solutions of Liquids in Liquids</span>, <a href="#page95">95</a>. Partial or limited
+miscibility, <a href="#page96">96</a>. Phenol and water, <a href="#page97">97</a>. Methylethylketone
+and water, <a href="#page100">100</a>. Triethylamine and water, <a href="#page101">101</a>. General form
+of concentration-temperature curve, <a href="#page101">101</a>. Pressure-concentration
+diagram, <a href="#page102">102</a>. Complete miscibility, <a href="#page104">104</a>. Pressure-concentration
+diagram, <a href="#page104">104</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER VII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Solutions of Solids in Liquids, only One of the Components being Volatile</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page106">106</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> General, <a href="#page106">106</a>. The saturated solution, <a href="#page108">108</a>. Form of the
+solubility curve, <a href="#page108">108</a>. A. <span class="sc">Anhydrous Salt and Water.</span>
+<!-- Page xiii --><span class="pagenum"><a name="pagexiii"></a>{xiii}</span>
+The solubility curve, <a href="#page111">111</a>. Suspended transformation and
+supersaturation, <a href="#page113">113</a>. Solubility curve at higher temperatures,
+<a href="#page114">114</a>. (1) <i>Complete miscibility of the fused components.</i> Ice as
+solid phase, <a href="#page116">116</a>. Cryohydrates, <a href="#page117">117</a>. Changes at the quadruple
+point, <a href="#page119">119</a>. Freezing mixtures, <a href="#page120">120</a>. (2) <i>Partial miscibility of
+the fused components.</i> Supersaturation, <a href="#page124">124</a>. Pressure-temperature
+diagram, <a href="#page126">126</a>. Vapour pressure of solid&mdash;solution&mdash;vapour,
+<a href="#page126">126</a>. Other univariant systems, <a href="#page127">127</a>. Bivariant systems, <a href="#page129">129</a>.
+Deliquescence, <a href="#page130">130</a>. Separation of salt on evaporation, <a href="#page130">130</a>.
+General summary, <a href="#page131">131</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER VIII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Solutions of Solids in Liquids, only One of the Components being Volatile</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page133">133</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> B. <span class="sc">Hydrated Salt and Water</span>, (1) <i>The compounds
+formed do not have a definite melting point.</i> Concentration-temperature
+diagram, <a href="#page133">133</a>. Sodium sulphate and water, <a href="#page134">134</a>.
+Suspended transformation, <a href="#page137">137</a>. Dehydration by means of
+anhydrous sodium sulphate, <a href="#page138">138</a>. Pressure-temperature diagram,
+<a href="#page138">138</a>. (2) <i>The compounds formed have a definite melting point.</i>
+Solubility curve of calcium chloride hexahydrate, <a href="#page145">145</a>. Pressure-temperature
+diagram, <a href="#page149">149</a>. The indifferent point, <a href="#page150">150</a>. The
+hydrates of ferric chloride, <a href="#page151">151</a>. Suspended transformation,
+<a href="#page155">155</a>. Evaporation of solutions at constant temperature, <a href="#page155">155</a>.
+Inevaporable solutions, <a href="#page157">157</a>. Illustration, <a href="#page158">158</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER IX</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Equilibria between Two Volatile Components</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page161">161</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> General, <a href="#page161">161</a>. Iodine and chlorine, <a href="#page161">161</a>. Concentration-temperature
+diagram, <a href="#page162">162</a>. Pressure-temperature diagram, <a href="#page165">165</a>.
+Bivariant systems, <a href="#page167">167</a>. Sulphur dioxide and water, <a href="#page169">169</a>.
+Pressure-temperature diagram, <a href="#page170">170</a>. Bivariant systems, <a href="#page173">173</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER X</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Solid Solutions. Mixed Crystals</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page175">175</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> General, <a href="#page175">175</a>. Solution of gases in solids, <a href="#page176">176</a>. Palladium
+and hydrogen, <a href="#page178">178</a>. Solutions of solids in solids. Mixed
+crystals, <a href="#page180">180</a>. Formation of mixed crystals of isomorphous
+substances, <a href="#page182">182</a>. I. The two components can form an unbroken
+series of mixed crystals. (<i>a</i>) <i>The freezing points of all mixtures
+lie between the freezing points of the pure components.</i> Examples,
+<a href="#page183">183</a>. Melting-point curve, <a href="#page183">183</a>. (<i>b</i>) <i>The freezing-point curve passes
+through a maximum.</i> Example, <a href="#page186">186</a>. (<i>c</i>) <i>The freezing-point
+curve passes through a minimum.</i> Example, <a href="#page188">188</a>. Fractional
+<!-- Page xiv --><span class="pagenum"><a name="pagexiv"></a>{xiv}</span>
+crystallization of mixed crystals, <a href="#page188">188</a>. II. The two components
+do not form a continuous series of mixed crystals. (<i>a</i>) <i>The
+freezing-point curve exhibits a transition point</i>, <a href="#page190">190</a>. Example,
+<a href="#page190">190</a>. (<i>b</i>) <i>The freezing-point curve exhibits a eutectic point</i>, <a href="#page191">191</a>.
+Examples, <a href="#page192">192</a>. Changes in mixed crystals with the temperature,
+<a href="#page192">192</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XI</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Equilibrium between Dynamic Isomerides</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page195">195</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Temperature-concentration diagram, <a href="#page196">196</a>. Transformation
+of the unstable into the stable form, <a href="#page201">201</a>. Examples, <a href="#page203">203</a>.
+<i>Benzaldoximes</i>, <a href="#page203">203</a>. <i>Acetaldehyde and paraldehyde</i>, <a href="#page204">204</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Summary.</span>&mdash;<span class="sc">Application of the Phase Rule to the Study of Systems of Two Components</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page207">207</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Summary of the different systems of two components, <a href="#page208">208</a>.
+(1) <i>Organic compounds</i>, <a href="#page212">212</a>. (2) <i>Optically active substances</i>,
+<a href="#page213">213</a>. Examples, <a href="#page216">216</a>. Transformations, <a href="#page217">217</a>. (3) <i>Alloys</i>, <a href="#page220">220</a>.
+Iron&mdash;carbon alloys, <a href="#page223">223</a>. Determination of the composition of
+compounds without analysis, <a href="#page228">228</a>. Formation of minerals, <a href="#page232">232</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XIII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Systems of Three Components</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page234">234</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> General, <a href="#page234">234</a>. Graphic representation, <a href="#page235">235</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XIV</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Solutions of Liquids in Liquids</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page240">240</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> 1. <i>The three components form only one pair of partially
+miscible liquids</i>, <a href="#page240">240</a>. Retrograde solubility, <a href="#page245">245</a>. The influence
+of temperature, <a href="#page247">247</a>. 2. <i>The three components can form two
+pairs of partially miscible liquids</i>, <a href="#page249">249</a>. 3. <i>The three components
+form three pairs of partially miscible liquids</i>, <a href="#page251">251</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XV</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Presence of Solid Phases</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page253">253</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> A. The ternary eutectic point, <a href="#page253">253</a>. Formation of compounds,
+<a href="#page255">255</a>. B. Equilibria at higher temperatures. Formation
+of double salts, <a href="#page258">258</a>. Transition point, <a href="#page258">258</a>. Vapour pressure.
+<!-- Page xv --><span class="pagenum"><a name="pagexv"></a>{xv}</span>
+Quintuple point, <a href="#page261">261</a>. Solubility curves at the transition point,
+<a href="#page264">264</a>. Decomposition of the double salt by water, <a href="#page267">267</a>. Transition
+interval, <a href="#page270">270</a>. Summary, <a href="#page271">271</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XVI</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Isothermal Curves and the Space Model</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page272">272</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Non-formation of double salts, <a href="#page272">272</a>. Formation of double
+salt, <a href="#page273">273</a>. Transition interval, <a href="#page277">277</a>. Isothermal evaporation,
+<a href="#page278">278</a>. Crystallization of double salt from solutions containing
+excess of one component, <a href="#page280">280</a>. Formation of mixed crystals,
+<a href="#page281">281</a>. Application to the characterization of racemates, <a href="#page282">282</a>.
+<i>Representation in space.</i> Space model for carnallite, <a href="#page284">284</a>.
+Summary and numerical data, <a href="#page287">287</a>. Ferric chloride&mdash;hydrogen
+chloride&mdash;water, <a href="#page290">290</a>. Ternary systems, <a href="#page291">291</a>. The isothermal
+curves, <a href="#page294">294</a>. Basic Salts, <a href="#page296">296</a>. Bi<sub><a href="#page2">2</a></sub>O<sub><a href="#page3">3</a></sub>&mdash;N<sub><a href="#page2">2</a></sub>O<sub><a href="#page5">5</a></sub>&mdash;H<sub><a href="#page2">2</a></sub>O, <a href="#page298">298</a>.
+Basic mercury salts, <a href="#page301">301</a>. Indirect determination of the composition
+of the solid phase, <a href="#page302">302</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XVII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Absence of Liquid Phase</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page305">305</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Iron, carbon monoxide, carbon dioxide, <a href="#page305">305</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> CHAPTER XVIII</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Systems of Four Components</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page312">312</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> Reciprocal salt-pairs. Choice of components, <a href="#page313">313</a>. Transition
+point, <a href="#page314">314</a>. Formation of double salts, <a href="#page315">315</a>. Transition
+interval, <a href="#page315">315</a>. Graphic representation, <a href="#page316">316</a>. Example, <a href="#page317">317</a>.
+Ammonia-soda process, <a href="#page320">320</a>. Preparation of barium nitrite, <a href="#page327">327</a>.
+Barium carbonate and potassium sulphate, <a href="#page328">328</a>.</td></tr>
+
+<tr><td class="spacsingle" style="text-align:center; padding-top:1em;"> APPENDIX</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Experimental Determination of the Transition Point</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page331">331</a></td></tr>
+
+<tr><td class="spacsingle" style="padding-left:2em"> I. The dilatometric method, <a href="#page331">331</a>. II. Measurement of
+the vapour pressure, <a href="#page334">334</a>. III. Solubility measurements, <a href="#page335">335</a>.
+IV. Thermometric method, <a href="#page337">337</a>. V. Optical method, <a href="#page338">338</a>.
+VI. Electrical methods, <a href="#page338">338</a>.</td></tr>
+
+<tr><td class="spacsingle">&nbsp;</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Name Index</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page341">341</a></td></tr>
+
+<tr><td class="spacsingle">&nbsp;</td></tr>
+
+<tr><td class="spacsingle"> <span class="sc">Subject Index</span> </td><td class="spacsingle" style="text-align:right"> <a href="#page345">345</a></td></tr>
+</table>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 1 --><span class="pagenum"><a name="page1"></a>{1}</span></p>
+
+<h2>THE PHASE RULE</h2>
+
+<h3>CHAPTER I</h3>
+
+<p class="cenhead">INTRODUCTION</p>
+
+  <p><b>General.</b>&mdash;Before proceeding to the more systematic
+  treatment of the Phase Rule, it may, perhaps, be not amiss to give first
+  a brief forecast of the nature of the subject we are about to study, in
+  order that we may gain some idea of what the Phase Rule is, of the kind
+  of problem which it enables us to solve, and of the scope of its
+  application.</p>
+
+  <p>It has long been known that if water is placed in a closed, exhausted
+  space, vapour is given off and a certain pressure is created in the
+  enclosing vessel. Thus, when water is placed in the Torricellian vacuum
+  of the barometer, the mercury is depressed, and the amount of depression
+  increases as the temperature is raised. But, although the pressure of the
+  vapour increases as the temperature rises, its value at any given
+  temperature is constant, no matter whether the amount of water present or
+  the volume of the vapour is great or small; if the pressure on the vapour
+  is altered while the temperature is maintained constant, either the water
+  or the vapour will ultimately disappear; the former by evaporation, the
+  latter by condensation. At any given temperature within certain limits,
+  therefore, water and vapour can exist permanently in contact with one
+  another&mdash;or, as it is said, be in equilibrium with one
+  another&mdash;only when the pressure has a certain definite value. The
+  same law of constancy of vapour pressure at a given <!-- Page 2 --><span
+  class="pagenum"><a name="page2"></a>{2}</span>temperature, quite
+  irrespective of the volumes of liquid and vapour,<a name="NtA1"
+  href="#Nt1"><sup>[1]</sup></a> holds good also in the case of alcohol,
+  ether, benzene, and other pure liquids. It is, therefore, not unnatural
+  to ask the question, Does it hold good for all liquids? Is it valid, for
+  example, in the case of solutions?</p>
+
+  <p>We can find the answer to these questions by studying the behaviour of
+  a solution&mdash;say, a solution of common salt in water&mdash;when
+  placed in the Torricellian vacuum. In this case, also, it is observed
+  that the pressure of the vapour increases as the temperature is raised,
+  but the pressure is no longer independent of the volume; as the volume
+  increases, the pressure slowly diminishes. If, however, solid salt is
+  present in contact with the solution, then the pressure again becomes
+  constant at constant temperature, even when the volume of the vapour is
+  altered. As we see, therefore, solutions do not behave in the same way as
+  pure liquids.</p>
+
+  <p>Moreover, on lowering the temperature of water, a point is reached at
+  which ice begins to separate out; and if heat be now added to the system
+  or withdrawn from it, no change will take place in the temperature or
+  vapour pressure of the latter until either the ice or the water has
+  disappeared.<a name="NtA2" href="#Nt2"><sup>[2]</sup></a> Ice, water, and
+  vapour, therefore, can be in equilibrium with one another only at one
+  definite temperature and one definite pressure.</p>
+
+  <p>In the case of a solution of common salt, however, we may have ice in
+  contact with the solution at different temperatures and pressures.
+  Further, it is possible to have a solution in equilibrium not only with
+  anhydrous salt (NaCl), but also with the hydrated salt (NaCl,
+  2H<sub>2</sub>O), as well as with ice, and the question, therefore,
+  arises: Is it possible to state in a general manner the conditions under
+  which such different systems can exist in equilibrium; or to obtain some
+  insight <!-- Page 3 --><span class="pagenum"><a
+  name="page3"></a>{3}</span>into the relations which exist between pure
+  liquids and solutions? As we shall learn, the Phase Rule enables us to
+  give an answer to this question.</p>
+
+  <p>The preceding examples belong to the class of so-called "physical"
+  equilibria, or equilibria depending on changes in the physical state.
+  More than a hundred years ago, however, it was shown by Wenzel and
+  Berthollet that "chemical" equilibria can also exist; that chemical
+  reactions do not always take place completely in one direction as
+  indicated by the usual chemical equation, but that before the reacting
+  substances are all used up the reaction ceases, and there is a condition
+  of equilibrium between the reacting substances and the products of
+  reaction. As an example of this, there may be taken the process of
+  lime-burning, which depends on the fact that when calcium carbonate is
+  heated, carbon dioxide is given off and quicklime is produced. If the
+  carbonate is heated in a closed vessel it will be found, however, not to
+  undergo entire decomposition. When the pressure of the carbon dioxide
+  reaches a certain value (which is found to depend on the temperature),
+  decomposition ceases, and calcium carbonate exists side by side with
+  calcium oxide and carbon dioxide. Moreover, at any given temperature the
+  pressure is constant and independent of the amount of carbonate or oxide
+  present, or of the volume of the gas; <i>nor does the addition of either
+  of the products of dissociation, carbon dioxide or calcium oxide, cause
+  any change in the equilibrium</i>. Here, then, we see that, although
+  there are three different substances present, and although the
+  equilibrium is no longer due to physical, but to chemical change, it
+  nevertheless obeys the same law as the vapour pressure of a pure volatile
+  liquid, such as water.</p>
+
+  <p>It might be supposed, now, that this behaviour would be shown by other
+  dissociating substances, <i>e.g.</i> ammonium chloride. When this
+  substance is heated it dissociates into ammonia and hydrogen chloride,
+  and at any given temperature the pressure of these gases is constant,<a
+  name="NtA3" href="#Nt3"><sup>[3]</sup></a> and is independent of the
+  amounts of solid and gas present. So far, therefore, ammonium chloride
+  behaves like calcium carbonate. If, however, one of the <!-- Page 4
+  --><span class="pagenum"><a name="page4"></a>{4}</span>products of
+  dissociation be added to the system, it is found that the pressure is no
+  longer constant at a given temperature, but varies with the amount of
+  gas, ammonia or hydrogen chloride, which is added. In the case of certain
+  dissociating substances, therefore, addition of one of the products of
+  dissociation alters the equilibrium, while in other cases it does not.
+  With the help of the Phase Rule, however, a general interpretation of
+  this difference of behaviour can be given&mdash;an interpretation which
+  can be applied not only to the two cases cited, but to all cases of
+  dissociation.</p>
+
+  <p>Again, it is well known that sulphur exists in two different
+  crystalline forms, octahedral and prismatic, each of which melts at a
+  different temperature. The problem here is, therefore, more complicated
+  than in the case of ice, for there is now a possibility not only of one
+  solid form, but of two different forms of the same substance existing in
+  contact with liquid. What are the conditions under which these two forms
+  can exist in contact with liquid, either singly or together, and under
+  what conditions can the two solid forms exist together without the
+  presence of liquid sulphur? To these questions an answer can also be
+  given with the help of the Phase Rule.</p>
+
+  <p>These cases are, however, comparatively simple; but when we come, for
+  instance, to study the conditions under which solutions are formed, and
+  especially when we inquire into the solubility relations of salts capable
+  of forming, perhaps, a series of crystalline hydrates; and when we seek
+  to determine the conditions under which these different forms can exist
+  in contact with the solution, the problem becomes more complicated, and
+  the necessity of some general guide to the elucidation of the behaviour
+  of these different systems becomes more urgent.</p>
+
+  <p>It is, now, to the study of such physical and chemical equilibria as
+  those above-mentioned that the Phase Rule finds application; to the
+  study, also, of the conditions regulating, for example, the formation of
+  alloys from mixtures of the fused metals, or of the various salts of the
+  Stassfurt deposits; the behaviour of iron and carbon in the formation of
+  steel and the <!-- Page 5 --><span class="pagenum"><a
+  name="page5"></a>{5}</span>separation of different minerals from a fused
+  rock-mass.<a name="NtA4" href="#Nt4"><sup>[4]</sup></a> With the help of
+  the Phase Rule we can group together into classes the large number of
+  different isolated cases of systems in equilibrium; with its aid we are
+  able to state, in a general manner at least, the conditions under which a
+  system can be in equilibrium, and by its means we can gain some insight
+  into the relations existing between different kinds of systems.</p>
+
+  <p><b>Homogeneous and Heterogeneous Equilibrium.</b>&mdash;Before passing
+  to the consideration of this generalization, it will be well to first
+  make mention of certain restrictions which must be placed on its
+  treatment, and also of the limitations to which it is subject. If a
+  system is uniform throughout its whole extent, and possesses in every
+  part identical physical properties and chemical composition, it is called
+  <i>homogeneous</i>. Such is, for example, a solution of sodium chloride
+  in water. An equilibrium occurring in such a homogeneous system (such as
+  the equilibrium occurring in the formation of an ester in alcoholic
+  solution) is called <i>homogeneous equilibrium</i>. If, however, the
+  system consists of parts which have different physical properties,
+  perhaps also different chemical properties, and which are marked off and
+  separated from one another by bounding surfaces, the system is said to be
+  <i>heterogeneous</i>. Such a system is formed by ice, water, and vapour,
+  in which the three portions, each in itself homogeneous, can be
+  mechanically separated from one another. When equilibrium exists between
+  different, physically distinct parts, it is known as <i>heterogeneous
+  equilibrium</i>. It is, now, with heterogeneous equilibria, with the
+  conditions under which a heterogeneous system can exist, that we shall
+  deal here.</p>
+
+  <p>Further, we shall not take into account changes of equilibrium due to
+  the action of electrical, magnetic, or capillary forces, or of gravity;
+  but shall discuss only those which are due to changes of pressure,
+  temperature, and volume (or concentration).</p>
+
+  <p><b>Real and Apparent Equilibrium.</b>&mdash;In discussing equilibria,
+  also, a distinction must be drawn between real and <!-- Page 6 --><span
+  class="pagenum"><a name="page6"></a>{6}</span>apparent equilibria. In the
+  former case there is a state of rest which undergoes continuous change
+  with change of the conditions (<i>e.g.</i> change of temperature or of
+  pressure), and for which the chief criterion is that <i>the same
+  condition of equilibrium is reached from whichever side it is
+  approached</i>. Thus in the case of a solution, if the temperature is
+  maintained constant, the same concentration will be obtained, no matter
+  whether we start with an unsaturated solution to which we add more solid,
+  or with a supersaturated solution from which we allow solid to
+  crystallize out; or, in the case of water in contact with vapour, the
+  same vapour pressure will be obtained, no matter whether we heat the
+  water up to the given temperature or cool it down from a higher
+  temperature. In this case, water and vapour are in <i>real</i>
+  equilibrium. On the other hand, water in contact with hydrogen and oxygen
+  at the ordinary temperature is a case only of <i>apparent</i>
+  equilibrium; on changing the pressure and temperature continuously within
+  certain limits there is no continuous change observed in the relative
+  amounts of the two gases. On heating beyond these limits there is a
+  sudden and not a continuous change, and the system no longer regains its
+  former condition on being cooled to the ordinary temperature. In all such
+  cases the system may be regarded as undergoing change and as tending
+  towards a state of true or real equilibrium, but with such slowness that
+  no change is observed.</p>
+
+  <p>Although the case of water in contact with hydrogen and oxygen is an
+  extreme one, it must be borne in mind that the condition of true
+  equilibrium may not be reached instantaneously or even with measurable
+  velocity, and in all cases it is necessary to be on one's guard against
+  mistaking apparent (or false) for real (or true) equilibrium. The
+  importance of this will be fully illustrated in the sequel.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 7 --><span class="pagenum"><a name="page7"></a>{7}</span></p>
+
+<h3>CHAPTER II</h3>
+
+<p class="cenhead">THE PHASE RULE</p>
+
+  <p>Although the fact that chemical reactions do not take place completely
+  in one direction, but proceed only to a certain point and there make a
+  halt, was known in the last quarter of the eighteenth century (Wenzel,
+  1777; Berthollet, 1799); and although the opening and subsequent decades
+  of the following century brought many further examples of such equilibria
+  to our knowledge, it was not until the last quarter of the nineteenth
+  century that a theorem, general in its application and with foundations
+  weakened by no hypothetical assumptions as to the nature or constitution
+  of matter, was put forward by Willard Gibbs;<a name="NtA5"
+  href="#Nt5"><sup>[5]</sup></a> a generalization which serves at once as a
+  golden rule by which the condition of equilibrium of a system can be
+  tested, and as a guide to the similarities and dissimilarities existing
+  in different systems.</p>
+
+  <p>Before that time, certainly, attempts had been made to bring the
+  different known cases of equilibria&mdash;chemical and
+  physical&mdash;under general laws. From the very first, both Wenzel<a
+  name="NtA6" href="#Nt6"><sup>[6]</sup></a> and Berthollet<a name="NtA7"
+  href="#Nt7"><sup>[7]</sup></a> recognized the influence exercised by the
+  <i>mass</i> of the substances on the equilibrium of the system. It was
+  reserved, however, for Guldberg and Waage, by their more general
+  statement and mathematical treatment of the Law of Mass Action,<a
+  name="NtA8" href="#Nt8"><sup>[8]</sup></a> to inaugurate the period of
+  quantitative study of equilibria. The law which these investigators
+  enunciated <!-- Page 8 --><span class="pagenum"><a
+  name="page8"></a>{8}</span>served satisfactorily to summarize the
+  conditions of equilibrium in many cases both of homogeneous and, with the
+  help of certain assumptions and additions, of heterogeneous equilibrium.
+  By reason, however, of the fact that it was developed on the basis of the
+  kinetic and molecular theories, and involved, therefore, certain
+  hypothetical assumptions as to the nature and condition of the substances
+  taking part in the equilibrium, the law of mass action failed, as it
+  necessarily must, when applied to those systems in which neither the
+  number of different molecular aggregates nor the degree of their
+  molecular complexity was known.</p>
+
+  <p>Ten years after the law of mass action was propounded by Guldberg and
+  Waage, Willard Gibbs,<a name="NtA9" href="#Nt9"><sup>[9]</sup></a>
+  Professor of Physics in Yale University, showed how, in a perfectly
+  general manner, free from all hypothetical assumptions as to the
+  molecular condition of the participating substances, all cases of
+  equilibrium could be surveyed and grouped into classes, and how
+  similarities in the behaviour of apparently different kinds of systems,
+  and differences in apparently similar systems, could be explained.</p>
+
+  <p>As the basis of his theory of equilibria, Gibbs adopted the laws of
+  thermodynamics,<a name="NtA10" href="#Nt10"><sup>[10]</sup></a> a method
+  of treatment which had first been employed by Horstmann.<a name="NtA11"
+  href="#Nt11"><sup>[11]</sup></a> In deducing the law of equilibrium,
+  Gibbs regarded a system as possessing only three independently variable
+  factors<a name="NtA12"
+  href="#Nt12"><sup>[12]</sup></a>&mdash;temperature, pressure, and the
+  concentration of the components of the system&mdash;and he enunciated the
+  general theorem now usually known as the <i>Phase Rule</i>, by which he
+  defined the conditions of equilibrium as a relationship between the
+  number of what are called the phases and the components of the
+  system.</p>
+
+  <p><b>Phases.</b>&mdash;Before proceeding farther we shall first consider
+  what exactly is meant by the terms <i>phase</i> and <i>component</i>. We
+  have already seen (p. <a href="#page5">5</a>) that a heterogeneous system
+  is made <!-- Page 9 --><span class="pagenum"><a
+  name="page9"></a>{9}</span>up of different portions, each in itself
+  homogeneous, but marked off in space and separated from the other
+  portions by bounding surfaces. These homogeneous, physically distinct and
+  mechanically separable portions are called <i>phases</i>. Thus ice,
+  water, and vapour, are three phases of the same chemical
+  substance&mdash;water. A phase, however, whilst it must be physically and
+  chemically homogeneous, need not necessarily be chemically simple. Thus,
+  a gaseous mixture or a solution may form a phase; but a heterogeneous
+  mixture of solid substances constitutes as many phases as there are
+  substances present. Thus when calcium carbonate dissociates under the
+  influence of heat, calcium oxide and carbon dioxide are formed. There are
+  then <i>two</i> solid phases present, viz. calcium carbonate and oxide,
+  and one gas phase, carbon dioxide.</p>
+
+  <p>The <i>number of phases</i> which can exist side by side may vary
+  greatly in different systems. In all cases, however, there can be but one
+  gas or vapour phase on the account of the fact that all gases are
+  miscible with one another in all proportions. In the case of liquid and
+  solid phases the number is indefinite, since the above property does not
+  apply to them. The number of phases which can be formed by any given
+  substance or group of substances also differs greatly, and in general
+  increases with the number of participating substances. Even in the case
+  of a single substance, however, the number may be considerable; in the
+  case of sulphur, for example, at least eight different solid phases are
+  known (<i>v.</i> Chap. III.).</p>
+
+  <p>It is of importance to bear in mind that equilibrium is <i>independent
+  of the amounts</i> of the phases present.<a name="NtA13"
+  href="#Nt13"><sup>[13]</sup></a> Thus it is a familiar fact that the
+  pressure of a vapour in contact with a <!-- Page 10 --><span
+  class="pagenum"><a name="page10"></a>{10}</span>liquid (<i>i.e.</i> the
+  pressure of the saturated vapour) is unaffected by the amounts, whether
+  relative or absolute, of the liquid and vapour; also the amount of a
+  substance dissolved by a liquid is independent of the amount of solid in
+  contact with the solution. It is true that deviations from this general
+  law occur when the amount of liquid or the size of the solid particles is
+  reduced beyond a certain point,<a name="NtA14"
+  href="#Nt14"><sup>[14]</sup></a> owing to the influence of surface
+  energy; but we have already (p.&nbsp;<a href="#page5">5</a>) excluded such
+  cases from consideration.</p>
+
+  <p><b>Components.</b>&mdash;Although the conception of phases is one
+  which is readily understood, somewhat greater difficulty is experienced
+  when we come to consider what is meant by the term <i>component</i>; for
+  the components of a system are not synonymous with the chemical elements
+  or compounds present, <i>i.e.</i> with the <i>constituents</i> of the
+  system, although both elements and compounds may be components. By the
+  latter term there are meant only those constituents the concentration of
+  which can undergo <i>independent</i> variation in the different phases,
+  and it is only with these that we are concerned here.<a name="NtA15"
+  href="#Nt15"><sup>[15]</sup></a></p>
+
+  <p>To understand the meaning of this term we shall consider briefly some
+  cases with which the reader will be familiar, and at the outset it must
+  be emphasized that the Phase Rule is concerned merely with those
+  constituents which take part in the state of real equilibrium (p.&nbsp;<a
+  href="#page5">5</a>); for it is only to the final state, not to the
+  processes by which that state is reached, that the Phase Rule
+  applies.</p>
+
+  <p>Consider now the case of the system water&mdash;vapour or
+  ice&mdash;water&mdash;vapour. The number of constituents taking part in
+  the equilibrium here is only one, viz. the chemical substance, water.
+  Hydrogen and oxygen, the constituents of water, are not to be regarded as
+  components, because, in the first place, they are <!-- Page 11 --><span
+  class="pagenum"><a name="page11"></a>{11}</span>not present in the system
+  in a state of real equilibrium (p.&nbsp;<a href="#page6">6</a>); in the second
+  place, they are combined in definite proportions to form water, and their
+  amounts, therefore, cannot be varied independently. A variation in the
+  amount of hydrogen necessitates a definite variation in the amount of
+  oxygen.</p>
+
+  <p>In the case, already referred to, in which hydrogen and oxygen are
+  present along with water at the ordinary temperature, we are not dealing
+  with a condition of true equilibrium. If, however, the temperature is
+  raised to a certain point, a state of true equilibrium between hydrogen,
+  oxygen, and water-vapour will be possible. In this case hydrogen and
+  oxygen will be components, because now they do take part in the
+  equilibrium; also, they need no longer be present in definite
+  proportions, but excess of one or the other may be added. Of course, if
+  the restriction be arbitrarily made that the free hydrogen and oxygen
+  shall be present always and only in the proportions in which they are
+  combined to form water, there will be, as before, only one component,
+  water. From this, then, we see that a change in the conditions of the
+  experiment (in the present case a rise of temperature) may necessitate a
+  change in the number of the components.</p>
+
+  <p>It is, however, only in the case of systems of more than one component
+  that any difficulty will be found; for only in this case will a choice of
+  components be possible. Take, for instance, the dissociation of calcium
+  carbonate into calcium oxide and carbon dioxide. At each temperature, as
+  we have seen, there is a definite state of equilibrium. When equilibrium
+  has been established, there are three different substances
+  present&mdash;calcium carbonate, calcium oxide, and carbon dioxide; and
+  these are the constituents of the system between which equilibrium
+  exists. Now, although these constituents take part in the equilibrium,
+  they are not all to be regarded as components, for they are not mutually
+  independent. On the contrary, the different phases are related to one
+  another, and if two of these are taken, the composition of the third is
+  defined by the equation</p>
+
+<p class="cenhead">CaCO<sub>3</sub> = CaO + CO<sub>2</sub></p>
+
+<p><!-- Page 12 --><span class="pagenum"><a name="page12"></a>{12}</span></p>
+
+  <p>Now, in deciding the number of components in any given system, not
+  only must the constituents chosen be capable of independent variation,
+  but a further restriction is imposed, and we obtain the following rule:
+  <i>As the components of a system there are to be chosen the</i>
+  <b>smallest number</b> <i>of independently variable constituents by means
+  of which the composition of each phase participating in the state of
+  equilibrium can be expressed in the form of a chemical equation.</i></p>
+
+  <p>Applying this rule to the case under consideration, we see that of the
+  three constituents present when the system is in a state of equilibrium,
+  only two, as already stated, are independently variable. It will further
+  be seen that in order to express the composition of each phase present,
+  two of these constituents are necessary. The system is, therefore, one of
+  <i>two components</i>, or a system of the second order.</p>
+
+  <p>When, now, we proceed to the actual choice of components, it is
+  evident that any two of the constituents can be selected. Thus, if we
+  choose as components CaCO<sub>3</sub> and CaO, the composition of each
+  phase can be expressed by the following equations:&mdash;</p>
+
+<p class="cenhead">CaCO<sub>3</sub> = CaCO<sub>3</sub> + 0CaO <br />CaO = CaO + 0CaCO<sub>3</sub><br />CO<sub>2</sub> = CaCO<sub>3</sub> - CaO</p>
+
+  <p>As we see, then, both zero and negative quantities of the components
+  have been introduced; and similar expressions would be obtained if
+  CaCO<sub>3</sub> and CO<sub>2</sub> were chosen as components. The matter
+  can, however, be simplified and the use of negative quantities avoided if
+  CaO and CO<sub>2</sub> are chosen; and it is, therefore, customary to
+  select these as the components.</p>
+
+  <p>While it is possible in the case of systems of the second order to
+  choose the two components in such a way that the composition of each
+  phase can be expressed by positive quantities of these, such a choice is
+  not always possible when dealing with systems of a higher order
+  (containing three or four components).</p>
+
+  <p>From the example which has just been discussed, it might <!-- Page 13
+  --><span class="pagenum"><a name="page13"></a>{13}</span>appear as if the
+  choice of the components was rather arbitrary. On examining the point,
+  however, it will be seen that the arbitrariness affects only the
+  <i>nature</i>, not the <i>number</i>, of the components; a choice could
+  be made with respect to which, not to how many, constituents were to be
+  regarded as components. As we shall see presently, however, it is only
+  the number, not the nature of the components that is of importance.</p>
+
+  <p>After the discussion of the conditions which the substances chosen as
+  components must satisfy, another method may be given by which the number
+  of components present in a system can be determined. Suppose a system
+  consisting of several phases in equilibrium, and the composition of each
+  phase determined by analysis. If each phase present, regarded as a whole,
+  has the same composition, the system contains only one component, or is
+  of the first order. If two phases must be mixed in suitable quantities in
+  order that the composition of a third phase may be obtained, the system
+  is one of two components or of the second order; and if three phases are
+  necessary to give the composition of a fourth coexisting phase, the
+  system is one of three components, or of the third order.<a name="NtA16"
+  href="#Nt16"><sup>[16]</sup></a></p>
+
+  <p>Although the examples to be considered in the sequel will afford
+  sufficient illustration of the application of the rules given above, one
+  case may perhaps be discussed to show the application of the method just
+  given for determining the number of components.</p>
+
+  <p>Consider the system consisting of Glauber's salt in equilibrium with
+  solution and vapour. If these three phases are analyzed, the composition
+  of the solid will be expressed by Na<sub>2</sub>SO<sub>4</sub>,
+  10H<sub>2</sub>O; that of the solution by Na<sub>2</sub>SO<sub>4</sub> +
+  <i>x</i>H<sub>2</sub>O, while the vapour phase will be H<sub>2</sub>O.
+  The system evidently cannot be a one-component system, for the phases
+  have not all the same composition. By varying the amounts of two phases,
+  however (<i>e.g.</i> Na<sub>2</sub>SO<sub>4</sub>, 10H<sub>2</sub>O and
+  H<sub>2</sub>O), the composition of the third phase&mdash;the
+  solution&mdash;can be obtained. The system is, therefore, one of <i>two
+  components</i>.</p>
+
+  <p>But sodium sulphate can also exist in the anhydrous form and as the
+  hydrate Na<sub>2</sub>SO<sub>4</sub>, 7H<sub>2</sub>O. In these cases
+  there may <!-- Page 14 --><span class="pagenum"><a
+  name="page14"></a>{14}</span>be chosen as components
+  Na<sub>2</sub>SO<sub>4</sub> and H<sub>2</sub>O, and
+  Na<sub>2</sub>SO<sub>4</sub>, 7H<sub>2</sub>O and H<sub>2</sub>O
+  respectively. In both cases, therefore, there are two components. But the
+  two systems (Na<sub>2</sub>SO<sub>4</sub>,
+  10H<sub>2</sub>O&mdash;H<sub>2</sub>O, and Na<sub>2</sub>SO<sub>4</sub>,
+  7H<sub>2</sub>O&mdash;H<sub>2</sub>O) can be regarded as special cases of
+  the system Na<sub>2</sub>SO<sub>4</sub>&mdash;H<sub>2</sub>O, and these
+  two components will apply to all systems made up of sodium sulphate and
+  water, no matter whether the solid phase is anhydrous salt or one of the
+  hydrates. In all three cases, of course, the <i>number</i> of components
+  is the same; but by choosing Na<sub>2</sub>SO<sub>4</sub> and
+  H<sub>2</sub>O as components, the possible occurrence of negative
+  quantities of components in expressing the composition of the phases is
+  avoided; and, further, these components apply over a much larger range of
+  experimental conditions. Again, therefore, we see that, although the
+  number of the components of a system is definite, a certain amount of
+  liberty is allowed in the choice of the substances; and we also see that
+  the choice will be influenced by the conditions of experiment.</p>
+
+  <p>Summing up, now, we may say&mdash;</p>
+
+  <p>(1) The components are to be chosen from among the constituents which
+  are present when the system is in a state of true equilibrium, and which
+  take part in that equilibrium.</p>
+
+  <p>(2) As components are to be chosen the <i>smallest number</i> of such
+  constituents necessary to express the composition of each phase
+  participating in the equilibrium, zero and negative quantities of the
+  components being permissible.</p>
+
+  <p>(3) In any given system the <i>number</i> of the components is
+  definite, but may alter with alteration of the conditions of experiment.
+  A certain freedom of choice, however, is allowed in the (qualitative, not
+  quantitative) selection of the components, the choice being influenced by
+  considerations of simplicity, suitability, or generality of
+  application.<a name="NtA17" href="#Nt17"><sup>[17]</sup></a></p>
+
+  <p><b>Degree of Freedom. Variability of a System.</b>&mdash;It is well
+  known that in dealing with a certain mass of gas or vapour, <i>e.g.</i>
+  water vapour, if only one of the independently variable
+  factors&mdash;temperature, pressure, and concentration (or
+  volume)&mdash;is fixed, the state of the gas or vapour is undefined;
+  while occupying the same volume (the concentration, therefore, remaining
+  <!-- Page 15 --><span class="pagenum"><a
+  name="page15"></a>{15}</span>unchanged), the temperature and the pressure
+  may be altered; at a given temperature, a gas can exist under different
+  pressures and occupy different volumes, and under any given pressure the
+  temperature and volume may vary. If, however, two of the factors are
+  arbitrarily fixed, then the third factor can only have a certain definite
+  value; at any given values of temperature and pressure a given mass of
+  gas can occupy only a definite volume.</p>
+
+  <p>Suppose, however, that the system consists of water in contact with
+  vapour. The condition of the system then becomes perfectly defined on
+  arbitrarily giving one of the variables a certain value. If the
+  temperature is fixed, the pressure under which water and water vapour can
+  coexist is also determined; and conversely, if a definite pressure is
+  chosen, the temperature is also defined. Water and vapour can coexist
+  under a given pressure only at a definite temperature.</p>
+
+  <p>Finally, let the water and vapour be cooled down until ice begins to
+  separate out. So soon as the third phase, ice, appears, the state of the
+  system as regards temperature and pressure of the vapour is perfectly
+  defined, and none of the variables can be arbitrarily changed without
+  causing the disappearance of one of the phases, ice, water, or
+  vapour.</p>
+
+  <p>We see, therefore, that in the case of some systems two, in other
+  cases, only one of the independent variables (temperature, pressure,
+  concentration) can be altered without destroying the nature of the
+  system; while in other systems, again, these variables have all fixed and
+  definite values. We shall therefore define the number of degrees of
+  freedom<a name="NtA18" href="#Nt18"><sup>[18]</sup></a> of a system as
+  the <i>number of the variable factors, temperature, pressure, and
+  concentration of the components, which must be arbitrarily fixed in order
+  that the condition of the system may be perfectly defined</i>. From what
+  has been said, therefore, we shall describe a gas or vapour as having two
+  degrees of freedom; the system water&mdash;vapour as having only one; and
+  the system ice&mdash;water&mdash;vapour as having no degrees of freedom.
+  We may also speak of the <!-- Page 16 --><span class="pagenum"><a
+  name="page16"></a>{16}</span><i>variability</i> or <i>variance</i> of a
+  system, and describe a system as being invariant, univariant, bivariant,
+  multivariant,<a name="NtA19" href="#Nt19"><sup>[19]</sup></a> according
+  as the number of degrees of freedom is nought, one, two, or more than
+  two.</p>
+
+  <p>A knowledge of its variability is, therefore, of essential importance
+  in studying the condition and behaviour of a system, and it is the great
+  merit of the Phase Rule that <i>the state of a system is defined entirely
+  by the relation existing between the number of the components and the
+  phases present</i>, no account being taken of the molecular complexity of
+  the participating substances, nor any assumption made with regard to the
+  constitution of matter. It is, further, as we see, quite immaterial
+  whether we are dealing with "physical" or "chemical" equilibrium; in
+  principle, indeed, no distinction need be drawn between the two classes,
+  although it is nevertheless often convenient to make use of the terms, in
+  spite of a certain amount of indefiniteness which attaches to
+  them&mdash;an indefiniteness, indeed, which attaches equally to the terms
+  "physical" and "chemical" process.<a name="NtA20"
+  href="#Nt20"><sup>[20]</sup></a></p>
+
+  <p><b>The Phase Rule.</b>&mdash;The Phase Rule of Gibbs, which defines
+  the condition of equilibrium by the relation between the number of
+  coexisting phases and the components, may be stated as follows: A system
+  consisting of n components can exist in <i>n</i> + 2 phases only when the
+  temperature, pressure, and concentration have fixed and definite values;
+  if there are <i>n</i> components in <i>n</i> + 1 phases, equilibrium can
+  exist while one of the factors varies, and if there are only <i>n</i>
+  phases, two of the varying factors may be arbitrarily fixed. This rule,
+  the application of which, it is hoped, will become clear in the sequel,
+  may be very concisely and conveniently summarized in the form of the
+  equation&mdash;</p>
+
+<p class="cenhead">P + F = C + 2, or F = C + 2 - P</p>
+
+  <p>where P denotes the number of the phases, F the degrees of freedom,
+  and C the number of components. From the second form of the equation it
+  can be readily seen that the greater the number of the phases, the fewer
+  are the degrees of freedom. With increase in the number of the phases,
+  therefore, the <!-- Page 17 --><span class="pagenum"><a
+  name="page17"></a>{17}</span>condition of the system becomes more and
+  more defined, or less and less variable.</p>
+
+  <p><b>Classification of Systems according to the Phase Rule.</b>&mdash;We
+  have already learned in the introductory chapter that systems which are
+  apparently quite different in character may behave in a very similar
+  manner. Thus it was stated that the laws which govern the equilibrium
+  between water and its vapour are quite analogous to those which are
+  obeyed by the dissociation of calcium carbonate into carbon dioxide and
+  calcium oxide; in each case a certain temperature is associated with a
+  definite pressure, no matter what the relative or absolute amounts of the
+  respective substances are. And other examples were given of systems which
+  were apparently similar in character, but which nevertheless behaved in a
+  different manner. The relations between the various systems, however,
+  become perfectly clear and intelligible in the light of the Phase Rule.
+  In the case first mentioned, that of water in equilibrium with its
+  vapour, we have one component&mdash;water&mdash;present in two phases,
+  <i>i.e.</i> in two physically distinct forms, viz. liquid and vapour.
+  According to the Phase Rule, therefore, since C = 1, and P = 2, the
+  degree of freedom F is equal to 1 + 2 - 2 = 1; the system possesses one
+  degree of freedom, as has already been stated. But in the case of the
+  second system mentioned above there are two components, viz. calcium
+  oxide and carbon dioxide (p.&nbsp;<a href="#page12">12</a>), and three phases,
+  viz. two solid phases, CaO and CaCO<sub>3</sub>, and the gaseous phase,
+  CO<sub>2</sub>. The number of degrees of freedom of the system,
+  therefore, is 2 + 2 - 3 = 1; this system, therefore, also possesses one
+  degree of freedom. We can now understand why these two systems behave in
+  a similar manner; both are univariant, or possess only one degree of
+  freedom. We shall therefore expect a similar behaviour in the case of all
+  univariant systems, no matter how dissimilar the systems may outwardly
+  appear. Similarly, all bivariant systems will exhibit analogous
+  behaviour; and generally, systems possessing the same degree of freedom
+  will show a like behaviour. In accordance with the Phase Rule, therefore,
+  we may classify the different systems which may be found into invariant,
+  univariant, bivariant, multivariant, <!-- Page 18 --><span
+  class="pagenum"><a name="page18"></a>{18}</span>according to the relation
+  which obtains between the number of the components and the number of
+  coexisting phases; and we shall expect that in each case the members of
+  any particular group will exhibit a uniform behaviour. By this means we
+  are enabled to obtain an insight into the general behaviour of any
+  system, so soon as we have determined the number of the components and
+  the number of the coexisting phases.</p>
+
+  <p>The adoption of the Phase Rule for the purposes of classification has
+  been of great importance in studying changes in the equilibrium existing
+  between different substances; for not only does it render possible the
+  grouping together of a large number of isolated phenomena, but the
+  guidance it affords has led to the discovery of new substances, has given
+  the clue to the conditions under which these substances can exist, and
+  has led to the recognition of otherwise unobserved resemblances existing
+  between different systems.</p>
+
+  <p><b>Deduction of the Phase Rule.</b>&mdash;In the preceding pages we
+  have restricted ourselves to the statement of the Phase Rule, without
+  giving any indication of how it has been deduced. At the close of this
+  chapter, therefore, the mathematical deduction of the generalization will
+  be given, but in brief outline only, the reader being referred to works
+  on Thermodynamics for a fuller treatment of the subject.<a name="NtA21"
+  href="#Nt21"><sup>[21]</sup></a></p>
+
+  <p>All forms of energy can be resolved into two factors, the
+  <i>capacity</i> factor and the <i>intensity</i> factor; but for the
+  production of equilibrium, only the intensity factor is of importance.
+  Thus, if two bodies having the same temperature are brought in contact
+  with each other, they will be in equilibrium as regards heat energy, no
+  matter what may be the amounts of heat (capacity factor) contained in
+  either, because the intensity factor&mdash;the temperature&mdash;is the
+  same. But if the temperature of the two bodies is different, <i>i.e.</i>
+  if the intensity factor of heat energy is different, the two bodies will
+  no longer be in equilibrium; but heat will pass from the hotter to the
+  colder until both have the same temperature.</p>
+
+  <p>As with heat energy, so with chemical energy. If we have a substance
+  existing in two different states, or in two different <!-- Page 19
+  --><span class="pagenum"><a name="page19"></a>{19}</span>phases of a
+  system, equilibrium can occur only when the intensity factor of chemical
+  energy is the same. This intensity factor may be called the <i>chemical
+  potential</i>; and we can therefore say that a system will be in
+  equilibrium when the chemical potential of each component is the same in
+  all the phases in which the component occurs. Thus, for example, ice,
+  water, and vapour have, at the triple point, the same chemical
+  potential.</p>
+
+  <p>The potential of a component in any phase depends not only on the
+  composition of the phase, but also on the temperature and the pressure
+  (or volume). If, therefore, we have a system of C components existing in
+  P phases, then, in order to fix the composition of unit mass of each
+  phase, it is necessary to know the masses of (C - 1) components in each
+  of the phases. As regards the composition, therefore, each phase
+  possesses (C - 1) variables. Since there are P phases, it follows that,
+  as regards composition, the whole system possesses P(C - 1) variables.
+  Besides these there are, however, two other variables, viz. temperature
+  and pressure, so that altogether a system of C components in P phases
+  possesses P(C - 1) + 2 variables.</p>
+
+  <p>In order to define the state of the system completely, it will be
+  necessary to have as many equations as there are variables. If,
+  therefore, there are fewer equations than there are variables, then,
+  according to the deficiency in the number of the equations, one or more
+  of the variables will have an undefined value; and values must be
+  assigned to these variables before the system is entirely defined. The
+  number of these undefined values gives us the variability or the degree
+  of freedom of the system.</p>
+
+  <p>The equations by which the system is to be defined are obtained from
+  the relationship between the potential of a component and the composition
+  of the phase, the temperature and the pressure. Further, as has already
+  been stated, equilibrium occurs when the potential of each component is
+  the same in the different phases in which it is present. If, therefore,
+  we choose as standard one of the phases in which all the components
+  occur, then in any other phase in equilibrium with <!-- Page 20 --><span
+  class="pagenum"><a name="page20"></a>{20}</span>it, the potential of each
+  component must be the same as in the standard phase. For each phase in
+  equilibrium with the standard phase, therefore, there will be a definite
+  equation of state for each component in the phase; so that, if there are
+  P phases, we obtain for each component (P - 1) equations; and for C
+  components, therefore, we obtain C(P - 1) equations.</p>
+
+  <p>But we have seen above that there are P(C - 1) + 2 variables, and as
+  we have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) =
+  C + 2 - P variables undefined. That is to say, the degree of freedom (F)
+  of a system consisting of C components in P phases is&mdash;</p>
+
+<p class="cenhead">F = C + 2 - P</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 21 --><span class="pagenum"><a name="page21"></a>{21}</span></p>
+
+<h3>CHAPTER III</h3>
+
+<p class="cenhead">TYPICAL SYSTEMS OF ONE COMPONENT</p>
+
+<p class="cenhead">A. <i>Water.</i></p>
+
+  <p>For the sake of rendering the Phase Rule more readily intelligible,
+  and at the same time also for the purpose of obtaining examples by which
+  we may illustrate the general behaviour of systems, we shall in this
+  chapter examine in detail the behaviour of several well-known systems
+  consisting of only one component.</p>
+
+  <p>The most familiar examples of equilibria in a one-component system are
+  those furnished by the three phases of water, viz. ice, water, water
+  vapour. The system consists of one component, because all three phases
+  have the same chemical composition, represented by the formula
+  H<sub>2</sub>O. As the criterion of equilibrium we shall choose a
+  definite pressure, and shall study the variation of the pressure with the
+  temperature; and for the purpose of representing the relationships which
+  we obtain we shall employ a temperature-pressure diagram, in which the
+  temperatures are measured as abscissæ and the pressures as ordinates. In
+  such a diagram invariant systems will be represented by points;
+  univariant systems by lines, and bivariant systems by areas.</p>
+
+  <p><b>Equilibrium between Liquid and Vapour. Vaporization
+  Curve.</b>&mdash;Consider in the first place the conditions for the
+  coexistence of liquid and vapour. According to the Phase Rule (p.&nbsp;<a
+  href="#page16">16</a>), a system consisting of one component in two
+  phases has one degree of freedom, or is univariant. We should therefore
+  expect that it will be possible for liquid water to coexist with water
+  vapour at different values of temperature and <!-- Page 22 --><span
+  class="pagenum"><a name="page22"></a>{22}</span>pressure, but that if we
+  arbitrarily fix one of the variable factors, pressure, temperature, or
+  volume (in the case of a given mass of substance), the state of the
+  system will then be defined. If we fix, say, the temperature, then the
+  pressure will have a definite value; or if we adopt a certain pressure,
+  the liquid and vapour can coexist only at a certain definite temperature.
+  Each temperature, therefore, will correspond to a definite pressure; and
+  if in our diagram we join by a continuous line all the points indicating
+  the values of the pressure corresponding to the different temperatures,
+  we shall obtain a curve (Fig. 1) representing the variation of the
+  pressure with the temperature. This is the curve of vapour pressure, or
+  the <i>vaporization curve</i> of water.</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig001.png"><img style="width:100%" src="images/fig001.png"
+      alt="Fig. 1" title="Fig. 1" /></a>
+    <span class="sc">Fig. 1.</span>
+  </div>
+
+  <p>Now, the results of experiment are quite in agreement with the
+  requirements of the Phase Rule, and at any given temperature the system
+  water&mdash;vapour can exist in equilibrium only under a definite
+  pressure.</p>
+
+  <p>The vapour pressure of water at different temperatures has been
+  subjected to careful measurement by Magnus,<a name="NtA22"
+  href="#Nt22"><sup>[22]</sup></a> Regnault,<a name="NtA23"
+  href="#Nt23"><sup>[23]</sup></a> Ramsay and Young,<a name="NtA24"
+  href="#Nt24"><sup>[24]</sup></a> Juhlin,<a name="NtA25"
+  href="#Nt25"><sup>[25]</sup></a> Thiesen and Scheel,<a name="NtA26"
+  href="#Nt26"><sup>[26]</sup></a> and others. In the following table the
+  values of the vapour pressure from -10° to +100° are those calculated
+  from the measurements of Regnault, corrected by the measurements of Wiebe
+  and Thiesen and Scheel;<a name="NtA27" href="#Nt27"><sup>[27]</sup></a>
+  those from 120° to 270° were determined <!-- Page 23 --><span
+  class="pagenum"><a name="page23"></a>{23}</span>by Ramsay and Young,
+  while the values of the critical pressure and temperature are those
+  determined by Battelli.<a name="NtA28"
+  href="#Nt28"><sup>[28]</sup></a></p>
+
+<p class="cenhead"><span class="sc">Vapour Pressure of Water.</span></p>
+
+<table class="nobctr" summary="Vapour Pressure of Water" title="Vapour Pressure of Water">
+
+<tr><td class="tobob" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure in cm.<br />mercury.
+</td><td class="allb" style="text-align:center"> Temperature.
+</td><td class="tobob" style="text-align:center"> Pressure in cm.<br />mercury. </td></tr>
+
+<tr><td class="rib" style="text-align:center"> &nbsp;-10° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; 0.213 </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;120° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp; 148.4</td></tr>
+<tr><td class="rib" style="text-align:center"> &nbsp; &nbsp; 0° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; 0.458<a name="NtA29" href="#Nt29"><sup>[29]</sup></a> </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;130° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp; 201.9</td></tr>
+<tr><td class="rib" style="text-align:center"> +20° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; 1.752 </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;150° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp; 356.8</td></tr>
+<tr><td class="rib" style="text-align:center"> &nbsp; 40° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; 5.516 </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;200° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; 1162.5</td></tr>
+<tr><td class="rib" style="text-align:center"> &nbsp; 60° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> 14.932 </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;250° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; 2973.4</td></tr>
+<tr><td class="rib" style="text-align:center"> &nbsp; 80° </td><td class="rib" style="text-align:left; padding-left:1.5em;"> 35.54 </td><td class="rib" style="text-align:left; padding-left:1.5em;"> &nbsp; &nbsp;270° </td><td class="nob" style="text-align:left; padding-left:1.5em;"> &nbsp; 4110.1</td></tr>
+
+<tr><td class="ribob" style="text-align:center; vertical-align:top"> 100° </td><td class="ribob" style="text-align:left; padding-left:1.5em; vertical-align:top;"> 76.00 </td><td class="ribob" style="text-align:left; padding-left:1.5em;"> 364.3° (critical<br />temperature) </td><td class="bob" style="text-align:left; padding-left:1.5em;"> 14790.4 (194.6 atm.)<br />(critical pressure).</td></tr>
+</table>
+
+  <p>The pressure is, of course, independent of the relative or absolute
+  volumes of the liquid and vapour; on increasing the volume at constant
+  temperature, a certain amount of the liquid will pass into vapour, and
+  the pressure will regain its former value. If, however, the pressure be
+  permanently maintained at a value different from that corresponding to
+  the temperature employed, then either all the liquid will pass into
+  vapour, or all the vapour will pass into liquid, and we shall have either
+  vapour alone or liquid alone.</p>
+
+  <p><b>Upper Limit of Vaporization Curve.</b>&mdash;On continuing to add
+  heat to water contained in a closed vessel, the pressure of the vapour
+  will gradually increase. Since with increase of pressure the density of
+  the vapour must increase, and since with rise of temperature the density
+  of the liquid must decrease, a point will be reached at which the density
+  of liquid and vapour become identical; the system ceases to be
+  heterogeneous, and passes into one homogeneous phase. The temperature at
+  which this occurs is called the <i>critical temperature</i>. To this
+  temperature there will, of course, correspond a certain definite
+  pressure, called the <i>critical pressure</i>. The curve representing the
+  <!-- Page 24 --><span class="pagenum"><a
+  name="page24"></a>{24}</span>equilibrium between liquid and vapour must,
+  therefore, end abruptly at the critical point. At temperatures above this
+  point no pressure, however great, can cause the formation of the liquid
+  phase; at temperatures above the critical point the vapour becomes a gas.
+  In the case of water, the critical temperature is 364.3°, and the
+  critical pressure 194.6 atm.; at the point representing these conditions
+  the vapour-pressure curve of water must cease.</p>
+
+  <p><b>Sublimation Curve of Ice.</b>&mdash;Vapour is given off not only by
+  liquid water, but also by solid water, or ice. That this is so is
+  familiar to every one through the fact that ice or snow, even at
+  temperatures below the melting point, gradually disappears in the form of
+  vapour. Even at temperatures considerably lower than 0°, the vapour
+  pressure of ice, although small, is quite appreciable; and it is
+  possible, therefore, to have ice and vapour coexisting in equilibrium.
+  When we inquire into the conditions under which such a system can exist,
+  we see again that we are dealing with a univariant system&mdash;one
+  component existing in two phases&mdash;and that, therefore, just as in
+  the case of the system water and vapour, there will be for each
+  temperature a certain definite pressure of the vapour, and this pressure
+  will be independent of the relative or absolute amounts of the solid or
+  vapour present, and will depend solely on the temperature. Further, just
+  as in the case of the vapour pressure of water, the condition of
+  equilibrium between ice and water vapour will be represented by a line or
+  curve showing the change of pressure with the temperature. Such a curve,
+  representing the conditions of equilibrium between a solid and its
+  vapour, is called a <i>sublimation curve</i>. At temperatures represented
+  by any point on this curve, the solid (ice) will sublime or pass into
+  vapour without previously fusing. Since ice melts at 0° (<i>vide
+  infra</i>), the sublimation curve must end at that temperature.</p>
+
+  <p>The following are the values of the vapour pressure of ice between 0°
+  and -50°.<a name="NtA30" href="#Nt30"><sup>[30]</sup></a></p>
+
+<p><!-- Page 25 --><span class="pagenum"><a name="page25"></a>{25}</span></p>
+
+<p class="cenhead"><span class="sc">Vapour Pressure of Ice.</span></p>
+
+<table class="nobctr" summary="Vapour Pressure of Ice" title="Vapour Pressure of Ice">
+<tr><td class="tobob" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure in mm.<br />mercury.
+</td><td class="allb" style="text-align:center"> Temperature.
+</td><td class="tobob" style="text-align:center"> Pressure in mm.<br />mercury. </td></tr>
+
+<tr><td class="rib" style="text-align:center"> -50° </td><td class="rib" style="text-align:center"> 0.050 </td><td class="rib" style="text-align:center"> -8° </td><td class="nob" style="text-align:center"> 2.379</td></tr>
+<tr><td class="rib" style="text-align:center"> -40° </td><td class="rib" style="text-align:center"> 0.121 </td><td class="rib" style="text-align:center"> -6° </td><td class="nob" style="text-align:center"> 2.821</td></tr>
+<tr><td class="rib" style="text-align:center"> -30° </td><td class="rib" style="text-align:center"> 0.312 </td><td class="rib" style="text-align:center"> -4° </td><td class="nob" style="text-align:center"> 3.334</td></tr>
+<tr><td class="rib" style="text-align:center"> -20° </td><td class="rib" style="text-align:center"> 0.806 </td><td class="rib" style="text-align:center"> -2° </td><td class="nob" style="text-align:center"> 3.925</td></tr>
+<tr><td class="rib" style="text-align:center"> -15° </td><td class="rib" style="text-align:center"> 1.279 </td><td class="rib" style="text-align:center"> &nbsp;0° </td><td class="nob" style="text-align:center"> 4.602</td></tr>
+<tr><td class="ribob" style="text-align:center"> -10° </td><td class="ribob" style="text-align:center"> 1.999 </td><td class="ribob" style="text-align:center"> </td><td class="bob" style="text-align:center"></td></tr>
+</table>
+
+  <p><b>Equilibrium between Ice and Water. Curve of Fusion.</b>&mdash;There
+  is still another univariant system of the one component water, the
+  existence of which, at definite values of temperature and pressure, the
+  Phase Rule allows us to predict. This is the system solid&mdash;liquid.
+  Ice on being heated to a certain temperature melts and passes into the
+  liquid state; and since this system solid&mdash;liquid is univariant,
+  there will be for each temperature a certain definite pressure at which
+  ice and water can coexist or be in equilibrium, independently of the
+  amounts of the two phases present. Since now the temperature at which the
+  solid phase is in equilibrium with the liquid phase is known as the
+  melting point or point of fusion of the solid, the curve representing the
+  temperatures and pressures at which the solid and liquid are in
+  equilibrium will represent the change of the melting point with the
+  pressure. Such a curve is called the <i>curve of fusion</i>, or the
+  melting-point curve.</p>
+
+  <p>It was not until the middle of the nineteenth century that this
+  connection between the pressure and the melting point, or the change of
+  the melting point with the pressure, was observed. The first to recognize
+  the existence of such a relationship was James Thomson,<a name="NtA31"
+  href="#Nt31"><sup>[31]</sup></a> who in 1849 showed that from theoretical
+  considerations such a relationship must exist, and predicted that in the
+  case of ice the melting point would be lowered by pressure. This
+  prediction was fully confirmed by his brother, W. Thomson<a name="NtA32"
+  href="#Nt32"><sup>[32]</sup></a> (Lord Kelvin), who found that under a
+  pressure <!-- Page 26 --><span class="pagenum"><a
+  name="page26"></a>{26}</span>of 8.1 atm. the melting point of ice was
+  -0.059°; under a pressure of 16.8 atm. the melting point was -0.129°.</p>
+
+  <p>The experiments which were first made in this connection were more of
+  a qualitative nature, but in recent years careful measurements of the
+  influence of pressure on the melting point of ice have been made more
+  especially by Tammann,<a name="NtA33" href="#Nt33"><sup>[33]</sup></a>
+  and the results obtained by him are given in the following table and
+  represented graphically in Fig. 2.</p>
+
+<p class="cenhead"><span class="sc">Fusion Pressure of Ice.</span></p>
+
+<table class="nobctr" summary="Fusion Pressure of Ice" title="Fusion Pressure of Ice">
+
+<tr><td class="tobob" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure in kilogms. per<br />sq. cm.<a name="NtA34" href="#Nt34"><sup>[34]</sup></a>
+</td><td class="tobob" style="text-align:center"> Change of melting point for an<br />increase of pressure of<br />1 kilogm. per sq. cm.</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em">-0°<br />-2.5°<br />-5°<br />-7.5°<br />-10.0°<br />-12.5°<br />-15.0°<br />-17.5°<br />-20.0°<br />-22.1°
+</td><td class="ribob" style="text-align:right; padding-right:5em">1<br />336<br />615<br />890<br />1155<br />1410<br />1625<br />1835<br />2042<br />2200
+</td><td class="bob" style="text-align:right; padding-right:5em">0.0074°<br />0.0090°<br />0.0091°<br />0.0094°<br />0.0100°<br />0.0116°<br />0.0119°<br />0.0121°<br />0.0133°</td></tr>
+
+</table>
+
+  <p>From the numbers in the table and from the figure we see that as the
+  pressure is increased the melting point of ice is lowered; but we also
+  observe that a very large change of pressure is required in order to
+  produce a very small change in the melting point. The curve, therefore,
+  is very steep. Increase of pressure by one atmosphere lowers the melting
+  point by only 0.0076°,<a name="NtA35" href="#Nt35"><sup>[35]</sup></a> or
+  an increase of pressure of 135 atm. is required to produce a lowering of
+  the melting point of 1°. We see further that the fusion curve bends
+  slightly as the pressure is increased, which signifies that the variation
+  of <!-- Page 27 --><span class="pagenum"><a
+  name="page27"></a>{27}</span>the melting point with the pressure changes;
+  at -15°, when the pressure is 1625 kilogm. per sq. cm., increase of
+  pressure by 1 kilogm. per sq. cm. lowers the melting point by 0.012°.
+  This curvature of the fusion curve we shall later (Chap. IV.) see to be
+  an almost universal phenomenon.</p>
+
+  <div class="figcenter" style="width:27%;">
+      <a href="images/fig002.png"><img style="width:100%" src="images/fig002.png"
+      alt="Fig. 2" title="Fig. 2" /></a>
+    <span class="sc">Fig. 2.</span>
+  </div>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig003.png"><img style="width:100%" src="images/fig003.png"
+      alt="Fig. 3" title="Fig. 3" /></a>
+    <span class="sc">Fig. 3.</span>
+  </div>
+
+  <p><b>Equilibrium between Ice, Water, and Vapour. The Triple
+  Point.</b>&mdash;On examining the vapour-pressure curves of ice and water
+  (Fig. 3), we see that at a temperature of about 0° and under a pressure
+  of about 4.6 mm. mercury, the two curves cut. At this point liquid water
+  and solid ice are each in equilibrium with vapour at the same pressure.
+  Since this is so, they must, of course, be in equilibrium <!-- Page 28
+  --><span class="pagenum"><a name="page28"></a>{28}</span>with one
+  another, as experiment also shows. At this point, therefore, ice, water,
+  and vapour can be in equilibrium, and as there are three phases present,
+  the point is called a <i>triple point</i>.<a name="NtA36"
+  href="#Nt36"><sup>[36]</sup></a></p>
+
+  <p>The triple point, however, does not lie exactly at 0° C., for this
+  temperature is defined as the melting point of ice under atmospheric
+  pressure. At the triple point, however, the pressure is equal to the
+  vapour pressure of ice and water, and this pressure, as we see from the
+  tables on pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less
+  than in the previous case. Now, we have just seen that a change of
+  pressure of 1 atm. corresponds to a change of the melting point of
+  0.0076°; the melting point of ice, therefore, when under the pressure of
+  its own vapour, will be very nearly +0.0076°, and the pressure of the
+  vapour will be very slightly greater than 4.579 mm., which is the
+  pressure at 0° (p.&nbsp;<a href="#page21">21</a>). The difference is, however,
+  slight, and may be neglected here. At the temperature, then, of +0.0076°,
+  and under a pressure of 4.6 mm. of mercury, ice, water, and vapour will
+  be in equilibrium; the point in our diagram representing this particular
+  temperature and pressure is, therefore, the triple point of the system
+  ice&mdash;water&mdash;vapour.</p>
+
+  <p>Since at the triple point we have three phases of one component, the
+  system at this point is invariant&mdash;it possesses no degrees of
+  freedom. If the temperature is changed, the system will undergo
+  alteration in such a way that one of the phases will disappear, and a
+  univariant system will result; if heat be added, ice will melt, and we
+  shall have left water and vapour; if heat be abstracted, water will
+  freeze, and we shall have left ice and vapour; if, when the temperature
+  is altered, the pressure is kept constant, then we shall ultimately
+  obtain only one phase (see Chap. IV.).</p>
+
+  <p>The triple point is not only the point of intersection of the
+  vaporization and sublimation curves, but it is also the end-point of the
+  fusion curve. The fusion curve, as we have seen, is the curve of
+  equilibrium between ice and water; and since at the triple point ice and
+  water are each in equilibrium with <!-- Page 29 --><span
+  class="pagenum"><a name="page29"></a>{29}</span>vapour of the same
+  pressure, they must, of course, also be in equilibrium with one
+  another.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig004.png"><img style="width:100%" src="images/fig004.png"
+      alt="Fig. 4" title="Fig. 4" /></a>
+    <span class="sc">Fig. 4.</span>
+  </div>
+
+  <p><b>Bivariant Systems of Water.</b>&mdash;If we examine Fig. 4, we see
+  that the curves OA, OB, OC, which represent diagrammatically the
+  conditions under which water and vapour, ice and vapour, and water and
+  ice are in equilibrium, form the boundaries of three "fields," or areas,
+  I., II., III. These areas, now, represent the conditions for the
+  existence of the single phases, solid, liquid, and vapour respectively.
+  At temperatures and pressures represented by any point in the field I.,
+  solid only can exist as a stable phase. Since we have here one component
+  in only one phase, the system is bivariant, and at any given temperature,
+  therefore, ice can exist under a series of pressures; and under any given
+  pressure, at a series of temperatures, these pressures and temperatures
+  being limited only by the curves OB, OC. Similarly also with the areas
+  II. and III.</p>
+
+  <p>We see, further, that the different areas are the regions of stability
+  of the phase common to the two curves by which the area is enclosed.<a
+  name="NtA37" href="#Nt37"><sup>[37]</sup></a> Thus, the phase common to
+  the two systems <!-- Page 30 --><span class="pagenum"><a
+  name="page30"></a>{30}</span>represented by BO (ice and vapour), and OA
+  (water and vapour) is the vapour phase; and the area BOA is therefore the
+  area of the vapour phase. Similarly, BOC is the area of the ice phase,
+  and COA the area of the water phase.</p>
+
+  <p><b>Supercooled Water. Metastable State.</b>&mdash;When heated under
+  the ordinary atmospheric pressure, ice melts when the temperature reaches
+  0°, and it has so far not been found possible to raise the temperature of
+  ice above this point without liquefaction taking place. On the other
+  hand, it has long been known that water can be cooled below zero without
+  solidification occurring. This was first discovered in 1724 by
+  Fahrenheit,<a name="NtA38" href="#Nt38"><sup>[38]</sup></a> who found
+  that water could be exposed to a temperature of -9.4° without
+  solidifying; so soon, however, as a small particle of ice was brought in
+  contact with the water, crystallization commenced. Superfused or
+  supercooled water&mdash;<i>i.e.</i> water cooled below 0°&mdash;is
+  unstable only in respect of the solid phase; so long as the presence of
+  the solid phase is carefully avoided, the water can be kept for any
+  length of time without solidifying, and the system supercooled water and
+  vapour behaves in every way like a stable system. A system, now, which in
+  itself is stable, and which becomes instable only in contact with a
+  particular phase, is said to be <i>metastable</i>, and the region
+  throughout which this condition exists is called the metastable region.
+  Supercooled water, therefore, is in a metastable condition. If the
+  supercooling be carried below a certain temperature, solidification takes
+  place spontaneously without the addition of the solid phase; the system
+  then ceases to be metastable, and becomes <i>instable</i>.</p>
+
+  <p>Not only has water been cooled to temperatures considerably below the
+  melting point of ice, but the vapour pressure of the supercooled water
+  has been measured. It is of interest and importance, now, to see what
+  relationship exists between the vapour pressure of ice and that of
+  supercooled water at the same temperature. This relationship is clearly
+  shown by the numbers in the following table,<a name="NtA39"
+  href="#Nt39"><sup>[39]</sup></a> and is represented in Fig. 3, <!-- Page
+  31 --><span class="pagenum"><a name="page31"></a>{31}</span><span
+  class="correction" title="Original reads 'p. 25.'.">p.&nbsp;<a
+  href="#page27">27</a>.</span>, and diagrammatically in Fig. 4, the vapour
+  pressures of supercooled water being represented by the curve OA&prime;,
+  which is the unbroken continuation of AO.</p>
+
+<p class="cenhead"><span class="sc">Vapour Pressure of Ice and of Supercooled Water.</span></p>
+
+<table class="nobctr" summary="Vapour Pressure of Ice and of Supercooled Water" title="Vapour Pressure of Ice and of Supercooled Water">
+<tr><td class="tobob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toleb" style="text-align:center" colspan="3"> Pressure in mm. mercury.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> Water.
+</td><td class="toboleb" style="text-align:center"> Ice.
+</td><td class="toboleb" style="text-align:center"> Difference.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:2em"> 0° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 4.618 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 4.602 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.016<a name="NtA40" href="#Nt40"><sup>[40]</sup></a></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em"> -2° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 3.995 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 3.925 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.070</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em"> -4° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 3.450 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 3.334 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.116</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em"> -8° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 2.558 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 2.379 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.179</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em"> -10° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 2.197 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 1.999 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.198</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em"> -15° </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 1.492 </td><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em"> 1.279 </td><td class="nob" style="text-align:left; padding-left:2em"> 0.213</td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:2em"> -20° </td><td class="ribob" style="text-align:center; padding-left:2em; padding-right:2em"> 1.005 </td><td class="ribob" style="text-align:center; padding-left:2em; padding-right:2em"> 0.806 </td><td class="bob" style="text-align:left; padding-left:2em"> 0.199</td></tr>
+</table>
+
+  <p>At all temperatures below 0° (more correctly +0.0076°), at which
+  temperature water and ice have the same vapour pressure, the vapour
+  pressure of supercooled water is <i>greater</i> than that of ice at the
+  same temperature.</p>
+
+  <p>From the relative positions of the curves OB and OA (Fig. 4) we see
+  that at all temperatures above 0°, the (metastable) sublimation curve of
+  ice, if it could be obtained, would be higher than the vaporization curve
+  of water. This shows, therefore, that at 0° a "break" must occur in the
+  curve of states, and that in the neighbourhood of this break the curve
+  above that point must ascend less rapidly than the curve below the break.
+  Since, however, the differences in the vapour pressures of supercooled
+  water and of ice are very small, the change in the direction of the
+  vapour-pressure curve on passing from ice to water was at first not
+  observed, and Regnault regarded the sublimation curve as passing
+  continuously into <!-- Page 32 --><span class="pagenum"><a
+  name="page32"></a>{32}</span>the vaporization curve. The existence of a
+  break was, however, shown by James Thomson<a name="NtA41"
+  href="#Nt41"><sup>[41]</sup></a> and by Kirchhoff<a name="NtA42"
+  href="#Nt42"><sup>[42]</sup></a> to be demanded by thermo-dynamical
+  considerations, and the prediction of theory was afterwards realized
+  experimentally by Ramsay and Young in their determinations of the vapour
+  pressure of water and ice, as well as in the case of other substances.<a
+  name="NtA43" href="#Nt43"><sup>[43]</sup></a></p>
+
+  <p>From what has just been said, we can readily understand why ice and
+  water cannot exist in equilibrium below 0°. For, suppose we have ice and
+  water in the same closed space, but not in contact with one another, then
+  since the vapour pressure of the supercooled water is higher than that of
+  ice, the vapour of the former must be supersaturated in contact with the
+  latter; vapour must, therefore, condense on the ice; and in this way
+  there will be a slow distillation from the water to the ice, until at
+  last all the water will have disappeared, and only ice and vapour
+  remain.<a name="NtA44" href="#Nt44"><sup>[44]</sup></a></p>
+
+  <p><b>Other Systems of the Substance Water.</b>&mdash;We have thus far
+  discussed only those systems which are constituted by the three
+  phases&mdash;ice, water, and water vapour. It has, however, been recently
+  found that at a low temperature and under a high pressure ordinary ice
+  can pass into two other crystalline varieties, called by Tammann<a
+  name="NtA45" href="#Nt45"><sup>[45]</sup></a> ice II. and ice III.,
+  ordinary ice being ice I. According to the Phase Rule, now, since each of
+  these solid forms constitutes a separate phase (p.&nbsp;<a
+  href="#page9">9</a>), it will be possible to have the following (and
+  more) systems of water, in addition to those already studied, viz. water,
+  ice I., ice II.; water, ice I., ice III.; water, ice II., ice III.,
+  forming invariant systems and existing in equilibrium only at a definite
+  triple point; further, water, ice II.; water, ice III.; ice I., ice II.;
+  ice I., ice III.; ice II., ice III., forming univariant systems,
+  existing, therefore, at definite corresponding values of <!-- Page 33
+  --><span class="pagenum"><a name="page33"></a>{33}</span>temperature and
+  pressure; and lastly, the bivariant systems, ice II. and ice III. Several
+  of these systems have been investigated by Tammann. The triple point for
+  water, ice I., ice III., lies at -22°, and a pressure of 2200 kilogms.
+  per sq. cm. (2130 atm.), as indicated in Fig. 2, <span class="correction"
+  title="Original reads 'p. 25.'.">p.&nbsp;<a href="#page27">27</a>.</span><a
+  name="NtA46" href="#Nt46"><sup>[46]</sup></a> In contrast with the
+  behaviour of ordinary ice, the temperature of equilibrium in the case of
+  water&mdash;ice II., and water&mdash;ice III., is <i>raised</i> by
+  increase of pressure.</p>
+
+<p class="cenhead">B. <i>Sulphur.</i></p>
+
+  <p><b>Polymorphism.</b>&mdash;Reference has just been made to the fact
+  that ice can exist not only in the ordinary form, but in at least two
+  other crystalline varieties. This phenomenon, the existence of a
+  substance in two or more different crystalline forms, is called
+  <i>polymorphism</i>. Polymorphism was first observed by Mitscherlich<a
+  name="NtA47" href="#Nt47"><sup>[47]</sup></a> in the case of sodium
+  phosphate, and later in the case of sulphur. To these two cases others
+  were soon added, at first of inorganic, and later of organic substances,
+  so that polymorphism is now recognized as of very frequent occurrence
+  indeed.<a name="NtA48" href="#Nt48"><sup>[48]</sup></a> These various
+  forms of a substance differ not only in crystalline shape, but also in
+  melting point, specific gravity, and other physical properties. In the
+  liquid state, however, the differences do not exist.</p>
+
+  <p>According to our definition of phases (p.&nbsp;<a href="#page9">9</a>),
+  each of these polymorphic forms constitutes a separate phase of the
+  particular substance. As is readily apparent, the number of possible
+  systems formed of one component may be considerably increased when that
+  component is capable of existing in different crystalline forms. We have,
+  therefore, to inquire what are the conditions under which different
+  polymorphic forms can coexist, either alone or in presence of the liquid
+  and vapour phase. For the purpose of illustrating the general behaviour
+  of such systems, we shall study the systems formed by the different
+  crystalline forms of sulphur, tin, and benzophenone.</p>
+
+<p><!-- Page 34 --><span class="pagenum"><a name="page34"></a>{34}</span></p>
+
+  <p><b>Sulphur</b> exists in two well-known crystalline
+  forms&mdash;rhombic, or octahedral, and monoclinic, or prismatic sulphur.
+  Of these, the former melts at 114.5°; the latter at 120°.<a name="NtA49"
+  href="#Nt49"><sup>[49]</sup></a> Further, at the ordinary temperature,
+  rhombic sulphur can exist unchanged, whereas, on being heated to
+  temperatures somewhat below the melting point, it passes into the
+  prismatic variety. On the other hand, at temperatures above 96°,
+  prismatic sulphur can remain unchanged, whereas at the ordinary
+  temperature it passes slowly into the rhombic form.</p>
+
+  <p>If, now, we examine the case of sulphur with the help of the Phase
+  Rule, we see that the following systems are theoretically
+  possible:&mdash;</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>I. <i>Bivariant Systems: One component in one phase.</i></p>
+      <p class="i2">(<i>a</i>) Rhombic sulphur.</p>
+      <p class="i2">(<i>b</i>) Monoclinic sulphur.</p>
+      <p class="i2">(<i>c</i>) Sulphur vapour.</p>
+      <p class="i2">(<i>d</i>) Liquid sulphur.</p>
+    </div>
+
+    <div class="stanza">
+      <p>II. <i>Univariant Systems: One component in two phases.</i></p>
+      <p class="i2">(<i>a</i>) Rhombic sulphur and vapour.</p>
+      <p class="i2">(<i>b</i>) Monoclinic sulphur and vapour.</p>
+      <p class="i2">(<i>c</i>) Rhombic sulphur and liquid.</p>
+      <p class="i2">(<i>d</i>) Monoclinic sulphur and liquid.</p>
+      <p class="i2">(<i>e</i>) Rhombic and monoclinic sulphur.</p>
+      <p class="i2">(<i>f</i>) Liquid and vapour.</p>
+    </div>
+
+    <div class="stanza">
+      <p>III. <i>Invariant Systems: One component in three phases.</i></p>
+      <p class="i2">(<i>a</i>) Rhombic and monoclinic sulphur and vapour.</p>
+      <p class="i2">(<i>b</i>) Rhombic sulphur, liquid and vapour.</p>
+      <p class="i2">(<i>c</i>) Monoclinic sulphur, liquid and vapour.</p>
+      <p class="i2">(<i>d</i>) Rhombic and monoclinic sulphur and liquid.</p>
+    </div>
+  </div>
+
+  <div class="figright" style="width:31%;">
+      <a href="images/fig005.png"><img style="width:100%" src="images/fig005.png"
+      alt="Fig. 5" title="Fig. 5" /></a>
+    <span class="sc">Fig.</span> 5.
+  </div>
+
+  <p><b>Triple Point&mdash;Rhombic and Monoclinic Sulphur and Vapour.
+  Transition Point.</b>&mdash;In the case of ice, water and vapour, we saw
+  that at the triple point the vapour pressures of ice and water are equal;
+  below this point, ice is stable; above this point, water is stable. We
+  saw, further, that below 0° the vapour pressure of the stable system is
+  lower than that of the metastable, and therefore that at the triple point
+  there is a break in the vapour pressure curve of such a kind that above
+  <!-- Page 35 --><span class="pagenum"><a name="page35"></a>{35}</span>the
+  triple point the vapour-pressure curve ascends more slowly than below it.
+  Now, although the vapour pressure of solid sulphur has not been
+  determined, we can nevertheless consider that it does possess a certain,
+  even if very small, vapour pressure,<a name="NtA50"
+  href="#Nt50"><sup>[50]</sup></a> and that at the temperature at which the
+  vapour pressures of rhombic and monoclinic sulphur become equal, we can
+  have these two solid forms existing in equilibrium with the vapour. Below
+  that point only one form, that with the lower vapour pressure, will be
+  stable; above that point only the other form will be stable. On passing
+  through the triple point, therefore, there will be a change of the one
+  form into the other. This point is represented in our diagram (Fig. 5) by
+  the point O, the two curves AO and OB representing diagrammatically the
+  vapour pressures of rhombic and monoclinic sulphur respectively. If the
+  vapour phase is absent and the system maintained under a constant
+  pressure, <i>e.g.</i> <!-- Page 36 --><span class="pagenum"><a
+  name="page36"></a>{36}</span>atmospheric pressure, there will also be a
+  definite temperature at which the two solid forms are in equilibrium, and
+  on passing through which complete and reversible transformation of one
+  form into the other occurs. This temperature, which refers to equilibrium
+  in absence of the vapour phase, is known as the <i>transition
+  temperature</i> or <i>inversion temperature</i>.</p>
+
+  <p>Were we dependent on measurements of pressure and temperature, the
+  determination of the transition point might be a matter of great
+  difficulty. When we consider, however, that the other physical properties
+  of the solid phases, <i>e.g.</i> the density, undergo an abrupt change on
+  passing through the transition point, owing to the transformation of one
+  form into the other, then any method by which this abrupt change in the
+  physical properties can be detected may be employed for determining the
+  transition point. A considerable number of such methods have been
+  devised, and a description of the most important of these is given in the
+  Appendix.</p>
+
+  <p>In the case of sulphur, the transition point of rhombic into
+  monoclinic sulphur was found by Reicher<a name="NtA51"
+  href="#Nt51"><sup>[51]</sup></a> to lie at 95.5°. Below this temperature
+  the octahedral, above it the monoclinic, is the stable form.</p>
+
+  <p><b>Condensed Systems.</b>&mdash;We have already seen that in the
+  change of the melting point of water with the pressure, a very great
+  increase of the latter was necessary in order to produce a comparatively
+  small change in the temperature of equilibrium. This is a characteristic
+  of all systems from which the vapour phase is absent, and which are
+  composed only of solid and liquid phases. Such systems are called
+  <i>condensed systems</i>,<a name="NtA52" href="#Nt52"><sup>[52]</sup></a>
+  and in determining the temperature of equilibrium of such systems,
+  practically the same point will be obtained whether the measurements are
+  carried out under atmospheric pressure or under the pressure of the
+  vapour of the solid or liquid phases. The transition point, therefore, as
+  determined in open vessels at atmospheric pressure, will differ only by a
+  very slight amount from the triple point, or point at which the two solid
+  or liquid phases are in equilibrium under the pressure of their vapour.
+  <!-- Page 37 --><span class="pagenum"><a name="page37"></a>{37}</span>The
+  determination of the transition point is thereby greatly simplified.</p>
+
+  <p><b>Suspended Transformation.</b>&mdash;In many respects the transition
+  point of two solid phases is analogous to the melting point of a solid,
+  or point at which the solid passes into a liquid. In both cases the
+  change of phase is associated with a definite temperature and pressure in
+  such a way that below the point the one phase, above the point the other
+  phase, is stable. The transition point, however, differs in so far from a
+  point of fusion, that while it is possible to supercool a liquid, no
+  definite case is known where the solid has been heated above the triple
+  point without passing into the liquid state. Transformation, therefore,
+  is suspended only on one side of the melting point. In the case of two
+  solid phases, however, the transition point can be overstepped in both
+  directions, so that each phase can be obtained in the metastable
+  condition. In the case of supercooled water, further, we saw that the
+  introduction of the stable, solid phase caused the speedy transformation
+  of the metastable to the stable condition of equilibrium; but in the case
+  of two solid phases the change from the metastable to the stable
+  modification may occur with great slowness, even in presence of the
+  stable form. This tardiness with which the stable condition of
+  equilibrium is reached greatly increases in many cases the difficulty of
+  accurately determining the transition point. The phenomena of suspended
+  transformation will, however, receive a fuller discussion later (p.&nbsp;<a
+  href="#page68">68</a>).</p>
+
+  <p><b>Transition Curve&mdash;Rhombic and Monoclinic
+  Sulphur.</b>&mdash;Just as we found the melting point of ice to vary with
+  the pressure, so also do we find that change of pressure causes an
+  alteration in the transition point. In the case of the transition point
+  of rhombic into monoclinic sulphur, increase of pressure by 1 atm. raises
+  the transition point by 0.04°-0.05°.<a name="NtA53"
+  href="#Nt53"><sup>[53]</sup></a> The transition curve, or curve
+  representing the change of the transition point with pressure, will
+  therefore slope to the right away from the pressure axis. This is curve
+  OC (Fig. 5).</p>
+
+<p><!-- Page 38 --><span class="pagenum"><a name="page38"></a>{38}</span></p>
+
+  <p><b>Triple Point&mdash;Monoclinic Sulphur, Liquid, and Vapour. Melting
+  Point of Monoclinic Sulphur.</b>&mdash;Above 95.5°, monoclinic sulphur
+  is, as we have seen, the stable form. On being heated to 120°, under
+  atmospheric pressure, it melts. This temperature is, therefore, the point
+  of equilibrium between monoclinic sulphur and liquid sulphur under
+  atmospheric pressure. Since we are dealing with a condensed system, this
+  temperature may be regarded as very nearly that at which the solid and
+  liquid are in equilibrium with their vapour, <i>i.e.</i> the triple
+  point, solid (monoclinic)&mdash;liquid&mdash;vapour. This point is
+  represented in the diagram by B.</p>
+
+  <p><b>Triple Point&mdash;Rhombic and Monoclinic Sulphur and
+  Liquid.</b>&mdash;In contrast with that of ice, the fusion point of
+  monoclinic sulphur is <i>raised</i> by increase of pressure, and the
+  fusion curve, therefore, slopes to the right. The transition curve of
+  rhombic and monoclinic sulphur, as we have seen, also slopes to the
+  right, and more so than the fusion curve of monoclinic sulphur. There
+  will, therefore, be a certain pressure and temperature at which the two
+  curves will cut. This point lies at 151°, and a pressure of 1320 kilogm.
+  per sq. cm., or about 1288 atm.<a name="NtA54"
+  href="#Nt54"><sup>[54]</sup></a> It, therefore, forms another triple
+  point, the existence of which had been predicted by Roozeboom,<a
+  name="NtA55" href="#Nt55"><sup>[55]</sup></a> at which rhombic and
+  monoclinic sulphur are in equilibrium with liquid sulphur. It is
+  represented in our diagram by the point C. <i>Beyond this point
+  monoclinic sulphur ceases to exist in a stable condition.</i> At
+  temperatures and pressures above this triple point, rhombic sulphur will
+  be the stable modification, and this fact is of mineralogical interest,
+  because it explains the occurrence in nature of well-formed rhombic
+  crystals. Under ordinary conditions, prismatic sulphur separates out on
+  cooling fused sulphur, but at temperatures above 151° and under pressures
+  greater than 1288 atm., the rhombic form would be produced.<a
+  name="NtA56" href="#Nt56"><sup>[56]</sup></a></p>
+
+  <p><b>Triple Point&mdash;Rhombic Sulphur, Liquid, and Vapour. Metastable
+  Triple Point.</b>&mdash;On account of the slowness with <!-- Page 39
+  --><span class="pagenum"><a name="page39"></a>{39}</span>which
+  transformation of one form into the other takes place on passing the
+  transition point, it has been found possible to heat rhombic sulphur up
+  to its melting point (114.5°). At this temperature, not only is rhombic
+  sulphur in a metastable condition, but the liquid is also metastable, its
+  vapour pressure being greater than that of solid monoclinic sulphur. This
+  point is represented in our diagram by the point <i>b</i>.</p>
+
+  <p>From the relative positions of the metastable melting point of rhombic
+  sulphur and the stable melting point of monoclinic sulphur at 120°, we
+  see that, of the two forms, the metastable form has the lower melting
+  point. This, of course, is valid only for the relative stability in the
+  neighbourhood of the melting point; for we have already learned that at
+  lower temperatures rhombic sulphur is the stable, monoclinic sulphur the
+  metastable (or unstable) form.</p>
+
+  <p><b>Fusion Curve of Rhombic Sulphur.</b>&mdash;Like any other melting
+  point, that of rhombic sulphur will be displaced by increase of pressure;
+  increase of pressure raises the melting point, and we can therefore
+  obtain a metastable fusion curve representing the conditions under which
+  rhombic sulphur is in equilibrium with liquid sulphur. This metastable
+  fusion curve must pass through the triple point for rhombic
+  sulphur&mdash;monoclinic sulphur&mdash;liquid sulphur, and on passing
+  this point it becomes a stable fusion curve. The continuation of this
+  curve, therefore, above 151° forms the stable fusion curve of rhombic
+  sulphur (curve CD).</p>
+
+  <p>These curves have been investigated at high pressures by Tammann, and
+  the results are represented according to scale in Fig. 6,<a name="NtA57"
+  href="#Nt57"><sup>[57]</sup></a> <i>a</i> being the curve for monoclinic
+  sulphur and liquid; <i>b</i>, that for rhombic sulphur and liquid; and
+  <i>c</i>, that for rhombic and monoclinic sulphur.</p>
+
+  <p><b>Bivariant Systems.</b>&mdash;Just as in the case of the diagram of
+  states of water, the areas in Fig. 5 represent the conditions for the
+  stable existence of the single phases: rhombic sulphur in the area to the
+  left of AOCD; monoclinic sulphur in the area OBC; liquid sulphur in the
+  area EBCD; sulphur vapour below the curves AOBE. As can be seen from the
+  diagram, <!-- Page 40 --><span class="pagenum"><a
+  name="page40"></a>{40}</span>the existence of monoclinic sulphur is
+  limited on all sides, its area being bounded by the curves OB, OC, BC. At
+  any point outside this area, monoclinic sulphur can exist only in a
+  metastable condition.</p>
+
+  <div class="figcenter" style="width:43%;">
+      <a href="images/fig006.png"><img style="width:100%" src="images/fig006.png"
+      alt="Fig. 6" title="Fig. 6" /></a>
+    <span class="sc">Fig. 6.</span>
+  </div>
+
+  <p>Other crystalline forms of sulphur have been obtained,<a name="NtA58"
+  href="#Nt58"><sup>[58]</sup></a> so that the existence of other systems
+  of the one-component sulphur besides those already described is possible.
+  Reference will be made to these later (p.&nbsp;<a href="#page51">51</a>).</p>
+
+<p><!-- Page 41 --><span class="pagenum"><a name="page41"></a>{41}</span></p>
+
+<p class="cenhead">C. <i>Tin.</i></p>
+
+  <p>Another substance capable of existing in more than one crystalline
+  form, is the metal tin, and although the general behaviour, so far as
+  studied, is analogous to that of sulphur, a short account of the two
+  varieties of tin may be given here, not only on account of their
+  metallurgical interest, but also on account of the importance which the
+  phenomena possess for the employment of this metal in everyday life.</p>
+
+  <p>After a winter of extreme severity in Russia (1867-1868), the somewhat
+  unpleasant discovery was made that a number of blocks of tin, which had
+  been stored in the Customs House at St. Petersburg, had undergone
+  disintegration and crumbled to a grey powder.<a name="NtA59"
+  href="#Nt59"><sup>[59]</sup></a> That tin undergoes change on exposure to
+  extreme cold was known, however, before that time, even as far back as
+  the time of Aristotle, who spoke of the tin as "melting."<a name="NtA60"
+  href="#Nt60"><sup>[60]</sup></a> Ludicrous as that term may now appear,
+  Aristotle nevertheless unconsciously employed a strikingly accurate
+  analogy, for the conditions under which ordinary white tin passes into
+  the grey modification are, in many ways, quite analogous to those under
+  which a substance passes from the solid to the liquid state. The
+  knowledge of this was, however, beyond the wisdom of the Greek
+  philosopher.</p>
+
+  <p>For many years there existed considerable confusion both as to the
+  conditions under which the transformation of white tin into its
+  allotropic modification occurs, and to the reason of the change. Under
+  the guidance of the Phase Rule, however, the confusion which obtained has
+  been cleared away, and the "mysterious" behaviour of tin brought into
+  accord with other phenomena of transformation.<a name="NtA61"
+  href="#Nt61"><sup>[61]</sup></a></p>
+
+  <p><b>Transition Point.</b>&mdash;Just as in the case of sulphur, so also
+  in the case of tin, there is a transition point above which the <!-- Page
+  42 --><span class="pagenum"><a name="page42"></a>{42}</span>one form,
+  ordinary white tin, and below which the other form, grey tin, is the
+  stable variety. In the case of this metal, the transition point was found
+  by Cohen and van Eyk, who employed both the dilatometric and the
+  electrical methods (Appendix) to be 20°. Below this temperature, grey tin
+  is the stable form. But, as we have seen in the case of sulphur, the
+  change of the metastable into the stable solid phase occurs with
+  considerable slowness, and this behaviour is found also in the case of
+  tin. Were it not so, we should not be able to use this metal for the many
+  purposes to which it is applied in everyday life; for, with the exception
+  of a comparatively small number of days in the year, the temperature of
+  our climate is below 20°, and <i>white tin is, therefore, at the ordinary
+  temperature, in a metastable condition</i>. The change, however, into the
+  stable form at the ordinary temperature, although slow, nevertheless
+  takes place, as is shown by the partial or entire conversion of articles
+  of tin which have lain buried for several hundreds of years.</p>
+
+  <p>On lowering the temperature, the velocity with which the
+  transformation of the tin occurs is increased, and Cohen and van Eyk
+  found that the temperature of maximum velocity is about -50°. Contact
+  with the stable form will, of course, facilitate the transformation.</p>
+
+  <p>The change of white tin into grey takes place also with increased
+  velocity in presence of a solution of tin ammonium chloride (pink salt),
+  which is able to dissolve small quantities of tin. In presence of such a
+  solution also, it was found that the temperature at which the velocity of
+  transformation was greatest was raised to 0°. At this temperature, white
+  tin in contact with a solution of tin ammonium chloride, and the grey
+  modification, undergoes transformation to an appreciable extent in the
+  course of a few days.</p>
+
+  <p>Fig. 7 is a photograph of a piece of white tin undergoing
+  transformation into the grey variety.<a name="NtA62"
+  href="#Nt62"><sup>[62]</sup></a> The bright surface of the tin becomes
+  covered with a number of warty masses, formed of the less dense grey
+  form, and the number and size of these continue to grow until the whole
+  of the white tin has passed <!-- Page 43 --><span class="pagenum"><a
+  name="page43"></a>{43}</span>into a grey powder. On account of the
+  appearance which is here seen, this transformation of tin has been called
+  by Cohen the "tin plague."</p>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/fig007.jpg"><img style="width:100%" src="images/fig007.jpg"
+      alt="Fig. 7" title="Fig. 7" /></a>
+    <span class="sc">Fig. 7.</span>
+  </div>
+
+<p><!-- Page 44 --><span class="pagenum"><a name="page44"></a>{44}</span></p>
+
+  <p><b>Enantiotropy and Monotropy.</b>&mdash;In the case of sulphur and
+  tin, we have met with two substances existing in polymorphic forms, and
+  we have also learned that these forms exhibit a definite transition point
+  at which their relative stability is reversed. Each form, therefore,
+  possesses a definite range of stable existence, and is capable of
+  undergoing transformation into the other, at temperatures above or below
+  that of the transition point.</p>
+
+  <p>Another class of dimorphous substances is, however, met with as, for
+  instance, in the case of the well-known compounds iodine monochloride and
+  benzophenone. Each crystalline form has its own melting point, the
+  dimorphous forms of iodine monochloride melting at 13.9° and 27.2°,<a
+  name="NtA63" href="#Nt63"><sup>[63]</sup></a> and those of benzophenone
+  at 26° and 48°.<a name="NtA64" href="#Nt64"><sup>[64]</sup></a> This
+  class of substance differs from that which we have already studied
+  (<i>e.g.</i> sulphur and tin), in that at all temperatures up to the
+  melting point, only one of the forms is stable, the other being
+  metastable. There is, therefore, no transition point, and transformation
+  of the crystalline forms can be observed <i>only in one direction</i>.
+  These two classes of phenomena are distinguished by the names
+  <i>enantiotropy</i> and <i>monotropy</i>; enantiotropic substances being
+  such that the change of one form into the other is a reversible process
+  (<i>e.g.</i> rhombic sulphur into monoclinic, and monoclinic sulphur into
+  rhombic), and monotropic substances, those in which the transformation of
+  the crystalline forms is irreversible.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/fig009.png"><img style="width:100%" src="images/fig009.png"
+      alt="Fig. 9" title="Fig. 9" /></a>
+    <span class="sc">Fig. 9.</span>
+  </div>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig008.png"><img style="width:100%" src="images/fig008.png"
+      alt="Fig. 8" title="Fig. 8" /></a>
+    <span class="sc">Fig. 8.</span>
+  </div>
+
+  <p>These differences in the behaviour can be explained very well in many
+  cases by supposing that in the case of enantiotropic substances the
+  transition point lies below the melting point, while in the case of
+  monotropic substances, it lies above the melting point.<a name="NtA65"
+  href="#Nt65"><sup>[65]</sup></a> These conditions would be represented by
+  the Figs. 8 and 9.</p>
+
+  <p>In these two figures, O<sub>3</sub> is the transition point,
+  O<sub>1</sub> and O<sub>2</sub> the melting points of the metastable and
+  stable forms <!-- Page 45 --><span class="pagenum"><a
+  name="page45"></a>{45}</span>respectively. From Fig. 9 we see that the
+  crystalline form I. at all temperatures up to its melting point is
+  metastable with respect to the form II. In such cases the transition
+  point could be reached only at higher pressures.</p>
+
+  <p>Although, as already stated, this explanation suffices for many cases,
+  it does not prove that in all cases of monotropy the transition point is
+  above the melting point of the two forms. It is also quite possible that
+  the transition point may lie below the melting points;<a name="NtA66"
+  href="#Nt66"><sup>[66]</sup></a> in this case we have what is known as
+  <i>pseudomonotropy</i>. It is possible that graphite and diamond,<a
+  name="NtA67" href="#Nt67"><sup>[67]</sup></a> perhaps also the two forms
+  of phosphorus, stand in the relation of pseudomonotropy (<i>v.</i> p.&nbsp;<a
+  href="#page49">49</a>).</p>
+
+  <p>The disposition of the curves in Figs. 8 and 9 also explains the
+  phenomenon sometimes met with, especially in organic chemistry, that the
+  substance first melts, then solidifies, and remelts at a higher
+  temperature. On again determining the melting point after
+  re-solidification, only the higher melting point is obtained.</p>
+
+  <p>The explanation of such a behaviour is, that if the determination of
+  the melting point is carried out rapidly, the point O<sub>1</sub>, the
+  melting point of the metastable solid form, may be realized. At this
+  temperature, however, the liquid is metastable with respect to the stable
+  solid form, and if the temperature is <!-- Page 46 --><span
+  class="pagenum"><a name="page46"></a>{46}</span>not allowed to rise above
+  the melting point of the latter, the liquid may solidify. The stable
+  solid modification thus obtained will melt only at a higher
+  temperature.</p>
+
+<p class="cenhead">D. <i>Phosphorus.</i></p>
+
+  <p>An interesting case of a monotropic dimorphous substance is found in
+  phosphorus, which occurs in two crystalline forms; white phosphorus
+  belonging to the regular system, and red phosphorus belonging to the
+  hexagonal system. From determinations of the vapour pressures of liquid
+  white phosphorus, and of solid red phosphorus,<a name="NtA68"
+  href="#Nt68"><sup>[68]</sup></a> it was found that the vapour pressure of
+  red phosphorus was considerably lower than that of liquid white
+  phosphorus at the same temperature, the values obtained being given in
+  the following table.</p>
+
+<p class="cenhead"><span class="sc">Vapour Pressures of White and Red Phosphorus.</span></p>
+
+<table class="nobctr" summary="Vapour Pressures of White and Red Phosphorus" title="Vapour Pressures of White and Red Phosphorus">
+<tr><td class="tobob" style="text-align:center" colspan="4"> Vapour pressure of liquid white phosphorus.
+</td><td class="toleb" style="text-align:center" colspan="2"> Vapour pressure of red<br />phosphorus.</td></tr>
+
+<tr><td class="tobob" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure<br />in cm.
+</td><td class="allb" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure<br />in atm.
+</td><td class="allb" style="text-align:center"> Temperature.
+</td><td class="tobob" style="text-align:center"> Pressure<br />in atm.</td></tr>
+
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 165° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 12 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 360° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; 3.2 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 360° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; 0.1</td></tr>
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 180° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 20.4 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 440° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; 7.5 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 440° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; 1.75</td></tr>
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 200° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 26.6 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 494° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 18.0 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 487° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; 6.8</td></tr>
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 219° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 35.9 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 503° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 21.9 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 510° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> 10.8</td></tr>
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 230° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 51.4 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 511° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 26.2 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 531° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> 16.0</td></tr>
+<tr><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 290° </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 76.0 </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="rib" style="text-align:left;padding-left:2em;padding-right:2em;"> 550° </td><td class="nob" style="text-align:left;padding-left:2em;padding-right:2em;"> 31.0</td></tr>
+<tr><td class="ribob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="ribob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="ribob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="ribob" style="text-align:left;padding-left:2em;padding-right:2em;"> &nbsp; &mdash; </td><td class="ribob" style="text-align:left;padding-left:2em;padding-right:2em;"> 577° </td><td class="bob" style="text-align:left;padding-left:2em;padding-right:2em;"> 56.0</td></tr>
+
+</table>
+
+  <p>These values are also represented graphically in Fig. 10.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/fig010.png"><img style="width:100%" src="images/fig010.png"
+      alt="Fig. 10" title="Fig. 10" /></a>
+    <span class="sc">Fig. 10.</span>
+  </div>
+
+  <p>At all temperatures above about 260°, transformation of the white into
+  the red modification takes place with appreciable velocity, and this
+  velocity increases as the temperature is raised. Even at lower
+  temperatures, <i>e.g.</i> at the ordinary temperature, the velocity of
+  transformation is increased under the influence <!-- Page 47 --><span
+  class="pagenum"><a name="page47"></a>{47}</span>of light,<a name="NtA69"
+  href="#Nt69"><sup>[69]</sup></a> or by the presence of certain
+  substances, <i>e.g.</i> iodine,<a name="NtA70"
+  href="#Nt70"><sup>[70]</sup></a> just as the velocity of transformation
+  of white tin into the grey modification was increased by the presence of
+  a solution of tin ammonium chloride (p.&nbsp;<a href="#page40">40</a>). At the
+  ordinary temperature, therefore, white phosphorus must be considered as
+  the less stable (metastable) form, for although it can exist in contact
+  with red phosphorus for a long period, its vapour pressure, as we have
+  seen, is greater than that of the red modification, and also, its
+  solubility in different solvents is greater<a name="NtA71"
+  href="#Nt71"><sup>[71]</sup></a> than that of the red modification; as we
+  shall find later, the solubility of the metastable form is always greater
+  than that of the stable.</p>
+
+  <p>The relationships which are met with in the case of phosphorus can be
+  best represented by the diagram, Fig. 11.<a name="NtA72"
+  href="#Nt72"><sup>[72]</sup></a></p>
+
+  <p>In this figure, BO<sub>1</sub> represents the conditions of
+  equilibrium of the univariant system red phosphorus and vapour, which
+  ends at O<sub>1</sub>, the melting point of red phosphorus. By heating in
+  capillary tubes of hard glass, Chapman<a name="NtA73"
+  href="#Nt73"><sup>[73]</sup></a> found that red phosphorus melts at the
+  melting point of potassium iodide, <i>i.e.</i> about 630°,<a name="NtA74"
+  href="#Nt74"><sup>[74]</sup></a> but the pressure at this temperature is
+  unknown.</p>
+
+  <p>At O<sub>1</sub>, then, we have the triple point, red phosphorus,
+  liquid, and vapour, and starting from it, we should have the <!-- Page 48
+  --><span class="pagenum"><a name="page48"></a>{48}</span>vaporization
+  curve of liquid phosphorus, O<sub>1</sub>A, and the fusion curve of red
+  phosphorus, O<sub>1</sub>F. Although these have not been determined, the
+  latter curve must, from theoretical considerations (<i>v.</i> p.&nbsp;<a
+  href="#page58">58</a>), slope slightly to the right; <i>i.e.</i> increase
+  of pressure raises the melting point of red phosphorus.</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/fig011.png"><img style="width:100%" src="images/fig011.png"
+      alt="Fig. 11" title="Fig. 11" /></a>
+    <span class="sc">Fig. 11.</span>
+  </div>
+
+  <p>When white phosphorus is heated to 44°, it melts. At this point,
+  therefore, we shall have another triple point, white
+  phosphorus&mdash;liquid&mdash;vapour; the pressure at this point has been
+  calculated to be 3 mm.<a name="NtA75" href="#Nt75"><sup>[75]</sup></a>
+  This point is the intersection of three curves, viz. sublimation curve,
+  vaporization curve, and the fusion curve of white phosphorus. The fusion
+  curve, O<sub>2</sub>E, has been determined by Tammann<a name="NtA76"
+  href="#Nt76"><sup>[76]</sup></a> and by G. A. Hulett,<a name="NtA77"
+  href="#Nt77"><sup>[77]</sup></a> and it was found that increase of
+  pressure by 1 atm. raises the melting point by 0.029°. The sublimation
+  curve of white phosphorus has not yet been determined.</p>
+
+  <p>As can be seen from the table of vapour pressures (p.&nbsp;<a
+  href="#page46">46</a>), the vapour pressure of white phosphorus has been
+  determined up to 500°; at temperatures above this, however, the velocity
+  with which transformation into red phosphorus takes place is so great as
+  to render the determination of the vapour pressure <!-- Page 49 --><span
+  class="pagenum"><a name="page49"></a>{49}</span>at higher temperatures
+  impossible. Since, however, the difference between white phosphorus and
+  red phosphorus disappears in the liquid state, the vapour pressure curve
+  of white phosphorus must pass through the point O<sub>1</sub>, the
+  melting point of red phosphorus, and must be continuous with the curve
+  O<sub>1</sub>A, the vapour pressure curve of liquid phosphorus (<i>vide
+  infra</i>). Since, as Fig. 10 shows, the vapour pressure curve of white
+  phosphorus ascends very rapidly at higher temperatures, the "break"
+  between BO<sub>1</sub> and O<sub>1</sub>A must be very slight.</p>
+
+  <p>As compared with monotropic substances like benzophenone, phosphorus
+  exhibits the peculiarity that transformation of the metastable into the
+  stable modification takes place with great slowness; and further, the
+  time required for the production of equilibrium between red phosphorus
+  and phosphorus vapour is great compared with that required for
+  establishing the same equilibrium in the case of white phosphorus. This
+  behaviour can be best explained by the assumption that change in the
+  molecular complexity (polymerization) occurs in the conversion of white
+  into red phosphorus, and when red phosphorus passes into vapour
+  (depolymerization).<a name="NtA78" href="#Nt78"><sup>[78]</sup></a></p>
+
+  <p>This is borne out by the fact that measurements of the vapour density
+  of phosphorus vapour at temperatures of 500° and more, show it to have
+  the molecular weight represented by P<sub>4</sub>,<a name="NtA79"
+  href="#Nt79"><sup>[79]</sup></a> and the same molecular weight has been
+  found for phosphorus in solution.<a name="NtA80"
+  href="#Nt80"><sup>[80]</sup></a> On the other hand, it has recently been
+  shown by R. Schenck,<a name="NtA81" href="#Nt81"><sup>[81]</sup></a> that
+  the molecular weight of red phosphorus is at least P<sub>8</sub>, and
+  very possibly higher.</p>
+
+  <p>In the case of phosphorus, therefore, it is more than possible that we
+  are dealing, not simply with two polymorphic <!-- Page 50 --><span
+  class="pagenum"><a name="page50"></a>{50}</span>forms of the same
+  substance, but with polymeric forms, and that there is no transition
+  point at temperatures above the absolute zero, unless we assume the
+  molecular complexity of the two forms to become the same. The curve for
+  red phosphorus would therefore lie below that of white phosphorus, for
+  the vapour pressure of the polymeric form, if produced from the simpler
+  form with evolution of heat, must be lower than that of the latter. A
+  transition point would, of course, become possible if the sign of the
+  heat effect in the transformation of the one modification into the other
+  should change. If, further, the liquid which is produced by the fusion of
+  red phosphorus at 630° under high pressure also exists in a polymeric
+  form, greater than P<sub>4</sub>, then the metastable vaporization curve
+  of white phosphorus would not pass through the melting point of red
+  phosphorus, as was assumed above.<a name="NtA82"
+  href="#Nt82"><sup>[82]</sup></a></p>
+
+  <p>We have already seen in the case of water (p.&nbsp;<a
+  href="#page31">31</a>) that the vapour pressure of supercooled water is
+  greater than that of ice, and that therefore it is possible,
+  theoretically at least, by a process of distillation, to transfer the
+  water from one end of a closed tube to the other, and to there condense
+  it as ice. On account of the very small difference between the vapour
+  pressure of supercooled water and ice, this distillation process has not
+  been experimentally realized. In the case of phosphorus, however, where
+  the difference in the vapour pressures is comparatively great, it has
+  been found possible to distil white phosphorus from one part of a closed
+  tube to another, and to there condense it as red phosphorus; and since
+  the vapour pressure of red phosphorus at 350° is less than the vapour
+  pressure of white phosphorus at 200°, it is possible to carry out the
+  distillation from a <i>colder</i> part of the tube to a <i>hotter</i>, by
+  having white phosphorus at the former and red phosphorus at the latter.
+  Such a process of distillation has been carried out by Troost and
+  Hautefeuille between 324° and 350°.<a name="NtA83"
+  href="#Nt83"><sup>[83]</sup></a></p>
+
+  <p>Relationships similar to those found in the case of phosphorus are
+  also met with in the case of cyanogen and <!-- Page 51 --><span
+  class="pagenum"><a name="page51"></a>{51}</span>paracyanogen, which have
+  been studied by Chappuis,<a name="NtA84" href="#Nt84"><sup>[84]</sup></a>
+  Troost and Hautefeuille,<a name="NtA85" href="#Nt85"><sup>[85]</sup></a>
+  and Dewar,<a name="NtA86" href="#Nt86"><sup>[86]</sup></a> and also in
+  the case of other organic substances.</p>
+
+  <p><b>Enantiotropy combined with Monotropy.</b>&mdash;Not only can
+  polymorphic substances exhibit enantiotropy or monotropy, but, if the
+  substance is capable of existing in more than two crystalline forms, both
+  relationships may be found, so that some of the forms may be
+  enantiotropic to one another, while the other forms exhibit only
+  monotropy. This behaviour is seen in the case of sulphur, which can exist
+  in as many as eight different crystalline varieties. Of these only
+  monoclinic and rhombic sulphur exhibit the relationship of enantiotropy,
+  <i>i.e.</i> they possess a definite transition point, while the other
+  forms are all metastable with respect to rhombic and monoclinic sulphur,
+  and remain so up to the melting point; that is to say, they are
+  monotropic modifications.<a name="NtA87"
+  href="#Nt87"><sup>[87]</sup></a></p>
+
+<p class="cenhead">E. <i>Liquid Crystals.</i></p>
+
+  <p><b>Phenomena observed.</b>&mdash;In 1888 it was discovered by
+  Reinitzer<a name="NtA88" href="#Nt88"><sup>[88]</sup></a> that the two
+  substances, cholesteryl acetate and cholesteryl benzoate, possess the
+  peculiar property of melting sharply at a definite temperature to milky
+  liquids; and that the latter, on being further heated, suddenly become
+  clear, also at a definite temperature. Other substances, more especially
+  <i>p</i>-azoxyanisole and <i>p</i>-azoxyphenetole, were, later, found to
+  possess the same property of having apparently a double melting point.<a
+  name="NtA89" href="#Nt89"><sup>[89]</sup></a> On cooling the clear
+  liquids, the reverse series of changes occurred.</p>
+
+  <p>The turbid liquids which were thus obtained were found to possess not
+  only the usual properties of liquids (such as the <!-- Page 52 --><span
+  class="pagenum"><a name="page52"></a>{52}</span>property of flowing and
+  of assuming a perfectly spherical shape when suspended in a liquid of the
+  same density), but also those properties which had hitherto been observed
+  only in the case of solid crystalline substances, viz. the property of
+  double refraction and of giving interference colours when examined by
+  polarized light; the turbid liquids are <i>anisotropic</i>. To such
+  liquids, the optical properties of which were discovered by O. Lehmann,<a
+  name="NtA90" href="#Nt90"><sup>[90]</sup></a> the name <i>liquid
+  crystals</i>, or crystalline liquids, was given.</p>
+
+  <p><b>Nature of Liquid Crystals.</b>&mdash;During the past ten years the
+  question as to the nature of liquid crystals has been discussed by a
+  number of investigators, several of whom have contended strongly against
+  the idea of the term "liquid" being applied to the crystalline condition;
+  and various attempts have been made to prove that the turbid liquids are
+  in reality heterogeneous and are to be classed along with emulsions.<a
+  name="NtA91" href="#Nt91"><sup>[91]</sup></a> This view was no doubt
+  largely suggested by the fact that the anisotropic liquids were turbid,
+  whereas the "solid" crystals were clear. Lehmann found, however, that,
+  when examined under the microscope, the "simple" liquid crystals were
+  also clear,<a name="NtA92" href="#Nt92"><sup>[92]</sup></a> the apparent
+  turbidity being due to the aggregation of a number of differently
+  oriented crystals, in the same way as a piece of marble does not appear
+  transparent although composed of transparent crystals.<a name="NtA93"
+  href="#Nt93"><sup>[93]</sup></a></p>
+
+  <p>Further, no proof of the heterogeneity of liquid crystals has yet been
+  obtained, but rather all chemical and physical investigations indicate
+  that they are homogeneous.<a name="NtA94"
+  href="#Nt94"><sup>[94]</sup></a> No separation <!-- Page 53 --><span
+  class="pagenum"><a name="page53"></a>{53}</span>of a solid substance from
+  the milky, anisotropic liquids has been effected; the anisotropic liquid
+  is in some cases less viscous than the isotropic liquid formed at a
+  higher temperature; and the temperature of liquefaction is constant, and
+  is affected by pressure and admixture with foreign substances exactly as
+  in the case of a pure substance.<a name="NtA95"
+  href="#Nt95"><sup>[95]</sup></a></p>
+
+  <div class="figcenter" style="width:29%;">
+      <a href="images/fig012.png"><img style="width:100%" src="images/fig012.png"
+      alt="Fig. 12" title="Fig. 12" /></a>
+    <span class="sc">Fig. 12.</span>
+  </div>
+
+  <p><b>Equilibrium Relations in the Case of Liquid
+  Crystals.</b>&mdash;Since, now, we have seen that we are dealing here
+  with substances in two crystalline forms (which we may call the solid and
+  liquid<a name="NtA96" href="#Nt96"><sup>[96]</sup></a> crystalline form),
+  which possess a definite transition point, at which, transformation of
+  the one form into the other occurs in both directions, we can represent
+  the conditions of equilibrium by a diagram in all respects similar to
+  that employed in the case of other enantiotropic substances, <i>e.g.</i>
+  sulphur (p.&nbsp;<a href="#page35">35</a>).</p>
+
+<p><!-- Page 54 --><span class="pagenum"><a name="page54"></a>{54}</span></p>
+
+  <p>In Fig. 12 there is given a diagrammatic representation of the
+  relationships found in the case of <i>p</i>-azoxyanisole.<a name="NtA97"
+  href="#Nt97"><sup>[97]</sup></a></p>
+
+  <p>Although the vapour pressure of the substance in the solid, or liquid
+  state, has not been determined, it will be understood from what we have
+  already learned, that the curves AO, OB, BC, representing the vapour
+  pressure of solid crystals, liquid crystals, isotropic liquid, must have
+  the relative positions shown in the diagram. Point O, the transition
+  point of the solid into the liquid crystals, lies at 118.27°, and the
+  change of the transition point with the pressure is +0.032° pro 1 atm.
+  The transition curve OE slopes, therefore, slightly to the right. The
+  point B, the melting point of the liquid crystals, lies at 135.85°, and
+  the melting point is raised 0.0485° pro 1 atm. The curve BD, therefore,
+  also slopes to the right, and more so than the transition curve. In this
+  respect azoxyanisole is different from sulphur.</p>
+
+  <p>The areas bounded by the curves represent the conditions for the
+  stable existence of the four single phases, solid crystals, liquid
+  crystals, isotropic liquid and vapour.</p>
+
+  <p>The most important substances hitherto found to form liquid crystals
+  are<a name="NtA98" href="#Nt98"><sup>[98]</sup></a>:&mdash;</p>
+
+<table class="nobctr" summary="Substances forming liquid crystals" title="Substances forming liquid crystals">
+<tr><td class="tobob" style="text-align:center"> Substance.
+</td><td class="allb" style="text-align:center"> Transition<br />point.
+</td><td class="tobob" style="text-align:center"> Melting<br />point.</td></tr>
+
+<tr><td class="nob" style="text-align:left"> Cholesteryl benzoate </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 145.5° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 178.5°</td></tr>
+<tr><td class="nob" style="text-align:left"> Azoxyanisole </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 118.3° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 135.9°</td></tr>
+<tr><td class="nob" style="text-align:left"> Azoxyphenetole </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 134.5° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 168.1°</td></tr>
+<tr><td class="nob" style="text-align:left"> Condensation product from benzaldehyde<br />&nbsp; &nbsp; and benzidine </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 234° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 260°</td></tr>
+<tr><td class="nob" style="text-align:left"> Azine of <i>p</i>-oxyethylbenzaldehyde </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 172° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 196°</td></tr>
+<tr><td class="nob" style="text-align:left"> Condensation product from <i>p</i>-tolylaldehyde<br />&nbsp; &nbsp; and benzidine </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> 231° </td><td class="leb" style="text-align:left; vertical-align:bottom; padding-left:2em; padding-right:2em;"> &nbsp; &mdash;</td></tr>
+
+<tr><td class="bob" style="text-align:left"> <i>p</i>-Methoxycinnamic acid </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 169° </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 185°</td></tr>
+</table>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 55 --><span class="pagenum"><a name="page55"></a>{55}</span></p>
+
+<h3>CHAPTER IV</h3>
+
+<p class="cenhead">GENERAL SUMMARY</p>
+
+  <p>In the preceding pages we have learned how the principles of the Phase
+  Rule can be applied to the elucidation of various systems consisting of
+  one component. In the present chapter it is proposed to give a short
+  summary of the relationships we have met with, and also to discuss more
+  generally how the Phase Rule applies to other one-component systems. On
+  account of the fact that beginners are sometimes inclined to expect too
+  much of the Phase Rule; to expect, for example, that it will inform them
+  as to the exact behaviour of a substance, it may here be emphasized that
+  the Phase Rule is a general rule; it informs us only as to the general
+  conditions of equilibrium, and leaves the determination of the definite,
+  numerical data to experiment.</p>
+
+  <p><b>Triple Point.</b>&mdash;We have already (p.&nbsp;<a
+  href="#page28">28</a>) defined a triple point in a one-component system,
+  as being that pressure and temperature at which three phases coexist in
+  equilibrium; it represents, therefore, an invariant system (p.&nbsp;<a
+  href="#page16">16</a>). At the triple point also, three curves cut, viz.
+  the curves representing the conditions of equilibrium of the three
+  univariant systems formed by the combination of the three phases in
+  pairs. The most common triple point of a one-component system is, of
+  course, the triple point, solid, liquid, vapour (S-L-V), but other triple
+  points<a name="NtA99" href="#Nt99"><sup>[99]</sup></a> are also possible
+  when, as in the case of <!-- Page 56 --><span class="pagenum"><a
+  name="page56"></a>{56}</span>sulphur or benzophenone, polymorphic forms
+  occur. Whether or not all the triple points can be experimentally
+  realized will, of course, depend on circumstances. We shall, in the first
+  place, consider only the triple point S-L-V.</p>
+
+  <p>As to the general arrangement of the three univariant curves around
+  the triple point, the following rules may be given. (1) The prolongation
+  of each of the curves beyond the triple point must lie between the other
+  two curves. (2) The middle position at one and the same temperature in
+  the neighbourhood of the triple point is taken by that curve (or its
+  metastable prolongation) which represents the two phases of most widely
+  differing specific volume.<a name="NtA100"
+  href="#Nt100"><sup>[100]</sup></a> That is to say, if a line of constant
+  temperature is drawn immediately above or below the triple point so as to
+  cut the three curves&mdash;two stable curves and the metastable
+  prolongation of the third&mdash;the position of the curves at that
+  temperature will be such that the middle position is occupied by that
+  curve (or its metastable prolongation) which represents the two phases of
+  most widely differing specific volume.</p>
+
+  <p>Now, although these rules admit of a considerable variety of possible
+  arrangements of curves around the triple point,<a name="NtA101"
+  href="#Nt101"><sup>[101]</sup></a> only two of these have been
+  experimentally obtained in the case of the triple point
+  solid&mdash;liquid&mdash;vapour. At present, therefore, we shall consider
+  only these two cases (Figs. 13 and 14).</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig014.png"><img style="width:100%" src="images/fig014.png"
+      alt="Fig. 14" title="Fig. 14" /></a>
+    <span class="sc">Fig. 14.</span>
+  </div>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/fig013.png"><img style="width:100%" src="images/fig013.png"
+      alt="Fig. 13" title="Fig. 13" /></a>
+    <span class="sc">Fig. 13.</span>
+  </div>
+
+  <p>An examination of these two figures shows that they satisfy the rules
+  laid down. Each of the curves on being prolonged passes between the other
+  two curves. In the case of substances of the first type (Fig. 13), the
+  specific volume of the solid is greater than that of the liquid (the
+  substance contracts on fusion); the difference of specific volume will,
+  therefore, be greatest between liquid and vapour. The curve, therefore,
+  for liquid and vapour (or its prolongation) must lie between the other
+  two curves; this is seen from the figure to be the case. Similarly, the
+  rule is satisfied by the arrangement of curves in Fig. 14, where the
+  difference of specific volumes is <!-- Page 57 --><span
+  class="pagenum"><a name="page57"></a>{57}</span>greatest between the
+  solid and vapour. In this case the curve S-V occupies the intermediate
+  position.</p>
+
+  <p>As we see, the two figures differ from one another only in that the
+  fusion curve OC in one case slopes to the right away from the pressure
+  axis, thus indicating that the melting point is raised by increase of
+  pressure; in the other case, to the left, indicating a lowering of the
+  melting point with the pressure. These conditions are found exemplified
+  in the case of sulphur and ice (<span class="correction" title="Original reads 'pp. 25 and 38'."
+  >pp. <a href="#page29">29</a> and <a href="#page35">35</a></span>). We
+  see further from the two figures, that O in Fig. 13 gives the highest
+  temperature at which the solid can exist, for the curve for
+  solid&mdash;liquid slopes back to regions of lower temperature; in Fig.
+  14, O gives the lowest temperature at which the liquid phase can exist as
+  stable phase.<a name="NtA102" href="#Nt102"><sup>[102]</sup></a></p>
+
+  <p><b>Theorems of van't Hoff and of Le Chatelier.</b>&mdash;So far we
+  have studied only the conditions under which various systems exist in
+  equilibrium; and we now pass to a consideration of the changes which take
+  place in a system when the external conditions of temperature and
+  pressure are altered. For all such changes there exist two theorems,
+  based on the laws of thermodynamics, by means of which the alterations in
+  a system can be qualitatively predicted.<a name="NtA103"
+  href="#Nt103"><sup>[103]</sup></a> The first of these, usually <!-- Page
+  58 --><span class="pagenum"><a name="page58"></a>{58}</span>known as
+  van't Hoff's <i>law of movable equilibrium</i>,<a name="NtA104"
+  href="#Nt104"><sup>[104]</sup></a> states: When the temperature of a
+  system in equilibrium is raised, that reaction takes place which is
+  accompanied by absorption of heat; and, conversely, when the temperature
+  is lowered, that reaction occurs which is accompanied by an evolution of
+  heat.</p>
+
+  <p>The second of the two theorems refers to the effect of change of
+  pressure, and states:<a name="NtA105" href="#Nt105"><sup>[105]</sup></a>
+  When the pressure on a system in equilibrium is increased, that reaction
+  takes place which is accompanied by a diminution of volume; and when the
+  pressure is diminished, a reaction ensues which is accompanied by an
+  increase of volume.</p>
+
+  <p>The demonstration of the universal applicability of these two theorems
+  is due chiefly to Le Chatelier, who showed that they may be regarded as
+  consequences of the general law of action and reaction. For this reason
+  they are generally regarded as special cases of the more general law,
+  known as the <i>theorem of Le Chatelier</i>, which may be stated in the
+  words of Ostwald, as follows:<a name="NtA106"
+  href="#Nt106"><sup>[106]</sup></a> <i>If a system in equilibrium is
+  subjected to a constraint by which the equilibrium is shifted, a reaction
+  takes place which opposes the constraint, </i>i.e.<i> one by which its
+  effect is partially destroyed.</i></p>
+
+  <p>This theorem of Le Chatelier is of very great importance, for it
+  applies to all systems and changes of the condition of equilibrium,
+  whether physical or chemical; to vaporization and fusion; to solution and
+  chemical action. In all cases, whenever changes in the external condition
+  of a system in equilibrium are produced, processes also occur within the
+  system which tend to counteract the effect of the external changes.</p>
+
+  <p><i>Changes at the Triple Point.</i>&mdash;If now we apply this theorem
+  to equilibria at the triple point S-L-V, and ask what changes will occur
+  in such a system when the external conditions of pressure and temperature
+  are altered, the general answer to the question will be: So long as the
+  three phases are present, no <!-- Page 59 --><span class="pagenum"><a
+  name="page59"></a>{59}</span>change in the temperature or pressure of the
+  system can occur, but <i>only changes in the relative amounts of the
+  phases</i>; that is to say, the effect on the system of change in the
+  external conditions is opposed by the reactions or changes which take
+  place within the system (according to the theorems of van't Hoff and Le
+  Chatelier). We now proceed to discuss what these changes are, and shall
+  consider first the effect of alteration of the temperature at constant
+  volume and constant pressure, and then the effect of alteration of the
+  pressure both when the temperature remains constant and when it
+  varies.</p>
+
+  <p>When the volume is kept constant, the effect of the addition of heat
+  to a system at the triple point S-L-V differs somewhat according as there
+  is an increase or diminution of volume when the solid passes into the
+  liquid state. In the former and most general case (Fig. 14), addition of
+  heat will cause a certain amount of the solid phase to melt, whereby the
+  heat which is added becomes latent; the temperature of the system
+  therefore does not rise. Since, however, the melting of the solid is
+  accompanied by an increase of volume, whereby an increase of pressure
+  would result, a certain portion of the vapour must condense to liquid, in
+  order that the pressure may remain constant. The total effect of addition
+  of heat, therefore, is to cause both solid and vapour to pass into
+  liquid, <i>i.e.</i> there occurs the change S + V <a
+  href="images/singlearrow.png"><img src="images/singlearrow.png"
+  class="middle" style="height:2ex" alt="arrow" /></a> L. It will,
+  therefore, depend on the relative quantities of solid and vapour, which
+  will disappear first. If the solid disappears first, then we shall pass
+  to the system L-V; if vapour disappears first, we shall obtain the system
+  S-L. Withdrawal of heat causes the reverse change, L <a
+  href="images/singlearrow.png"><img src="images/singlearrow.png"
+  class="middle" style="height:2ex" alt="arrow" /></a> S + V; at all
+  temperatures below the triple point the liquid is unstable or metastable
+  (p.&nbsp;<a href="#page30">30</a>).</p>
+
+  <p>When fusion is accompanied by a diminution of volume (<i>e.g.</i> ice,
+  Fig. 13), then, since the melting of the solid phase would decrease the
+  total volume, <i>i.e.</i> would lower the pressure, a certain quantity of
+  the solid must also pass into vapour in order that the pressure may be
+  maintained constant. On addition of heat, therefore, there occurs the
+  reaction S <a href="images/singlearrow.png"><img
+  src="images/singlearrow.png" class="middle" style="height:2ex"
+  alt="arrow" /></a> L + V; withdrawal of heat causes the reverse change L
+  + V <a href="images/singlearrow.png"><img src="images/singlearrow.png"
+  class="middle" style="height:2ex" alt="arrow" /></a> S. Above the
+  temperature of the triple point the <!-- Page 60 --><span
+  class="pagenum"><a name="page60"></a>{60}</span>solid cannot exist; below
+  the triple point both systems, S-L and S-V, can exist, and it will
+  therefore depend on the relative amounts of liquid and vapour which of
+  these two systems is obtained on withdrawing heat from the system at
+  constant volume.</p>
+
+  <p>The same changes in the phases occur when heat is added or withdrawn
+  at constant pressure, so long as the three phases are present. Continued
+  addition of heat, however, at constant pressure will ultimately cause the
+  formation of the bivariant system vapour alone; continued withdrawal of
+  heat will ultimately cause the formation of solid alone. This will be
+  readily understood from Fig. 15. The dotted line D&prime;OD is a line of
+  constant pressure; on adding heat, the system passes along the line OD
+  into the region of vapour; on heat being withdrawn, the system passes
+  along OD&prime; into the area of solid.</p>
+
+  <div class="figcenter" style="width:18%;">
+      <a href="images/fig015.png"><img style="width:100%" src="images/fig015.png"
+      alt="Fig. 15" title="Fig. 15" /></a>
+    <span class="sc">Fig. 15.</span>
+  </div>
+
+  <p>Similar changes are produced when the volume of the system is altered.
+  Alteration of volume may take place either while transference of heat to
+  or from the system is cut off (adiabatic change), or while such
+  transference may occur (isothermal change). In the latter case, the
+  temperature of the system will remain constant; in the former case, since
+  at the triple point the pressure must be constant so long as the three
+  phases are present, increase of volume must be compensated by the
+  evaporation of liquid. This, however, would cause the temperature to fall
+  (since communication of heat from the outside is supposed to be cut off),
+  and a portion of the liquid must therefore freeze. In this way the latent
+  heat of evaporation is counterbalanced by the latent heat of fusion. As
+  the result of increase of volume, therefore, the process occurs L <a
+  href="images/singlearrow.png"><img src="images/singlearrow.png"
+  class="middle" style="height:2ex" alt="arrow" /></a> S + V. Diminution of
+  volume, without transference of heat, will bring about the opposite
+  change, S + V <a href="images/singlearrow.png"><img
+  src="images/singlearrow.png" class="middle" style="height:2ex"
+  alt="arrow" /></a> L. In the former case there is ultimately obtained the
+  univariant system S-V; in the latter case there will be <!-- Page 61
+  --><span class="pagenum"><a name="page61"></a>{61}</span>obtained either
+  S-L or L-V according as the vapour or solid phase disappears first.</p>
+
+  <p>This argument holds good for both types of triple point shown in Figs.
+  13 and 14 (p.&nbsp;<a href="#page57">57</a>). A glance at these figures will
+  show that increase of volume (diminution of pressure) will lead
+  ultimately to the system S-V, for at pressures lower than that of the
+  triple point, the liquid phase cannot exist. Decrease of volume (increase
+  of pressure), on the other hand, will lead either to the system S-L or
+  L-V, because these systems can exist at pressures higher than that of the
+  triple point. If the vapour phase disappears and we pass to the curve
+  S-L, continued diminution of volume will be accompanied by a fall in
+  temperature in the case of systems of the first type (Fig. 13), and by a
+  rise in temperature in the case of systems of the second type (Fig.
+  14).</p>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig017.png"><img style="width:100%" src="images/fig017.png"
+      alt="Fig. 17" title="Fig. 17" /></a>
+    <span class="sc">Fig. 17.</span>
+  </div>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig016.png"><img style="width:100%" src="images/fig016.png"
+      alt="Fig. 16" title="Fig. 16" /></a>
+    <span class="sc">Fig. 16.</span>
+  </div>
+
+  <p>Lastly, if the temperature is maintained constant, <i>i.e.</i> if heat
+  can pass into or out of the system, then on changing the volume the same
+  changes in the phases will take place as described above until one of the
+  phases has disappeared. Continued increase of volume (decrease of
+  pressure) will then cause the disappearance of a second phase, the system
+  passing along the dotted line OE&prime; (Figs. 16, 17), so that
+  ultimately there remains only the vapour phase. Conversely, diminution of
+  volume (increase of pressure) will ultimately lead either to solid (Fig.
+  16) or to liquid alone (Fig. 17), the system passing along the dotted
+  line OE. <!-- Page 62 --><span class="pagenum"><a
+  name="page62"></a>{62}</span></p>
+
+  <p>In discussing the alterations which may take place at the triple point
+  with change of temperature and pressure, we have considered only the
+  triple point S-L-V. The same reasoning, however, applies, <i>mutatis
+  mutandis</i>, to all other triple points, so that if the specific volumes
+  of the phases are known, and the sign of the heat effects which accompany
+  the transformation of one phase into the other, it is possible to predict
+  (by means of the theorem of Le Chatelier) the changes which will be
+  produced in the system by alteration of the pressure and temperature.</p>
+
+  <p>In all cases of transformation at the triple point, it should be noted
+  that all <i>three phases are involved in the change</i>,<a name="NtA107"
+  href="#Nt107"><sup>[107]</sup></a> and not two only; the fact that in the
+  case, say, of the transformation from solid to liquid, or liquid to
+  solid, at the melting point with change of temperature, only these two
+  phases appear to be affected, is due to there generally being a large
+  excess of the vapour phase present and to the prior disappearance
+  therefore of the solid or liquid phase.</p>
+
+  <p>In the case of triple points at which two solid phases are in
+  equilibrium with liquid, other arrangements of the curves around the
+  triple point are found. It is, however, unnecessary to give a general
+  treatment of these here, since the principles which have been applied to
+  the triple point S-L-V can also be applied to the other triple points.<a
+  name="NtA108" href="#Nt108"><sup>[108]</sup></a></p>
+
+  <p><b>Triple Point Solid&mdash;Solid&mdash;Vapour.</b>&mdash;The triple
+  point solid&mdash;solid&mdash;vapour is one which is of considerable
+  importance. Examples of such a triple point have already been given in
+  sulphur and tin, and a list of other substances capable of yielding two
+  solid phases is given below. The triple point S-S-V is not precisely the
+  same as the transition point, but is very nearly so. The transition point
+  is the temperature at which the relative stability of the two solid
+  phases undergoes change, when the vapour phase is absent and the pressure
+  is 1 atm.; whereas at the triple point the pressure is that of the system
+  itself. The transition point, therefore, bears the same relation to the
+  triple point S-S-V as the melting point to the triple point S-L-V.</p>
+
+<p><!-- Page 63 --><span class="pagenum"><a name="page63"></a>{63}</span></p>
+
+  <p>In the following table is given a list of the most important
+  polymorphous substances, and the temperatures of the transition point.<a
+  name="NtA109" href="#Nt109"><sup>[109]</sup></a></p>
+
+<table class="nobctr" summary="Polymorphous substances" title="Polymorphous substances">
+
+<tr><td class="tobob" style="text-align:center"> Substance.
+</td><td class="toboleb" style="text-align:center"> Transition<br />temperature.</td></tr>
+
+<tr><td class="nob" style="text-align:left"> Ammonium nitrate&mdash; </td><td class="leb" style="text-align:left; padding-left:2em"></td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; <span class="grk">&beta;</span>-rhombic <a href="images/singlearrow.png"><img src="images/singlearrow.png" class="middle" style="height:2ex" alt="arrow" /></a> <span class="grk">&alpha;</span>-rhombic </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 35°</td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; <span class="grk">&alpha;</span>-rhombic <a href="images/singlearrow.png"><img src="images/singlearrow.png" class="middle" style="height:2ex" alt="arrow" /></a> rhombohedral </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 83°</td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; Rhombohedral <a href="images/singlearrow.png"><img src="images/singlearrow.png" class="middle" style="height:2ex" alt="arrow" /></a> regular </td><td class="leb" style="text-align:left; padding-left:2em"> 125°</td></tr>
+<tr><td class="nob" style="text-align:left"> Mercuric iodide </td><td class="leb" style="text-align:left; padding-left:2em"> 126°</td></tr>
+<tr><td class="nob" style="text-align:left"> Potassium nitrate </td><td class="leb" style="text-align:left; padding-left:2em"> 129°</td></tr>
+<tr><td class="nob" style="text-align:left"> Silver iodide </td><td class="leb" style="text-align:left; padding-left:2em"> 145°</td></tr>
+<tr><td class="nob" style="text-align:left"> Silver nitrate </td><td class="leb" style="text-align:left; padding-left:2em"> 160°</td></tr>
+<tr><td class="nob" style="text-align:left"> Sulphur </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 95.5°</td></tr>
+<tr><td class="nob" style="text-align:left"> Tetrabrommethane </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 46.8°</td></tr>
+<tr><td class="nob" style="text-align:left"> Thallium nitrate&mdash; </td><td class="leb" style="text-align:left; padding-left:2em"></td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; Rhombic <a href="images/singlearrow.png"><img src="images/singlearrow.png" class="middle" style="height:2ex" alt="arrow" /></a> rhombohedral </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 80°</td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; Rhombohedral <a href="images/singlearrow.png"><img src="images/singlearrow.png" class="middle" style="height:2ex" alt="arrow" /></a> regular </td><td class="leb" style="text-align:left; padding-left:2em"> 142.5°</td></tr>
+<tr><td class="nob" style="text-align:left"> Thallium picrate </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; 46°</td></tr>
+<tr><td class="ribob" style="text-align:left"> Tin </td><td class="bob" style="text-align:left; padding-left:2em"> &nbsp; 20°</td></tr>
+
+</table>
+
+  <p><b>Sublimation and Vaporization Curves.</b>&mdash;We have already
+  seen, in the case of ice and liquid water, that the vapour pressure
+  increases as the temperature rises, the increase of pressure per degree
+  being greater the higher the temperature. The sublimation and
+  vaporization curves, therefore, are not straight lines, but are bent, the
+  convex side of the curve being towards the temperature axis in the
+  ordinary <i>pt</i>-diagram.</p>
+
+  <p>In the case of sulphur and of tin, we assumed vapour to be given off
+  by the solid substance, although the pressure of the vapour has not
+  hitherto been measured. The assumption, however, is entirely justified,
+  not only on theoretical grounds, but also because the existence of a
+  vapour pressure has been observed in the case of many solid substances at
+  temperatures much below the melting point,<a name="NtA110"
+  href="#Nt110"><sup>[110]</sup></a> and in some cases, <i>e.g.</i>
+  camphor,<a name="NtA111" href="#Nt111"><sup>[111]</sup></a> the vapour
+  pressure is considerable.</p>
+
+<p><!-- Page 64 --><span class="pagenum"><a name="page64"></a>{64}</span></p>
+
+  <p>As the result of a large number of determinations, it has been found
+  that all vapour pressure curves have the same general form alluded to
+  above. Attempts have also been made to obtain a general expression for
+  the quantitative changes in the vapour pressure with change of
+  temperature, but without success. Nevertheless, the <i>qualitative</i>
+  changes, or the general direction of the curves, can be predicted by
+  means of the theorem of Le Chatelier.</p>
+
+  <p>As we have already learned (p.&nbsp;<a href="#page16">16</a>), the Phase
+  Rule takes no account of the molecular complexity of the substances
+  participating in an equilibrium. A dissociating substance, therefore, in
+  contact with its vaporous products of dissociation (<i>e.g.</i> ammonium
+  chloride in contact with ammonia and hydrogen chloride), will likewise
+  constitute a univariant system of one component, provided the composition
+  of the vapour phase as a whole is the same as that of the solid or liquid
+  phase (p.&nbsp;<a href="#page13">13</a>). For all such substances, therefore,
+  the conditions of equilibrium will be represented by a curve of the same
+  general form as the vapour pressure curve of a non-dissociating
+  substance.<a name="NtA112" href="#Nt112"><sup>[112]</sup></a> The same
+  behaviour is also found in the case of substances which polymerize on
+  passing into the solid or liquid state (<i>e.g.</i> red phosphorus).
+  Where such changes in the molecular state occur, however, the time
+  required for equilibrium to be established is, as a rule, greater than
+  when the molecular state is the same in both phases.</p>
+
+  <p>From an examination of Figs. 13 and 14, it will be easy to predict the
+  effect of change of pressure and temperature on the univariant systems
+  S-V or L-V. If the volume is kept constant, addition of heat will cause
+  an increase of pressure, the system S-V moving along the curve AO until
+  at the triple point the liquid phase is formed, and the system L-V moving
+  along the curve OB; so long as two phases are present, the condition of
+  the system must be represented by these two curves. Conversely,
+  withdrawal of heat will cause condensation of vapour, and therefore
+  diminution of pressure; the system will therefore move along the
+  vaporization or sublimation curve to lower temperatures and pressures, so
+  long as the system remains univariant.</p>
+
+<p><!-- Page 65 --><span class="pagenum"><a name="page65"></a>{65}</span></p>
+
+  <p>If transference of heat to or from the system is prevented, increase
+  of volume (diminution of pressure) will cause the system L-V to pass
+  along the curve BO; liquid will pass into vapour and the temperature will
+  fall.<a name="NtA113" href="#Nt113"><sup>[113]</sup></a> At O solid may
+  appear, and the temperature of the system will then remain constant until
+  the liquid phase has disappeared (<span class="correction"
+  title="Original reads 'p. 60'.">p.&nbsp;<a href="#page57">57</a></span>); the
+  system will then follow the curve OA until the solid phase disappears,
+  and we are ultimately left with vapour. On the other hand, diminution of
+  volume (increase of pressure) will cause condensation of vapour, and the
+  system S-V will pass along the curve AO to higher temperatures and
+  pressures; at O the solid will melt, and the system will ultimately pass
+  to the curve OB or to OC (<span class="correction" title="Original reads 'p. 60'."
+  >p.&nbsp;<a href="#page57">57</a></span>).</p>
+
+  <p>Addition or withdrawal of heat at constant pressure, and increase or
+  diminution of the pressure at constant temperature, will cause the system
+  to pass along lines parallel to the temperature and the pressure axis
+  respectively; the working out of these changes may be left to the reader,
+  guided by what has been said on pp. 60 and 61.</p>
+
+  <p>The sublimation curve of all substances, so far as yet found, has its
+  upper limit at the melting point (triple point), although the possibility
+  of the existence of a superheated solid is not excluded. The lower limit
+  is, theoretically at least, at the absolute zero, provided no new phase,
+  <i>e.g.</i> a different crystalline modification, is formed. If the
+  sublimation pressure of a substance is greater than the atmospheric
+  pressure at any temperature below the point of fusion, then the substance
+  will <i>sublime without melting</i> when heated in an open vessel; and
+  fusion will be possible only at a pressure higher than the atmospheric.
+  This is found, for example, in the case of red phosphorus (p.&nbsp;<a
+  href="#page47">47</a>). If, however, the sublimation pressure of a
+  substance at its triple point S-L-V is less than one atmosphere, then the
+  substance will melt when heated in an open vessel.</p>
+
+  <p>In the case of the vaporization curve, the upper limit lies at the
+  critical point where the liquid ceases to exist;<a name="NtA114"
+  href="#Nt114"><sup>[114]</sup></a> the <!-- Page 66 --><span
+  class="pagenum"><a name="page66"></a>{66}</span>lower limit is determined
+  by the range of the metastable state of the supercooled liquid.</p>
+
+  <p>The interpolation and extrapolation of vapour-pressure curves is
+  rendered very easy by means of a relationship which Ramsay and Young<a
+  name="NtA115" href="#Nt115"><sup>[115]</sup></a> found to exist between
+  the vapour-pressure curves of different substances. It was observed that
+  in the case of closely related substances, the ratio of the absolute
+  temperatures corresponding to equal vapour pressures is constant,
+  <i>i.e.</i> T<sub>1</sub>/T&prime;<sub>1</sub> =
+  T<sub>2</sub>/T&prime;<sub>2</sub>. When the two substances are not
+  closely related, it was found that the relationship could be expressed by
+  the equation T<sub>1</sub>/T&prime;<sub>1</sub> =
+  T<sub>2</sub>/T&prime;<sub>2</sub> + <i>c</i>(<i>t</i>&prime; - <i>t</i>)
+  where <i>c</i> is a constant having a small positive or negative value,
+  and <i>t</i>&prime; and <i>t</i> are the temperatures at which one of the
+  substances has the two values of the vapour pressure in question. By
+  means of this equation, if the vapour-pressure curve of one substance is
+  known, the vapour-pressure curve of any other substance can be calculated
+  from the values at any two temperatures of the vapour pressure of that
+  substance.</p>
+
+  <p><b>Fusion Curve&mdash;Transition Curve.</b>&mdash;The fusion curve
+  represents the conditions of equilibrium between the solid and liquid
+  phase; it shows the change of the melting point of a substance with
+  change of pressure.</p>
+
+  <p>As shown in Figs. 13 and 14, the fusion curve is inclined either
+  towards the pressure axis or away from it; that is, increase of pressure
+  can either lower or raise the melting point. It is easy to predict in a
+  qualitative manner the different effect of pressure on the melting point
+  in the two cases mentioned, if we consider the matter in the light of the
+  theorem of Le Chatelier (p.&nbsp;<a href="#page58">58</a>). Water, on passing
+  into ice, expands; therefore, if the pressure on the system
+  ice&mdash;water be increased, a reaction will take place which is
+  accompanied by a diminution in volume, <i>i.e.</i> the ice will melt.
+  Consequently, a lower temperature will be required in order to counteract
+  the effect of increase of pressure; or, in other words, the melting point
+  will <!-- Page 67 --><span class="pagenum"><a
+  name="page67"></a>{67}</span>be lowered by pressure.<a name="NtA116"
+  href="#Nt116"><sup>[116]</sup></a> In the second case, the passage of the
+  liquid to the solid state is accompanied by a diminution of volume; the
+  effect of increase of pressure will therefore be the reverse of that in
+  the previous case.</p>
+
+  <p>If the value of the heat of fusion and the alteration of volume
+  accompanying the change of state are known, it is possible to calculate
+  <i>quantitatively</i> the effect of pressure.<a name="NtA117"
+  href="#Nt117"><sup>[117]</sup></a></p>
+
+  <p>We have already seen (p.&nbsp;<a href="#page25">25</a>) that the effect of
+  pressure on the melting point of a substance was predicted as the result
+  of theoretical considerations, and was first proved experimentally in the
+  case of ice. Soon after, Bunsen<a name="NtA118"
+  href="#Nt118"><sup>[118]</sup></a> showed that the melting point of other
+  substances is also affected by pressure; and in more recent years, ample
+  experimental proof of the change of the melting point with the pressure
+  has been obtained. The change of the melting point is, however, small; as
+  a rule, increase of pressure by 1 atm. changes the melting point by about
+  0.03°, but in the case of water the change is much less (0.0076°), and in
+  the case of camphor much more (0.13°). In other words, if we take the
+  mean case, an increase of pressure of more than 30 atm. is required to
+  produce a change in the melting point of 1°.</p>
+
+  <p>Investigations which were made of the influence of pressure on the
+  melting-point, showed that up to pressures of several hundred atmospheres
+  the fusion curve is a straight line.<a name="NtA119"
+  href="#Nt119"><sup>[119]</sup></a> Tammann<a name="NtA120"
+  href="#Nt120"><sup>[120]</sup></a> has, however, found that on increasing
+  the pressure the fusion curve no longer remains straight, but bends
+  towards the pressure axis, so that, on sufficiently increasing the
+  pressure, a maximum temperature might at length be reached. This maximum
+  has, so far, however, not been attained, although the melting point
+  curves of various substances have been studied up to pressures of 4500
+  atm. This is to be accounted for partly <!-- Page 68 --><span
+  class="pagenum"><a name="page68"></a>{68}</span>by the fact that the
+  probable maximum temperature in the case of most substances lies at very
+  great pressures, and also by the fact that other solid phases make their
+  appearance, as, for example, in the case of ice (p.&nbsp;<a
+  href="#page32">32</a>).</p>
+
+  <p>As to the upper limit of the fusion curve, the view has been
+  expressed<a name="NtA121" href="#Nt121"><sup>[121]</sup></a> that just as
+  in the case of liquid and vapour, so also in the case of solid and
+  liquid, there exists a critical point at which the solid and the liquid
+  phase become identical. Experimental evidence, however, does not appear
+  to favour this view.<a name="NtA122"
+  href="#Nt122"><sup>[122]</sup></a></p>
+
+  <p>The <i>transition point</i>, like the melting point, is also
+  influenced by the pressure, and in this case also it is found that
+  pressure may either raise or lower the transition point, so that the
+  transition curve may be inclined either away from or towards the pressure
+  axis. The direction of the transition curve can also be predicted if the
+  change of volume accompanying the passage of one form into the other is
+  known. In the case of sulphur, we saw that the transition point is raised
+  by increase of pressure; in the case of the transition of rhombohedral
+  into <span class="grk">&alpha;</span>-rhombic form of ammonium nitrate,
+  however, the transition point is lowered by pressure, as shown by the
+  following table.<a name="NtA123" href="#Nt123"><sup>[123]</sup></a></p>
+
+<table class="nobctr" summary="Transition point of ammonium nitrate" title="Transition point of ammonium nitrate">
+
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Pressure.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 85.85° </td><td class="leb" style="text-align:left"> &nbsp; &nbsp; 1 atm.</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 84.38° </td><td class="leb" style="text-align:left"> 100 &nbsp; "</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 83.03° </td><td class="leb" style="text-align:left"> 200 &nbsp; "</td></tr>
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 82.29° </td><td class="boleb" style="text-align:left"> 250 &nbsp; "</td></tr>
+
+</table>
+
+  <p>So far as investigations have been carried out, it appears that in
+  most cases the transition curve is practically a straight line.</p>
+
+  <p>It has, however, been found in the case of Glauber's salt, that with
+  increase of pressure the transition curve passes through a point of
+  maximum temperature, and exhibits, therefore, a form similar to that
+  assumed by Tammann for the fusion curve.<a name="NtA124"
+  href="#Nt124"><sup>[124]</sup></a></p>
+
+<p><!-- Page 69 --><span class="pagenum"><a name="page69"></a>{69}</span></p>
+
+  <p><b>Suspended Transformation. Metastable Equilibria.</b>&mdash;Hitherto
+  we have considered only systems in stable equilibrium. We have, however,
+  already seen, in the case of water, that on cooling the liquid down to
+  the triple point, solidification did not necessarily take place, although
+  the conditions were such as to allow of its formation. Similarly, we saw
+  that rhombic sulphur can be heated above the transition point, and
+  monoclinic sulphur can be obtained at temperatures below the transition
+  point, although in both cases transformation into a more stable form is
+  possible; the system becomes metastable.</p>
+
+  <p>The same reluctance to form a new phase is observed also in the
+  phenomena of superheating of liquids, and in the "hanging" of mercury in
+  barometers, in which case the vapour phase is not formed. In general,
+  then, we may say that <i>a new phase will not necessarily be formed
+  immediately the system passes into such a condition that the existence of
+  that phase is possible</i>; but rather, instead of the system undergoing
+  transformation so as to pass into the most stable condition under the
+  existing pressure and temperature, this transformation will be
+  "suspended" or delayed, and the system will become metastable. Only in
+  the case of the formation of the liquid from the solid phase, in a
+  one-component system, has this reluctance to form a new phase not been
+  observed.</p>
+
+  <p><i>To ensure the formation of the new phase, it is necessary to have
+  that phase present.</i> The presence of the solid phase will prevent the
+  supercooling of the liquid; and the presence of the vapour phase will
+  prevent the superheating of the liquid. However, even in the presence of
+  the more stable phase, transformation of the metastable phase occurs with
+  very varying velocity; in some cases so quickly as to appear almost
+  instantaneous; while in other cases, the change takes place so slowly as
+  to require hundreds of years for its achievement. It is this slow rate of
+  transformation that renders the existence of metastable forms possible,
+  when in contact with the more stable phase. Thus, for example, although
+  calcite is the most stable form of calcium carbonate at the ordinary
+  temperature,<a name="NtA125" href="#Nt125"><sup>[125]</sup></a> the less
+  stable <!-- Page 70 --><span class="pagenum"><a
+  name="page70"></a>{70}</span>modification, aragonite, nevertheless exists
+  under the ordinary conditions in an apparently very stable state.</p>
+
+  <p>As to the amount of the new phase required to bring about the
+  transformation of the metastable phase, quantitative measurements have
+  been carried out only in the case of the initiation of crystallization in
+  a supercooled liquid.<a name="NtA126" href="#Nt126"><sup>[126]</sup></a>
+  As the result of these investigations, it was found that, in the case of
+  superfused salol, the very small amount of 1 × 10<sup>-7</sup> gm. of the
+  solid phase was sufficient to induce crystallization. Crystallization of
+  a supercooled liquid, however, can be initiated only by a "nucleus" of
+  the same substance in the solid state, or, as has also been found, by a
+  nucleus of an isomorphous solid phase; it is not brought about by the
+  presence of any chance solid.</p>
+
+  <p><b>Velocity of Transformation.</b>&mdash;Attention has already been
+  drawn to the sluggishness with which reciprocal transformation of the
+  polymorphic forms of a substance may occur. In the case of tin, for
+  example, it was found that the white modification, although apparently
+  possessing permanence, is in reality in a metastable state, under the
+  ordinary conditions of temperature and pressure. This great degree of
+  stability is due to the tardiness with which transformation into the grey
+  form occurs.</p>
+
+  <p>What was found in the case of tin, is met with also in the case of all
+  transformations in the solid state, but the velocity of the change is
+  less in some cases than in others, and appears to decrease with increase
+  of the valency of the element.<a name="NtA127"
+  href="#Nt127"><sup>[127]</sup></a> To this fact van't Hoff attributes the
+  great permanence of many really unstable (or metastable) carbon
+  compounds.</p>
+
+  <p>Reference has been made to the fact that the velocity of
+  transformation can be accelerated by various means. One of the most
+  important of these is the employment of a liquid which has a solvent
+  action on the solid phases. Just as we have seen that at any given
+  temperature the less stable form has the higher vapour pressure, but that
+  at the transition point the vapour pressure of both forms becomes
+  identical, so also it can be proved theoretically, and be shown
+  experimentally, that <!-- Page 71 --><span class="pagenum"><a
+  name="page71"></a>{71}</span>at a given temperature the solubility of the
+  less stable form is greater than that of the more stable, but that at the
+  transition point the solubility of the two forms becomes identical.<a
+  name="NtA128" href="#Nt128"><sup>[128]</sup></a></p>
+
+  <p>If, then, the two solid phases are brought into contact with a
+  solvent, the less stable phase will dissolve more abundantly than the
+  more stable; the solution will therefore become supersaturated with
+  respect to the latter, which will be deposited. A gradual change of the
+  less stable form, therefore, takes place through the medium of the
+  solvent. In this way the more rapid conversion of white tin into grey in
+  presence of a solution of tin ammonium chloride (p.&nbsp;<a
+  href="#page42">42</a>) is to be explained. Although, as a rule, solvents
+  accelerate the transformation of one solid phase into the other, they may
+  also have a retarding influence on the velocity of transformation, as was
+  found by Reinders in the case of mercuric iodide.<a name="NtA129"
+  href="#Nt129"><sup>[129]</sup></a></p>
+
+  <p>The velocity of inversion, also, is variously affected by different
+  solvents, and in some cases, at least, it appears to be slower the more
+  viscous the solvent;<a name="NtA130" href="#Nt130"><sup>[130]</sup></a>
+  indeed, Kastle and Reed state that yellow crystals of mercuric iodide,
+  which, ordinarily, change with considerable velocity into the red
+  modification, have been preserved for more than a year under
+  vaseline.</p>
+
+  <p>Temperature, also, has a very considerable influence on the velocity
+  of transformation. The higher the temperature, and the farther it is
+  removed from the equilibrium point (transition point), the greater is the
+  velocity of change. Above the transition point, these two factors act in
+  the same direction, and the velocity of transformation will therefore go
+  on increasing indefinitely the higher the temperature is raised. Below
+  the transition point, however, the two factors act in opposite
+  directions, and the more the temperature is lowered, the more is the
+  effect of removal from the equilibrium point counteracted. A point will
+  therefore be reached at which the velocity is a maximum. Reduction of the
+  temperature <!-- Page 72 --><span class="pagenum"><a
+  name="page72"></a>{72}</span>below this point causes a rapid falling off
+  in the velocity of change. The point of maximum velocity, however, is not
+  definite, but may be altered by various causes. Thus, Cohen found that in
+  the case of tin, the point of maximum velocity was altered if the metal
+  had already undergone transformation; and also by the presence of
+  different liquids.<a name="NtA131" href="#Nt131"><sup>[131]</sup></a></p>
+
+  <p>Lastly, the presence of small quantities of different
+  substances&mdash;catalytic agents or catalyzers&mdash;has a great
+  influence on the velocity of transformation. Thus, <i>e.g.</i>, the
+  conversion of white to red phosphorus is accelerated by the presence of
+  iodine (p.&nbsp;<a href="#page47">47</a>).</p>
+
+  <p>Greater attention, however, has been paid to the study of the velocity
+  of crystallization of a supercooled liquid, the first experiments in this
+  direction having been made by Gernez<a name="NtA132"
+  href="#Nt132"><sup>[132]</sup></a> on the velocity of crystallization of
+  phosphorus and sulphur. Since that time, the velocity of crystallization
+  of other supercooled liquids has been investigated; such as acetic acid
+  and phenol by Moore;<a name="NtA133" href="#Nt133"><sup>[133]</sup></a>
+  supercooled water by Tumlirz;<a name="NtA134"
+  href="#Nt134"><sup>[134]</sup></a> and a number of organic substances by
+  Tammann,<a name="NtA135" href="#Nt135"><sup>[135]</sup></a> Friedländer
+  and Tammann,<a name="NtA136" href="#Nt136"><sup>[136]</sup></a> and by
+  Bogojawlenski.<a name="NtA137" href="#Nt137"><sup>[137]</sup></a></p>
+
+  <p>In measuring the velocity of crystallization, the supercooled liquids
+  were contained in narrow glass tubes, and the time required for the
+  crystallization to advance along a certain length of the tube was
+  determined, the velocity being expressed in millimetres per minute. The
+  results which have so far been obtained may be summarized as follows. For
+  any given degree of supercooling of a substance, the velocity of
+  crystallization is constant. As the degree of supercooling increases, the
+  velocity of crystallization also increases, until a certain point is
+  reached at which the velocity is a maximum, which has a definite
+  characteristic value for each substance. This maximum velocity remains
+  constant over a certain range of <!-- Page 73 --><span class="pagenum"><a
+  name="page73"></a>{73}</span>temperature; thereafter, the velocity
+  diminishes fairly rapidly, and, with sufficient supercooling, may become
+  zero. The liquid then passes into a glassy mass, which will remain
+  (practically) permanent even in contact with the crystalline solid.</p>
+
+  <p>In ordinary glass we have a familiar example of a liquid which has
+  been cooled to a temperature at which crystallization takes place with
+  very great slowness. If, however, glass is heated, a temperature is
+  reached, much below the melting point of the glass, at which
+  crystallization occurs with appreciable velocity, and we observe the
+  phenomenon of devitrification.<a name="NtA138"
+  href="#Nt138"><sup>[138]</sup></a></p>
+
+  <p>When the velocity of crystallization is studied at temperatures above
+  the maximum point, it is found that the velocity is diminished by the
+  addition of foreign substances; and in many cases, indeed, it has been
+  found that the diminution is the same for equimolecular quantities of
+  different substances. It would hence appear possible to utilize this
+  behaviour as a method for determining molecular weights.<a name="NtA139"
+  href="#Nt139"><sup>[139]</sup></a> The rule is, however, by no means a
+  universal one. Thus it has been found by F. Dreyer,<a name="NtA140"
+  href="#Nt140"><sup>[140]</sup></a> in studying the velocity of
+  crystallization of formanilide, that the diminution in the velocity
+  produced by equivalent amounts of different substances is not the same,
+  but that the foreign substances exercise a specific influence. Further,
+  von Pickardt's rule does not hold when the foreign substance forms mixed
+  crystals (Chap. X.) with the crystallizing substance.<a name="NtA141"
+  href="#Nt141"><sup>[141]</sup></a></p>
+
+  <p><b>Law of Successive Reactions.</b>&mdash;When sulphur vapour is
+  cooled at the ordinary temperature, it first of all condenses to drops of
+  liquid, which solidify in an amorphous form, and only after some time
+  undergo crystallization; or, when phosphorus vapour is condensed, white
+  phosphorus is first formed, and not the more stable form&mdash;red
+  phosphorus. It has also been observed that even at the ordinary
+  temperature (therefore much below the transition point) sulphur may
+  crystallize out from solution in benzene, alcohol, carbon disulphide, and
+  other <!-- Page 74 --><span class="pagenum"><a
+  name="page74"></a>{74}</span>solvents, in the prismatic form, the less
+  stable prismatic crystals then undergoing transformation into the rhombic
+  form;<a name="NtA142" href="#Nt142"><sup>[142]</sup></a> a similar
+  behaviour has also been observed in the transformation of the monotropic
+  crystalline forms of sulphur.<a name="NtA143"
+  href="#Nt143"><sup>[143]</sup></a></p>
+
+  <p>Many other examples might be given. In organic chemistry, for
+  instance, it is often found that when a substance is thrown out of
+  solution, it is first deposited as a liquid, which passes later into the
+  more stable crystalline form. In analysis, also, rapid precipitation from
+  concentrated solution often causes the separation of a less stable and
+  more soluble amorphous form.</p>
+
+  <p>On account of the great frequency with which the prior formation of
+  the less stable form occurs, Ostwald<a name="NtA144"
+  href="#Nt144"><sup>[144]</sup></a> has put forward the <i>law of
+  successive reactions</i>, which states that when a system passes from a
+  less stable condition it does not pass directly into the most stable of
+  the possible states; but into the next more stable, and so step by step
+  into the most stable. This law explains the formation of the metastable
+  forms of monotropic substances, which would otherwise not be obtainable.
+  Although it is not always possible to observe the formation of the least
+  stable form, it should be remembered that that may quite conceivably be
+  due to the great velocity of transformation of the less stable into the
+  more stable form. From what we have learned about the velocity of
+  transformation of metastable phases, we can understand that rapid cooling
+  to a low temperature will tend to preserve the less stable form; and, on
+  account of the influence of temperature in increasing the velocity of
+  change, it can be seen that the formation of the less stable form will be
+  more difficult to observe in superheated than in supercooled systems. The
+  factors, however, which affect the readiness with which <!-- Page 75
+  --><span class="pagenum"><a name="page75"></a>{75}</span>the less stable
+  modification is produced, appear to be rather various.<a name="NtA145"
+  href="#Nt145"><sup>[145]</sup></a></p>
+
+  <p>Although a number of at least apparent exceptions to Ostwald's law
+  have been found, it may nevertheless be accepted as a very useful
+  generalization which sums up very frequently observed phenomena.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 76 --><span class="pagenum"><a name="page76"></a>{76}</span></p>
+
+<h3>CHAPTER V</h3>
+
+<p class="cenhead">SYSTEMS OF TWO COMPONENTS&mdash;PHENOMENA OF
+DISSOCIATION</p>
+
+  <p>In the preceding pages we have studied the behaviour of systems
+  consisting of only one component, or systems in which all the phases,
+  whether solid, liquid, or vapour, had the same chemical composition
+  (p.&nbsp;<a href="#page13">13</a>). In some cases, as, for example, in the
+  case of phosphorus and sulphur, the component was an elementary
+  substance; in other cases, however, <i>e.g.</i> water, the component was
+  a compound. The systems which we now proceed to study are characterized
+  by the fact that the different phases have no longer all the same
+  chemical composition, and cannot, therefore, according to definition, be
+  considered as one-component systems.</p>
+
+  <p>In most cases, little or no difficulty will be experienced in deciding
+  as to the <i>number</i> of the components, if the rules given on pp. <a
+  href="#page12">12</a> and <a href="#page13">13</a> are borne in mind. If
+  the composition of all the phases, each regarded as a whole, is the same,
+  the system is to be regarded as of the first order, or a one-component
+  system; if the composition of the different phases varies, the system
+  must contain more than one component. If, in order to <i>express</i> the
+  composition of all the phases present when the system is in equilibrium,
+  two of the constituents participating in the equilibrium are necessary
+  and sufficient, the system is one of two components. Which two of the
+  possible substances are to be regarded as components will, however, be to
+  a certain extent a matter of arbitrary choice.</p>
+
+  <p>The principles affecting the choice of components will best be learned
+  by a study of the examples to be discussed in the sequel. <!-- Page 77
+  --><span class="pagenum"><a name="page77"></a>{77}</span></p>
+
+  <p><b>Different Systems of Two Components.</b>&mdash;Applying the Phase
+  Rule</p>
+
+<p class="cenhead">P + F = C + 2</p>
+
+  <p>to systems of two components, we see that in order that the system may
+  be invariant, there must be four phases in equilibrium together; two
+  components in three phases constitute a univariant, two components in two
+  phases a bivariant system. In the case of systems of one component, the
+  highest degree of variability found was two (one component in one phase);
+  but, as is evident from the formula, there is a higher degree of freedom
+  possible in the case of two-component systems. Two components existing in
+  only one phase constitute a tervariant system, or a system with three
+  degrees of freedom. In addition to the pressure and temperature,
+  therefore, a third variable factor must be chosen, and as such there is
+  taken the <i>concentration of the components</i>. In systems of two
+  components, therefore, not only may there be change of pressure and
+  temperature, as in the case of one-component systems, but the
+  concentration of the components in the different phases may also alter; a
+  variation which did not require to be considered in the case of
+  one-component systems.</p>
+
+  <div class="figcenter" style="width:33%;">
+      <a href="images/fig018.png"><img style="width:100%" src="images/fig018.png"
+      alt="Fig. 18" title="Fig. 18" /></a>
+    <span class="sc">Fig.</span> 18.
+  </div>
+
+  <p>Since a two-component system may undergo three possible <!-- Page 78
+  --><span class="pagenum"><a name="page78"></a>{78}</span>independent
+  variations, we should require for the graphic representation of all the
+  possible conditions of equilibrium a system of three co-ordinates in
+  space, three axes being chosen, say, at right angles to one another, and
+  representing the three variables&mdash;pressure, temperature, and
+  concentration of components (Fig. 18). A curve (<i>e.g.</i> AB) in the
+  plane containing the pressure and temperature axes would then represent
+  the change of pressure with the temperature, the concentration remaining
+  unaltered (<i>pt</i>-diagram); one in the plane containing the pressure
+  and concentration axes (<i>e.g.</i> AF or DF), the change of pressure
+  with the concentration, the temperature remaining constant
+  (<i>pc</i>-diagram), while in the plane containing the concentration and
+  the temperature axes, the simultaneous change of these two factors at
+  constant pressure would be represented (<i>tc</i>-diagram). If the points
+  on these three curves are joined together, a surface, ABDE, will be
+  formed, and any line on that surface (<i>e.g.</i> FG, or GH, or GI) would
+  represent the simultaneous variation of the three factors&mdash;pressure,
+  temperature, concentration. Although we shall at a later point make some
+  use of these solid figures, we shall for the present employ the more
+  readily intelligible plane diagram.</p>
+
+  <p>The number of different systems which can be formed from two
+  components, as well as the number of the different phenomena which can
+  there be observed, is much greater than in the case of one component. In
+  the case of no two substances, however, have all the possible
+  relationships been studied; so that for the purpose of gaining an insight
+  into the very varied behaviour of two-component systems, a number of
+  different examples will be discussed, each of which will serve to give a
+  picture of some of the relationships.</p>
+
+  <p>Although the strict classification of the different systems according
+  to the Phase Rule would be based on the variability of the systems, the
+  study of the many different phenomena, and the correlation of the
+  comparatively large number of different systems, will probably be
+  rendered easiest by grouping these different phenomena into classes, each
+  of these classes being studied with the help of one or more typical
+  examples. The order of treatment adopted here is, of course, quite
+  arbitrary; <!-- Page 79 --><span class="pagenum"><a
+  name="page79"></a>{79}</span>but has been selected from considerations of
+  simplicity and clearness.</p>
+
+<p class="cenhead"><span class="sc">Phenomena of Dissociation.</span></p>
+
+  <p><b>Bivariant Systems.</b>&mdash;As the first examples of the
+  equilibria between a substance and its products of dissociation, we shall
+  consider very briefly those cases in which there is one solid phase in
+  equilibrium with vapour. Reference has already been made to such systems
+  in the case of ammonium chloride. On being heated, ammonium chloride
+  dissociates into ammonia and hydrogen chloride. Since, however, in that
+  case the vapour phase has the same total composition as the solid phase,
+  viz. NH<sub>3</sub> + HCl = NH<sub>4</sub>Cl, the system consists of only
+  one component existing in two phases; it is therefore univariant, and to
+  each temperature there will correspond a definite vapour pressure
+  (dissociation pressure).<a name="NtA146"
+  href="#Nt146"><sup>[146]</sup></a></p>
+
+  <p>If, however, excess of one of the products of dissociation be added,
+  the system becomes one of two components.</p>
+
+  <p>In the first place, analysis of each of the two phases yields as the
+  composition of each, solid: NH<sub>4</sub>Cl (= NH<sub>3</sub> + HCl);
+  vapour: <i>m</i>NH<sub>3</sub> + <i>n</i>HCl. Obviously the smallest
+  number of substances by which the composition of the two phases can be
+  expressed is two; that is, the number of components is two. What, then,
+  are the components? The choice lies between NH<sub>3</sub> + HCl,
+  NH<sub>4</sub>Cl + NH<sub>3</sub>, and NH<sub>4</sub>Cl + HCl; for the
+  three substances, ammonium chloride, ammonia, hydrogen chloride, are the
+  only ones taking part in the equilibrium of the system.</p>
+
+  <p>Of these three pairs of components, we should obviously choose as the
+  most simple NH<sub>3</sub> and HCl, for we can then represent the
+  composition of the two phases as the <i>sum</i> of the two components. If
+  one of the other two possible pairs of components be chosen, we should
+  have to introduce negative quantities of one of the components, in order
+  to represent the composition of the vapour phase. Although it must be
+  allowed that the introduction of negative quantities of a component in
+  such cases is quite permissible, still it will be <!-- Page 80 --><span
+  class="pagenum"><a name="page80"></a>{80}</span>better to adopt the
+  simpler and more direct choice, whereby the composition of each of the
+  phases is represented as a sum of two components in varying proportions
+  (p.&nbsp;<a href="#page12">12</a>).</p>
+
+  <p>If, therefore, we have a solid substance, such as ammonium chloride,
+  which dissociates on volatilization, and if the products of dissociation
+  are added in varying amounts to the system, we shall have, in the sense
+  of the Phase Rule, a <i>two-component system existing in two phases</i>.
+  Such a system will possess two degrees of freedom. At any given
+  temperature, not only the pressure, but also the composition, of the
+  vapour-phase, <i>i.e.</i> the concentration of the components, can vary.
+  Only after one of these independent variables, pressure or composition,
+  has been arbitrarily fixed does the system become univariant, and exhibit
+  a definite, constant pressure at a given temperature.</p>
+
+  <p>Now, although the Phase Rule informs us that at a given temperature
+  change of composition of the vapour phase will be accompanied by change
+  of pressure, it does not cast any light on the relation between these two
+  variables. This relationship, however, can be calculated theoretically by
+  means of the Law of Mass Action.<a name="NtA147"
+  href="#Nt147"><sup>[147]</sup></a> From this we learn that in the case of
+  a substance which dissociates into equivalent quantities of two gases,
+  the product of the partial pressures of the gases is constant at a given
+  temperature.</p>
+
+  <p>This has been proved experimentally in the case of ammonium
+  hydrosulphide, ammonium cyanide, phosphonium bromide, and other
+  substances.<a name="NtA148" href="#Nt148"><sup>[148]</sup></a></p>
+
+  <p><b>Univariant Systems.</b>&mdash;In order that a system of two
+  components shall possess only one degree of freedom, three phases must be
+  present. Of such systems, there are seven possible, viz. S-S-S, S-S-L,
+  S-S-V, L-L-L, S-L-L, L-L-V, S-L-V; S denoting solid, L liquid, and V
+  vapour. In the present chapter we shall consider only the systems S-S-V,
+  <i>i.e.</i> those systems in which there are two solid phases and a
+  vapour phase present.</p>
+
+<p><!-- Page 81 --><span class="pagenum"><a name="page81"></a>{81}</span></p>
+
+  <p>As an example of this, we may first consider the well-known case of
+  the dissociation of calcium carbonate. This substance on being heated
+  dissociates into calcium oxide, or quick-lime, and carbon dioxide, as
+  shown by the equation CaCO<sub>3</sub> <a
+  href="images/doublearrow.png"><img src="images/doublearrow.png"
+  class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CaO +
+  CO<sub>2</sub>. In accordance with our definition (p.&nbsp;<a
+  href="#page9">9</a>), we have here two solid phases, the carbonate and
+  the quick-lime, and one vapour phase; the system is therefore univariant.
+  To each temperature, therefore, there will correspond a certain, definite
+  maximum pressure of carbon dioxide (dissociation pressure), and this will
+  follow the same law as the vapour pressure of a pure liquid (p.&nbsp;<a
+  href="#page21">21</a>). More particularly, it will be independent of the
+  relative or absolute amounts of the two solid phases, and of the volume
+  of the vapour phase. If the temperature is maintained constant, increase
+  of volume will cause the dissociation of a further amount of the
+  carbonate until the pressure again reaches its maximum value
+  corresponding to the given temperature. Diminution of volume, on the
+  other hand, will bring about the combination of a certain quantity of the
+  carbon dioxide with the calcium oxide until the pressure again reaches
+  its original value.</p>
+
+  <p>The dissociation pressure of calcium carbonate was first studied by
+  Debray,<a name="NtA149" href="#Nt149"><sup>[149]</sup></a> but more exact
+  measurements have been made by Le Chatelier,<a name="NtA150"
+  href="#Nt150"><sup>[150]</sup></a> who found the following corresponding
+  values of temperature and pressure:&mdash;</p>
+
+<table class="nobctr" summary="Dissociation pressure of calcium carbonate" title="Dissociation pressure of calcium carbonate">
+
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Pressure in cm. mercury.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 547° </td><td class="leb" style="text-align:right; padding-right:4em"> 2.7</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 610° </td><td class="leb" style="text-align:right; padding-right:4em"> 4.6</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 625° </td><td class="leb" style="text-align:right; padding-right:4em"> 5.6</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 740° </td><td class="leb" style="text-align:right; padding-right:4em"> 25.5</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 745° </td><td class="leb" style="text-align:right; padding-right:4em"> 28.9</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 810° </td><td class="leb" style="text-align:right; padding-right:4em"> 67.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 812° </td><td class="leb" style="text-align:right; padding-right:4em"> 76.3</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 865° </td><td class="boleb" style="text-align:right; padding-right:4em"> 133.3</td></tr>
+
+</table>
+
+  <p>From this table we see that it is only at a temperature of about 812°
+  that the pressure of the carbon dioxide becomes equal to atmospheric
+  pressure. In a vessel open to <!-- Page 82 --><span class="pagenum"><a
+  name="page82"></a>{82}</span>the air, therefore, the complete
+  decomposition of the calcium carbonate would not take place below this
+  temperature by the mere heating of the carbonate. If, however, the carbon
+  dioxide is removed as quickly as it is formed, say by a current of air,
+  then the entire decomposition can be made to take place at a much lower
+  temperature. For the dissociation equilibrium of the carbonate depends
+  only on the partial pressure of the carbon dioxide, and if this is kept
+  small, then the decomposition can proceed, even at a temperature below
+  that at which the pressure of the carbon dioxide is less than atmospheric
+  pressure.</p>
+
+  <p><b>Ammonia Compounds of Metal Chlorides.</b>&mdash;Ammonia possesses
+  the property of combining with various substances, chiefly the halides of
+  metals, to form compounds which again yield up the ammonia on being
+  heated. Thus, for example, on passing ammonia over silver chloride,
+  absorption of the gas takes place with formation of the substances
+  AgCl,3NH<sub>3</sub> and 2AgCl,3NH<sub>3</sub>, according to the
+  conditions of the experiment. These were the first known substances
+  belonging to this class, and were employed by Faraday in his experiments
+  on the liquefaction of ammonia. Similar compounds have also been obtained
+  by the action of ammonia on silver bromide, iodide, cyanide, and nitrate;
+  and with the halogen compounds of calcium, zinc, and magnesium, as well
+  as with other salts. The behaviour of the ammonia compounds of silver
+  chloride is typical for the compounds of this class, and may be briefly
+  considered here.</p>
+
+  <p>It was found by Isambert<a name="NtA151"
+  href="#Nt151"><sup>[151]</sup></a> that at temperatures below 15°, silver
+  chloride combined with ammonia to form the compound AgCl,3NH<sub>3</sub>,
+  while at temperatures above 20° the compound 2AgCl,3NH<sub>3</sub> was
+  produced. On heating these substances, ammonia was evolved, and the
+  pressure of this gas was found in the case of both compounds to be
+  constant at a given temperature, but was greater in the case of the
+  former than in the case of the latter substance; the pressure, further,
+  was independent of the amount decomposed. The behaviour of these two
+  substances is, therefore, exactly analogous to that shown by calcium
+  carbonate, and the explanation is also similar.</p>
+
+<p><!-- Page 83 --><span class="pagenum"><a name="page83"></a>{83}</span></p>
+
+  <p>Regarded from the point of view of the Phase Rule, we see that we are
+  here dealing with two components, AgCl and NH<sub>3</sub>. On being
+  heated, the compounds decompose according to the equations:&mdash;</p>
+
+<p class="cenhead">2(AgCl,3NH<sub>3</sub>) <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> 2AgCl,3NH<sub>3</sub> + 3NH<sub>3</sub>.<br />
+  2AgCl,3NH<sub>3</sub> <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> 2AgCl + 3NH<sub>3</sub>. &nbsp; &nbsp; &nbsp;</p>
+
+  <p>There are, therefore, three phases, viz. AgCl,3NH<sub>3</sub>;
+  2AgCl,3NH<sub>3</sub>, and NH<sub>3</sub>, in the one case; and
+  2AgCl,3NH<sub>3</sub>; AgCl, and NH<sub>3</sub> in the other. These two
+  systems are therefore univariant, and to each temperature there must
+  correspond a definite pressure of dissociation, quite irrespective of the
+  amounts of the phases present. Similarly, if, at constant temperature,
+  the volume is increased (or if the ammonia which is evolved is pumped
+  off), the pressure will remain constant so long as two solid phases,
+  AgCl,3NH<sub>3</sub> and 2AgCl,3NH<sub>3</sub>, are present, <i>i.e.</i>
+  until the compound richer in ammonia is completely decomposed, when there
+  will be a sudden fall in the pressure to the value corresponding to the
+  system 2AgCl,3NH<sub>3</sub>&mdash;AgCl&mdash;NH<sub>3</sub>. The
+  pressure will again remain constant at constant temperature, until all
+  the ammonia has been pumped off, when there will again be a sudden fall
+  in the pressure to that of the system formed by solid silver chloride in
+  contact with its vapour.</p>
+
+  <p>The reverse changes take place when the pressure of the ammonia is
+  gradually increased. If the volume is continuously diminished, the
+  pressure will first increase until it has reached a certain value; the
+  compound 2AgCl,3NH<sub>3</sub> can then be formed, and the pressure will
+  now remain constant until all the silver chloride has disappeared. The
+  pressure will again rise, until it has reached the value at which the
+  compound AgCl,3NH<sub>3</sub> can be formed, when it will again remain
+  constant until the complete disappearance of the lower compound. <i>There
+  is no gradual change of pressure</i> on passing from one system to
+  another; but the changes are abrupt, as is demanded by the Phase Rule,
+  and as experiment has conclusively proved.<a name="NtA152"
+  href="#Nt152"><sup>[152]</sup></a></p>
+
+  <p>The dissociation pressures of the two compounds of silver <!-- Page 84
+  --><span class="pagenum"><a name="page84"></a>{84}</span>chloride and
+  ammonia, as determined by Isambert,<a name="NtA153"
+  href="#Nt153"><sup>[153]</sup></a> are given in the following
+  table:&mdash;</p>
+
+<table class="nobctr" summary="Dissociation pressures of compounds of silver chloride and ammonia" title="Dissociation pressures of compounds of silver chloride and ammonia">
+
+<tr><td class="toribob" style="text-align:center" colspan="2"> AgCl,3NH<sub>3</sub>.
+</td><td class="toboleb" style="text-align:center" colspan="2"> 2AgCl,3NH<sub>3</sub>.</td></tr>
+
+<tr><td class="tobob" style="text-align:center"> Temperature.
+</td><td class="allb" style="text-align:center"> Pressure.
+</td><td class="allb" style="text-align:center"> Temperature.
+</td><td class="tobob" style="text-align:center"> Pressure.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; &nbsp; &nbsp;0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 29.3 cm. </td><td class="leb" style="text-align:left; padding-left:2em"> 20.0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; &nbsp; 9.3 cm.</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 10.6° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 50.5 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 31.0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 12.5 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 17.5° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 65.5 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 47.0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 26.8 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 24.0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 93.7 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 58.5° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 52.8 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 28.0° </td><td class="leb" style="text-align:left; padding-left:2em"> 135.5 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 69.0° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 78.6 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 34.2° </td><td class="leb" style="text-align:left; padding-left:2em"> 171.3 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 71.5° </td><td class="leb" style="text-align:left; padding-left:2em">&nbsp; 94.6 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 48.5° </td><td class="leb" style="text-align:left; padding-left:2em"> 241.4 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 77.5° </td><td class="leb" style="text-align:left; padding-left:2em"> 119.8 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 51.5° </td><td class="leb" style="text-align:left; padding-left:2em"> 413.2 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 83.5° </td><td class="leb" style="text-align:left; padding-left:2em"> 159.3 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 54.0° </td><td class="leb" style="text-align:left; padding-left:2em"> 464.1 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 86.1° </td><td class="leb" style="text-align:left; padding-left:2em"> 181.3 &nbsp; ,,</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> </td><td class="boleb" style="text-align:left; padding-left:2em"> </td><td class="boleb" style="text-align:left; padding-left:2em"> 88.5° </td><td class="boleb" style="text-align:left; padding-left:2em"> 201.3 &nbsp; ,,</td></tr>
+
+</table>
+
+  <p>The conditions for the formation of these two compounds, by passing
+  ammonia over silver chloride, to which reference has already been made,
+  will be readily understood from the above tables. In the case of the
+  triammonia mono-chloride, the dissociation pressure becomes equal to
+  atmospheric pressure at a temperature of about 20°; above this
+  temperature, therefore, it cannot be formed by the action of ammonia at
+  atmospheric pressure on silver chloride. The triammonia dichloride can,
+  however, be formed, for its dissociation pressure at this temperature
+  amounts to only 9 cm., and becomes equal to the atmospheric pressure only
+  at a temperature of about 68°; and this temperature, therefore,
+  constitutes the limit above which no combination can take place between
+  silver chloride and ammonia under atmospheric pressure.</p>
+
+  <p>Attention may be here drawn to the fact, to which reference will also
+  be made later, that <i>two</i> solid phases are necessary in order that
+  the dissociation pressure at a given temperature shall be definite;
+  <i>and for the exact definition of this pressure it is necessary to know,
+  not merely what is the substance undergoing dissociation, but also what
+  is the solid product of dissociation formed</i>. For the definition of
+  the equilibrium, the latter is as important as the former. We shall
+  presently find proof of this in the case <!-- Page 85 --><span
+  class="pagenum"><a name="page85"></a>{85}</span>of an analogous class of
+  phenomena, viz. the dissociation of salt hydrates.</p>
+
+  <p><b>Salts with Water of Crystallization.</b>&mdash;In the case of the
+  dehydration of crystalline salts containing water of crystallization, we
+  meet with phenomena which are in all respects similar to those just
+  studied. A salt hydrate on being heated dissociates into a lower hydrate
+  (or anhydrous salt) and water vapour. Since we are dealing with two
+  components&mdash;salt and water<a name="NtA154"
+  href="#Nt154"><sup>[154]</sup></a>&mdash;in three phases, viz. hydrate
+  <i>a</i>, hydrate <i>b</i> (or anhydrous salt), and vapour, the system is
+  univariant, and to each temperature there will correspond a certain,
+  definite vapour pressure (the dissociation pressure), which will be
+  independent of the relative or absolute amounts of the phases,
+  <i>i.e.</i> of the amount of hydrate which has already undergone
+  dissociation or dehydration.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig019.png"><img style="width:100%" src="images/fig019.png"
+      alt="Fig. 19" title="Fig. 19" /></a>
+    <span class="sc">Fig.</span> 19.
+  </div>
+
+  <p>The constancy of the dissociation pressure had been proved
+  experimentally by several investigators<a name="NtA155"
+  href="#Nt155"><sup>[155]</sup></a> a number of years before the
+  theoretical basis for its necessity had been given. In the case of salts
+  capable of forming more than one hydrate, we should obtain a series of
+  dissociation curves (<i>pt</i>-curves), as in the case of the different
+  hydrates of copper sulphate. In Fig. 19 there are represented
+  diagrammatically the vapour-pressure curves of the following univariant
+  systems of copper sulphate and water:&mdash;</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>Curve OA: CuSO<sub>4</sub>,5H<sub>2</sub>O <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CuSO<sub>4</sub>,3H<sub>2</sub>O + 2H<sub>2</sub>O.</p>
+      <p>Curve OB: CuSO<sub>4</sub>,3H<sub>2</sub>O <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CuSO<sub>4</sub>,H<sub>2</sub>O + 2H<sub>2</sub>O.</p>
+      <p>Curve OC: CuSO<sub>4</sub>,H<sub>2</sub>O &nbsp; <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CuSO<sub>4</sub> + H<sub>2</sub>O.</p>
+    </div>
+  </div>
+
+  <p>Let us now follow the changes which take place on <!-- Page 86
+  --><span class="pagenum"><a name="page86"></a>{86}</span>increasing the
+  pressure of the aqueous vapour in contact with anhydrous copper sulphate,
+  the temperature being meanwhile maintained constant. If, starting from
+  the point D, we slowly add water vapour to the system, the pressure will
+  gradually rise, without formation of hydrate taking place; for at
+  pressures below the curve OC only the anhydrous salt can exist. At E,
+  however, the hydrate CuSO<sub>4</sub>,H<sub>2</sub>O will be formed, and
+  as there are now three phases present, viz. CuSO<sub>4</sub>,
+  CuSO<sub>4</sub>,H<sub>2</sub>O, and vapour, the system becomes
+  <i>univariant</i>; and since the temperature is constant, the pressure
+  must also be constant. Continued addition of vapour will result merely in
+  an increase in the amount of the hydrate, and a decrease in the amount of
+  the anhydrous salt. When the latter has entirely disappeared, <i>i.e.</i>
+  has passed into hydrated salt, the system again becomes <i>bivariant</i>,
+  and passes along the line EF; the pressure gradually increases,
+  therefore, until at F the hydrate 3H<sub>2</sub>O is formed, and the
+  system again becomes univariant; the three phases present are
+  CuSO<sub>4</sub>,H<sub>2</sub>O, CuSO<sub>4</sub>,3H<sub>2</sub>O,
+  vapour. The pressure will remain constant, therefore, until the hydrate
+  1H<sub>2</sub>O has disappeared, when it will again increase till G is
+  reached; here the hydrate 5H<sub>2</sub>O is formed, and the pressure
+  once more remains constant until the complete disappearance of the
+  hydrate 3H<sub>2</sub>O has taken place.</p>
+
+  <p>Conversely, on dehydrating CuSO<sub>4</sub>,5H<sub>2</sub>O at
+  constant temperature, we should find that the pressure would maintain the
+  value corresponding to the dissociation pressure of the system
+  CuSO<sub>4</sub>,5H<sub>2</sub>O&mdash;CuSO<sub>4</sub>,3H<sub>2</sub>O&mdash;vapour,
+  until all the hydrate 5H<sub>2</sub>O had disappeared; further removal of
+  water would then cause the pressure to fall <i>abruptly</i> to the
+  pressure of the system
+  CuSO<sub>4</sub>,3H<sub>2</sub>O&mdash;CuSO<sub>4</sub>,H<sub>2</sub>O&mdash;vapour,
+  at which value it would again remain constant until the tri-hydrate had
+  passed into the monohydrate, when a further sudden diminution of the
+  pressure would occur. This behaviour is represented diagrammatically in
+  Fig. 20, the values of the pressure being those at 50°.</p>
+
+  <p><b>Efflorescence.</b>&mdash;From Fig. 19 we are enabled to predict the
+  conditions under which a given hydrated salt will effloresce when exposed
+  to the air. We have just learned that copper <!-- Page 87 --><span
+  class="pagenum"><a name="page87"></a>{87}</span>sulphate pentahydrate,
+  for example, will not be formed unless the pressure of the aqueous vapour
+  reaches a certain value; and that conversely, if the vapour pressure
+  falls below the dissociation pressure of the pentahydrate, this salt will
+  undergo dehydration. From this, then, it is evident that a crystalline
+  salt hydrate will effloresce when exposed to the air, if the partial
+  pressure of the water vapour in the air is lower than the dissociation
+  pressure of the hydrate. At the ordinary temperature the dissociation
+  pressure of copper sulphate is less than the pressure of water vapour in
+  the air, and therefore copper sulphate does not effloresce. In the case
+  of sodium sulphate decahydrate, however, the dissociation pressure is
+  greater than the normal vapour pressure in a room, and this salt
+  therefore effloresces.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig020.png"><img style="width:100%" src="images/fig020.png"
+      alt="Fig. 20" title="Fig. 20" /></a>
+    <span class="sc">Fig. 20.</span>
+  </div>
+
+  <p><b>Indefiniteness of the Vapour Pressure of a
+  Hydrate.</b>&mdash;Reference has already been made (p.&nbsp;<a
+  href="#page84">84</a>), in the case of the ammonia compounds of the metal
+  chlorides, to the importance of the solid product of dissociation for the
+  definition of the dissociation pressure. Similarly also in the case of a
+  hydrated salt. A salt hydrate in contact with vapour constitutes only a
+  bivariant system, and can exist therefore at different values of
+  temperature and pressure of vapour, as is seen from the diagram, Fig. 19.
+  Anhydrous copper sulphate can exist in contact with water vapour at all
+  values of temperature and pressure lying in the field below the curve OC;
+  and the hydrate CuSO<sub>4</sub>,H<sub>2</sub>O can exist in contact with
+  vapour at all values of temperature and pressure in the field BOC.
+  Similarly, each of the other hydrates can exist in contact with vapour at
+  different values of temperature and pressure.</p>
+
+  <p>From the Phase Rule, however, we learn that, in order that at a given
+  temperature the pressure of a two-component system <!-- Page 88 --><span
+  class="pagenum"><a name="page88"></a>{88}</span>may be constant, there
+  must be three phases present. Strictly, therefore, we can speak only of
+  the vapour pressure of a <i>system</i>; and since, in the cases under
+  discussion, the hydrates dissociate into a solid and a vapour, any
+  statement as to the vapour pressure of a hydrate has a definite meaning
+  <i>only when the second solid phase produced by the dissociation is
+  given</i>. The everyday custom of speaking of the vapour pressure of a
+  hydrated salt acquires a meaning only through the assumption, tacitly
+  made, that the second solid phase, or the solid produced by the
+  dehydration of the hydrate, is the <i>next lower</i> hydrate, where more
+  hydrates than one exist. That a hydrate always dissociates in such a way
+  that the next lower hydrate is formed is, however, by no means certain;
+  indeed, cases have been met with where apparently the anhydrous salt, and
+  not the lower hydrate (the existence of which was possible), was produced
+  by the dissociation of the higher hydrate.<a name="NtA156"
+  href="#Nt156"><sup>[156]</sup></a></p>
+
+  <p>That a salt hydrate can exhibit different vapour pressures according
+  to the solid product of dissociation, can not only be proved
+  theoretically, but it has also been shown experimentally to be a fact.
+  Thus CaCl<sub>2</sub>,6H<sub>2</sub>O can dissociate into water vapour
+  and either of two lower hydrates, each containing four molecules of water
+  of crystallization, and designated respectively as
+  CaCl<sub>2</sub>,4H<sub>2</sub>O<span class="grk">&alpha;</span>, and
+  CaCl<sub>2</sub>,4H<sub>2</sub>O<span class="grk">&beta;</span>.
+  Roozeboom<a name="NtA157" href="#Nt157"><sup>[157]</sup></a> has shown
+  that the vapour pressure which is obtained differs according to which of
+  these two hydrates is formed, as can be seen from the following
+  figures:&mdash;</p>
+
+<table class="nobctr" summary="Dissociation pressures of calcium chloride hydrate" title="Dissociation pressures of calcium chloride hydrate">
+
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Pressure of System.</td></tr>
+
+<tr><td class="tobob" style="text-align:center"> CaCl<sub>2</sub>,6H<sub>2</sub>O; CaCl<sub>2</sub>,<br />4H<sub>2</sub>O<span class="grk">&alpha;</span>; vapour.
+</td><td class="toboleb" style="text-align:center"> CaCl<sub>2</sub>,6H<sub>2</sub>O; CaCl<sub>2</sub>,<br />4H<sub>2</sub>O<span class="grk">&beta;</span>; vapour.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp;-15° </td><td class="leb" style="text-align:left; padding-left:2em"> 0.027 cm. </td><td class="leb" style="text-align:left; padding-left:2em"> 0.022 cm.</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; &nbsp; 0 </td><td class="leb" style="text-align:left; padding-left:2em"> 0.092 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 0.076 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> +10 </td><td class="leb" style="text-align:left; padding-left:2em"> 0.192 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 0.162 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 20 </td><td class="leb" style="text-align:left; padding-left:2em"> 0.378 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 0.315 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 25 </td><td class="leb" style="text-align:left; padding-left:2em"> 0.508 &nbsp; ,, </td><td class="leb" style="text-align:left; padding-left:2em"> 0.432 &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 29.2 </td><td class="leb" style="text-align:left; padding-left:2em"> &nbsp; &nbsp; &nbsp;&mdash; </td><td class="leb" style="text-align:left; padding-left:2em"> 0.567 &nbsp; ,,</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> &nbsp; 29.8 </td><td class="boleb" style="text-align:left; padding-left:2em"> 0.680 &nbsp; ,, </td><td class="boleb" style="text-align:left; padding-left:2em"> &nbsp; &nbsp; &nbsp;&mdash;</td></tr>
+
+</table>
+
+<p><!-- Page 89 --><span class="pagenum"><a name="page89"></a>{89}</span></p>
+
+  <p>By reason of the non-recognition of the importance of the solid
+  dissociation product for the definition of the dissociation pressure of a
+  salt hydrate, many of the older determinations lose much of their
+  value.</p>
+
+  <p><b>Suspended Transformation.</b>&mdash;Just as in systems of one
+  component we found that a new phase was not necessarily formed when the
+  conditions for its existence were established, so also we find that even
+  when the vapour pressure is lowered below the dissociation pressure of a
+  system, dissociation does not necessarily occur. This is well known in
+  the case of Glauber's salt, first observed by Faraday. Undamaged crystals
+  of Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O could be kept unchanged
+  in the open air, although the vapour pressure of the system
+  Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O&mdash;Na<sub>2</sub>SO<sub>4</sub>&mdash;vapour
+  is greater than the ordinary pressure of aqueous vapour in the air. That
+  is to say, the possibility of the formation of the new phase
+  Na<sub>2</sub>SO<sub>4</sub> was given; nevertheless this new phase did
+  not appear, and the system therefore became metastable, or unstable with
+  respect to the anhydrous salt. When, however, a trace of the new
+  phase&mdash;the anhydrous salt&mdash;was brought in contact with the
+  hydrate, transformation occurred; the hydrate effloresced.</p>
+
+  <p>The possibility of suspended transformation or the non-formation of
+  the new phases must also be granted in the case where the vapour pressure
+  is raised above that corresponding to the system hydrate&mdash;anhydrous
+  salt (or lower hydrate)&mdash;vapour; in this case the formation of the
+  higher hydrate becomes a possibility, but not a certainty. Although there
+  is no example of this known in the case of hydrated salts, the suspension
+  of the transformation has been observed in the case of the compounds of
+  ammonia with the metal chlorides (p.&nbsp;<a href="#page82">82</a>).
+  Horstmann,<a name="NtA158" href="#Nt158"><sup>[158]</sup></a> for
+  example, found that the pressure of ammonia in contact with
+  2AgCl,3NH<sub>3</sub> could be raised to a value higher than the
+  dissociation pressure of AgCl,3NH<sub>3</sub> without this compound being
+  formed. We see, therefore, that even when the existence of the higher
+  compound in contact with the lower became possible, the higher compound
+  was not immediately formed.</p>
+
+  <p><b>Range of Existence of Hydrates.</b>&mdash;In Fig. 19 the vapour
+  <!-- Page 90 --><span class="pagenum"><a
+  name="page90"></a>{90}</span>pressure curves of the different hydrates of
+  copper sulphate are represented as maintaining their relative positions
+  throughout the whole range of temperatures. But this is not necessarily
+  the case. It is possible that at some temperature the vapour pressure
+  curve of a lower hydrate may cut that of a higher hydrate. At
+  temperatures above the point of intersection, the lower hydrate would
+  have a higher vapour pressure than the higher hydrate, and would
+  therefore be metastable with respect to the latter. The range of stable
+  existence of the lower hydrate would therefore end at the point of
+  intersection. This appears to be the case with the two hydrates of sodium
+  sulphate, to which reference will be made later.<a name="NtA159"
+  href="#Nt159"><sup>[159]</sup></a></p>
+
+  <p><b>Constancy of Vapour Pressure and the Formation of
+  Compounds.</b>&mdash;We have seen in the case of the salt hydrates that
+  the continued addition of the vapour phase to the system caused an
+  increase in the pressure until at a definite value of the pressure a
+  hydrate is formed; the pressure then becomes constant, and remains so,
+  until one of the solid phases has disappeared. Conversely, on withdrawing
+  the vapour phase, the pressure remained constant so long as any of the
+  dissociating compound was present, independently of the degree of the
+  decomposition (p.&nbsp;<a href="#page86">86</a>). This behaviour, now, has
+  been employed for the purpose of determining whether or not definite
+  chemical compounds are formed. Should compounds be formed between the
+  vapour phase and the solid, then, on continued addition or withdrawal of
+  the vapour phase, it will be found that the vapour pressure remains
+  constant for a certain time, and will then suddenly assume a new value,
+  at which it will again remain constant. By this method, Ramsay<a
+  name="NtA160" href="#Nt160"><sup>[160]</sup></a> found that no definite
+  hydrates were formed in the case of ferric and aluminium oxides, but that
+  two are formed in the case of lead oxide, viz. 2PbO,H<sub>2</sub>O and
+  3PbO,H<sub>2</sub>O.</p>
+
+  <p>The method has also been applied to the investigation of the so-called
+  palladium hydride,<a name="NtA161" href="#Nt161"><sup>[161]</sup></a> and
+  the results obtained appear to show that no compound is formed. Reference
+  will, however, be made to this case later (Chap. X.).</p>
+
+<p><!-- Page 91 --><span class="pagenum"><a name="page91"></a>{91}</span></p>
+
+  <p><b>Measurement of the Vapour Pressure of Hydrates.</b>&mdash;For the
+  purpose of measuring the small pressures exerted by the vapour of salt
+  hydrates, use is very generally made of a differential manometer called
+  the <i>Bremer-Frowein tensimeter</i>.<a name="NtA162"
+  href="#Nt162"><sup>[162]</sup></a></p>
+
+  <p>This apparatus has the form shown in Fig. 21. It consists of a
+  <b>U</b>-tube, the limbs of which are bent close together, and placed in
+  front of a millimetre scale. The bend of the tube is filled with oil or
+  other suitable liquid, <i>e.g.</i> bromonaphthalene. If it is desired to
+  measure the dissociation pressure of, say, a salt hydrate, concentrated
+  sulphuric acid is placed in the flask <i>e</i>, and a quantity of the
+  hydrate, well dried and powdered,<a name="NtA163"
+  href="#Nt163"><sup>[163]</sup></a> in the bulb <i>d</i>. The necks of the
+  bulbs <i>d</i> and <i>e</i> are then sealed off. Since, as we have
+  learned, suspended transformation may occur, it is advisable to first
+  partially dehydrate the salt, in order to ensure the presence of the
+  second solid product of dissociation; the value of the dissociation
+  pressure being independent of the degree of dissociation of the hydrate
+  (p.&nbsp;<a href="#page86">86</a>). The small bulbs <i>d</i> and <i>e</i>
+  having been filled, the apparatus is placed on its side, so as to allow
+  the liquid to run from the bend of the tube into the bulbs <i>a</i> and
+  <i>b</i>; it is then exhausted through <i>f</i> by means of a mercury
+  pump, and sealed off. The apparatus is now placed in a perpendicular
+  position in a thermostat, and kept at constant temperature until
+  equilibrium is established. Since the vapour pressure on the side
+  containing the sulphuric acid may be regarded as zero, the difference in
+  level of the two surfaces of liquid in the <b>U</b>-tube gives directly
+  the dissociation pressure of the hydrate in terms of the particular
+  liquid employed; if the density of the latter is known, the pressure can
+  then be calculated to cm. of mercury.</p>
+
+  <div class="figcenter" style="width:11%;">
+      <a href="images/fig021.png"><img style="width:100%" src="images/fig021.png"
+      alt="Fig. 21" title="Fig. 21" /></a>
+    <span class="sc">Fig.</span> 21.
+  </div>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 92 --><span class="pagenum"><a name="page92"></a>{92}</span></p>
+
+<h3>CHAPTER VI</h3>
+
+<p class="cenhead">SOLUTIONS</p>
+
+  <p><b>Definition.</b>&mdash;In all the cases which have been considered
+  in the preceding pages, the different phases&mdash;with the exception of
+  the vapour phase&mdash;consisted of a single substance of definite
+  composition, or were definite chemical individuals.<a name="NtA164"
+  href="#Nt164"><sup>[164]</sup></a> But this invariability of the
+  composition is by no means imposed by the Phase Rule; on the contrary, we
+  shall find in the examples which we now proceed to study, that the
+  participation of phases of variable composition in the equilibrium of a
+  system is in no way excluded. To such phases of variable composition
+  there is applied the term <i>solution</i>. A solution, therefore, is to
+  be defined as <i>a homogeneous mixture, the composition of which can
+  undergo continuous variation within certain limits</i>; the limits,
+  namely, of its existence.<a name="NtA165"
+  href="#Nt165"><sup>[165]</sup></a></p>
+
+  <p>From this definition we see that the term solution is not restricted
+  to any particular physical state of substances, but includes within its
+  range not only the liquid, but also the gaseous and solid states. We may
+  therefore have solutions of gases in liquids, and of gases in solids; of
+  liquids in liquids or in solids; of solids in liquids, or of solids in
+  solids. Solutions of gases in gases are, of course, also possible; since,
+  however, gas solutions never give rise to more than one phase, their <!--
+  Page 93 --><span class="pagenum"><a
+  name="page93"></a>{93}</span>treatment does not come within the scope of
+  the Phase Rule, which deals with heterogeneous equilibria.</p>
+
+  <p>It should also be emphasized that the definition of solution given
+  above, neither creates nor recognizes any distinction between solvent and
+  dissolved substance (solute); and, indeed, a too persistent use of these
+  terms and the attempt to permanently label the one or other of two
+  components as the solvent or the solute, can only obscure the true
+  relationships and aggravate the difficulty of their interpretation. In
+  all cases it should be remembered that we are dealing with equilibria
+  between two components (we confine our attention in the first instance to
+  such), the solution being constituted of these components in variable and
+  varying amounts. The change from the case where the one component is in
+  great excess (ordinarily called the solvent) to that in which the other
+  component predominates, may be quite gradual, so that it is difficult or
+  impossible to say at what point the one component ceases to be the
+  solvent and becomes the solute. The adoption of this standpoint need not,
+  however, preclude one from employing the conventional terms solvent and
+  solute in ordinary language, especially when reference is made only to
+  some particular condition of equilibrium of the system, when the
+  concentration of the two components in the solution is widely
+  different.</p>
+
+<p class="cenhead"><span class="sc">Solutions of Gases in Liquids.</span></p>
+
+  <p>As the first class of solutions to which we shall turn our attention,
+  there may be chosen the solutions of gases in liquids, or the equilibria
+  between a liquid and a gas. These equilibria really constitute a part of
+  the equilibria to be studied more fully in Chapter VIII.; but since the
+  two-phase systems formed by the solutions of gases in liquids are among
+  the best-known of the two-component systems, a short section may be here
+  allotted to their treatment.</p>
+
+  <p>When a gas is passed into a liquid, absorption takes place to a
+  greater or less extent, and a point is at length reached when the liquid
+  absorbs no more of the gas; a condition of equilibrium is attained, and
+  the liquid is said to be saturated <!-- Page 94 --><span
+  class="pagenum"><a name="page94"></a>{94}</span>with the gas. In the
+  light of the Phase Rule, now, such a system is bivariant (two components
+  in two phases); and two of the variable factors, pressure, temperature,
+  and concentration of the components, must therefore be chosen in order
+  that the condition of the system may be defined. If the concentration and
+  the temperature are fixed, then the pressure is also defined; or under
+  given conditions of temperature and pressure, the concentration of the
+  gas in the solution must have a definite value. If, however, the
+  temperature alone is fixed, the concentration and the pressure can alter;
+  a fact so well known that it does not require to be further insisted
+  on.</p>
+
+  <p>As to the way in which the solubility of a gas in a liquid varies with
+  the pressure, the Phase Rule of course does not state; but guidance on
+  this point is again yielded by the theorem of van't Hoff and Le
+  Chatelier. Since the absorption of a gas is in all cases accompanied by a
+  diminution of the total volume, this process must take place with
+  increase of pressure. This, indeed, is stated in a quantitative manner in
+  the law of Henry, according to which the amount of a gas absorbed is
+  proportional to the pressure. But this law must be modified in the case
+  of gases which are very readily absorbed; the <i>direction of change</i>
+  of concentration with the pressure will, however, still be in accordance
+  with the theorem of Le Chatelier.</p>
+
+  <p>If, on the other hand, the pressure is fixed, then the concentration
+  will vary with the temperature; and since the absorption of gases is in
+  all cases accompanied by the evolution of heat, the solubility is found,
+  in accordance with the theorem of Le Chatelier, to diminish with rise of
+  temperature.</p>
+
+  <p>In considering the changes of pressure accompanying changes of
+  concentration and temperature, a distinction must be drawn between the
+  total pressure and the partial pressure of the dissolved gas, in cases
+  where the solvent is volatile. In these cases, the law of Henry applies
+  not to the total pressure of the vapour, but only to the partial pressure
+  of the dissolved gas. <!-- Page 95 --><span class="pagenum"><a
+  name="page95"></a>{95}</span></p>
+
+<p class="cenhead"><span class="sc">Solutions of Liquids in Liquids.</span></p>
+
+  <p>When mercury and water are brought together, the two liquids remain
+  side by side without mixing. Strictly speaking, mercury undoubtedly
+  dissolves to a certain extent in the water, and water no doubt dissolves,
+  although to a less extent, in the mercury; the amount of substance
+  passing into solution is, however, so minute, that it may, for all
+  practical purposes, be left out of account, so long as the temperature
+  does not rise much above the ordinary.<a name="NtA166"
+  href="#Nt166"><sup>[166]</sup></a> On the other hand, if alcohol and
+  water be brought together, complete miscibility takes place, and one
+  homogeneous solution is obtained. Whether water be added in increasing
+  quantities to pure alcohol, or pure alcohol be added in increasing amount
+  to water, at no point, at no degree of concentration, is a system
+  obtained containing more than one liquid phase. At the ordinary
+  temperature, water and alcohol can form only two phases, liquid and
+  vapour. If, however, water be added to ether, or if ether be added to
+  water, solution will not occur to an indefinite extent; but a point will
+  be reached when the water or the ether will no longer dissolve more of
+  the other component, and a further addition of water on the one hand, or
+  ether on the other, will cause the formation of two liquid layers, one
+  containing excess of water, the other excess of ether. We shall,
+  therefore, expect to find all grades of miscibility, from almost perfect
+  immiscibility to perfect miscibility, or miscibility in all proportions.
+  In cases of perfect immiscibility, the components do not affect one
+  another, and the system therefore remains unchanged. Such cases do not
+  call for treatment here. We have to concern ourselves here only with the
+  second and third cases, viz. with cases of complete and of partial
+  miscibility. There is no essential difference between the two classes,
+  for, as we shall see, <!-- Page 96 --><span class="pagenum"><a
+  name="page96"></a>{96}</span>the one passes into the other with change of
+  temperature. The formal separation into two groups is based on the
+  miscibility relations at ordinary temperatures.</p>
+
+  <p><b>Partial or Limited Miscibility.</b>&mdash;In accordance with the
+  Phase Rule, a pure liquid in contact with its vapour constitutes a
+  univariant system. If, however, a small quantity of a second substance is
+  added, which is capable of dissolving in the first, a bivariant system
+  will be obtained; for there are now two components and, as before, only
+  two phases&mdash;the homogeneous liquid solution and the vapour. At
+  constant temperature, therefore, both the composition of the solution and
+  the pressure of the vapour can undergo change; or, if the composition of
+  the solution remains unchanged, the pressure and the temperature can
+  alter. If the second (liquid) component is added in increasing amount,
+  the liquid will at first remain homogeneous, and its composition and
+  pressure will undergo a continuous change; when, however, the
+  concentration has reached a definite value, solution no longer takes
+  place; two liquid phases are produced. Since there are now three phases
+  present, two liquids and vapour, the system is univariant; at a given
+  temperature, therefore, the concentration of the components in the two
+  liquid phases, as well as the vapour pressure, must have definite values.
+  Addition of one of the components, therefore, cannot alter the
+  concentrations or the pressure, but can only cause a change in the
+  relative amounts of the phases.</p>
+
+  <p>The two liquid phases can be regarded, the one as a solution of the
+  component I. in component II., the other as a solution of component II.
+  in component I. If the pressure is maintained constant, then to each
+  temperature there will correspond a definite concentration of the
+  components in the two liquid phases; and addition of excess of one will
+  merely alter the relative amounts of the two solutions. As the
+  temperature changes, the composition of the two solutions will change,
+  and there will therefore be obtained two solubility curves, one showing
+  the solubility of component I. in component II., the other showing the
+  solubility of component II. in component I. Since heat may be either
+  evolved or absorbed when one liquid dissolves in another, the solubility
+  may diminish or increase <!-- Page 97 --><span class="pagenum"><a
+  name="page97"></a>{97}</span>with rise of temperature. The two solutions
+  which at a given temperature correspond to one another are known as
+  <i>conjugate solutions</i>.</p>
+
+  <p>The solubility relations of partially miscible liquids have been
+  studied by Guthrie,<a name="NtA167" href="#Nt167"><sup>[167]</sup></a>
+  and more especially by Alexejeff<a name="NtA168"
+  href="#Nt168"><sup>[168]</sup></a> and by Rothmund.<a name="NtA169"
+  href="#Nt169"><sup>[169]</sup></a> A considerable variety of curves have
+  been obtained, and we shall therefore discuss only a few of the different
+  cases which may be taken as typical of the rest.</p>
+
+  <p><b>Phenol and Water.</b>&mdash;When phenol is added to water at the
+  ordinary temperature, solution takes place, and a homogeneous liquid is
+  produced. When, however, the concentration of the phenol in the solution
+  has risen to about 8 per cent., phenol ceases to be dissolved; and a
+  further addition of it causes the formation of a second liquid phase,
+  which consists of excess of phenol and a small quantity of water. In
+  ordinary language it may be called a solution of water in phenol. If now
+  the temperature is raised, this second liquid phase will disappear, and a
+  further amount of phenol must be added in order to produce a separation
+  of the liquid into two layers. In this way, by increasing the amount of
+  phenol and noting the temperature at which the two layers disappear, the
+  so-called solubility curve of phenol in water can be obtained. By noting
+  the change of the solubility with the temperature in this manner, it is
+  found that at all temperatures below 68.4°, the addition of more than a
+  certain amount of phenol causes the formation of two layers; at
+  temperatures above this, however, two layers cannot be formed, no matter
+  how much phenol is added. At temperatures above 68.4°, therefore, water
+  and phenol are miscible in all proportions.</p>
+
+  <p>On the other hand, if water is added to phenol at the ordinary
+  temperature, a liquid is produced which consists chiefly of phenol, and
+  on increasing the amount of water beyond a certain point, two layers are
+  formed. On raising the temperature these two layers disappear, and a
+  homogeneous solution is again obtained. The phenomena are exactly
+  analogous to those already described. Since, now, in the second <!-- Page
+  98 --><span class="pagenum"><a name="page98"></a>{98}</span>case the
+  concentration of the phenol in the solution gradually decreases, while in
+  the former case it gradually increases, a point must at length be reached
+  at which the composition of the two solutions becomes the same. On mixing
+  the two solutions, therefore, one homogeneous liquid will be obtained.
+  But the point at which two phases become identical is called a critical
+  point, so that, in accordance with this definition, the temperature at
+  which the two solutions of phenol and water become identical may be
+  called the <i>critical solution temperature</i>, and the concentration at
+  this point may be called the <i>critical concentration</i>.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig022.png"><img style="width:100%" src="images/fig022.png"
+      alt="Fig. 22" title="Fig. 22" /></a>
+    <span class="sc">Fig. 22.</span>
+  </div>
+
+  <p>From what has been said above, it will be seen that at any temperature
+  below the critical solution temperature, two conjugate solutions
+  containing water and phenol in different concentration can exist
+  together, one containing excess of water, the other excess of phenol. The
+  following table gives the composition of the two layers, and the values
+  are represented graphically in Fig. 22.<a name="NtA170"
+  href="#Nt170"><sup>[170]</sup></a></p>
+
+<p class="cenhead"><span class="sc">Phenol and Water.</span></p>
+
+<p class="cenhead">C<sub>1</sub> is the percentage amount of phenol in the first layer.&nbsp; &nbsp; &nbsp;<br />
+C<sub>2</sub> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; second layer.</p>
+<table class="nobctr" summary="Composition of phenol and water layers" title="Composition of phenol and water layers">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> C<sub>1</sub>.
+</td><td class="toboleb" style="text-align:center"> C<sub>2</sub>.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 8.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 72.2</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 30° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 8.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 69.9</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 40° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 9.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 66.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 50° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 12.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 62.7</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 55° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 14.2 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 60.0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 60° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 17.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 56.2</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 65° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 22.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 49.7</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 68.4° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> 36.1 </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> 36.1</td></tr>
+</table>
+
+<p><!-- Page 99 --><span class="pagenum"><a name="page99"></a>{99}</span></p>
+
+  <p>The critical solution temperature for phenol and water is 68.4°, the
+  critical concentration 36.1 per cent. of phenol. At all temperatures
+  above 68.4°, only homogeneous solutions of phenol and water can be
+  obtained; water and phenol are then miscible in all proportions.</p>
+
+  <p>At the critical solution point the system exists in only two
+  phases&mdash;liquid and vapour. It ought, therefore, to possess two
+  degrees of freedom. The restriction is, however, imposed that the
+  composition of the two liquid phases, coexisting at a point infinitely
+  near to the critical point, becomes the same, and this disposes of one of
+  the degrees of freedom. The system is therefore univariant; and at a
+  given temperature the pressure will have a definite value. Conversely, if
+  the pressure is fixed (as is the case when the system is under the
+  pressure of its own vapour), then the temperature will also be fixed;
+  that is, the critical solution temperature has a definite value depending
+  only on the substances. If the vapour phase is omitted, the temperature
+  will alter with the pressure; in this case, however, as in the case of
+  other condensed systems, the effect of pressure is slight.</p>
+
+  <p>From Fig. 22 it is easy to predict the effect of bringing together
+  water and phenol in any given quantities at any temperature. Start with a
+  solution of phenol and water having the composition represented by the
+  point <i>x</i>. If to this solution phenol is added at constant
+  temperature, it will dissolve, and the composition of the solution will
+  gradually change, as shown by the dotted line <i>xy</i>. When, however,
+  the concentration has reached the value represented by the point
+  <i>y</i>, two liquid layers will be formed, the one solution having the
+  composition represented by <i>y</i>, the other that represented by
+  <i>y&prime;</i>. The system is now univariant, and on further addition of
+  phenol, the composition of the two liquid phases will remain unchanged,
+  but their relative amounts will alter. The phase richer in phenol will
+  increase in amount; that richer in water will decrease, and ultimately
+  disappear, and there will remain the solution <i>y&prime;</i>. Continued
+  addition of phenol will then lead to the point <i>x&prime;</i>, there
+  being now only one liquid phase present.</p>
+
+  <p>Since the critical solution point represents the highest temperature
+  at which two liquid phases consisting of phenol and <!-- Page 100
+  --><span class="pagenum"><a name="page100"></a>{100}</span>water can
+  exist together, these two substances can be brought together in any
+  amount whatever at temperatures higher than 68.4°, without the formation
+  of two layers. It will therefore be possible to pass from a system
+  represented by <i>x</i> to one represented by <i>x&prime;</i>, without at
+  any time two liquid phases appearing. Starting with <i>x</i>, the
+  temperature is first raised above the critical solution temperature;
+  phenol is then added until the concentration reaches the point
+  <i>x</i><sub>2</sub>. On allowing the temperature to fall, the system
+  will then pass into the condition represented by <i>x&prime;</i>.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig023.png"><img style="width:100%" src="images/fig023.png"
+      alt="Fig. 23" title="Fig. 23" /></a>
+    <span class="sc">Fig. 23.</span>
+  </div>
+
+  <p><b>Methylethylketone and Water.</b>&mdash;In the case just described,
+  the solubility of each component in the other increased continuously with
+  the temperature. There are, however, cases where a maximum or minimum of
+  solubility is found, <i>e.g.</i> methylethylketone and water. The curve
+  which represents the equilibria between these two substances is given in
+  Fig. 23, the concentration values being contained in the following
+  table:<a name="NtA171" href="#Nt171"><sup>[171]</sup></a>&mdash;</p>
+
+<p class="cenhead"><span class="sc">Methylethylketone and Water.</span></p>
+
+<table class="nobctr" summary="Composition of methylethylketone and water layers" title="Composition of methylethylketone and water layers">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> C<sub>1</sub> per cent.
+</td><td class="toboleb" style="text-align:center"> C<sub>2</sub> per cent.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp;-10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 34.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 89.7</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> +10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 26.1 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 90.0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 30° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 21.9 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 89.9</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 50° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 17.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 89.0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 70° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 16.2 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 85.7</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 90° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 16.1 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 84.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 110° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 17.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 80.0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 130° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 21.8 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 71.9</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 140° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 26.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 64.0</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 151.8° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> 44.2 </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> 44.2</td></tr>
+
+</table>
+
+<p><!-- Page 101 --><span class="pagenum"><a name="page101"></a>{101}</span></p>
+
+  <p>These numbers and Fig. 23 show clearly the occurrence of a minimum in
+  the solubility of the ketone in water, and also a minimum (at about 10°)
+  in the solubility of water in methylethylketone. Minima of solubility
+  have also been found in other cases.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig024.png"><img style="width:100%" src="images/fig024.png"
+      alt="Fig. 24" title="Fig. 24" /></a>
+    <span class="sc">Fig. 24.</span>
+  </div>
+
+  <p><b>Triethylamine and Water.</b>&mdash;Although in most of the cases
+  studied the solubility of one liquid in another increases with rise of
+  temperature, this is not so in all cases. Thus, at temperatures below
+  18°, triethylamine and water mix together in all proportions; but, on
+  raising the temperature, the homogeneous solution becomes turbid and
+  separates into two layers. In this case, therefore, the critical solution
+  temperature is found in the direction of lower temperature, not in the
+  direction of higher.<a name="NtA172" href="#Nt172"><sup>[172]</sup></a>
+  This behaviour is clearly shown by the graphic representation in Fig. 24,
+  and also by the numbers in the following table:&mdash;</p>
+
+<p class="cenhead"><span class="sc">Triethylamine and Water.</span></p>
+
+<table class="nobctr" summary="Composition of triethylamine and water layers" title="Composition of triethylamine and water layers">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> C<sub>1</sub> per cent.
+</td><td class="toboleb" style="text-align:center"> C<sub>2</sub> per cent.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 70° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 1.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 50° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 2.9 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 30° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 5.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 96&nbsp; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 7.3 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 95.5</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 15.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 73&nbsp; &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> ±18.5° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> ±30&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em"> ±30&nbsp; &nbsp;</td></tr>
+</table>
+
+  <p><b>General Form of Concentration-Temperature Curve.</b>&mdash;From the
+  preceding figures it will be seen that the general <!-- Page 102 --><span
+  class="pagenum"><a name="page102"></a>{102}</span>form of the solubility
+  curve is somewhat parabolic in shape; in the case of triethylamine and
+  water, the closed end of the curve is very flat. Since for all liquids
+  there is a point (critical point) at which the liquid and gaseous states
+  become identical, and since all gases are miscible in all proportions, it
+  follows that there must be some temperature at which the liquids become
+  perfectly miscible. In the case of triethylamine and water, which has
+  just been considered, there must therefore be an upper critical solution
+  temperature, so that the complete solubility relations would be
+  represented by a closed curve of an ellipsoidal aspect. An example of
+  such a curve is furnished by nicotine and water. At temperatures below
+  60° and above 210°, nicotine and water mix in all proportions.<a
+  name="NtA173" href="#Nt173"><sup>[173]</sup></a> Although it is possible
+  that this is the general form of the curve for all pairs of liquids,
+  there are as yet insufficient data to prove it.</p>
+
+  <p>With regard to the closed end of the curve it may be said that it is
+  continuous; the critical solution point is not the intersection of two
+  curves, for such a break in the continuity of the curve could occur only
+  if there were some discontinuity in one of the phases. No such
+  discontinuity exists. The curve is, therefore, not to be considered as
+  two solubility curves cutting at a point; it is a curve of equilibrium
+  between two components, and so long as the phases undergo continuous
+  change, the curve representing the equilibrium must also be continuous.
+  As has already been emphasized, a distinction between solvent and solute
+  is merely conventional (p.&nbsp;<a href="#page93">93</a>).</p>
+
+  <p><b>Pressure-Concentration Diagram.</b>&mdash;In considering the
+  pressure-concentration diagram of a system of two liquid components, a
+  distinction must be drawn between the total pressure of the system and
+  the partial pressures of the components. On studying the total pressure
+  of a system, it is found that two cases can be obtained.<a name="NtA174"
+  href="#Nt174"><sup>[174]</sup></a></p>
+
+  <p>So long as there is only one liquid phase, the system is bivariant.
+  The pressure therefore can change with the concentration and the
+  temperature. If the temperature is maintained <!-- Page 103 --><span
+  class="pagenum"><a name="page103"></a>{103}</span>constant, the pressure
+  will vary only with the concentration, and this variation can therefore
+  be represented by a curve. If, however, two liquid phases are formed, the
+  system becomes univariant: and if one of the variables, say the
+  temperature, is arbitrarily fixed, the system no longer possesses any
+  degree of freedom. <i>When two liquid phases are formed, therefore, the
+  concentrations and the vapour pressure have definite values, which are
+  maintained so long as the two liquid phases are present</i>; the
+  temperature being supposed constant.</p>
+
+  <p>In Fig. 25 is given a diagrammatic representation of the two kinds of
+  pressure-concentration curves which have so far been obtained. In the one
+  case, the vapour pressure of the invariant system (at constant
+  temperature) lies higher than the vapour pressure of either of the pure
+  components; a phenomenon which is very generally found in the case of
+  partially miscible liquids, <i>e.g.</i> ether and water.<a name="NtA175"
+  href="#Nt175"><sup>[175]</sup></a> Accordingly, by the addition of water
+  to ether, or of ether to water, there is an increase in the <i>total</i>
+  vapour pressure of the system.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig025.png"><img style="width:100%" src="images/fig025.png"
+      alt="Fig. 25" title="Fig. 25" /></a>
+    <span class="sc">Fig. 25.</span>
+  </div>
+
+  <p>With regard to the second type, the vapour pressure of the systems
+  with two liquid phases lies between that of the two single components. An
+  example of this is found in sulphur dioxide and water.<a name="NtA176"
+  href="#Nt176"><sup>[176]</sup></a> On adding sulphur dioxide to water
+  there is an increase of the total vapour pressure; but on adding water to
+  liquid sulphur dioxide, the total vapour pressure is diminished.</p>
+
+  <p>The case that the vapour pressure of the system with two <!-- Page 104
+  --><span class="pagenum"><a name="page104"></a>{104}</span>liquid phases
+  is <i>less</i> than that of each of the components is not possible.</p>
+
+  <p>With regard to the <i>partial pressure</i> of the components, the
+  behaviour is more uniform. The partial pressure of one component is in
+  all cases lowered by the addition of the other component, the diminution
+  being approximately proportional to the amount added. If two liquid
+  phases are present, the partial pressure of the components, as well as
+  the total pressure, is constant, and is the same for both phases. That is
+  to say, in the case of the two liquids, saturated solution of water in
+  ether, and of ether in water, the partial pressure of the ether in the
+  vapour in contact with the one solution is the same as that in the vapour
+  over the other solution.<a name="NtA177"
+  href="#Nt177"><sup>[177]</sup></a></p>
+
+  <p><b>Complete Miscibility.</b>&mdash;Although the phenomena of complete
+  miscibility are here treated under a separate heading, it must not be
+  thought that there is any essential difference between those cases where
+  the liquids exhibit limited miscibility and those in which only one
+  homogeneous solution is formed. As has been already pointed out, the
+  solubility relations alter with the temperature; and liquids which at one
+  temperature can dissolve in one another only to a limited extent, are
+  found at some other temperature to possess the property of complete
+  miscibility. Conversely, we may expect that liquids which at one
+  temperature, say at the ordinary temperature, are miscible in all
+  proportions, will be found at some other temperature to be only partially
+  miscible. Thus, for example, it was found by Guthrie that ethyl alcohol
+  and carbon disulphide, which are miscible in all proportions at the
+  ordinary temperature, possess only limited miscibility at temperatures
+  below -14.4°.<a name="NtA178" href="#Nt178"><sup>[178]</sup></a>
+  Nevertheless, it is doubtful if the critical solution temperature is in
+  all cases experimentally realizable.</p>
+
+  <p><b>Pressure-Concentration Diagram.</b>&mdash;Since, in the cases of
+  complete miscibility of two liquid components, there are never more than
+  two phases present, the system must always be bivariant; and two of the
+  variables pressure, temperature or concentration of the components, must
+  be arbitrarily chosen <!-- Page 105 --><span class="pagenum"><a
+  name="page105"></a>{105}</span>before the system becomes defined. For
+  this reason the Phase Rule affords only a slight guidance in the study of
+  such equilibria; and we shall therefore not enter in detail into the
+  behaviour of these homogeneous mixtures. All that the Phase Rule can tell
+  us in connection with these solutions, is that at constant temperature
+  the vapour pressure of the solution varies with the composition of the
+  liquid phase; and if the composition of the liquid phase remains
+  unchanged, the pressure also must remain unchanged. This constancy of
+  composition is exhibited not only by pure liquids, but also by liquid
+  solutions in all cases where the vapour pressure of the solution reaches
+  a maximum or minimum value. This is the case, for example, with mixtures
+  of constant boiling point.<a name="NtA179"
+  href="#Nt179"><sup>[179]</sup></a></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 106 --><span class="pagenum"><a name="page106"></a>{106}</span></p>
+
+<h3>CHAPTER VII</h3>
+
+<p class="cenhead">SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE
+COMPONENTS BEING VOLATILE</p>
+
+  <p><b>General.</b>&mdash;When a solid is brought into contact with a
+  liquid in which it can dissolve, a certain amount of it passes into
+  solution; and the process continues until the concentration reaches a
+  definite value independent of the amount of solid present. A condition of
+  equilibrium is established between the solid and the solution; the
+  solution becomes <i>saturated</i>. Since the number of components is two,
+  and the number of phases three, viz. solid, liquid solution, vapour, the
+  system is univariant. If, therefore, one of the factors, pressure,
+  temperature, or concentration of the components (in the solution<a
+  name="NtA180" href="#Nt180"><sup>[180]</sup></a>), is arbitrarily fixed,
+  the state of the system becomes perfectly defined. Thus, at any given
+  temperature, the vapour pressure of the system and the concentration of
+  the components have a definite value. If the temperature is altered, the
+  vapour pressure and also, in general, the concentration will undergo
+  change. Likewise, if the pressure varies, while the system is isolated so
+  that no heat can pass between it and its surroundings, the concentration
+  and the temperature must also undergo variation until they attain values
+  corresponding to the particular pressure.</p>
+
+  <p>That the temperature has an influence, sometimes a very considerable
+  influence, on the amount of substance passing into solution, is
+  sufficiently well known; the effect of pressure, although less apparent,
+  is no less certain. If at any given temperature the volume of the vapour
+  phase is diminished, <!-- Page 107 --><span class="pagenum"><a
+  name="page107"></a>{107}</span>vapour will condense to liquid, in order
+  that the pressure may remain constant, and so much of the solid will pass
+  into solution that the concentration may remain unchanged; for, so long
+  as the three phases are present, the state of the system cannot alter.
+  If, however, one of the phases, <i>e.g.</i> the vapour phase, disappears,
+  the system becomes bivariant; at any given temperature, therefore, there
+  may be different values of concentration and pressure.</p>
+
+  <p>The direction in which change of concentration will occur with change
+  of pressure can be predicted by means of the theorem of Le Chatelier, if
+  it is known whether solution is accompanied by increase or diminution of
+  the total volume. If diminution of the total volume of the system occurs
+  on solution, increase of pressure will increase the solubility; in the
+  reverse case, increase of pressure will diminish the solubility.</p>
+
+  <p>This conclusion has also been verified by experiment, as is shown by
+  the following figures.<a name="NtA181"
+  href="#Nt181"><sup>[181]</sup></a></p>
+
+<table class="nobctr" summary="Solubility of salts" title="Solubility of salts">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Salt.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Change of<br />volume by<br />dissolving 1 gm.<br />of salt in the<br />saturated<br />solution.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Solubility (at 18°) (grams salt in<br />1 gram of solution).</td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Pressure<br />= 1 atm.
+</td><td class="toboleb" style="text-align:center"> Pressure<br />= 500 atm.</td></tr>
+
+<tr><td class="rib" style="text-align:left;"> Sodium chloride </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> -0.07 &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.264 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.270</td></tr>
+<tr><td class="rib" style="text-align:left;"> Ammonium chloride </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> +0.10 &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.272 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.258</td></tr>
+<tr><td class="rib" style="text-align:left;"> Alum </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> -0.067 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.115 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em"> 0.142</td></tr>
+<tr><td class="ribob" style="text-align:left;"> </td><td class="boleb" style="text-align:left;"> </td><td class="boleb" style="text-align:left;"> </td><td class="boleb" style="text-align:left;">(<i>p</i> = 400 atm.)</td></tr>
+
+</table>
+
+  <p>As can be seen, a large increase of the pressure brings about a no
+  more than appreciable alteration of the solubility; a result which is
+  due, as in the case of the alteration of the fusion point with the
+  pressure, to the small change in volume accompanying solution or increase
+  of pressure. For all practical purposes, therefore, the solubility as
+  determined under atmospheric pressure may be taken as equal to the true
+  <!-- Page 108 --><span class="pagenum"><a
+  name="page108"></a>{108}</span>solubility, that is, the solubility when
+  the system is under the pressure of its own vapour.</p>
+
+  <p><b>The Saturated Solution.</b>&mdash;From what has been said above, it
+  will be seen that the condition of saturation of a solution can be
+  defined only with respect to a certain solid phase; if no solid is
+  present, the system is undefined, for it then consists of only two
+  phases, and is therefore bivariant. Under such circumstances not only can
+  there be at one given temperature solutions of different concentration,
+  all containing less of one of the components than when that component is
+  present in the solid form, but there can also exist solutions containing
+  more of that component than corresponds to the equilibrium when the solid
+  is present. In the former case the solutions are <i>unsaturated</i>, in
+  the latter case they are <i>supersaturated with respect to a certain
+  solid phase</i>; in themselves, the solutions are stable, and are neither
+  unsaturated nor supersaturated. Further, if the solid substance can exist
+  in different allotropic modifications, the particular form of the
+  substance which is in equilibrium with the solution must be known, in
+  order that the statement of the solubility may be definite; for each form
+  has its own solubility, and, as we shall see presently, the less stable
+  form has the greater solubility (cf. p.&nbsp;<a href="#page47">47</a>). In all
+  determinations of the solubility, therefore, not only must the
+  concentration of the components in the solution be determined, but equal
+  importance should be attached to the characterisation of the solid phase
+  present.</p>
+
+  <p>In this connection, also, one other point may be emphasised. For the
+  production of the equilibrium between a solid and a liquid, time is
+  necessary, and this time not only varies with the state of division of
+  the solid and the efficiency of the stirring, but is also dependent on
+  the nature of the substance.<a name="NtA182"
+  href="#Nt182"><sup>[182]</sup></a> Considerable care must therefore be
+  taken that sufficient time is allowed for equilibrium to be established.
+  Such care is more especially needful when changes may occur in the solid
+  phase, and neglect of it has greatly diminished the value of many of the
+  older determinations of solubility.</p>
+
+  <p><b>Form of the Solubility Curve.</b>&mdash;The solubility
+  curve&mdash;that <!-- Page 109 --><span class="pagenum"><a
+  name="page109"></a>{109}</span>is, the curve representing the change of
+  concentration of the components in the solution with the
+  temperature&mdash;differs markedly from the curve of vapour pressure
+  (p.&nbsp;<a href="#page63">63</a>), in that it possesses no general form, but
+  may vary in the most diverse manner. Not only may the curve have an
+  almost straight and horizontal course, or slope or curve upwards at
+  varying angles; but it may even slope downwards, corresponding to a
+  decrease in the solubility with rise of temperature; may exhibit maxima
+  or minima of solubility, or may, as in the case of some hydrated salts,
+  pass through a point of maximum temperature. In the latter case the salt
+  may possess two values of solubility at the same temperature. We shall
+  consider these cases in the following chapter.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig026.png"><img style="width:100%" src="images/fig026.png"
+      alt="Fig. 26" title="Fig. 26" /></a>
+    <span class="sc">Fig.</span> 26.
+  </div>
+
+  <p>The great variety of form shown by solubility curves is at once
+  apparent from Fig. 26, in which the solubility curves of various
+  substances (not, however, drawn to scale) are reproduced.<a name="NtA183"
+  href="#Nt183"><sup>[183]</sup></a></p>
+
+  <p>Varied as is the form of the solubility curve, its <i>direction</i>,
+  nevertheless, can be predicted by means of the theorem of van't Hoff and
+  Le Chatelier; for in accordance with that theorem (p.&nbsp;<a
+  href="#page57">57</a>) increase of solubility with the temperature must
+  occur in those cases where the process of solution is accompanied by an
+  <i>absorption</i> of heat; and a decrease in the solubility with rise of
+  temperature will be found in cases where solution occurs with
+  <i>evolution</i> of heat. Where there is no heat effect accompanying
+  solution, <!-- Page 110 --><span class="pagenum"><a
+  name="page110"></a>{110}</span>change of temperature will be without
+  influence on the solubility; and if the sign of the heat of solution
+  changes, the direction of the solubility curve must also change,
+  <i>i.e.</i> must show a maximum or minimum point. This has in all cases
+  been verified by experiment.<a name="NtA184"
+  href="#Nt184"><sup>[184]</sup></a></p>
+
+  <p>In applying the theorem of Le Chatelier to the course of the
+  solubility curve, it should be noted that by heat of solution there is
+  meant, not the heat effect produced on dissolving the salt in a large
+  amount of solvent (which is the usual signification of the expression),
+  but the heat which is absorbed or evolved when the salt is dissolved in
+  the almost saturated solution (the so-called last heat of solution). Not
+  only does the heat effect in the two cases have a different value, but it
+  may even have a different sign. A striking example of this is afforded by
+  cupric chloride, as the following figures show:<a name="NtA185"
+  href="#Nt185"><sup>[185]</sup></a>&mdash;</p>
+
+<table class="nobctr" summary="Heat of solution, cupric chloride" title="Heat of solution, cupric chloride">
+<tr><td class="toribob" style="text-align:center"> Number of gram-molecules of<br />CuCl<sub>2</sub>, 2H<sub>2</sub>O dissolved in 198<br />gram-molecules of water.
+</td><td class="toboleb" style="text-align:center"> Heat effect.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:5em"> &nbsp; 1 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp; +37 K</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> &nbsp; 2.02 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp; +66 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> &nbsp; 4.15 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> +105 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> &nbsp; 7.07 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> +117 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> &nbsp; 9.95 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> +117 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> 11 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp; +91 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> 18.8 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp; &nbsp;-10 &nbsp;,,</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:5em"> 19.6 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp; &nbsp;-31 &nbsp;,,</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:5em"> 24.75 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em"> &nbsp;-198 &nbsp;,,</td></tr>
+
+</table>
+
+  <p>In the above table the positive sign indicates evolution of heat, the
+  negative sign, absorption of heat; and the values of the heat effect are
+  expressed in centuple calories. Judging from the heat effect produced on
+  dissolving cupric chloride in a large bulk of water, we should predict
+  that the solubility of that salt would diminish with rise of temperature;
+  as a matter of fact, it increases. This is in accordance with the fact
+  that <!-- Page 111 --><span class="pagenum"><a
+  name="page111"></a>{111}</span>the last heat of solution is
+  <i>negative</i> (as expressed above), <i>i.e.</i> solution of the salt in
+  the almost saturated solution is accompanied by absorption of heat. We
+  are led to expect this from the fact that the heat of solution changes
+  sign from positive to negative as the concentration increases; experiment
+  also showed it to be the case.</p>
+
+  <p>Despite its many forms, it should be particularly noted that the
+  solubility curve of any substance is <i>continuous</i>, so long as the
+  solid phase, or solid substance in contact with the solution, remains
+  unchanged. If any "break" or discontinuous change in the direction of the
+  curve occurs, it is a sign that the <i>solid phase has undergone
+  alteration</i>. Conversely, if it is known that a change takes place in
+  the solid phase, a break in the solubility curve can be predicted. We
+  shall presently meet with examples of this.<a name="NtA186"
+  href="#Nt186"><sup>[186]</sup></a></p>
+
+<p class="cenhead">A.&mdash;<span class="sc">Anhydrous Salt and Water</span>.</p>
+
+  <p><b>The Solubility Curve.</b>&mdash;In studying the equilibria in those
+  systems of two components in which the liquid phase is a solution or
+  phase of varying composition, we shall in the present chapter limit the
+  discussion to those cases where no compounds are formed, but where the
+  components crystallise out in the pure state. Since some of the
+  best-known examples of such systems are yielded by the solutions of
+  anhydrous salts in water, we shall first of all briefly consider some of
+  the results which have been obtained with them.</p>
+
+  <p>For the most part the solubility curves have been studied only at
+  temperatures lying between 0° and 100°, the solid phase in contact with
+  the solution being the anhydrous salt. For the representation of these
+  equilibria, the concentration-temperature <!-- Page 112 --><span
+  class="pagenum"><a name="page112"></a>{112}</span>diagram is employed,
+  the concentration being expressed as the number of grams of the salt
+  dissolved in 100 grams of water, or as the number of gram-molecules of
+  salt in 100 gram-molecules of water. The curves thus obtained exhibit the
+  different forms to which reference has already been made. So long as the
+  salt remains unchanged the curve will be continuous, but if the salt
+  alters its form, then the solubility curve will show a break.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig027.png"><img style="width:100%" src="images/fig027.png"
+      alt="Fig. 27" title="Fig. 27" /></a>
+    <span class="sc">Fig.</span> 27.
+  </div>
+
+  <p>Now, we have already seen in Chapter III. that certain substances are
+  capable of existing in various crystalline forms, and these forms are so
+  related to one another that at a given temperature the relative stability
+  of each pair of polymorphic forms undergoes change. Since each
+  crystalline variety of a substance must have its own solubility, there
+  must be a break in the solubility curve at the temperature of transition
+  of the two enantiotropic forms. At this point the two solubility curves
+  must cut, for since the two forms are in equilibrium with respect to
+  their vapour, they must also be in equilibrium with respect to their
+  solutions. From the table on p.&nbsp;<a href="#page63">63</a> it is seen that
+  potassium nitrate, ammonium nitrate, silver nitrate, thallium nitrate,
+  thallium picrate, are capable of existing in two or more different
+  enantiotropic crystalline forms, the range of stability of these forms
+  being limited by definite temperatures (transition temperature). Since
+  the transition point is not altered by a solvent (provided the latter is
+  not absorbed by the solid phase), we should find on studying the
+  solubility of these substances in water that the solubility curve would
+  exhibit a change in direction at the temperature of transition. As a
+  matter of fact this has been verified, more especially in the case of
+  ammonium nitrate<a name="NtA187" href="#Nt187"><sup>[187]</sup></a> <!--
+  Page 113 --><span class="pagenum"><a name="page113"></a>{113}</span>and
+  thallium picrate.<a name="NtA188" href="#Nt188"><sup>[188]</sup></a> The
+  following table contains the values of the solubility of ammonium nitrate
+  obtained by Müller and Kaufmann, the solubility being expressed in
+  gram-molecules NH<sub>4</sub>NO<sub>3</sub> in 100 gram-molecules of
+  water. In Fig. 27 these results are represented graphically. The
+  equilibrium point was approached both from the side of unsaturation and
+  of supersaturation, and the condition of equilibrium was controlled by
+  determinations of the density of the solution.</p>
+
+<p class="cenhead"><span class="sc">Solubility of Ammonium Nitrate.</span></p>
+
+<table class="nobctr" summary="Solubility of Ammonium Nitrate" title="Solubility of Ammonium Nitrate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.
+</td><td class="toboleb" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 12.2° </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.50 </td><td class="leb" style="text-align:left; padding-left:2em;"> 32.7° </td><td class="leb" style="text-align:left; padding-left:2em;"> 57.90</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 20.2° </td><td class="leb" style="text-align:left; padding-left:2em;"> 43.30 </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 58.89</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 25.05° </td><td class="leb" style="text-align:left; padding-left:2em;"> 48.19 </td><td class="leb" style="text-align:left; padding-left:2em;"> 35.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 59.80</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 28.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 51.86 </td><td class="leb" style="text-align:left; padding-left:2em;"> 36.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 61.00</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 30.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 54.40 </td><td class="leb" style="text-align:left; padding-left:2em;"> 37.5° </td><td class="leb" style="text-align:left; padding-left:2em;"> 62.90</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 30.2° </td><td class="leb" style="text-align:left; padding-left:2em;"> 54.61 </td><td class="leb" style="text-align:left; padding-left:2em;"> 38.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 63.60</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 31.9° </td><td class="leb" style="text-align:left; padding-left:2em;"> 57.20 </td><td class="leb" style="text-align:left; padding-left:2em;"> 39.0° </td><td class="leb" style="text-align:left; padding-left:2em;"> 65.09</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 32.1° </td><td class="boleb" style="text-align:left; padding-left:2em;"> 57.60 </td><td class="boleb" style="text-align:left; padding-left:2em;"> 40.0° </td><td class="boleb" style="text-align:left; padding-left:2em;"> 66.80</td></tr>
+
+</table>
+
+  <p>From the graphic representation of the solubility given in Fig. 27,
+  there is seen to be a distinct change in the direction of the curve at a
+  temperature of 32°; and this break in the curve corresponds to the
+  transition of the <span class="grk">&beta;</span>-rhombic into the <span
+  class="grk">&alpha;</span>-rhombic form of ammonium nitrate (p.&nbsp;<a
+  href="#page63">63</a>).</p>
+
+  <p><b>Suspended Transformation and Supersaturation.</b>&mdash;As has
+  already been learned, the transformation of the one crystalline form into
+  the other does not necessarily take place immediately the transition
+  point has been passed; and it has therefore been found possible in a
+  number of cases to follow the solubility curve of a given crystalline
+  form beyond the point at which it ceases to be the most stable
+  modification. Now, it will be readily seen from Fig. 27 that if the two
+  solubility curves be prolonged beyond the point of intersection, the
+  solubility of the less stable form is greater than that of the more
+  stable. A solution, therefore, which is saturated with respect to the
+  less stable form, <i>i.e.</i> which is in equilibrium with that form, is
+  <i>supersaturated with respect to the more stable modification</i>. If,
+  <!-- Page 114 --><span class="pagenum"><a
+  name="page114"></a>{114}</span>therefore, a small quantity of the more
+  stable form is introduced into the solution, the latter must deposit such
+  an amount of the more stable form that the concentration of the solution
+  corresponds to the solubility of the stable form at the particular
+  temperature. Since, however, the solution is now <i>unsaturated</i> with
+  respect to the less stable variety, the latter, if present, must pass
+  into solution; and the two processes, deposition of the stable and
+  solution of the metastable form, must go on until the latter form has
+  entirely disappeared and a saturated solution of the stable form is
+  obtained. There will thus be a conversion, through the medium of the
+  solvent, of the less stable into the more stable modification. This
+  behaviour is of practical importance in the determination of transition
+  points (<i>v.</i> Appendix).</p>
+
+  <p>From the above discussion it will be seen how important is the
+  statement of the solid phase for the definition of saturation and
+  supersaturation.<a name="NtA189" href="#Nt189"><sup>[189]</sup></a></p>
+
+  <p><b>Solubility Curve at Higher Temperatures.</b>&mdash;On passing to
+  the consideration of the solubility curves at higher temperatures, two
+  chief cases must be distinguished.</p>
+
+<blockquote class="b1n">
+
+  <p>(1) The two components in the fused state can mix in all
+  proportions.</p>
+
+  <p>(2) The two components in the fused state cannot mix in all
+  proportions.</p>
+
+</blockquote>
+
+<p class="cenhead">1. <i>Complete Miscibility of the Fused Components.</i></p>
+
+  <div class="figcenter" style="width:35%;">
+      <a href="images/fig028.png"><img style="width:100%" src="images/fig028.png"
+      alt="Fig. 28" title="Fig. 28" /></a>
+    <span class="sc">Fig.</span> 28.
+  </div>
+
+  <p>The best example of this which has been studied, so far as anhydrous
+  salts and water are concerned, is that of silver nitrate and water. The
+  solubility of this salt at temperatures <!-- Page 115 --><span
+  class="pagenum"><a name="page115"></a>{115}</span>above 100° has been
+  studied chiefly by Etard<a name="NtA190"
+  href="#Nt190"><sup>[190]</sup></a> and by Tilden and Shenstone.<a
+  name="NtA191" href="#Nt191"><sup>[191]</sup></a> The values obtained by
+  Etard are given in the following table, and represented graphically in
+  Fig. 28.</p>
+
+<p class="cenhead"><span class="sc">Solubility of Silver Nitrate.</span></p>
+
+<table class="nobctr" summary="Solubility of Silver Nitrate" title="Solubility of Silver Nitrate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Parts of dry salt in 100 parts<br />of solution.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; &nbsp;-7° </td><td class="leb" style="text-align:left; padding-left:4em;"> 46.2</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; &nbsp;-1° </td><td class="leb" style="text-align:left; padding-left:4em;"> 52.1</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; +5° </td><td class="leb" style="text-align:left; padding-left:4em;"> 56.3</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 10° </td><td class="leb" style="text-align:left; padding-left:4em;"> 61.2</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 20° </td><td class="leb" style="text-align:left; padding-left:4em;"> 67.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 40.5° </td><td class="leb" style="text-align:left; padding-left:4em;"> 76.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> &nbsp; 73° </td><td class="leb" style="text-align:left; padding-left:4em;"> 84.0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em"> 135° </td><td class="leb" style="text-align:left; padding-left:4em;"> 92.8</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em"> 182° </td><td class="boleb" style="text-align:left; padding-left:4em;"> 96.9</td></tr>
+
+</table>
+
+  <p>In this figure the composition of the solution is expressed in parts
+  of silver nitrate in 100 parts by weight of the solution, so that 100 per
+  cent. represents pure silver nitrate. As can be seen, the solubility
+  increases with the temperature. At a temperature of about 160° there
+  should be a break in the curve due to change of crystalline form (p.&nbsp;<a
+  href="#page63">63</a>). Such a change in the direction of the solubility
+  curve, however, does not in any way alter the essential nature of the
+  relationships discussed here, and may for the present be left out of
+  account. On following the solubility curve of silver nitrate to higher
+  temperatures, therefore, the concentration of silver nitrate in the
+  solution gradually increases, until at last, at a temperature of 208°,<a
+  name="NtA192" href="#Nt192"><sup>[192]</sup></a> the melting point of
+  pure silver nitrate is reached, and the concentration of the water has
+  become zero. The curve throughout its whole extent represents the
+  equilibrium between silver nitrate, solution, and vapour. Conversely,
+  starting with pure silver nitrate in contact with the fused salt,
+  addition of water will lower the melting point, <i>i.e.</i> will lower
+  the temperature at which the solid salt can exist in contact with the
+  liquid; <!-- Page 116 --><span class="pagenum"><a
+  name="page116"></a>{116}</span>and the depression will be all the greater
+  the larger the amount of water added. As the concentration of the water
+  in the liquid phase is increased, therefore, the system will pass back
+  along the curve from higher to lower temperatures, and from greater to
+  smaller concentrations of silver nitrate in the liquid phase. The curve
+  in Fig. 28 may, therefore, be regarded either as the solubility curve of
+  silver nitrate in water, or as the freezing point curve for silver
+  nitrate in contact with a solution consisting of that salt and water.</p>
+
+  <p>As the temperature of the saturated solution falls, silver nitrate is
+  deposited, and on lowering the temperature sufficiently a point will at
+  last be reached at which ice also begins to separate out. Since there are
+  now four phases co-existing, viz. silver nitrate, ice, solution, vapour,
+  the system is invariant, and the point is a <i>quadruple point</i>. This
+  quadruple point, therefore, forms the lower limit of the solubility curve
+  of silver nitrate. Below this point the solution becomes metastable.</p>
+
+  <p><b>Ice as Solid Phase.</b>&mdash;Ice melts or is in equilibrium with
+  water at a temperature of 0°. The melting point, will, however, be
+  lowered by the solution of silver nitrate in the water; and the greater
+  the concentration of the salt in the solution the greater will be the
+  depression of the temperature of equilibrium. On continuing the addition
+  of silver nitrate, a point will at length be reached at which the salt is
+  no longer dissolved, but remains in the solid form along with the ice. We
+  again obtain, therefore, the invariant system
+  ice&mdash;salt&mdash;solution&mdash;vapour. The temperature at which this
+  invariant system can exist has been found by Middelberg<a name="NtA193"
+  href="#Nt193"><sup>[193]</sup></a> to be -7.3°, the solution at this
+  point containing 47.1 per cent. of silver nitrate.</p>
+
+  <p>The same general behaviour will be found in the case of all other
+  systems of two components belonging to this class; that is, in the case
+  of systems from which the components crystallise out in the pure state,
+  and in which the fused components are miscible in all proportions. In all
+  such cases, therefore, the solubility curves (curves of equilibrium) can
+  be represented diagrammatically as in Fig. 29. In this figure OA
+  represents the solubility curve of the salt, and OB the freezing <!--
+  Page 117 --><span class="pagenum"><a name="page117"></a>{117}</span>point
+  curve of ice. O is the quadruple point at which the invariant system
+  exists, and may be regarded as the point of intersection of the
+  solubility curve with the freezing-point curve. Since this point is
+  fixed, the condition of the system as regards temperature, vapour
+  pressure, and concentration of the components (or composition of the
+  solution), is perfectly definite. From the way, also, in which the
+  condition is attained, it is evident that the quadruple point is the
+  lowest temperature that can be obtained with mixtures of the two
+  components in presence of vapour. It is known as the <i>cryohydric
+  point</i>, or, generally, the <i>eutectic point</i>.<a name="NtA194"
+  href="#Nt194"><sup>[194]</sup></a></p>
+
+  <div class="figcenter" style="width:27%;">
+      <a href="images/fig029.png"><img style="width:100%" src="images/fig029.png"
+      alt="Fig. 29" title="Fig. 29" /></a>
+    <span class="sc">Fig.</span> 29.
+  </div>
+
+  <p><b>Cryohydrates.</b><a name="NtA195"
+  href="#Nt195"><sup>[195]</sup></a>&mdash;On cooling a solution of common
+  salt in water to a temperature of -3°, Guthrie observed that the hydrate
+  NaCl,2H<sub>2</sub>O separated out. This salt continued to be deposited
+  until at a temperature of -22° opaque crystals made their appearance, and
+  the liquid passed into the solid state without change of temperature. A
+  similar behaviour was found by Guthrie in the case of a large number of
+  other salts, a temperature below that of the melting point of ice being
+  reached at which on continued withdrawal of heat, the solution solidified
+  at a constant temperature. When the system had attained this minimum
+  temperature, it was found that the composition of the solid and the
+  liquid phases was the same, and remained unchanged throughout the period
+  of solidification. This is shown by the following figures, which give the
+  composition of different samples of the solid phase deposited from the
+  solution at constant temperature.<a name="NtA196"
+  href="#Nt196"><sup>[196]</sup></a></p>
+
+<p><!-- Page 118 --><span class="pagenum"><a name="page118"></a>{118}</span></p>
+
+<table class="nobctr" summary="Cryohydrates of sodium chloride" title="Cryohydrates of sodium chloride">
+<tr><td class="toribob" style="text-align:center; padding-left:2em; padding-right:2em"> No.
+</td><td class="toboleb" style="text-align:center"> Temperature of<br />solidification.
+</td><td class="toboleb" style="text-align:center"> NaCl.<br />Per cent.</td></tr>
+
+<tr><td class="rib" style="text-align:center"> 1 </td><td class="leb" style="text-align:center;"> -21° to -22° </td><td class="leb" style="text-align:center;"> 23.72</td></tr>
+<tr><td class="rib" style="text-align:center"> 2 </td><td class="leb" style="text-align:center;"> -22° </td><td class="leb" style="text-align:center;"> 23.66</td></tr>
+<tr><td class="rib" style="text-align:center"> 3 </td><td class="leb" style="text-align:center;"> -22° </td><td class="leb" style="text-align:center;"> 23.73</td></tr>
+<tr><td class="rib" style="text-align:center"> 4 </td><td class="leb" style="text-align:center;"> -23° </td><td class="leb" style="text-align:center;"> 23.82</td></tr>
+<tr><td class="rib" style="text-align:center"> 5 </td><td class="leb" style="text-align:center;"> -23° </td><td class="leb" style="text-align:center;"> 23.34</td></tr>
+<tr><td class="rib" style="text-align:center"> 6 </td><td class="leb" style="text-align:center;"> -23° </td><td class="leb" style="text-align:center;"> 23.35</td></tr>
+
+<tr><td class="tobob" style="text-align:right;" colspan="2"> Mean </td><td class="toboleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 23.6</td></tr>
+
+</table>
+
+  <p>Conversely, a mixture of ice and salt containing 23.6 per cent. of
+  sodium chloride will melt at a definite and constant temperature, and
+  exhibit, therefore, a behaviour supposed to be characteristic of a pure
+  chemical compound. This, then, combined with the fact that the solid
+  which was deposited was crystalline, and that the same constant
+  temperature was attained, no matter with what proportions of water and
+  salt one started, led Guthrie to the belief that the solids which thus
+  separated at constant temperature were definite chemical compounds, to
+  which he gave the general name <i>cryohydrate</i>. A large number of such
+  cryohydrates were prepared and analysed by Guthrie, and a few of these
+  are given in the following table, together with the temperature of the
+  cryohydric point:<a name="NtA197"
+  href="#Nt197"><sup>[197]</sup></a>&mdash;</p>
+
+<p class="cenhead"><span class="sc">Cryohydrates.</span></p>
+
+<table class="nobctr" summary="Cryohydrates" title="Cryohydrates">
+<tr><td class="toribob" style="text-align:center"> Salt.
+</td><td class="toboleb" style="text-align:center"> Cryohydric point.
+</td><td class="toboleb" style="text-align:center"> Percentage of anhydrous<br />salt in the cryohydrate.</td></tr>
+
+<tr><td class="rib" style="text-align:left;"> Sodium bromide </td><td class="leb" style="text-align:left; padding-left:2em;"> -24° </td><td class="leb" style="text-align:left; padding-left:4em;"> 41.33</td></tr>
+<tr><td class="rib" style="text-align:left;"> Sodium chloride </td><td class="leb" style="text-align:left; padding-left:2em;"> -22° </td><td class="leb" style="text-align:left; padding-left:4em;"> 23.60</td></tr>
+<tr><td class="rib" style="text-align:left;"> Potassium iodide </td><td class="leb" style="text-align:left; padding-left:2em;"> -22° </td><td class="leb" style="text-align:left; padding-left:4em;"> 52.07</td></tr>
+<tr><td class="rib" style="text-align:left;"> Sodium nitrate </td><td class="leb" style="text-align:left; padding-left:2em;"> -17.5° </td><td class="leb" style="text-align:left; padding-left:4em;"> 40.80</td></tr>
+<tr><td class="rib" style="text-align:left;"> Ammonium sulphate </td><td class="leb" style="text-align:left; padding-left:2em;"> -17° </td><td class="leb" style="text-align:left; padding-left:4em;"> 41.70</td></tr>
+<tr><td class="rib" style="text-align:left;"> Ammonium chloride </td><td class="leb" style="text-align:left; padding-left:2em;"> -15° </td><td class="leb" style="text-align:left; padding-left:4em;"> 19.27</td></tr>
+<tr><td class="rib" style="text-align:left;"> Sodium iodide </td><td class="leb" style="text-align:left; padding-left:2em;"> -15° </td><td class="leb" style="text-align:left; padding-left:4em;"> 59.45</td></tr>
+<tr><td class="rib" style="text-align:left;"> Potassium bromide </td><td class="leb" style="text-align:left; padding-left:2em;"> -13° </td><td class="leb" style="text-align:left; padding-left:4em;"> 32.15</td></tr>
+<tr><td class="rib" style="text-align:left;"> Potassium chloride </td><td class="leb" style="text-align:left; padding-left:2em;"> -11.4° </td><td class="leb" style="text-align:left; padding-left:4em;"> 20.03</td></tr>
+<tr><td class="rib" style="text-align:left;"> Magnesium sulphate &nbsp;</td><td class="leb" style="text-align:left; padding-left:2em;"> -&nbsp; 5° </td><td class="leb" style="text-align:left; padding-left:4em;"> 21.86</td></tr>
+<tr><td class="rib" style="text-align:left;"> Potassium nitrate </td><td class="leb" style="text-align:left; padding-left:2em;"> -2.6° </td><td class="leb" style="text-align:left; padding-left:4em;"> 11.20</td></tr>
+
+<tr><td class="ribob" style="text-align:left;"> Sodium sulphate </td><td class="boleb" style="text-align:left; padding-left:2em;"> -0.7° </td><td class="boleb" style="text-align:left; padding-left:4em;"> &nbsp; 4.55</td></tr>
+
+</table>
+
+<p><!-- Page 119 --><span class="pagenum"><a name="page119"></a>{119}</span></p>
+
+  <p>The chemical individuality of these cryohydrates was, however, called
+  in question by Pfaundler,<a name="NtA198"
+  href="#Nt198"><sup>[198]</sup></a> and disproved by Offer,<a
+  name="NtA199" href="#Nt199"><sup>[199]</sup></a> who showed that in spite
+  of the constancy of the melting point, the cryohydrates had the
+  properties, not of definite chemical compounds, but of mixtures; the
+  arguments given being that the heat of solution and the specific volume
+  are the same for the cryohydrate as for a mixture of ice and salt of the
+  same composition; and it was further shown that the cryohydrate had not a
+  definite crystalline form, but separated out as an opaque mass containing
+  the two components in close juxtaposition. The heterogeneous nature of
+  cryohydrates can also be shown by a microscopical examination.</p>
+
+  <p>At the cryohydric point, therefore, we are not dealing with a single
+  solid phase, but with two solid phases, ice and salt; the cryohydric
+  point, therefore, as already stated, is a quadruple point and represents
+  an invariant system.</p>
+
+  <p>Although on cooling a solution to the cryohydric point, separation of
+  ice may occur, it will not necessarily take place; the system may become
+  metastable. Similarly, separation of salt may not take place immediately
+  the cryohydric point is reached. It will, therefore, be possible to
+  follow the curves BO and AO beyond the quadruple point,<a name="NtA200"
+  href="#Nt200"><sup>[200]</sup></a> which is thereby clearly seen to be
+  the point of intersection of the solubility curve of the salt and the
+  freezing-point curve of ice. At this point, also, the curves of the
+  univariant systems ice&mdash;salt&mdash;vapour and
+  ice&mdash;salt&mdash;solution intersect.</p>
+
+  <p><b>Changes at the Quadruple Point.</b>&mdash;Since the invariant
+  system ice&mdash;salt&mdash;solution&mdash;vapour can exist only at a
+  definite temperature, addition or withdrawal of heat must cause the
+  disappearance of one of the phases, whereby the system will become
+  univariant. So long as all four phases are present the temperature,
+  pressure, and concentration of the components in the solution must remain
+  constant. When, therefore, heat is added to or withdrawn from the system,
+  mutually compensatory changes will take place within the system whereby
+  the <!-- Page 120 --><span class="pagenum"><a
+  name="page120"></a>{120}</span>condition of the latter is preserved.
+  These changes can in all cases be foreseen with the help of the theorem
+  of van't Hoff and Le Chatelier; and, after what was said in Chap. IV.,
+  need only be briefly referred to here. In the first place, addition of
+  heat will cause ice to melt, and the concentration of the solution will
+  be thereby altered; salt must therefore dissolve until the original
+  concentration is reached, and the heat of fusion of ice will be
+  counteracted by the heat of solution of the salt. Changes of volume of
+  the solid and liquid phases must also be taken into account; an
+  alteration in the volume of these phases being compensated by
+  condensation or evaporation. All four phases will therefore be involved
+  in the change, and the final state of the system will be dependent on the
+  amounts of the different phases present; the ultimate result of addition
+  or withdrawal of heat or of change of pressure at the quadruple point
+  will be one of the four univariant systems:
+  ice&mdash;solution&mdash;vapour; salt&mdash;solution&mdash;vapour;
+  ice&mdash;salt&mdash;vapour; ice&mdash;salt&mdash;solution. If the vapour
+  phase disappear, there will be left the univariant system
+  ice&mdash;salt&mdash;solution, and the temperature at which this system
+  can exist will alter with the pressure. Since in this case the influence
+  of pressure is comparatively slight, the temperature of the quadruple
+  point will differ only slightly from that of the cryohydric point as
+  determined under atmospheric pressure.</p>
+
+  <p><b>Freezing Mixtures.</b>&mdash;Not only will the composition of a
+  univariant system undergo change when the temperature is varied, but,
+  conversely, if the <i>composition</i> of the system is caused to change,
+  corresponding changes of temperature must ensue. Thus, if ice is added to
+  the univariant system salt&mdash;solution&mdash;vapour, the ice must melt
+  and the temperature fall; and if sufficient ice is added, the temperature
+  of the cryohydric point must be at length reached, for it is only at this
+  temperature that the four phases
+  ice&mdash;salt&mdash;solution&mdash;vapour can coexist. Or, on the other
+  hand, if salt is added to the system ice&mdash;solution&mdash;vapour, the
+  concentration of the solution will increase, ice must melt, and the
+  temperature must thereby fall; and this process also will go on until the
+  cryohydric point is reached. In both cases ice melts and there is a
+  change in the <!-- Page 121 --><span class="pagenum"><a
+  name="page121"></a>{121}</span>composition of the solution; in the former
+  case, salt will be deposited<a name="NtA201"
+  href="#Nt201"><sup>[201]</sup></a> because the solubility diminishes as
+  the temperature falls; in the latter, salt will pass into solution. This
+  process may be accompanied either by an evolution or, more generally, by
+  absorption of heat; in the former case the effect of the addition of ice
+  will be partially counteracted; in the latter case it will be
+  augmented.</p>
+
+  <p>These principles are made use of in the preparation of <i>freezing
+  mixtures</i>. The lowest temperature which can be reached by means of
+  these (under atmospheric pressure) is the cryohydric point. This
+  temperature-minimum is, however, not always attained in the preparation
+  of a freezing mixture, and that for various reasons. The chief of these
+  are radiation and the heat absorbed in cooling the solution produced. The
+  lower the temperature falls, the more rapid does the radiation become;
+  and the rate at which the temperature sinks decreases as the amount of
+  solution increases. Both these factors counteract the effect of the
+  latent heat of fusion and the heat of solution, so that a point is
+  reached (which may lie considerably above the cryohydric point) at which
+  the two opposing influences balance. The absorption of heat by the
+  solution can be diminished by allowing the solution to drain off as fast
+  as it is produced; and the effect of radiation can be partially annulled
+  by increasing the rate of cooling. This can be done by the more intimate
+  mixing of the components. Since, under atmospheric pressure, the
+  temperature of the cryohydric point is constant, the cryohydrates are
+  very valuable for the production of baths of constant low
+  temperature.</p>
+
+<p class="cenhead">2. <i>Partial Miscibility of the Fused Components.</i></p>
+
+  <p>On passing to the study of the second class of systems of two
+  components belonging to this group, namely, those in which the fused
+  components are not miscible in all proportions, we find that the
+  relationships are not quite so simple as <!-- Page 122 --><span
+  class="pagenum"><a name="page122"></a>{122}</span>in the case of silver
+  nitrate and water. In the latter case, only one liquid phase was
+  possible; in the cases now to be studied, two liquid phases can be
+  formed, and there is a marked discontinuity in the solubility curve on
+  passing from the cryohydric point to the melting point of the second
+  (non-volatile) component.</p>
+
+  <p>Paratoluidine dissolves in water, and the solubility increases as the
+  temperature rises.<a name="NtA202" href="#Nt202"><sup>[202]</sup></a> At
+  44.2°, however, paratoluidine in contact with water melts, and two liquid
+  phases are formed, viz. a solution of water in fused paratoluidine and a
+  solution of fused paratoluidine in water. We have, therefore, the
+  phenomenon of <i>melting under the solvent</i>. This melting point will,
+  of course, be lower than the melting point of the pure substance, because
+  the solid is now in contact with a solution, and, as we have already
+  seen, addition of a foreign substance lowers the melting point. Such
+  cases of melting under the solvent are by no means rare, and a review of
+  the relationships met with may, therefore, be undertaken here. As an
+  example, there may be chosen the equilibrium between succinic nitrile,
+  C<sub>2</sub>H<sub>4</sub>(CN)<sub>2</sub> and water, which has been
+  fully studied by Schreinemakers.<a name="NtA203"
+  href="#Nt203"><sup>[203]</sup></a></p>
+
+  <div class="figcenter" style="width:29%;">
+      <a href="images/fig030.png"><img style="width:100%" src="images/fig030.png"
+      alt="Fig. 30" title="Fig. 30" /></a>
+    <span class="sc">Fig.</span> 30.
+  </div>
+
+  <p>If to the system ice&mdash;water at 0° succinic nitrile is added, the
+  temperature will fall; and continued addition of the nitrile will lead at
+  last to the cryohydric point <i>b</i> (Fig. 30), at which solid nitrile,
+  ice, solution, and vapour can coexist. The temperature of the cryohydric
+  point is -1.2°, and the composition of the solution is 1.29 mol. of
+  nitrile in 100 mol. of solution. From <i>a</i> to <i>b</i> the solid
+  phase in contact with the solution is ice. <!-- Page 123 --><span
+  class="pagenum"><a name="page123"></a>{123}</span>If the temperature be
+  now raised so as to cause the disappearance of the ice, and the addition
+  of nitrile be continued, the concentration of the nitrile in the solution
+  will increase as represented by the curve <i>bc</i>. At the point
+  <i>c</i> (18.5°), when the concentration of the nitrile in the solution
+  has increased to 2.5 molecules per cent., the nitrile melts and two
+  liquid phases are formed; the concentration of the nitrile in these two
+  phases is given by the points <i>c</i> and <i>c&prime;</i>. As there are
+  now four phases present, viz. solid nitrile, solution of fused nitrile in
+  water, solution of water in fused nitrile, and vapour, the system is
+  <i>invariant</i>. Since at this point the concentration, temperature, and
+  pressure are completely defined, addition or withdrawal of heat can only
+  cause a change in the relative amounts of the phases, <i>but no variation
+  of the concentrations</i> of the respective phases. As a matter of fact,
+  continued addition of nitrile and addition of heat will cause an increase
+  in the amount of the liquid phase containing excess of nitrile
+  (<i>i.e.</i> the solution of water in fused nitrile), whereas the other
+  liquid phase, the solution of fused nitrile in water, will gradually
+  disappear. When it has completely disappeared, the system will be
+  represented by the point <i>c&prime;</i>, where the molecular
+  concentration of nitrile is now 75 per cent., and again becomes
+  univariant, the three phases being solid nitrile, liquid phase containing
+  excess of nitrile, and vapour; and as the amount of the water is
+  diminished the temperature of equilibrium rises, until at 54° the melting
+  point of the pure nitrile is reached.</p>
+
+  <p>Return now to the point <i>c</i>. At this point there exists the
+  invariant system solid nitrile, two liquid phases, vapour. If heat be
+  added, the solid nitrile will disappear, and there will be left the
+  univariant system, consisting of two liquid phases and vapour.<a
+  name="NtA204" href="#Nt204"><sup>[204]</sup></a> Such a system will
+  exhibit relationships similar to those already studied in the previous
+  chapter. As the temperature rises, the mutual solubility of the two fused
+  components becomes greater, until at <i>d</i> (55.5°) the critical
+  solution temperature is reached, and the fused components become miscible
+  in all proportions.</p>
+
+  <p>At all temperatures and concentrations lying to the right <!-- Page
+  124 --><span class="pagenum"><a name="page124"></a>{124}</span>of the
+  curve <i>abcdc&prime;e</i> there can be only one liquid phase; in the
+  field <i>cdc&prime;</i> there are two liquid phases.</p>
+
+  <p>From the figure it will be easy to see what will be the result of
+  bringing together succinic nitrile and water at different temperatures
+  and in different amounts. Since <i>b</i> is the lowest temperature at
+  which liquid can exist in stable equilibrium with solid, ice and succinic
+  nitrile can be mixed in any proportions at temperatures below <i>b</i>
+  without undergoing change. Between <i>b</i> and <i>c</i> succinic nitrile
+  will be dissolved until the concentration reaches the value on the curve
+  <i>bc</i>, corresponding to the given temperature. On adding the nitrile
+  to water at temperatures between <i>c</i> and <i>d</i>, it will dissolve
+  until a concentration lying on the curve <i>cd</i> is attained; at this
+  point two liquid phases will be formed, and further addition of nitrile
+  will cause the one liquid phase (that containing excess of nitrile) to
+  increase, while the other liquid phase will decrease, until it finally
+  disappears and there is only one liquid phase left, that containing
+  excess of nitrile. This can dissolve further quantities of the nitrile,
+  and the concentration will increase until the curve <i>c&prime;e</i> is
+  reached, when the concentration will remain unchanged, and addition of
+  solid will merely increase the amount of the solid phase.</p>
+
+  <p>If a solution represented by any point in the field lying below the
+  curve <i>bcd</i> is heated to a temperature above <i>d</i>, the critical
+  solution temperature, then the concentration of the nitrile can be
+  increased to any desired amount without at any time two liquid phases
+  making their appearance; the system can then be cooled down to a
+  temperature represented by any point between the curves
+  <i>dc&prime;e</i>. In this way it is possible to pass continuously from a
+  solution containing excess of one component to solutions containing
+  excess of the other, as represented by the dotted line <i>xxxx</i>
+  (<i>v.</i> p.&nbsp;<a href="#page100">100</a>). At no point is there formation
+  of two liquid phases.</p>
+
+  <p><b>Supersaturation.</b>&mdash;Just as suspended transformation is
+  rarely met with in the passage from the solid to the liquid state, so
+  also it is found in the case of the melting of substances under the
+  solvent that suspended fusion does not occur; but that when the
+  temperature of the invariant point is reached at which, therefore, the
+  formation of two liquid layers is possible, <!-- Page 125 --><span
+  class="pagenum"><a name="page125"></a>{125}</span>these two liquid
+  layers, as a matter of fact, make their appearance. Suspended
+  transformation can, however, take place from the side of the liquid
+  phase, just as water or other liquid can be cooled below the normal
+  freezing point without solidification occurring. The question, therefore,
+  arises as to the relative solubilities of the solid and the supercooled
+  liquid at the same temperature.</p>
+
+  <div class="figcenter" style="width:31%;">
+      <a href="images/fig031.png"><img style="width:100%" src="images/fig031.png"
+      alt="Fig. 31" title="Fig. 31" /></a>
+    <span class="sc">Fig.</span> 31.
+  </div>
+
+  <p>The answer to this question can at once be given from what we have
+  already learned (p.&nbsp;<a href="#page113">113</a>), if we recollect that at
+  temperatures below the point of fusion under the solvent, the solid form,
+  at temperatures above that point, the liquid form, is the more stable; at
+  this temperature, therefore, the relative stability of the solid and
+  liquid forms changes. Since, as we have already seen, the less stable
+  form has the greater solubility, it follows that the supercooled liquid,
+  being the less stable form, must have the greater solubility. This was
+  first proved experimentally by Alexejeff<a name="NtA205"
+  href="#Nt205"><sup>[205]</sup></a> in the case of benzoic acid and water,
+  the solubility curves for which are given in Fig. 31. As can be seen from
+  the figure, the prolongation of the curve for liquid&mdash;liquid, which
+  represents the solubility of the supercooled liquid benzoic acid, lies
+  above that for the solubility of the <!-- Page 126 --><span
+  class="pagenum"><a name="page126"></a>{126}</span>solid benzoic acid in
+  water; the solution saturated with respect to the supercooled liquid is
+  therefore supersaturated with respect to the solid form. A similar
+  behaviour has been found in the case of other substances.<a name="NtA206"
+  href="#Nt206"><sup>[206]</sup></a></p>
+
+  <p><b>Pressure-Temperature Diagram.</b>&mdash;Having considered the
+  changes which occur in the concentration of the components in a solution
+  with the temperature, we may conclude the discussion of the equilibrium
+  between a salt and water by studying the variation of the vapour
+  pressure.</p>
+
+  <p>Since in systems of two components the two phases, solution and
+  vapour, constitute a bivariant system, the vapour pressure is undefined,
+  and may have different values at the same temperature, depending on the
+  concentration. In order that there may be for each temperature a definite
+  corresponding pressure of the vapour, a third phase must be present. This
+  condition is satisfied by the system solid&mdash;liquid
+  (solution)&mdash;vapour; that is, by the saturated solution (p.&nbsp;<a
+  href="#page108">108</a>). In the case of a saturated solution, therefore,
+  the pressure of the vapour at any given temperature is constant.</p>
+
+  <p><b>Vapour Pressure of Solid&mdash;Solution&mdash;Vapour.</b>&mdash;It
+  has long been known that the addition of a non-volatile solid to a liquid
+  in which it is soluble lowers the vapour pressure of the solvent; and the
+  diminution of the pressure is approximately proportional to the amount of
+  substance dissolved (Law of Babo). The vapour-pressure curve, therefore,
+  of a solution of a salt in water must lie below that for pure water.
+  Further, in the case of a pure liquid, the vaporization curve is a
+  function only of the temperature (p.&nbsp;<a href="#page63">63</a>), whereas,
+  in the case of a solution, the pressure varies both with the temperature
+  and the <i>concentration</i>. These two factors, however, act in opposite
+  directions; for although the vapour pressure in all cases increases as
+  the temperature rises, increase of concentration, as we have seen, lowers
+  the vapour pressure. Again, since the concentration itself varies with
+  the temperature, two cases have to be considered, viz. where the
+  concentration increases with rise of <!-- Page 127 --><span
+  class="pagenum"><a name="page127"></a>{127}</span>temperature, and where
+  the concentration diminishes with rise of temperature.</p>
+
+  <p>The relations which are found here will be best understood with the
+  help of Fig. 32.<a name="NtA207" href="#Nt207"><sup>[207]</sup></a> In
+  this figure, OB represents the sublimation curve of ice, and BC the
+  vaporization curve of water; the curve for the solution must lie below
+  this, and must cut the sublimation curve of ice at some temperature below
+  the melting point. The point of intersection A is the cryohydric point.
+  If the solubility increases with rise of temperature, the increase of the
+  vapour pressure due to the latter will be partially annulled. Since at
+  first the effect of increase of temperature more than counteracts the
+  depressing action of increase of concentration, the vapour pressure will
+  increase on raising the temperature above the cryohydric point. If the
+  elevation of temperature is continued, however, to the melting point of
+  the salt, the effect of increasing concentration makes itself more and
+  more felt, so that the vapour-pressure curve of the solution falls more
+  and more below that of the pure liquid, and the pressure will ultimately
+  become equal to that of the pure salt; that is to say, practically equal
+  to zero. The curve will therefore be of the general form AMF shown in
+  Fig. 32. If the solubility should diminish with rise of temperature, the
+  two factors, temperature and concentration, will act in the same
+  direction, and the vapour-pressure curve will rise relatively more rapid
+  than that of the pure liquid; since, however, the pure salt is ultimately
+  obtained, the vapour-pressure curve must in this case also finally
+  approach the value zero.</p>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/fig032.png"><img style="width:100%" src="images/fig032.png"
+      alt="Fig. 32" title="Fig. 32" /></a>
+    <span class="sc">Fig.</span> 32.
+  </div>
+
+  <p><b>Other Univariant Systems.</b>&mdash;Besides the univariant system
+  <!-- Page 128 --><span class="pagenum"><a
+  name="page128"></a>{128}</span>salt&mdash;solution&mdash;vapour already
+  considered, three others are possible, viz.
+  ice&mdash;solution&mdash;vapour, ice&mdash;salt&mdash;solution, and
+  ice&mdash;salt&mdash;vapour.</p>
+
+  <p>The fusion point of a substance is lowered, as we have seen, by the
+  addition of a foreign substance, and the depression is all the greater
+  the larger the quantity of substance added. The vapour pressure of the
+  water, also, is lowered by the solution in it of other substances, so
+  that the vapour pressure of the system ice&mdash;solution&mdash;vapour
+  must decrease as the temperature falls from the fusion point of ice to
+  the cryohydric point. This curve is represented by BA (Fig. 32), and is
+  coincident with the sublimation curve of ice.</p>
+
+  <p>This, at first sight, strange fact will be readily understood when we
+  consider that since ice and solution are together in equilibrium with the
+  same vapour, they must have the same vapour pressure. For suppose at any
+  given temperature equilibrium to have been established in the system
+  ice&mdash;solution&mdash;vapour, removal of the ice will not alter this
+  equilibrium. Suppose, now, the ice and the solution placed under a
+  bell-jar so that they have a common vapour, but are not themselves in
+  contact; then, if they do not have the same vapour pressure, distillation
+  must take place and the solution will become more dilute or more
+  concentrated. Since, at the completion of this process, the ice and
+  solution are now in equilibrium when they are not in contact, they must
+  also be in equilibrium when they are in contact (p.&nbsp;<a
+  href="#page32">32</a>). But if distillation has taken place the
+  concentration of the solution must have altered, so that the ice will now
+  be in equilibrium with a solution of a different concentration from
+  before. But according to the Phase Rule ice cannot at one and the same
+  temperature be in equilibrium with two solutions of different
+  concentration, for the system ice&mdash;solution&mdash;vapour is
+  univariant, and at any given temperature, therefore, not only the
+  pressure but also the <i>concentration of the components in the solution
+  must be constant</i>. Distillation could not, therefore, take place from
+  the ice to the solution or <i>vice versâ</i>; that is to say, the
+  solution and the ice must have the same vapour pressure&mdash;the
+  sublimation pressure of ice. The reason of the coincidence is the
+  non-volatility of the salt: had <!-- Page 129 --><span class="pagenum"><a
+  name="page129"></a>{129}</span>the salt a measurable vapour pressure
+  itself, the sublimation curve of ice and the curve for
+  ice&mdash;solution&mdash;vapour would no longer fall together.</p>
+
+  <p>The curve AO represents the pressures of the system
+  ice&mdash;salt&mdash;vapour. This curve will also be coincident with the
+  sublimation curve of ice, on account of the non-volatility of the
+  salt.</p>
+
+  <p>The equilibria of the fourth univariant system
+  ice&mdash;salt&mdash;solution are represented by AE. Since this is a
+  condensed system, the effect of a small change of temperature will be to
+  cause a large change of pressure, as in the case of the fusion point of a
+  pure substance. The direction of this curve will depend on whether there
+  is an increase or diminution of volume on solidification; but the effect
+  in any given case can be predicted with the help of the theorem of Le
+  Chatelier.</p>
+
+  <p>Since the cryohydric point is a quadruple point in a two-component
+  system, it represents an invariant system. The condition of the system
+  is, therefore, completely defined; the four phases, ice, salt, solution,
+  vapour, can co-exist only when the temperature, pressure, and
+  concentration of the solution have constant and definite values. Addition
+  or withdrawal of heat, therefore, can cause no alteration of the
+  condition of the system except a variation of the relative amounts of the
+  phases. Addition of heat at constant volume will ultimately lead to the
+  system salt&mdash;solution&mdash;vapour or the system
+  ice&mdash;solution&mdash;vapour, according as ice or salt disappears
+  first. This is readily apparent from the diagram (Fig. 32), for the
+  systems ice&mdash;salt&mdash;solution and ice&mdash;salt&mdash;vapour can
+  exist only at temperatures below the cryohydric point (provided the curve
+  for ice&mdash;salt&mdash;solution slopes towards the pressure axis).</p>
+
+  <p><b>Bivariant Systems.</b>&mdash;Besides the univariant systems already
+  discussed, various bivariant systems are possible, the conditions for the
+  existence of which are represented by the different areas of Fig. 32.
+  They are as follows:&mdash;</p>
+
+<table class="nobctr" summary="Bivariant Systems in Fig. 32" title="Bivariant Systems in Fig. 32">
+<tr><td class="nob" style="text-align:center"> <i>Area.</i>
+</td><td class="nob" style="text-align:center"> <i>System.</i></td></tr>
+
+<tr><td class="nob" style="text-align:left"> OAMF </td><td class="nob" style="text-align:left"> Salt&mdash;vapour.</td></tr>
+<tr><td class="nob" style="text-align:left"> CBAMF </td><td class="nob" style="text-align:left"> Solution&mdash;vapour; salt&mdash;solution.</td></tr>
+<tr><td class="nob" style="text-align:left"> EABD </td><td class="nob" style="text-align:left"> Salt&mdash;solution; ice&mdash;solution.</td></tr>
+<tr><td class="nob" style="text-align:left"> EAO </td><td class="nob" style="text-align:left"> Ice&mdash;salt.</td></tr>
+
+</table>
+
+<p><!-- Page 130 --><span class="pagenum"><a name="page130"></a>{130}</span></p>
+
+  <p><b>Deliquescence.</b>&mdash;As is evident from Fig. 32, salt can exist
+  in contact with water vapour at pressures under those represented by
+  OAMF. If, however, the pressure of the vapour is increased until it
+  reaches a value lying on this curve at temperatures above the cryohydric
+  point, solution will be formed; for the curve AMF represents the
+  equilibria between salt&mdash;solution&mdash;vapour. From this,
+  therefore, it is clear that if the pressure of the aqueous vapour in the
+  atmosphere is greater than that of the saturated solution of a salt, that
+  salt will, on being placed in the air, form a solution; it will
+  <i>deliquesce</i>.</p>
+
+  <p><b>Separation of Salt on Evaporation.</b>&mdash;With the help of Fig.
+  32 it is possible to state in a general manner whether or not salt will
+  be deposited when a solution is evaporated under a constant pressure.<a
+  name="NtA208" href="#Nt208"><sup>[208]</sup></a></p>
+
+  <p>The curve AMF (Fig. 32) is the vapour-pressure curve of the saturated
+  solutions of the salt, <i>i.e.</i> it represents, as we have seen, the
+  maximum vapour pressure at which salt can exist in contact with solution
+  and vapour. The dotted line <i>aa</i> represents atmospheric pressure.
+  If, now, an unsaturated solution, the composition of which is represented
+  by the point <i>x</i>, is heated in an open vessel, the temperature will
+  rise, and the vapour pressure of the solution will increase. The system
+  will, therefore, pass along a line represented diagrammatically by
+  <i>xx&prime;</i>. At the point <i>x&prime;</i> the vapour pressure of the
+  system becomes equal to 1 atm.; and as the vessel is open to the air, the
+  pressure cannot further rise; the solution boils. If the heating is
+  continued, water passes off, the concentration increases, and the boiling
+  point rises. The system will therefore pass along the line
+  <i>x&prime;m</i>, until at the point <i>m</i> solid salt separates out
+  (provided supersaturation is excluded). The system is now univariant, and
+  continued heating will no longer cause an alteration of the
+  concentration; as water passes off, solid salt will be deposited, and the
+  solution will evaporate to dryness.</p>
+
+  <p>If, however, the atmospheric pressure is represented not by <i>aa</i>
+  but by <i>bb</i>, then, as Fig. 32 shows, the maximum vapour <!-- Page
+  131 --><span class="pagenum"><a name="page131"></a>{131}</span>pressure
+  of the system salt&mdash;solution&mdash;vapour never reaches the pressure
+  of 1 atm. Further, since the curve <i>bb</i> lies in the area of the
+  bivariant system solution&mdash;vapour there can at no point be a
+  separation of the solid form; for the system
+  solid&mdash;solution&mdash;vapour can exist only along the curve AMF.</p>
+
+  <p>On evaporating the solution of a salt in an open vessel, therefore,
+  salt can be deposited only if at some temperature the pressure of the
+  saturated solution is equal to the atmospheric pressure. This is found to
+  be the case with most salts. In the case of aqueous solutions of sodium
+  and potassium hydroxide, however, the vapour pressure of the saturated
+  solution never reaches the value of 1 atm., and on evaporating these
+  solutions, therefore, in an open vessel, there is no separation of the
+  solid. Only a homogeneous fused mass is obtained. If, however, the
+  evaporation be carried out under a pressure which is lower than the
+  maximum pressure of the saturated solution, separation of the solid
+  substance will be possible.</p>
+
+  <p><b>General Summary.</b>&mdash;The systems which have been discussed in
+  the present chapter contained water as one of their components, and an
+  anhydrous salt as the other. It will, however, be clear that the
+  relationships which were found in the case of these will be found also in
+  other cases where it is a question of the equilibria between two
+  components, which crystallize out in the pure state, and only one of
+  which possesses a measurable vapour pressure. A similar behaviour will,
+  for example, be found in the case of many pairs of organic substances;
+  and in all cases the equilibria will be represented by a diagram of the
+  general appearance of Fig. 29 or Fig. 30. That is to say: Starting from
+  the fusion point of component I., the system will pass, by progressive
+  addition of component II., to regions of lower temperature, until at last
+  the cryohydric or eutectic point is reached. On further addition of
+  component II., the system will pass to regions of higher temperature, the
+  solid phase now being component II. If the fused components are miscible
+  with one another in all proportions a continuous curve will be obtained
+  leading up to the point of fusion of component II. Slight changes of
+  direction, it is true, due to changes in the crystalline form, may be
+  found along this curve, <!-- Page 132 --><span class="pagenum"><a
+  name="page132"></a>{132}</span>but throughout its whole course there will
+  be but one liquid phase. If, on the other hand, the fused components are
+  not miscible in all proportions, then the second curve will exhibit a
+  marked discontinuity, and two liquid phases will make their
+  appearance.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 133 --><span class="pagenum"><a name="page133"></a>{133}</span></p>
+
+<h3>CHAPTER VIII</h3>
+
+<p class="cenhead">SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF
+THE COMPONENTS BEING VOLATILE</p>
+
+<p class="cenhead">B.&mdash;<span class="sc">Hydrated Salt and Water.</span></p>
+
+  <p>In the preceding chapter we discussed the behaviour of systems formed
+  of two components, only one of which was volatile, in those cases where
+  the two components separated from solution in the pure state. In the
+  present chapter we shall consider those systems in which combination
+  between the components can occur with the formation of definite
+  compounds; such as are found in the case of crystalline salt hydrates.
+  Since a not inconsiderable amount of study has been devoted to the
+  systems formed by hydrated salts and water, systems which are of great
+  chemical interest and importance, the behaviour of these will first call
+  for discussion in some detail, and it will be found later that the
+  relationships which exist in such systems appear also in a large number
+  of other two-component systems.</p>
+
+  <p>The systems belonging to this group may be divided into two classes
+  according as the compounds formed possess a definite melting point,
+  <i>i.e.</i> form a liquid phase of the same composition, or do not do so.
+  We shall consider the latter first.</p>
+
+<p class="cenhead">1. <i>The Compounds formed do not have a Definite Melting Point.</i></p>
+
+  <p><b>Concentration-Temperature Diagram.</b>&mdash;In the case of salts
+  which can form crystalline hydrates, the temperature-concentration
+  diagram, representing the equilibria of the <!-- Page 134 --><span
+  class="pagenum"><a name="page134"></a>{134}</span>different possible
+  systems, must necessarily be somewhat more complicated than where no such
+  combination of the components occurs. For, as has already been pointed
+  out, each substance has its own solubility curve; and there will
+  therefore be as many solubility curves as there are solid phases
+  possible, <i>the curve for each particular solid phase being continuous
+  so long as it remains unchanged in contact with the solution</i>. As an
+  example of the relationships met with in such cases, we shall first of
+  all consider the systems formed of sodium sulphate and water.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig033.png"><img style="width:100%" src="images/fig033.png"
+      alt="Fig. 33" title="Fig. 33" /></a>
+    <span class="sc">Fig.</span> 33.
+  </div>
+
+  <p><b>Sodium Sulphate and Water.</b>&mdash;At the ordinary temperatures,
+  sodium sulphate crystallises from water with ten molecules of water of
+  crystallisation, forming Glauber's salt. On determining the solubility of
+  this salt in water, it is found that the solubility increases as the
+  temperature rises, the values of the solubility, represented graphically
+  by the curve AC (Fig. 33), being given in the following table.<a
+  name="NtA209" href="#Nt209"><sup>[209]</sup></a> The numbers denote grams
+  of sodium sulphate, calculated as anhydrous salt, dissolved by 100 grams
+  of water.</p>
+
+<p class="cenhead"><span class="sc">Solubility of</span> Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O.</p>
+
+<table class="nobctr" summary="Solubility of sodium sulphate decahydrate" title="Solubility of sodium sulphate decahydrate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 5.02</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 9.00</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 15° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 13.20</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 18° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 16.80</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 19.40</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 28.00</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 30° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 40.00</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 33° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 50.76</td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 34° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 55.00</td></tr>
+</table>
+
+<p><!-- Page 135 --><span class="pagenum"><a name="page135"></a>{135}</span></p>
+
+  <p>On continuing the investigation at higher temperatures, it was found
+  that the solubility no longer increased, but <i>decreased with rise of
+  temperature</i>. At the same time, it was observed that the solid phase
+  was now different from that in contact with the solution at temperatures
+  below 33°; for whereas in the latter case the solid phase was sodium
+  sulphate decahydrate, at temperatures above 33° the solid phase was the
+  anhydrous salt. The course of the solubility curve of anhydrous sodium
+  sulphate is shown by BD, and the values of the solubility are given in
+  the following table:&mdash;<a name="NtA210"
+  href="#Nt210"><sup>[210]</sup></a></p>
+
+<p class="cenhead"><span class="sc">Solubility of Anhydrous Sodium Sulphate.</span></p>
+<table class="nobctr" summary="Solubility of anhydrous sodium sulphate" title="Solubility of anhydrous sodium sulphate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 18° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 53.25</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 52.76</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 51.53</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 30° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 50.37</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 33° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 49.71</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 34° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 49.53</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 36° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 49.27</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 40.15° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 48.78</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 50.40° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 46.82</td></tr>
+
+</table>
+
+  <p>As is evident from the figure, the solubility curve which is obtained
+  when anhydrous sodium sulphate is present as the solid phase, cuts the
+  curve representing the solubility of the decahydrate, at a temperature of
+  about 33°.</p>
+
+  <p>If a solution of sodium sulphate which has been saturated at a
+  temperature of about 34° be cooled down to a temperature below 17°, while
+  care is taken that the solution is protected against access of particles
+  of Glauber's salt, crystals of a second hydrate of sodium sulphate,
+  having the composition Na<sub>2</sub>SO<sub>4</sub>,7H<sub>2</sub>O,
+  separate out. On determining the composition of the solutions in
+  equilibrium with this hydrate at different temperatures, the following
+  values were obtained, these values being represented by the curve FE
+  (Fig. 33):&mdash;</p>
+
+<p><!-- Page 136 --><span class="pagenum"><a name="page136"></a>{136}</span></p>
+
+<p class="cenhead"><span class="sc">Solubility of</span> Na<sub>2</sub>SO<sub>4</sub>,7H<sub>2</sub>O.</p>
+
+<table class="nobctr" summary="Solubility of sodium sulphate heptahydrate" title="Solubility of sodium sulphate heptahydrate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp; 0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 19.62</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 30.49</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 15° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 37.43</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 18° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 41.63</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 44.73</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 52.94</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 26° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 54.97</td></tr>
+
+</table>
+
+  <p>Since, as has already been stated, each solid substance has its own
+  solubility curve, there are three separate curves to be considered in the
+  case of sodium sulphate and water. Where two curves cut, the solution
+  must be saturated with respect to two solid phases; at the point B,
+  therefore, the point of intersection of the solubility curve of anhydrous
+  sodium sulphate with that of the decahydrate, the solution must be
+  saturated with respect to these two solid substances. But a system of two
+  components existing in four phases, anhydrous salt&mdash;hydrated
+  salt&mdash;solution&mdash;vapour, is invariant; and this invariability
+  will remain even if only three phases are present, provided that one of
+  the factors, pressure, temperature, or concentration of components
+  retains a constant value. This is the case when solubilities are
+  determined in open vessels; the pressure is then equal to atmospheric
+  pressure. Under these circumstances, then, the system, anhydrous sodium
+  sulphate&mdash;decahydrate&mdash;solution, will possess no degree of
+  freedom, and can exist, therefore, only at one definite temperature and
+  when the solution has a certain definite composition. The temperature of
+  this point is 32.482° on a mercury thermometer, or 32.379° on the
+  hydrogen thermometer.<a name="NtA211"
+  href="#Nt211"><sup>[211]</sup></a></p>
+
+<p><!-- Page 137 --><span class="pagenum"><a name="page137"></a>{137}</span></p>
+
+  <p><b>Suspended Transformation.</b>&mdash;Although it is possible for the
+  anhydrous salt to make its appearance at the temperature of the quadruple
+  point, it will not necessarily do so; and it is therefore possible to
+  follow the solubility curve of sodium sulphate decahydrate to a higher
+  temperature. Since, however, the solubility of the decahydrate at
+  temperatures above the quadruple point is greater than that of the
+  anhydrous salt, the solution which is <i>saturated</i> with respect to
+  the former will be <i>supersaturated</i> with respect to the latter. On
+  bringing a small quantity of the anhydrous salt in contact with the
+  solution, therefore, anhydrous salt will be deposited; and all the
+  hydrated salt present will ultimately undergo conversion into the
+  anhydrous salt, through the medium of the solution. In this case, as in
+  all cases, the solid phase, which is the most stable at the temperature
+  of the experiment, has at that temperature the least solubility.</p>
+
+  <p>Similarly, the solubility curve of anhydrous sodium sulphate has been
+  followed to temperatures below 32.5°. Below this temperature, however,
+  the solubility of this salt is greater than that of the decahydrate, and
+  the saturated solution of the anhydrous salt will therefore be
+  supersaturated for the decahydrate, and will deposit this salt if a
+  "nucleus" is added to the solution. From this we see that at temperatures
+  above 32.5° the anhydrous salt is the stable form, while the decahydrate
+  is unstable (or metastable); at temperatures below 32.5° the decahydrate
+  is stable. This temperature, therefore, is the <i>transition
+  temperature</i> for decahydrate and anhydrous salt.</p>
+
+  <p>From Fig. 33 we see further that the solubility curve of the anhydrous
+  salt (which at all temperatures below 32.5° is metastable) is cut by the
+  solubility curve of the heptahydrate; and this point of intersection (at
+  a temperature of 24.2°) must be the <i>transition point</i> for
+  heptahydrate and anhydrous salt. Since at all temperatures the solubility
+  of the heptahydrate is greater than that of the decahydrate, the former
+  hydrate must be metastable with respect to the latter; so that throughout
+  its whole course the solubility curve of the heptahydrate <!-- Page 138
+  --><span class="pagenum"><a name="page138"></a>{138}</span>represents
+  only metastable equilibria. Sodium sulphate, therefore, forms only one
+  stable hydrate, the decahydrate.</p>
+
+  <p>The solubility relations of sodium sulphate illustrate very clearly
+  the importance of the solid phase for the definition of saturation and
+  supersaturation. Since the solubility curve of the anhydrous salt has
+  been followed backwards to a temperature of about 18°, it is readily
+  seen, from Fig. 33, that at a temperature of, say, 20° three different
+  <i>saturated</i> solutions of sodium sulphate are possible, according as
+  the anhydrous salt, the heptahydrate or the decahydrate, is present as
+  the solid phase. Two of these solutions, however, would be metastable and
+  <i>supersaturated with respect to the decahydrate</i>.</p>
+
+  <p>Further, the behaviour of sodium sulphate and water furnishes a very
+  good example of the fact that a "break" in the solubility curve occurs
+  when, and only when, the solid phase undergoes change. So long as the
+  decahydrate, for example, remained unaltered in contact with the
+  solution, the solubility curve was continuous; but when the anhydrous
+  salt appeared in the solid phase, a distinct change in the direction of
+  the solubility curve was observed.</p>
+
+  <p><b>Dehydration by Means of Anhydrous Sodium Sulphate.</b>&mdash;The
+  change in the relative stability of sodium sulphate decahydrate and
+  anhydrous salt in presence of water at a temperature of 32.5° explains
+  why the latter salt cannot be employed for dehydration purposes at
+  temperatures above the transition point. The dehydrating action of the
+  anhydrous salt depends on the formation of the decahydrate; but since at
+  temperatures above 33° the latter is unstable, and cannot be formed in
+  presence of the anhydrous salt, this salt cannot, of course, effect a
+  dehydration above that temperature.</p>
+
+  <p><b>Pressure-Temperature Diagram.</b>&mdash;The consideration of the
+  pressure-temperature relations of the two components, sodium sulphate and
+  water, must include not only the vapour pressure of the saturated
+  solutions, but also that of the crystalline hydrates. The vapour
+  pressures of salt hydrates have already been treated in a general manner
+  (Chap. V.), so that it is only necessary here to point out the connection
+  between the two classes of systems. <!-- Page 139 --><span
+  class="pagenum"><a name="page139"></a>{139}</span></p>
+
+  <p>In most cases the vapour pressure of a salt hydrate, <i>i.e.</i> the
+  vapour pressure of the system hydrate&mdash;anhydrous salt (or lower
+  hydrate)&mdash;vapour, is at all temperatures lower than that of the
+  system anhydrous salt (or lower hydrate)&mdash;solution&mdash;vapour.
+  This, however, is not a necessity; and cases are known where the vapour
+  pressure of the former system is, under certain circumstances, equal to
+  or higher than that of the latter. An example of this is found in sodium
+  sulphate decahydrate.</p>
+
+  <p>On heating Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O, a point is
+  reached at which the dissociation pressure into anhydrous salt and water
+  vapour becomes equal to the vapour pressure of the saturated solution of
+  the anhydrous salt, as is apparent from the following measurements;<a
+  name="NtA212" href="#Nt212"><sup>[212]</sup></a> the differences in
+  pressure being expressed in millimetres of a particular oil.</p>
+
+<table class="nobctr" summary="Dissociation pressure of sodium sulphate decahydrate" title="Dissociation pressure of sodium sulphate decahydrate">
+
+<tr><td class="nob" style="text-align:left"> Temperature: </td><td class="nob" style="text-align:center"> 29.0°</td><td class="nob" style="text-align:center"> 30.83°</td><td class="nob" style="text-align:center"> 31.79°</td><td class="nob" style="text-align:center"> 32.09°</td><td class="nob" style="text-align:center"> 32.35°</td><td class="nob" style="text-align:center"> 32.6°</td></tr>
+<tr><td class="nob" style="text-align:left"> Difference of pressure:</td><td class="nob" style="text-align:center"> 23.8 </td><td class="nob" style="text-align:center"> 10.8 </td><td class="nob" style="text-align:center"> 5.6 </td><td class="nob" style="text-align:center"> 3.6 </td><td class="nob" style="text-align:center"> 1.6 </td><td class="nob" style="text-align:center"> 0</td></tr>
+
+</table>
+
+  <p>At 32.6°, therefore, the vapour pressures of the two systems</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O&mdash;Na<sub>2</sub>SO<sub>4</sub>&mdash;vapour</p>
+      <p>Na<sub>2</sub>SO<sub>4</sub>&mdash;solution&mdash;vapour</p>
+    </div>
+  </div>
+
+  <p>are equal; at this temperature the four phases,
+  Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O;
+  Na<sub>2</sub>SO<sub>4</sub>; solution; vapour, can coexist. From this it
+  is evident that when sodium sulphate decahydrate is heated to 32.6°, the
+  two new phases anhydrous salt and solution will be formed (suspended
+  transformation being supposed excluded), and the hydrate will appear to
+  undergo <i>partial fusion</i>; and during the process of "melting" the
+  vapour pressure and temperature will remain constant.<a name="NtA213"
+  href="#Nt213"><sup>[213]</sup></a> This is, however, not a true but a
+  so-called <i>incongruent</i> melting point; for the composition of the
+  liquid phase is not the same as that of the solid. As has already been
+  pointed out (p.&nbsp;<a href="#page137">137</a>), we are dealing here with the
+  <i>transition point</i> of the decahydrate and anhydrous salt,
+  <i>i.e.</i> with the reaction
+  Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O <a
+  href="images/doublearrow.png"><img src="images/doublearrow.png"
+  class="middle" style="height:2.5ex" alt="reversible arrow" /></a>
+  Na<sub>2</sub>SO<sub>4</sub> + 10H<sub>2</sub>O.</p>
+
+  <p>Since at the point of partial fusion of the decahydrate four <!-- Page
+  140 --><span class="pagenum"><a name="page140"></a>{140}</span>phases can
+  coexist, the point is a quadruple point in a two-component system, and
+  the system at this point is therefore invariant. The temperature of this
+  point is therefore perfectly definite, and on this account the proposal
+  has been made to adopt this as a fixed point in thermometry.<a
+  name="NtA214" href="#Nt214"><sup>[214]</sup></a> The temperature is, of
+  course, practically the same as that at which the two solubility curves
+  intersect (p.&nbsp;<a href="#page112">112</a>). If, however, the vapour phase
+  disappears, the system becomes univariant, and the equilibrium
+  temperature undergoes change with change of pressure. The transition
+  curve has been determined by Tammann,<a name="NtA215"
+  href="#Nt215"><sup>[215]</sup></a> and shown to pass through a point of
+  maximum temperature.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig034.png"><img style="width:100%" src="images/fig034.png"
+      alt="Fig. 34" title="Fig. 34" /></a>
+    <span class="sc">Fig.</span> 34.
+  </div>
+
+  <p>The vapour pressure of the different systems of sodium sulphate and
+  water can best be studied with the help of the diagram in Fig. 34.<a
+  name="NtA216" href="#Nt216"><sup>[216]</sup></a> The curve ABCD
+  represents the vapour-pressure curve of the saturated solution of
+  anhydrous sodium sulphate. GC is the pressure curve of decahydrate +
+  anhydrous salt, which, as we have seen, cuts the curve ABCD at the
+  transition temperature, 32.6°. Since at this point the solution is
+  saturated with respect to both the anhydrous salt and the decahydrate,
+  the vapour-pressure curve of the saturated solution of the latter must
+  also pass through the point C.<a name="NtA217"
+  href="#Nt217"><sup>[217]</sup></a> As at temperatures below this point
+  the solubility of the decahydrate is less than that of the anhydrous
+  salt, the vapour pressure of the solution will, in accordance with Babo's
+  law (p.&nbsp;<a href="#page126">126</a>), be higher than that of the solution
+  of the anhydrous salt; which was also found experimentally to be the case
+  (curve HC).</p>
+
+<p><!-- Page 141 --><span class="pagenum"><a name="page141"></a>{141}</span></p>
+
+  <p>In connection with the vapour pressure of the saturated solutions of
+  the anhydrous salt and the decahydrate, attention must be drawn to a
+  conspicuous deviation from what was found to hold in the case of
+  one-component systems in which a vapour phase was present (p.&nbsp;<a
+  href="#page31">31</a>). There, it was seen that the vapour pressure of
+  the more stable system was always <i>lower</i> than that of the less
+  stable; in the present case, however, we find that this is no longer so.
+  We have already learned that at temperatures below 32.5° the system
+  decahydrate&mdash;solution&mdash;vapour is more stable than the system
+  anhydrous salt&mdash;solution&mdash;vapour; but the vapour pressure of
+  the latter system is, as has just been stated, lower than that of the
+  former. At temperatures above the transition point the vapour pressure of
+  the saturated solution of the decahydrate will be lower than that of the
+  saturated solution of the anhydrous salt.</p>
+
+  <p>This behaviour depends on the fact that the less stable form is the
+  more soluble, and that the diminution of the vapour pressure increases
+  with the amount of salt dissolved.</p>
+
+  <p>With regard to sodium sulphate heptahydrate the same considerations
+  will hold as in the case of the decahydrate. Since at 24° the four phases
+  heptahydrate, anhydrous salt, solution, vapour can coexist, the
+  vapour-pressure curves of the systems hydrate&mdash;anhydrous
+  salt&mdash;vapour (curve EB) and hydrate&mdash;solution&mdash;vapour
+  (curve FB) must cut the pressure curve of the saturated solution of the
+  anhydrous salt at the above temperature, as represented in Fig. 34 by the
+  point B. This constitutes, therefore, a second quadruple point, which is,
+  however, metastable.</p>
+
+  <p>From the diagram it is also evident that the dissociation pressure of
+  the heptahydrate is higher than that of the decahydrate, although it
+  contains less water of crystallization. The system
+  heptahydrate&mdash;anhydrous salt&mdash;vapour must be metastable with
+  respect to the system decahydrate&mdash;anhydrous salt&mdash;vapour, and
+  will pass into the latter.<a name="NtA218"
+  href="#Nt218"><sup>[218]</sup></a> Whether or not there is a temperature
+  at which the vapour-pressure curves of the two systems intersect, and
+  below which the heptahydrate becomes the more stable form, is not
+  known.</p>
+
+<p><!-- Page 142 --><span class="pagenum"><a name="page142"></a>{142}</span></p>
+
+  <p>In the case of sodium sulphate there is only one stable hydrate. Other
+  salts are known which exhibit a similar behaviour; and we shall therefore
+  expect that the solubility relationships will be represented by a diagram
+  similar to that for sodium sulphate. A considerable number of such cases
+  have, indeed, been found,<a name="NtA219"
+  href="#Nt219"><sup>[219]</sup></a> and in some cases there is more than
+  one metastable hydrate. This is found, for example, in the case of nickel
+  iodate,<a name="NtA220" href="#Nt220"><sup>[220]</sup></a> the solubility
+  curves for which are given in Fig. 35. As can be seen from the figure,
+  suspended transformation occurs, the solubility curves having in some
+  cases been followed to a considerable distance beyond the transition
+  point. One of the most brilliant examples, however, of suspended
+  transformation in the case of salt hydrates, and the sluggish transition
+  from the less stable to the more stable form, is found in the case of the
+  hydrates of calcium chromate.<a name="NtA221"
+  href="#Nt221"><sup>[221]</sup></a></p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig035.jpg"><img style="width:100%" src="images/fig035.jpg"
+      alt="Fig. 35" title="Fig. 35" /></a>
+    <span class="sc">Fig.</span> 35.
+  </div>
+
+  <p>In the preceding cases, the dissociation-pressure curve of the
+  hydrated salt cuts the vapour-pressure curve of the saturated <!-- Page
+  143 --><span class="pagenum"><a name="page143"></a>{143}</span>solution
+  of the anhydrous salt. It can, however, happen that the
+  dissociation-pressure curve of one hydrate cuts the solubility curve, not
+  of the anhydrous salt, but of a lower hydrate; in this case there will be
+  more than one stable hydrate, each having a stable solubility curve; and
+  these curves will intersect at the temperature of the transition point.
+  Various examples of this behaviour are known, and we choose for
+  illustration the solubility relationships of barium acetate and its
+  hydrates<a name="NtA222" href="#Nt222"><sup>[222]</sup></a> (Fig.
+  36).</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/fig036.png"><img style="width:100%" src="images/fig036.png"
+      alt="Fig. 36" title="Fig. 36" /></a>
+    <span class="sc">Fig.</span> 36.
+  </div>
+
+  <p>At temperatures above 0°, barium acetate can form two stable hydrates,
+  a trihydrate and a monohydrate. The solubility of the trihydrate
+  increases very rapidly with rise of temperature, and has been determined
+  up to 26.1°. At temperatures above 24.7°, however, the trihydrate is
+  metastable with respect to the monohydrate; for at this temperature the
+  solubility curve of the latter hydrate cuts that of the former. This is,
+  therefore, the transition temperature for the trihydrate and monohydrate.
+  The solubility curve of the monohydrate succeeds that of the trihydrate,
+  and exhibits a conspicuous point of minimum solubility at about 30°.
+  Below 24.7° the <!-- Page 144 --><span class="pagenum"><a
+  name="page144"></a>{144}</span>monohydrate is the less stable hydrate,
+  but its solubility has been determined to a temperature of 22°. At 41°
+  the solubility curve of the monohydrate intersects that of the anhydrous
+  salt, and this is therefore the transition temperature for the
+  monohydrate and anhydrous salt. Above this temperature the anhydrous salt
+  is the stable solid phase. Its solubility curve also passes through a
+  minimum.</p>
+
+  <p>The diagram of solubilities of barium acetate not only illustrates the
+  way in which the solubility curves of the different stable hydrates of a
+  salt succeed one another, but it has also an interest and importance from
+  another point of view. In Fig. 36 there is also shown a faintly drawn
+  curve which is continuous throughout its whole course. This curve
+  represents the solubility of barium acetate as determined by Krasnicki.<a
+  name="NtA223" href="#Nt223"><sup>[223]</sup></a> Since, however, three
+  different solid phases can exist under the conditions of experiment, it
+  is evident, from what has already been stated (p.&nbsp;<a
+  href="#page111">111</a>), that the different equilibria between barium
+  acetate and water could not be represented by one <i>continuous</i>
+  curve.</p>
+
+  <p>Another point which these experiments illustrate and which it is of
+  the highest importance to bear in mind is, that in making determinations
+  of the solubility of salts which are capable of forming hydrates, it is
+  not only necessary to determine the composition of the solution, but
+  <i>it is of equal importance to determine the composition of the solid
+  phase in contact with it</i>. In view of the fact, also, that the
+  solution equilibrium is in many cases established with comparative
+  slowness, it is necessary to confirm the point of equilibrium, either by
+  approaching it from higher as well as from lower temperatures, or by
+  actually determining the rate with which the condition of equilibrium is
+  attained. This can be accomplished by actual weighing of the dissolved
+  salt or by determinations of the density of the solution, as well as by
+  other methods.</p>
+
+<p><!-- Page 145 --><span class="pagenum"><a name="page145"></a>{145}</span></p>
+
+<p class="cenhead">2. <i>The Compounds formed have a Definite Melting Point.</i></p>
+
+  <p>In the cases which have just been considered we saw that the salt
+  hydrates on being heated did not undergo complete fusion, but that a
+  solid was deposited consisting of a lower hydrate or of the anhydrous
+  salt. It has, however, been long known that certain crystalline salt
+  hydrates (<i>e.g.</i> sodium thiosulphate,
+  Na<sub>2</sub>S<sub>2</sub>O<sub>3</sub>,5H<sub>2</sub>O, sodium acetate,
+  NaC<sub>2</sub>H<sub>3</sub>O<sub>2</sub>,3H<sub>2</sub>O) melt
+  completely in their water of crystallization, and yield a liquid of the
+  <i>same composition</i> as the crystalline salt. In the case of sodium
+  thiosulphate pentahydrate the temperature of liquefaction is 56°; in the
+  case of sodium acetate trihydrate, 58°. These two salts, therefore, have
+  a definite melting point. For the purpose of studying the behaviour of
+  such salt hydrates, we shall choose not the cases which have just been
+  mentioned, but two others which have been more fully studied, viz. the
+  hydrates of calcium chloride and of ferric chloride.</p>
+
+  <p><b>Solubility Curve of Calcium Chloride Hexahydrate.</b><a
+  name="NtA224" href="#Nt224"><sup>[224]</sup></a>&mdash;Although calcium
+  chloride forms several hydrates, each of which possesses its own
+  solubility, it is nevertheless the solubility curve of the hexahydrate
+  which will chiefly interest us at present, and we shall therefore first
+  discuss that curve by itself.</p>
+
+  <div class="figcenter" style="width:21%;">
+      <a href="images/fig037.png"><img style="width:100%" src="images/fig037.png"
+      alt="Fig. 37" title="Fig. 37" /></a>
+    <span class="sc">Fig.</span> 37.
+  </div>
+
+  <p>The solubility of this salt has been determined from the cryohydric
+  point, which lies at about -55°, up to the melting point of the salt.<a
+  name="NtA225" href="#Nt225"><sup>[225]</sup></a> The solubility increases
+  with rise of temperature, as is shown by the figures in the following
+  table, and by the (diagrammatic) curve AB in Fig. 37. In the table, the
+  numbers under the heading "solubility" denote the number of grams of
+  CaCl<sub>2</sub> dissolved in 100 grams <!-- Page 146 --><span
+  class="pagenum"><a name="page146"></a>{146}</span>of water; those under
+  the heading "composition," the number of gram-molecules of water in the
+  solution to one gram-molecule of CaCl<sub>2</sub>.</p>
+
+<p class="cenhead"><span class="sc">Solubility of Calcium Chloride Hexahydrate.</span></p>
+
+<table class="nobctr" summary="Solubility of calcium chloride hexahydrate" title="Solubility of calcium chloride hexahydrate">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility.
+</td><td class="toboleb" style="text-align:center"> Composition.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> -55° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 42.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 14.5 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> -25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 50.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 12.3 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> -10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 55.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 11.2 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp; &nbsp;0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 59.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 10.37</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;10° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 65.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 9.49</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;20° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 74.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 8.28</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;25° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 82.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 7.52</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;28.5° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 90.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 6.81</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;29.5° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 95.5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 6.46</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;30.2° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 102.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 6.00</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> &nbsp;29.6° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 109.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 5.70</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> &nbsp;29.2° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 112.8 </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 5.41</td></tr>
+
+</table>
+
+  <p>So far as the first portion of the curve is concerned, it resembles
+  the most general type of solubility curve. In the present case the
+  solubility is so great and increases so rapidly with rise of temperature,
+  that a point is reached at which the water of crystallization of the salt
+  is sufficient for its complete solution. This temperature is 30.2°; and
+  since the composition of the solution is the same as that of the solid
+  salt, viz. 1 mol. of CaCl<sub>2</sub> to 6 mols. of water, this
+  temperature must be the melting point of the hexahydrate. At this point
+  the hydrate will fuse or the solution will solidify without change of
+  temperature and without change of composition. Such a melting point is
+  called a <i>congruent</i> melting point.</p>
+
+  <p>But the solubility curve of calcium chloride hexahydrate differs
+  markedly from the other solubility curves hitherto considered in that it
+  possesses a <i>retroflex portion</i>, represented in the figure by BC. As
+  is evident from the figure, therefore, calcium chloride hexahydrate
+  exhibits the peculiar and, as it was at first thought, impossible
+  behaviour that it can be in equilibrium at one and the same temperature
+  with two different solutions, one of which contains more, the other less,
+  water than the solid hydrate; for it must be remembered that <!-- Page
+  147 --><span class="pagenum"><a name="page147"></a>{147}</span>throughout
+  the whole course of the curve ABC the solid phase present in equilibrium
+  with the solution is the hexahydrate.</p>
+
+  <p>Such a behaviour, however, on the part of calcium chloride hexahydrate
+  will appear less strange if one reflects that the melting point of the
+  hydrate will, like the melting point of other substances, be lowered by
+  the addition of a second substance. If, therefore, water is added to the
+  hydrate at its melting point, the temperature at which the solid hydrate
+  will be in equilibrium with the liquid phase (solution) will be lowered;
+  or if, on the other hand, anhydrous calcium chloride is added to the
+  hydrate at its melting point (or what is the same thing, if water is
+  removed from the solution), the temperature at which the hydrate will be
+  in equilibrium with the liquid will also be lowered; <i>i.e.</i> the
+  hydrate will melt at a lower temperature. In the former case we have the
+  hydrate in equilibrium with a solution containing more water, in the
+  latter case with a solution containing less water than is contained in
+  the hydrate itself.</p>
+
+  <p>It has already been stated (p.&nbsp;<a href="#page109">109</a>) that the
+  solubility curve (in general, the equilibrium curve) is continuous so
+  long as the solid phase remains unchanged; and we shall therefore expect
+  that the curve ABC will be continuous. Formerly, however, it was
+  considered by some that the curve was not continuous, but that the
+  melting point is the point of intersection of two curves, a solubility
+  curve and a fusion curve. Although the earlier solubility determinations
+  were insufficient to decide this point conclusively, more recent
+  investigation has proved beyond doubt that the curve is continuous and
+  exhibits no break.<a name="NtA226" href="#Nt226"><sup>[226]</sup></a></p>
+
+<p><!-- Page 148 --><span class="pagenum"><a name="page148"></a>{148}</span></p>
+
+  <p>Although in taking up the discussion of the equilibria between calcium
+  chloride and water, it was desired especially to call attention to the
+  form of the solubility curve in the case of salt hydrates possessing a
+  definite melting point, nevertheless, for the sake of completeness, brief
+  mention may be made of the other systems which these two components can
+  form.</p>
+
+  <div class="figcenter" style="width:43%;">
+      <a href="images/fig038.png"><img style="width:100%" src="images/fig038.png"
+      alt="Fig. 38" title="Fig. 38" /></a>
+    <span class="sc">Fig.</span> 38.
+  </div>
+
+  <p>Besides the hexahydrate, the solubility curve of which has already
+  been described, calcium chloride can also crystallize in two different
+  forms, each of which contains four molecules <!-- Page 149 --><span
+  class="pagenum"><a name="page149"></a>{149}</span>of water of
+  crystallization; these are distinguished as <span
+  class="grk">&alpha;</span>-tetrahydrate, and <span
+  class="grk">&beta;</span>-tetrahydrate. Two other hydrates are also
+  known, viz. a dihydrate and a monohydrate. The solubility curves of these
+  different hydrates are given in Fig. 38.</p>
+
+  <p>On following the solubility curve of the hexahydrate from the ordinary
+  temperature upwards, it is seen that at a temperature of 29.8°
+  represented by the point H, it cuts the solubility curve of the <span
+  class="grk">&alpha;</span>-tetrahydrate. This point is therefore a
+  quadruple point at which the four phases hexahydrate, <span
+  class="grk">&alpha;</span>-tetrahydrate, solution, and vapour can
+  coexist. It is also the transition point for these two hydrates. Since,
+  at temperatures above 29.8°, the <span
+  class="grk">&alpha;</span>-tetrahydrate is the stable form, it is evident
+  from the data given before (p.&nbsp;<a href="#page146">146</a>), as also from
+  Fig. 38, that the portion of the solubility curve of the hexahydrate
+  lying above this temperature represents <i>metastable</i> equilibria. The
+  realization of the metastable melting point of the hexahydrate is,
+  therefore, due to suspended transformation. At the transition point,
+  29.8°, the solubility of the hexahydrate and <span
+  class="grk">&alpha;</span>-tetrahydrate is 100.6 parts of
+  CaCl<sub>2</sub> in 100 parts of water.</p>
+
+  <p>The retroflex portion of the solubility curve of the hexahydrate
+  extends to only 1° below the melting point of the hydrate. At 29.2°
+  crystals of a new hydrate, <span class="grk">&beta;</span>-tetrahydrate,
+  separate out, and the solution, which now contains 112.8 parts of
+  CaCl<sub>2</sub> to 100 parts of water, is saturated with respect to the
+  two hydrates. Throughout its whole extent the solubility curve EDF of the
+  <span class="grk">&beta;</span>-tetrahydrate represents metastable
+  equilibria. The upper limit of the solubility curve of <span
+  class="grk">&beta;</span>-tetrahydrate is reached at 38.4° (F), the point
+  of intersection with the curve for the dihydrate.</p>
+
+  <p>Above 29.8° the stable hydrate is the <span
+  class="grk">&alpha;</span>-tetrahydrate; and its solubility curve extends
+  to 45.3° (K), at which temperature it cuts the solubility curve of the
+  dihydrate. The curve of the latter hydrate extends to 175.5° (L), and is
+  then succeeded by the curve for the monohydrate. The solubility curve of
+  the anhydrous salt does not begin until a temperature of about 260°. The
+  whole diagram, therefore, shows a succession of stable hydrates, a
+  metastable hydrate, a metastable melting point and retroflex solubility
+  curve. <!-- Page 150 --><span class="pagenum"><a
+  name="page150"></a>{150}</span></p>
+
+  <p><b>Pressure-Temperature Diagram.</b>&mdash;The complete study of the
+  equilibria between the two components calcium chloride and water would
+  require the discussion of the vapour pressure of the different systems,
+  and its variation with the temperature. For our present purpose, however,
+  such a discussion would not be of great value, and will therefore be
+  omitted here; in general, the same relationships would be found as in the
+  case of sodium sulphate (p.&nbsp;<a href="#page138">138</a>), except that the
+  rounded portion of the solubility curve of the hexahydrate would be
+  represented by a similar rounded portion in the pressure curve.<a
+  name="NtA227" href="#Nt227"><sup>[227]</sup></a> As in the case of sodium
+  sulphate, the transition points of the different hydrates would be
+  indicated by breaks in the curve of pressures. Finally, mention may again
+  be made of the difference of the pressure of dissociation of the
+  hexahydrate according as it becomes dehydrated to the <span
+  class="grk">&alpha;</span>- or the <span
+  class="grk">&beta;</span>-tetrahydrate (p.&nbsp;<a href="#page88">88</a>).</p>
+
+  <p><b>The Indifferent Point.</b>&mdash;We have already seen that at 30.2°
+  calcium chloride hexahydrate melts congruently, and that, provided the
+  pressure is maintained constant, addition or withdrawal of heat will
+  cause the complete liquefaction or solidification, without the
+  temperature of the system undergoing change. This behaviour, therefore,
+  is similar to, but is not quite the same as the fusion of a simple
+  substance such as ice; and the difference is due to the fact that in the
+  case of the hexahydrate the emission of vapour by the liquid phase causes
+  an alteration in the composition of the latter, owing to the
+  non-volatility of the calcium chloride; whereas in the case of ice this
+  is, of course, not so.</p>
+
+  <p>Consider, however, for the present that the vapour phase is absent,
+  and that we are dealing with the two-phase system solid&mdash;solution.
+  Then, since there are two components, the system is bivariant. For any
+  given value of the pressure, therefore, we should expect that the system
+  could exist at different temperatures; which, indeed, is the case. It
+  has, however, already been noted that when the composition of the liquid
+  phase becomes the same as that of the solid, the system then behaves as a
+  <i>univariant</i> system; for, at a given pressure, the system
+  solid&mdash;solution can exist only at <i>one</i> temperature, change of
+  temperature producing complete transformation in <!-- Page 151 --><span
+  class="pagenum"><a name="page151"></a>{151}</span>one or other direction.
+  <i>The variability of the system has therefore been diminished.</i></p>
+
+  <p>This behaviour will perhaps be more clearly understood when one
+  reflects that since the composition of the two phases is the same, the
+  system may be regarded as being formed of <i>one component</i>, just as
+  the system NH<sub>4</sub>Cl <a href="images/doublearrow.png"><img
+  src="images/doublearrow.png" class="middle" style="height:2.5ex"
+  alt="reversible arrow" /></a> NH<sub>3</sub> + HCl was regarded as being
+  composed of one component when the vapour had the same total composition
+  as the solid (p.&nbsp;<a href="#page13">13</a>). One component in two phases,
+  however, constitutes a univariant system, and we can therefore see that
+  calcium chloride hexahydrate in contact with solution of the same
+  composition will constitute a univariant system. The temperature of
+  equilibrium will, however, vary with the pressure;<a name="NtA228"
+  href="#Nt228"><sup>[228]</sup></a> if the latter is constant, the
+  temperature will also be constant.</p>
+
+  <p>A point such as has just been referred to, which represents the
+  special behaviour of a system of two (or more) components, in which the
+  composition of two phases becomes identical, is known as an
+  <i>indifferent point</i>,<a name="NtA229"
+  href="#Nt229"><sup>[229]</sup></a> and it has been shown<a name="NtA230"
+  href="#Nt230"><sup>[230]</sup></a> that at a given pressure the
+  temperature in the indifferent point is the <i>maximum</i> or
+  <i>minimum</i> temperature possible at the particular pressure<a
+  name="NtA231" href="#Nt231"><sup>[231]</sup></a> (cf. critical solution
+  temperature). At such a point a system loses one degree of freedom, or
+  behaves like a system of the next lower order.</p>
+
+  <p><b>The Hydrates of Ferric Chloride.</b>&mdash;A better illustration of
+  the formation of compounds possessing a definite melting point, and of
+  the existence of retroflex solubility curves, is afforded by the hydrates
+  of ferric chloride, which not only possess definite points of fusion, but
+  these melting points are stable. A very brief description of the
+  relations met with will suffice.<a name="NtA232"
+  href="#Nt232"><sup>[232]</sup></a></p>
+
+<p><!-- Page 152 --><span class="pagenum"><a name="page152"></a>{152}</span></p>
+
+  <p>Ferric chloride can form no less than four stable hydrates, viz.
+  Fe<sub>2</sub>Cl<sub>6</sub>,12H<sub>2</sub>O,
+  Fe<sub>2</sub>Cl<sub>6</sub>,7H<sub>2</sub>O,
+  Fe<sub>2</sub>Cl<sub>6</sub>,5H<sub>2</sub>O, and
+  Fe<sub>2</sub>Cl<sub>6</sub>,4H<sub>2</sub>O, and each of these hydrates
+  possesses a definite, stable melting point. On analogy with the behaviour
+  of calcium chloride, therefore, we shall expect that the solubility
+  curves of these different hydrates will exhibit a series of
+  <i>temperature maxima</i>; the points of maximum temperature representing
+  systems in which the composition of the solid and liquid phases is the
+  same. A graphical representation of the solubility relations is given in
+  Fig. 39, and the composition of the different saturated solutions which
+  can be formed is given in the following tables, the composition being
+  expressed in molecules of Fe<sub>2</sub>Cl<sub>6</sub> to 100 molecules
+  of water. The figures printed in thick type refer to transition and
+  melting points.</p>
+
+  <div class="figcenter" style="width:38%;">
+      <a href="images/fig039.png"><img style="width:100%" src="images/fig039.png"
+      alt="Fig. 39" title="Fig. 39" /></a>
+    <span class="sc">Fig.</span> 39.
+  </div>
+
+<p><!-- Page 153 --><span class="pagenum"><a name="page153"></a>{153}</span></p>
+
+<p class="cenhead"><span class="sc">Composition of the Saturated Solutions of Ferric Chloride and its Hydrates.</span></p>
+
+<p class="cenhead">(<i>The name placed at the head of each table is the solid phase.</i>)</p>
+
+<table class="nobctr" summary="Composition of saturated solutions of ferric chloride hydrates" title="Composition of saturated solutions of ferric chloride hydrates">
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> <span class="sc">Ice.</span></td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>±-55°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>±2.75</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -40°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 2.37</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -27.5° </td><td class="leb" style="text-align:right; padding-right:2em;"> 1.90</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -20.5° </td><td class="leb" style="text-align:right; padding-right:2em;"> 1.64</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -10°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 1.00</td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 0°&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> 0&nbsp; &nbsp; &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> Fe<sub>2</sub>Cl<sub>6</sub>,12H<sub>2</sub>O.</td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>-55°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>±2.75</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -41°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 2.81</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> -27°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 2.98</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 0°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 4.13</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 10°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 4.54</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 20°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 5.10</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 30°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 5.93</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 35°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 6.78</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 36.5° </td><td class="leb" style="text-align:right; padding-right:2em;"> 7.93</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>37°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>8.33</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 36°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 9.29</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 33°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 10.45</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 30°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 11.20</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>27·4°</b> </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>12.15</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 20°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 12.83</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 10°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 13.20</td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 8°&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> 13.70</td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> Fe<sub>2</sub>Cl<sub>6</sub>,7H<sub>2</sub>O.</td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 20°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 11.35</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>27·4°</b> </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>12.15</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 32°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 13.55</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>32.5°</b> </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>14.29</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>30°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>15.12</b></td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 25°&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> 15.54</td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> Fe<sub>2</sub>Cl<sub>6</sub>,5H<sub>2</sub>O.</td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 12°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 12.87</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 20°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 13.95</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 27°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 14.85</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>30°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>15.12</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 35°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 15.64</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 50°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 17.50</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 55°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 19.15</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>56°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>20.00</b></td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> <b>55°</b>&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> <b>20.32</b></td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> Fe<sub>2</sub>Cl<sub>6</sub>,4H<sub>2</sub>O </td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 50°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 19.96</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>55°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>20.32</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 60°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 20.70</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 69°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 21.53</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 72.5° </td><td class="leb" style="text-align:right; padding-right:2em;"> 23.35</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>73.5°</b> </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>25.00</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 72.5° </td><td class="leb" style="text-align:right; padding-right:2em;"> 26.15</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 70°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 27.90</td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> <b>66°</b>&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> <b>29.20</b></td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em;" colspan="2"> Fe<sub>2</sub>Cl<sub>6</sub> (<span class="scac">ANHYDROUS</span>).</td></tr>
+<tr><td class="toribob" style="text-align:center;"> Temperature. </td><td class="toboleb" style="text-align:center;"> Composition.</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> <b>66°</b>&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> <b>29.20</b></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 70°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 29.42</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 75°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 28.92</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 80°&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:2em;"> 29.20</td></tr>
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 100°&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-right:2em;"> 29.75</td></tr>
+
+</table>
+
+  <p>The lowest portion of the curve, AB, represents the equilibria between
+  ice and solutions containing ferric chloride. It represents, in other
+  words, the lowering of the fusion point of ice by addition of ferric
+  chloride. At the point B (-55°), the cryohydric point (p.&nbsp;<a
+  href="#page117">117</a>) is reached, at which the solution is in
+  equilibrium with ice and ferric chloride dodecahydrate. As <!-- Page 154
+  --><span class="pagenum"><a name="page154"></a>{154}</span>has already
+  been shown, such a point represents an invariant system; and the liquid
+  phase will, therefore, solidify to a mixture of ice and hydrate without
+  change of temperature. If heat is added, ice will melt and the system
+  will pass to the curve BCDN, which is the solubility curve of the
+  dodecahydrate. At C (37°), the point of maximum temperature, the hydrate
+  melts completely. The retroflex portion of this curve can be followed
+  backwards to a temperature of 8°, but below 27.4° (D), the solutions are
+  supersaturated with respect to the heptahydrate; point D is the eutectic
+  point for dodecahydrate and heptahydrate. The curve DEF is the solubility
+  curve of the heptahydrate, E being the melting point, 32.5°. On further
+  increasing the quantity of ferric chloride, the temperature of
+  equilibrium is lowered until at F (30°) another eutectic point is
+  reached, at which the heptahydrate and pentahydrate can co-exist with
+  solution. Then follow the solubility curves for the pentahydrate, the
+  tetrahydrate, and the anhydrous salt; G (56°) is the melting point of the
+  former hydrate, J (73.5°) the melting point of the latter. H and K, the
+  points at which the curves intersect, represent eutectic points; the
+  temperature of the former is 55°, that of the latter 66°. The dotted
+  portions of the curves represent metastable equilibria.</p>
+
+  <p>As is seen from the diagram, a remarkable series of solubility curves
+  is obtained, each passing through a point of maximum temperature, the
+  whole series of curves forming an undulating "festoon." To the right of
+  the series of curves the diagram represents unsaturated solutions; to the
+  left, supersaturated.</p>
+
+  <p>If an unsaturated solution, the composition of which is represented by
+  a point in the field to the right of the solubility curves, is cooled
+  down, the result obtained will differ according as the composition of the
+  solution is the same as that of a cryohydric point, or of a melting
+  point, or has an intermediate value. Thus, if a solution represented by
+  <i>x</i><sub>1</sub> is cooled down, the composition will remain
+  unchanged as indicated by the horizontal dotted line, until the point D
+  is reached. At this point, dodecahydrate and heptahydrate will separate
+  out, and the liquid will ultimately solidify completely to a mixture or
+  "conglomerate" of these two hydrates; the temperature of <!-- Page 155
+  --><span class="pagenum"><a name="page155"></a>{155}</span>the system
+  remaining constant until complete solidification has taken place. If, on
+  the other hand, a solution of the composition <i>x</i><sub>3</sub> is
+  cooled down, ferric chloride dodecahydrate will be formed when the
+  temperature has fallen to that represented by C, and the solution will
+  completely solidify, without alteration of temperature, with formation of
+  this hydrate. In both these cases, therefore, a point is reached at which
+  complete solidification occurs without change of temperature.</p>
+
+  <p>Somewhat different, however, is the result when the solution has an
+  intermediate composition, as represented by <i>x</i><sub>2</sub> or
+  <i>x</i><sub>4</sub>. In the former case the dodecahydrate will first of
+  all separate out, but on further withdrawal of heat the temperature will
+  fall, the solution will become relatively richer in ferric chloride,
+  owing to separation of the hydrate, and ultimately the eutectic point D
+  will be reached, at which complete solidification will occur. Similarly
+  with the second solution. Ferric chloride dodecahydrate will first be
+  formed, and the temperature will gradually fall, the composition of the
+  solution following the curve CB until the cryohydric point B is reached,
+  when the whole will solidify to a conglomerate of ice and
+  dodecahydrate.</p>
+
+  <p><b>Suspended Transformation.</b>&mdash;Not only can the upper branch
+  of the solubility curve of the dodecahydrate be followed backwards to a
+  temperature of 8°, or about 19° below the temperature of transition to
+  the heptahydrate; but suspended transformation has also been observed in
+  the case of the heptahydrate and the pentahydrate. To such an extent is
+  this the case that the solubility curve of the latter hydrate has been
+  followed downwards to its point of intersection with the curve for the
+  dodecahydrate. This point of intersection, represented in Fig. 39 by M,
+  lies at a temperature of about 15°; and at this temperature, therefore,
+  it is possible for the two solid phases dodecahydrate and pentahydrate to
+  coexist, so that M is a eutectic point for the dodecahydrate and the
+  pentahydrate. It is, however, a metastable eutectic point, for it lies in
+  the region of supersaturation with respect to the heptahydrate; and it
+  can be realized only because of the fact that the latter hydrate is not
+  readily formed.</p>
+
+  <p><b>Evaporation of Solutions at Constant Temperature.</b>&mdash;On <!--
+  Page 156 --><span class="pagenum"><a
+  name="page156"></a>{156}</span>evaporating dilute solutions of ferric
+  chloride at constant temperature, a remarkable series of changes is
+  observed, which, however, will be understood with the help of Fig. 40.
+  Suppose an unsaturated solution, the composition of which is represented
+  by the point <i>x</i><sub>1</sub>, is evaporated at a temperature of
+  about 17° - 18°. As water passes off, the composition of the solution
+  will follow the dotted line of constant temperature, until at the point
+  where it cuts the curve BC the solid hydrate
+  Fe<sub>2</sub>Cl<sub>6</sub>,12H<sub>2</sub>O separates out. As water
+  continues to be removed, the hydrate must be deposited (in order that the
+  solution shall remain saturated), until finally the solution dries up to
+  the hydrate. As dehydration proceeds, the heptahydrate can be formed, and
+  the dodecahydrate will finally pass into the heptahydrate; and this, in
+  turn, into the pentahydrate.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig040.png"><img style="width:100%" src="images/fig040.png"
+      alt="Fig. 40" title="Fig. 40" /></a>
+    <span class="sc">Fig.</span> 40.
+  </div>
+
+  <p>But the heptahydrate is not always formed by the dehydration of the
+  dodecahydrate, and the behaviour on evaporation is therefore somewhat
+  perplexing at first sight. After the solution has dried to the
+  dodecahydrate, as explained above, further removal of water causes
+  liquefaction, and the system is now represented by the point of
+  intersection at <i>a</i>; at this point the solid hydrate is in
+  equilibrium with a solution containing relatively more ferric chloride.
+  If, therefore, evaporation is continued, the solid hydrate must <i>pass
+  into solution</i> in order that the composition of the latter may remain
+  unchanged, so that ultimately a liquid will again be obtained. A very
+  slight further dehydration will bring the solution into the state
+  represented by <i>b</i>, at which the pentahydrate is formed, and the
+  solution will at last disappear and leave this hydrate alone.</p>
+
+  <p>Without the information to be obtained from the curves in Figs. 39 and
+  40, the phenomena which would be observed on carrying out the evaporation
+  at a temperature of about 31 - 32° <!-- Page 157 --><span
+  class="pagenum"><a name="page157"></a>{157}</span>would be still more
+  bewildering. The composition of the different solutions formed will be
+  represented by the perpendicular line <i>x</i><sub>2</sub>12345.
+  Evaporation will first cause the separation of the dodecahydrate, and
+  then total disappearance of the liquid phase. Then liquefaction will
+  occur, and the system will now be represented by the point 2, in which
+  condition it will remain until the solid hydrate has disappeared.
+  Following this there will be deposition of the heptahydrate (point 3),
+  with subsequent disappearance of the liquid phase. Further dehydration
+  will again cause liquefaction, when the concentration of the solution
+  will be represented by the point 4; the heptahydrate will ultimately
+  disappear, and then will ensue the deposition of the pentahydrate, and
+  complete solidification will result. On evaporating a solution,
+  therefore, of the composition <i>x</i><sub>2</sub>, the following series
+  of phenomena will be observed: solidification to dodecahydrate;
+  liquefaction; solidification to heptahydrate; liquefaction;
+  solidification to pentahydrate.<a name="NtA233"
+  href="#Nt233"><sup>[233]</sup></a></p>
+
+  <p>Although ferric chloride and water form the largest and best-studied
+  series of hydrates possessing definite melting points, examples of
+  similar hydrates are not few in number; and more careful investigation is
+  constantly adding to the list.<a name="NtA234"
+  href="#Nt234"><sup>[234]</sup></a> In all these cases the solubility
+  curve will show a point of maximum temperature, at which the hydrate
+  melts, and will end, above and below, in a cryohydric point. Conversely,
+  if such a curve is found in a system of two components, we can argue that
+  a definite compound of the components possessing a definite melting point
+  is formed.</p>
+
+  <p><b>Inevaporable Solutions.</b>&mdash;If a saturated solution in
+  contact with two hydrates, or with a hydrate and anhydrous salt is
+  heated, the temperature and composition of the solution will, of course,
+  remain unchanged so long as the two solid phases are present, for such a
+  system is invariant. In addition to this, however, the <i>quantity</i> of
+  the solution will also remain unchanged, the water which evaporates being
+  supplied by the higher hydrate. The same phenomenon is also observed in
+  the case of cryohydric points when ice is a solid phase; so long as the
+  latter is present, evaporation will be accompanied <!-- Page 158 --><span
+  class="pagenum"><a name="page158"></a>{158}</span>by fusion of the ice,
+  and the quantity of solution will remain constant. Such solutions are
+  called <i>inevaporable</i>.<a name="NtA235"
+  href="#Nt235"><sup>[235]</sup></a></p>
+
+  <div class="figcenter" style="width:38%;">
+      <a href="images/fig041.png"><img style="width:100%" src="images/fig041.png"
+      alt="Fig. 41" title="Fig. 41" /></a>
+    <span class="sc">Fig.</span> 41.
+  </div>
+
+  <p><b>Illustration.</b>&mdash;In order to illustrate the application of
+  the principles of the Phase Rule to the study of systems formed by a
+  volatile and a non-volatile component, a brief description may be given
+  of the behaviour of sulphur dioxide and potassium iodide, which has
+  formed the subject of a recent investigation. After it had been found<a
+  name="NtA236" href="#Nt236"><sup>[236]</sup></a> that liquid sulphur
+  dioxide has the property of dissolving potassium iodide, and that the
+  solutions thus obtained present certain peculiarities of behaviour, the
+  question arose as to whether or not compounds are formed between the
+  sulphur dioxide and the potassium iodide, and if so, what these compounds
+  are. To find an answer to this question, Walden and Centnerszwer<a
+  name="NtA237" href="#Nt237"><sup>[237]</sup></a> made a complete
+  investigation of the solubility curves (equilibrium curves) of these two
+  components, the investigation extending from the freezing point to the
+  critical point of sulphur dioxide. For convenience of reference, the
+  results which they obtained are represented diagrammatically in Fig. 41.
+  The freezing point (A) of pure sulphur dioxide was found to be -72.7°.
+  Addition of potassium iodide lowered the freezing point, but the maximum
+  depression obtained was very small, and was reached when the
+  concentration of the potassium iodide in the solution was only 0.336
+  mols. per cent. Beyond this point, an increase in the concentration of
+  the iodide was accompanied by an elevation of the freezing point, the
+  change of the freezing point with the concentration being represented by
+  the curve BC. The solid <!-- Page 159 --><span class="pagenum"><a
+  name="page159"></a>{159}</span>which separated from the solutions
+  represented by BC was a bright <i>yellow</i> crystalline substance. At
+  the point C (-23.4°) a temperature-maximum was reached; and as the
+  concentration of the potassium iodide was continuously increased, the
+  temperature of equilibrium first fell and then slowly rose, until at
+  +0.26° (E) a second temperature-maximum was registered. On passing the
+  point D, the solid which was deposited from the solution was a <i>red</i>
+  crystalline substance. On withdrawing sulphur dioxide from the system,
+  the solution became turbid, and the temperature remained constant. The
+  investigation was not pursued farther at this point, the attention being
+  then directed to the equilibria at higher temperatures.</p>
+
+  <p>When a solution of potassium iodide in liquid sulphur dioxide
+  containing 1.49 per cent. of potassium iodide was heated, solid
+  (potassium iodide) was deposited at a temperature of 96.4°. Solutions
+  containing more than about 3 per cent. of the iodide separated, on being
+  heated, into two layers, and the temperature at which the liquid became
+  heterogeneous fell as the concentration was increased; a
+  temperature-minimum being obtained with solutions containing 12 per cent.
+  of potassium iodide. On the other hand, solutions containing 30.9 per
+  cent. of the iodide, on being heated, deposited potassium iodide; while a
+  solution containing 24.5 per cent. of the salt first separated into two
+  layers at 89.3°, and then, on cooling, solid was deposited and one of the
+  liquid layers disappeared.</p>
+
+  <p>Such are, in brief, the results of experiment; their interpretation in
+  the light of the Phase Rule is the following:&mdash;</p>
+
+  <p>The curve AB is the freezing-point curve of solid sulphur dioxide in
+  contact with solutions of potassium iodide. BCD is the solubility curve
+  of the yellow crystalline solid which is deposited from the solutions. C,
+  the temperature-maximum, is the melting point of this <i>yellow</i>
+  solid, and the composition of the latter must be the same as that of the
+  solution at this point (p.&nbsp;<a href="#page145">145</a>), which was found
+  to be that represented by the formula KI,14SO<sub>2</sub>. B is therefore
+  the eutectic point, at which solid sulphur dioxide and the compound
+  KI,14SO<sub>2</sub> can exist together in equilibrium with solution and
+  vapour. The curve DE is the solubility curve of the <i>red</i>
+  crystalline solid, and the <!-- Page 160 --><span class="pagenum"><a
+  name="page160"></a>{160}</span>point E, at which the composition of
+  solution and solid is the same, is the melting point of the solid. The
+  composition of this substance was found to be KI,4SO<sub>2</sub>.<a
+  name="NtA238" href="#Nt238"><sup>[238]</sup></a> D is, therefore, the
+  eutectic point at which the compounds KI,14SO<sub>2</sub> and
+  KI,4SO<sub>2</sub> can coexist in equilibrium with solution and vapour.
+  The curve DE does not exhibit a retroflex portion; on the contrary, on
+  attempting to obtain more concentrated solutions in equilibrium with the
+  compound KI,4SO<sub>2</sub>, a new solid phase (probably potassium
+  iodide) was formed. Since at this point there are four phases in
+  equilibrium, viz. the compound KI,4SO<sub>2</sub>, potassium iodide,
+  solution, and vapour, the system is invariant. E is, therefore, the
+  <i>transition point</i> for KI,4SO<sub>2</sub> and KI.</p>
+
+  <p>Passing to higher temperatures, FG is the solubility curve of
+  potassium iodide in sulphur dioxide; at G two liquid phases are formed,
+  and the system therefore becomes invariant (cf. p.&nbsp;<a
+  href="#page121">121</a>). The curve GHK is the solubility curve for two
+  partially miscible liquids; and since complete miscibility occurs on
+  <i>lowering</i> the temperature, the curve is similar to that obtained
+  with triethylamine and water (p.&nbsp;<a href="#page101">101</a>). K is also
+  an invariant point at which potassium iodide is in equilibrium with two
+  liquid phases and vapour.</p>
+
+  <p>The complete investigation of the equilibria between sulphur dioxide
+  and potassium iodide, therefore, shows that these two components form the
+  compounds KI,14SO<sub>2</sub> and KI,4SO<sub>2</sub>; and that when
+  solutions having a concentration between those represented by the points
+  G and K are heated, separation into two layers occurs. The temperatures
+  and concentrations of the different characteristic points are as
+  follows:&mdash;</p>
+
+<table class="nobctr" summary="Composition of potassium iodide solution in sulphur dioxide" title="Composition of potassium iodide solution in sulphur dioxide">
+<tr><td class="toribob" style="text-align:center"> Point.
+</td><td class="toboleb" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Composition of<br />the solution<br />per cent. KI.</td></tr>
+
+<tr><td class="rib" style="text-align:left;"> A (m.p. of SO<sub>2</sub>) </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> -72.7° &nbsp;</td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:left;"> B (eutectic point) </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 0.86</td></tr>
+<tr><td class="rib" style="text-align:left;"> C (m.p. of KI,14SO<sub>2</sub>) </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> -23.4° &nbsp;</td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 17.63</td></tr>
+<tr><td class="rib" style="text-align:left;"> E (m.p. of KI,4SO<sub>2</sub>) </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> +0.26° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 39.33</td></tr>
+<tr><td class="rib" style="text-align:left;"> G (KI + two liquid phases) &nbsp; &nbsp;</td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> (about) 88° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 24.0 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> H (critical solution point)</td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 77.3° &nbsp;</td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 12&nbsp; &nbsp; &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:left;"> K (KI + two liquid phases) </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> (about) 88° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 2.7 &nbsp;</td></tr>
+
+</table>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 161 --><span class="pagenum"><a name="page161"></a>{161}</span></p>
+
+<h3>CHAPTER IX</h3>
+
+<p class="cenhead">EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS</p>
+
+  <p><b>General.</b>&mdash;In the two preceding chapters certain
+  restrictions were imposed on the discussion of the equilibria between two
+  components; but in the present chapter the restriction that only one of
+  the components is volatile will be allowed to fall, and the general
+  behaviour of two volatile<a name="NtA239"
+  href="#Nt239"><sup>[239]</sup></a> components, each of which is capable
+  of forming a liquid solution with the other, will be studied. As we shall
+  see, however, the removal of the previous restriction produces no
+  alteration in the general aspect of the equilibrium curves for
+  concentration and temperature, but changes to some extent the appearance
+  of the pressure-temperature diagram. The latter would become still more
+  complicated if account were taken not only of the total pressure but also
+  of the partial pressures of the two components in the vapour phase; this
+  complication, however, will not be introduced in the present
+  discussion.<a name="NtA240" href="#Nt240"><sup>[240]</sup></a> In this
+  chapter we shall consider the systems formed by the two components iodine
+  and chlorine, and sulphur dioxide and water.</p>
+
+  <p><b>Iodine and Chlorine.</b>&mdash;The different systems furnished by
+  iodine and chlorine, rendered classical by the studies of Stortenbeker,<a
+  name="NtA241" href="#Nt241"><sup>[241]</sup></a> form a very complete
+  example of equilibria in a two-component system. We shall first of all
+  consider the <!-- Page 162 --><span class="pagenum"><a
+  name="page162"></a>{162}</span>relations between concentration and
+  temperature, with the help of the accompanying diagram, Fig. 42.</p>
+
+  <div class="figcenter" style="width:33%;">
+      <a href="images/fig042.png"><img style="width:100%" src="images/fig042.png"
+      alt="Fig. 42" title="Fig. 42" /></a>
+    <span class="sc">Fig. 42.</span>
+  </div>
+
+  <p><b>Concentration-Temperature Diagram.</b>&mdash;In this diagram the
+  temperatures are taken as the abscissæ, and the composition of the
+  solution, expressed in atoms of chlorine to one atom of iodine,<a
+  name="NtA242" href="#Nt242"><sup>[242]</sup></a> is represented by the
+  ordinates. In the diagram, A represents the melting point of pure iodine,
+  114°. If chlorine is added to the system, a solution of chlorine in
+  liquid iodine is obtained, and the temperature at which solid iodine is
+  in equilibrium with the liquid solution will be all the lower the greater
+  the concentration of the chlorine. We therefore obtain the curve ABF,
+  which represents the composition of the solution <!-- Page 163 --><span
+  class="pagenum"><a name="page163"></a>{163}</span>with which solid iodine
+  is in equilibrium at different temperatures. This curve can be followed
+  down to 0°, but at temperatures below 7.9° (B) it represents metastable
+  equilibria. At B iodine monochloride can be formed, and if present the
+  system becomes invariant; B is therefore a quadruple point at which the
+  four phases, iodine, iodine monochloride, solution, and vapour, can
+  coexist. Continued withdrawal of heat at this point will therefore lead
+  to the complete solidification of the solution to a mixture or
+  conglomerate of iodine and iodine monochloride, while the temperature
+  remains constant during the process. B is the eutectic point for iodine
+  and iodine monochloride.</p>
+
+  <p>Just as we found in the case of aqueous salt solutions that at
+  temperatures above the cryohydric or eutectic point, two different
+  solutions could exist, one in equilibrium with ice, the other in
+  equilibrium with the salt (or salt hydrate), so in the case of iodine and
+  chlorine there can be two solutions above the eutectic point B, one
+  containing a lower proportion of chlorine in equilibrium with iodine, the
+  other containing a higher proportion of chlorine in equilibrium with
+  iodine monochloride. The composition of the latter solution is
+  represented by the curve BCD. As the concentration of chlorine is
+  increased, the temperature at which there is equilibrium between iodine
+  monochloride and solution rises until a point is reached at which the
+  composition of the solution is the same as that of the solid. At this
+  point (C), iodine monochloride melts. Addition of one of the components
+  will lower the temperature of fusion, and a continuous curve,<a
+  name="NtA243" href="#Nt243"><sup>[243]</sup></a> exhibiting a retroflex
+  portion as in the case of CaCl<sub>2</sub>,6H<sub>2</sub>O, will be
+  obtained. At temperatures below its melting point, therefore, iodine
+  monochloride can be in equilibrium with two different solutions.</p>
+
+  <p>The upper portion of this curve, CD, can be followed downwards to a
+  temperature of 22.7°. At this temperature iodine trichloride can separate
+  out, and a second quadruple <!-- Page 164 --><span class="pagenum"><a
+  name="page164"></a>{164}</span>point (D) is obtained. This is the
+  eutectic point for iodine monochloride and iodine trichloride.</p>
+
+  <p>By addition of heat and increase in the amount of chlorine, the iodine
+  monochloride disappears, and the system passes along the curve DE, which
+  represents the composition of the solutions in equilibrium with solid
+  iodine trichloride. The concentration of chlorine in the solution
+  increases as the temperature is raised, until at the point E, where the
+  solution has the same composition as the solid, the maximum temperature
+  is reached; the iodine trichloride melts. On increasing still further the
+  concentration of chlorine in the solution, the temperature of equilibrium
+  falls, and a continuous curve, similar to that for the monochloride, is
+  obtained. The upper branch of this curve has been followed down to a
+  temperature of 30°, the solution at this point containing 99.6 per cent.
+  of chlorine.<a name="NtA244" href="#Nt244"><sup>[244]</sup></a> The very
+  rounded form of the curve is due to the trichloride being largely
+  dissociated in the liquid state.</p>
+
+  <p>One curve still remains to be considered. As has already been
+  mentioned, iodine monochloride can exist in two crystalline forms, only
+  one of which, however, is stable at temperatures below the melting point;
+  the two forms are <i>monotropic</i> (p.&nbsp;<a href="#page44">44</a>). The
+  stable form which melts at 27.2°, is called the <span
+  class="grk">&alpha;</span>-form, while the less stable variety, melting
+  at 13.9°, is known as the <span class="grk">&beta;</span>-form. If, now,
+  the presence of <span class="grk">&alpha;</span>-ICl is excluded, it is
+  possible to obtain the <span class="grk">&beta;</span>-form, and to study
+  the conditions of equilibrium between it and solutions of iodine and
+  chlorine, from the eutectic point F to the melting point G. As the <span
+  class="grk">&beta;</span>-ICl becomes less stable in presence of excess
+  of chlorine, it has not been possible to study the retroflex portion of
+  the curve represented by the dotted continuation of FG.</p>
+
+  <p>The following table gives some of the numerical data from which Fig.
+  42 was constructed.<a name="NtA245"
+  href="#Nt245"><sup>[245]</sup></a></p>
+
+<p><!-- Page 165 --><span class="pagenum"><a name="page165"></a>{165}</span></p>
+
+<p class="cenhead"><span class="sc">Iodine and Chlorine.</span></p>
+
+<p class="cenhead">I. <i>Invariant systems.</i></p>
+
+<table class="nobctr" summary="Invariant systems, iodine and chlorine" title="Invariant systems, iodine and chlorine">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="allb" style="text-align:center" rowspan="2"> Pressure.
+</td><td class="toboleb" style="text-align:center" colspan="3"> Phases present.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> Solid.
+</td><td class="toboleb" style="text-align:center"> Liquid.
+</td><td class="toboleb" style="text-align:center"> Vapour.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 7.9° </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 11 mm. </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> I<sub>2</sub>,<span class="grk">&alpha;</span>-ICl </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> I<a href="images/wavy.png"><img src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy" /></a>Cl<sub>0.66</sub> </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> I + Cl<sub>0.92</sub></td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 0.9° </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> &mdash; </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> I<sub>2</sub>,<span class="grk">&beta;</span>-ICl </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> I<a href="images/wavy.png"><img src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy" /></a>Cl<sub>0.72</sub> </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 22.7° </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 42 mm. </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> <span class="grk">&alpha;</span>-ICl,ICl<sub>3</sub> </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> I<a href="images/wavy.png"><img src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy" /></a>Cl<sub>1.19</sub> </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> I + Cl<sub>1.75</sub></td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> [-102° </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> &lt;1 atm. </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> ICl<sub>3</sub>,Cl<sub>2</sub> </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> I<a href="images/wavy.png"><img src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy" /></a>Cl<sub>m</sub> </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> I + Cl<sub>n</sub>]</td></tr>
+
+</table>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>II. <i>Melting points.</i></p>
+    </div>
+
+    <div class="stanza">
+      <p>A. Iodine,<a name="NtA246" href="#Nt246"><sup>[246]</sup></a> 114.15° (pressure 89.8 mm.).</p>
+      <p>C. <span class="grk">&alpha;</span>-Iodine monochloride, 27.2° (pressure 37 mm.).</p>
+      <p>E. Iodine trichloride, 101° (pressure 16 atm.).</p>
+      <p>G. <span class="grk">&beta;</span>-Iodine monochloride, 13.9°.</p>
+    </div>
+  </div>
+
+  <p>Since the vapour pressure at the melting point of iodine trichloride
+  amounts to 16 atm., the experiments must of course be carried out in
+  closed vessels. At 63.7° the vapour pressure of the system
+  trichloride&mdash;solution&mdash;vapour is equal to 1 atm.</p>
+
+  <p><b>Pressure-Temperature Diagram.</b>&mdash;In this diagram there are
+  represented the values of the vapour pressure of the saturated solutions
+  of chlorine and iodine. To give a complete picture of the relations
+  between pressure, temperature, and concentration, a solid model would be
+  required, with three axes at right angles to one another along which
+  could be measured the values of pressure, temperature, and concentration
+  of the components in the solution. Instead of this, however, there may be
+  employed the accompanying projection figure<a name="NtA247"
+  href="#Nt247"><sup>[247]</sup></a> (Fig. 43), the lower portion of which
+  shows the projection of the equilibrium curve on the surface containing
+  the concentration and temperature axes, while the upper portion is the
+  projection on the plane containing the pressure and temperature axes. The
+  lower portion is therefore a concentration-temperature diagram; <!-- Page
+  166 --><span class="pagenum"><a name="page166"></a>{166}</span>the upper
+  portion, a pressure-temperature diagram. The corresponding points of the
+  two diagrams are joined by dotted lines.</p>
+
+  <div class="figcenter" style="width:52%;">
+      <a href="images/fig043.png"><img style="width:100%" src="images/fig043.png"
+      alt="Fig. 43" title="Fig. 43" /></a>
+    <span class="sc">Fig. 43.</span>
+  </div>
+
+  <p>Corresponding to the point C, the melting point of pure iodine, there
+  is the point <span class="correction" title="Original reads C'."
+  >C<sub>1</sub></span>, which represents the vapour pressure of iodine at
+  its melting point. At this point three curves cut: 1, the sublimation
+  curve of iodine; 2, the vaporization curve of fused iodine; 3,
+  C<sub>1</sub>B<sub>1</sub>, the vapour-pressure curve of the saturated
+  solutions in equilibrium with solid iodine. Starting, therefore, with the
+  system solid iodine&mdash;liquid iodine, addition of chlorine will cause
+  the temperature of equilibrium to fall continuously, while the vapour
+  pressure will first increase, pass through a maximum and then fall
+  continuously <!-- Page 167 --><span class="pagenum"><a
+  name="page167"></a>{167}</span>until the eutectic point, B
+  (B<sub>1</sub>), is reached.<a name="NtA248"
+  href="#Nt248"><sup>[248]</sup></a> At this point the system is invariant,
+  and the pressure will therefore remain constant until all the iodine has
+  disappeared. As the concentration of the chlorine increases in the manner
+  represented by the curve B<i>f</i>H, the pressure of the vapour also
+  increases as represented by the curve
+  B<sub>1</sub><i>f</i><sub>1</sub>H<sub>1</sub>. At H<sub>1</sub>, the
+  eutectic point for iodine monochloride and iodine trichloride, the
+  pressure again remains constant until all the monochloride has
+  disappeared. As the concentration of the solution passes along the curve
+  HF, the pressure of the vapour increases as represented by the curve
+  H<sub>1</sub>F<sub>1</sub>; F<sub>1</sub> represents the pressure of the
+  vapour at the melting point of iodine trichloride. If the concentration
+  of the chlorine in the solution is continuously increased from this
+  point, the vapour pressure first increases and then decreases, until the
+  eutectic point for iodine trichloride and solid chlorine is reached
+  (D<sub>1</sub>). Curves Cl<sub>2</sub> solid and Cl<sub>2</sub> liquid
+  represent the sublimation and vaporization curves of chlorine, the
+  melting point of chlorine being -102°.</p>
+
+  <p>Although complete measurements of the vapour pressure of the different
+  systems of pure iodine to pure chlorine have not been made, the
+  experimental data are nevertheless sufficient to allow of the general
+  form of the curves being indicated with certainty.</p>
+
+  <p><b>Bivariant Systems.</b>&mdash;To these, only a brief reference need
+  be made. Since there are two components, two phases will form a bivariant
+  system. The fields in which these systems can exist are shown in Fig. 43
+  and Fig. 44, which is a more diagrammatic representation of a portion of
+  Fig. 43.</p>
+
+  <div class="contents">
+    <div class="stanza">
+      <p>&nbsp; I. Iodine&mdash;vapour.</p>
+      <p>&nbsp;II. Solution&mdash;vapour.</p>
+      <p>III. Iodine trichloride&mdash;vapour.</p>
+      <p>IV. Iodine monochloride&mdash;vapour.</p>
+    </div>
+  </div>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/fig044.png"><img style="width:100%" src="images/fig044.png"
+      alt="Fig. 44" title="Fig. 44" /></a>
+    <span class="sc">Fig. 44.</span>
+  </div>
+
+  <p>The conditions for the existence of these systems will probably be
+  best understood from Fig. 44. Since the curve B&prime;A&prime; <!-- Page
+  168 --><span class="pagenum"><a name="page168"></a>{168}</span>represents
+  the pressures under which the system iodine&mdash;solution&mdash;vapour
+  can exist, increase of volume (diminution of pressure) will cause the
+  volatilization of the solution, and the system iodine&mdash;vapour will
+  remain. If, therefore, we start with a system represented by <i>a</i>,
+  diminution of pressure at constant temperature will lead to the condition
+  represented by <i>x</i>. On the other hand, increase of pressure at
+  <i>a</i> will lead to the condensation of a portion of the vapour phase.
+  Since, now, the concentration of chlorine in the vapour is greater than
+  in the solution, condensation of vapour would increase the concentration
+  of chlorine in the solution; a certain amount of iodine must therefore
+  pass into solution in order that the composition of the latter shall
+  remain unchanged.<a name="NtA249" href="#Nt249"><sup>[249]</sup></a> If,
+  therefore, the volume of vapour be sufficiently great, continued
+  diminution of volume will ultimately lead to the disappearance of all the
+  iodine, and there will remain only solution and vapour (field II.). As
+  the diminution of volume is continued, the vapour pressure and the
+  concentration of the chlorine in the solution will increase, until when
+  the pressure has reached the value <i>b</i>, iodine monochloride can
+  separate out. The system, therefore, again becomes univariant, and at
+  constant temperature the pressure and composition of the phases must
+  remain unchanged. Diminution of volume will therefore not effect an
+  increase of pressure, but a condensation of the vapour; and since this is
+  richer in chlorine than the <!-- Page 169 --><span class="pagenum"><a
+  name="page169"></a>{169}</span>solution, solid iodine monochloride must
+  separate out in order that the concentration of the solution remain
+  unchanged.<a name="NtA250" href="#Nt250"><sup>[250]</sup></a> As the
+  result, therefore, we obtain the bivariant system iodine
+  monochloride&mdash;vapour.</p>
+
+  <p>A detailed discussion of the effect of a continued increase of
+  pressure will not be necessary. From what has already been said and with
+  the help of Fig. 44, it will readily be understood that this will lead
+  successively to the univariant system (<i>c</i>), iodine
+  monochloride&mdash;solution&mdash;vapour; the bivariant system
+  solution&mdash;vapour (field II.); the univariant system (<i>d</i>),
+  iodine trichloride&mdash;solution&mdash;vapour; and the bivariant system
+  <i>x&prime;</i>, iodine trichloride&mdash;vapour. If the temperature of
+  the experiment is above the melting point of the monochloride, then the
+  systems in which this compound occurs will not be formed.</p>
+
+  <p><b>Sulphur Dioxide and Water.</b>&mdash;In the case just studied we
+  have seen that the components can combine to form definite compounds
+  possessing stable melting points. The curves of equilibrium, therefore,
+  resemble in their general aspect those of calcium chloride and water, or
+  of ferric chloride and water. In the case of sulphur dioxide and water,
+  however, the melting point of the compound formed cannot be realized,
+  because transition to another system occurs; retroflex
+  concentration-temperature curves are therefore not found here, but the
+  curves exhibit breaks or sudden changes in direction at the transition
+  points, as in the case of the systems formed by sodium sulphate and
+  water. The case of sulphur dioxide and water is also of interest from the
+  fact that two liquid phases can be formed.</p>
+
+  <p>The phases which occur are&mdash;Solid: ice, sulphur dioxide hydrate,
+  SO<sub>2</sub>,7H<sub>2</sub>O. Liquid: two solutions, the one containing
+  excess of sulphur dioxide, the other excess of water, and represented by
+  the symbols SO<sub>2</sub> <a href="images/wavy.png"><img
+  src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy"
+  /></a> <i>x</i>H<sub>2</sub>O (solution I.), and H<sub>2</sub>O <a
+  href="images/wavy.png"><img src="images/wavy.png" class="middle"
+  style="height:1.5ex" alt="wavy" /></a> <i>y</i>SO<sub>2</sub> (solution
+  II.). Vapour: a mixture of sulphur dioxide and water vapour in varying
+  proportions. Since there are two components, sulphur dioxide and water,
+  the number of <!-- Page 170 --><span class="pagenum"><a
+  name="page170"></a>{170}</span>possible systems is considerable. Only the
+  following, however, have been studied:&mdash;</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>I. <i>Invariant Systems: Four co-existing phases.</i></p>
+      <p class="i2">(<i>a</i>) Ice, hydrate, solution, vapour.</p>
+      <p class="i2">(<i>b</i>) Hydrate, solution I., solution II., vapour.</p>
+    </div>
+
+    <div class="stanza">
+      <p>II. <i>Univariant Systems: Three co-existing phases.</i></p>
+      <p class="i2">(<i>a</i>) Hydrate, solution I., vapour.</p>
+      <p class="i2">(<i>b</i>) Hydrate, solution II., vapour.</p>
+      <p class="i2">(<i>c</i>) Solution I., solution II., vapour.</p>
+      <p class="i2">(<i>d</i>) Hydrate, solution I., solution II.</p>
+      <p class="i2">(<i>e</i>) Hydrate, ice, vapour.</p>
+      <p class="i2">(<i>f</i>) Ice, solution II., vapour.</p>
+      <p class="i2">(<i>g</i>) Ice, hydrate, solution II.</p>
+    </div>
+
+    <div class="stanza">
+      <p>III. <i>Bivariant Systems: Two co-existing phases.</i></p>
+      <p class="i2">(<i>a</i>) Hydrate, solution I.</p>
+      <p class="i2">(<i>b</i>) Hydrate, solution II.</p>
+      <p class="i2">(<i>c</i>) Hydrate, vapour.</p>
+      <p class="i2">(<i>d</i>) Hydrate, ice.</p>
+      <p class="i2">(<i>e</i>) Solution I., solution II.</p>
+      <p class="i2">(<i>f</i>) Solution I., vapour.</p>
+      <p class="i2">(<i>g</i>) Solution I., ice.</p>
+      <p class="i2">(<i>h</i>) Solution II., vapour.</p>
+      <p class="i2">(<i>i</i>) Solution II., ice.</p>
+      <p class="i2">(<i>j</i>) Ice, vapour.</p>
+    </div>
+  </div>
+
+  <div class="figright" style="width:33%;">
+      <a href="images/fig045.png"><img style="width:100%" src="images/fig045.png"
+      alt="Fig. 45" title="Fig. 45" /></a>
+    <span class="sc">Fig.</span> 45.
+  </div>
+
+  <p><b>Pressure-Temperature Diagram.</b><a name="NtA251"
+  href="#Nt251"><sup>[251]</sup></a>&mdash;If sulphur dioxide is passed
+  into water at 0°, a solution will be formed and the temperature at which
+  ice can exist in equilibrium with this solution will fall more and more
+  as the concentration of the sulphur dioxide increases. At -2.6°, however,
+  a cryohydric point is reached at which solid hydrate separates out, and
+  the system becomes invariant. The curve AB (Fig. 45) therefore represents
+  the pressure of the system ice&mdash;solution II.&mdash;vapour, and B
+  represents the temperature and pressure at which the invariant system
+  ice&mdash;hydrate&mdash;solution II.&mdash;vapour can exist. At this
+  point the temperature is -2.6°, and the pressure 21.1 cm. If heat is
+  withdrawn from this system, the solution will ultimately <!-- Page 171
+  --><span class="pagenum"><a name="page171"></a>{171}</span>solidify to a
+  mixture of ice and hydrate, and there will be obtained the univariant
+  system ice&mdash;hydrate&mdash;vapour. The vapour pressure of this system
+  has been determined down to a temperature of -9.5°, at which temperature
+  the pressure amounts to 15 cm. The pressures for this system are
+  represented by the curve BC. If at the point B the volume is diminished,
+  the pressure must remain constant, but the relative amounts of the
+  different phases will undergo change. If suitable quantities of these are
+  present, diminution of volume will ultimately lead to the total
+  condensation of the vapour phase, and there will remain the univariant
+  system ice&mdash;hydrate&mdash;solution. The temperature of equilibrium
+  of this system will alter with the pressure, but, as in the case of the
+  melting point of a simple substance, great differences of pressure will
+  cause only comparatively small changes in the temperature of equilibrium.
+  The change of the cryohydric point with the pressure is represented by
+  the line BE; the actual values have not been determined, but the curve
+  must slope towards the pressure axis because fusion is accompanied by
+  diminution of volume, as in the case of pure ice. <!-- Page 172 --><span
+  class="pagenum"><a name="page172"></a>{172}</span></p>
+
+  <p>A fourth univariant system can be formed at B. This is the system
+  hydrate&mdash;solution II.&mdash;vapour. The conditions for the existence
+  of this system are represented by the curve BF, which may therefore be
+  regarded as the vapour-pressure curve of the saturated solution of
+  sulphur dioxide heptahydrate in water. Unlike the curve for iodine
+  trichloride&mdash;solution&mdash;vapour, this curve cannot be followed to
+  the melting point of the hydrate. Before this point is reached, a second
+  liquid phase appears, and an invariant system consisting of
+  hydrate&mdash;solution I.&mdash;solution II.&mdash;vapour is formed. We
+  have here, therefore, the phenomenon of melting under the solution as in
+  the case of succinic nitrile and water (p.&nbsp;<a href="#page122">122</a>).
+  This point is represented in the diagram by F; the temperature at this
+  point is 12.1°, and the pressure 177.3 cm. The range of stable existence
+  of the hydrate is therefore from -2.6° to 12.1°; nevertheless, the curve
+  FB has been followed down to a temperature of -6°, at which point ice
+  formed spontaneously.</p>
+
+  <p>So long as the four phases hydrate, two liquid phases, and vapour are
+  present, the condition of the system is perfectly defined. By altering
+  the conditions, however, one of the phases can be made to disappear, and
+  a univariant system will then be obtained. Thus, if the vapour phase is
+  made to disappear, the univariant system solution I.&mdash;solution
+  II.&mdash;hydrate, will be left, and the temperature at which this system
+  is in equilibrium will vary with the pressure. This is represented by the
+  curve FI; under a pressure of 225 atm. the temperature of equilibrium is
+  17.1°. Increase of pressure, therefore, raises the temperature at which
+  the three phases can coexist.</p>
+
+  <p>Again, addition of heat to the invariant system at F will cause the
+  disappearance of the solid phase, and there will be formed the univariant
+  system solution I.&mdash;solution II.&mdash;vapour. In the case of this
+  system the vapour pressure increases as the temperature rises, as
+  represented by the curve FG. Such a system is analogous to the case of
+  ether and water, or other two partially miscible liquids (p.&nbsp;<a
+  href="#page103">103</a>). As the temperature changes, the composition of
+  the two liquid phases will undergo change; but this system has not been
+  studied fully.</p>
+
+  <p>The fourth curve, which ends at the quadruple point F, is <!-- Page
+  173 --><span class="pagenum"><a name="page173"></a>{173}</span>that
+  representing the vapour pressure of the system hydrate&mdash;solution
+  I.&mdash;vapour (FH). This curve has been followed to a temperature of
+  0°, the pressure at this point being 113 cm. The metastable prolongation
+  of GF has also been determined. Although, theoretically, this curve must
+  lie below FH, it was found that the difference in the pressure for the
+  two curves was within the error of experiment.</p>
+
+  <p><b>Bivariant Systems.</b>&mdash;The different bivariant systems,
+  consisting of two phases, which can exist within the range of temperature
+  and pressure included in Fig. 45, were given on p.&nbsp;<a
+  href="#page170">170</a>. The conditions under which these systems can
+  exist are represented by the areas in the diagram, and the fields of the
+  different bivariant systems are indicated by letters, corresponding to
+  the letters on p.&nbsp;<a href="#page170">170</a>. Just as in the case of
+  one-component systems (p.&nbsp;<a href="#page29">29</a>), we found that the
+  field lying between any two curves gave the conditions of existence of
+  that phase which was common to the two curves, so also in the case of
+  two-component systems, a bivariant two-phase system occurs in the field
+  enclosed<a name="NtA252" href="#Nt252"><sup>[252]</sup></a> by the two
+  curves to which the two phases are common. As can be seen, the same
+  bivariant system can occur in more than one field.</p>
+
+  <p>As is evident from Fig. 45, three different bivariant systems are
+  capable of existing in the area HFI; which of these will be obtained will
+  depend on the relative masses of the different phases in the univariant
+  or invariant system. Thus, starting with a system represented by a point
+  on the curve HF, diminution of volume at constant temperature will cause
+  the condensation of a portion of the vapour, which is rich in sulphur
+  dioxide; since this would increase the concentration of sulphur dioxide
+  in the solution, it must be counteracted by the passage of a portion of
+  the hydrate (which is relatively poor in sulphur dioxide) into the
+  solution. If, therefore, the amount of hydrate present is relatively very
+  small, the final result of the compression will be the production of the
+  system <i>f</i>, solution I.&mdash;vapour. On the other hand, if the
+  vapour is present in relatively small amount, it will be the first phase
+  to disappear, <!-- Page 174 --><span class="pagenum"><a
+  name="page174"></a>{174}</span>and the bivariant system <i>a</i>,
+  hydrate&mdash;solution I., will be obtained. Finally, if we start with
+  the invariant system at F, compression will cause the condensation of
+  vapour, while the composition of the two solutions will remain unchanged.
+  When all the vapour has disappeared, the univariant system
+  hydrate&mdash;solution I.&mdash;solution II. will be left. If, now, the
+  pressure is still further increased, while the temperature is kept below
+  12°, more and more hydrate must be formed at the expense of the two
+  liquid phases (because 12° is the lower limit for the coexistence of the
+  two liquid phases), and if the amount of the solution I. (containing
+  excess of sulphur dioxide) is relatively small, it will disappear before
+  solution II., and there will be obtained the bivariant system
+  hydrate&mdash;solution II. (bivariant system <i>b</i>).</p>
+
+  <p>In a similar manner, account can be taken of the formation of the
+  other bivariant systems.</p>
+
+  <p>A behaviour similar to that of sulphur dioxide and water is shown by
+  chlorine and water and by bromine and water, although these have not been
+  so fully studied.<a name="NtA253" href="#Nt253"><sup>[253]</sup></a> In
+  the case of hydrogen bromide and water, and of hydrogen chloride and
+  water, a hydrate, viz. HBr,2H<sub>2</sub>O and HCl,2H<sub>2</sub>O, is
+  formed which possesses a definite melting point, as in the case of iodine
+  trichloride. In these cases, therefore, a retroflex curve is obtained.
+  Further, just as in the case of the chlorides of iodine the upper branch
+  of the retroflex curve ended in a eutectic point, so also in the case of
+  the hydrate HBr,2H<sub>2</sub>O the upper branch of the curve ends in a
+  eutectic point at which the system
+  dihydrate&mdash;monohydrate&mdash;solution&mdash;vapour can exist. Before
+  the melting point of the monohydrate is reached, two liquid phases are
+  formed, as in the case of sulphur dioxide and water.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 175 --><span class="pagenum"><a name="page175"></a>{175}</span></p>
+
+<h3>CHAPTER X</h3>
+
+<p class="cenhead">SOLID SOLUTIONS. MIXED CRYSTALS</p>
+
+  <p><b>General.</b>&mdash;With the conception of gaseous and liquid
+  solutions, every one is familiar. Gases can mix in all proportions to
+  form homogeneous solutions. Gases can dissolve in or be "absorbed" by
+  liquids; and solids, also, when brought in contact with liquids, "pass
+  into solution" and yield a homogeneous liquid phase. On the other hand,
+  the conception of a <i>solid solution</i> is one which in many cases is
+  found more difficult to appreciate; and the existence and behaviour of
+  solid solutions, in spite of their not uncommon occurrence and
+  importance, are in general comparatively little known.</p>
+
+  <p>The reason of this is to be found, to some extent, no doubt, in the
+  fact that the term "solid solution" was introduced at a comparatively
+  recent date,<a name="NtA254" href="#Nt254"><sup>[254]</sup></a> but it is
+  probably also due in some measure to a somewhat hazy comprehension of the
+  definition of the term "solution" itself. As has already been said (p.&nbsp;<a
+  href="#page92">92</a>), a solution is a homogeneous phase, the
+  composition of which can vary continuously within certain limits; the
+  definition involves, therefore, no condition as to the physical state of
+  the substances. Accordingly, solid solutions are homogeneous solid
+  phases, the composition of which can undergo continuous variation within
+  certain limits. Just as we saw that the range of variation of composition
+  is more limited in the case of liquids than in the case of gases, so also
+  we find that the limits of miscibility are in general still more
+  restricted in the case of solids. Examples of complete miscibility are,
+  however, not unknown even in the case of solid substances.</p>
+
+  <p>Solid solutions have long been known, although, of course, <!-- Page
+  176 --><span class="pagenum"><a name="page176"></a>{176}</span>they were
+  not defined as such. Thus, the phenomena of "occlusion" of gases by
+  metals and other substances (occlusion of hydrogen by palladium;
+  occlusion of hydrogen by iron) are due to the formation of solid
+  solutions. The same is probably also true of the phenomena of
+  "adsorption," as in the removal of organic colouring matter by charcoal,
+  although, in this case, surface tension no doubt plays a considerable
+  part.<a name="NtA255" href="#Nt255"><sup>[255]</sup></a></p>
+
+  <p>As examples of the solution of gases in solids there may be cited (in
+  addition to the phenomena of occlusion already mentioned), the hydrated
+  silicates and the zeolites. During dehydration these crystalline
+  substances remain clear and transparent, and the pressure of the water
+  vapour which they emit varies with the degree of hydration or the
+  concentration of water in the mineral.<a name="NtA256"
+  href="#Nt256"><sup>[256]</sup></a> As examples of the solution of solids
+  in solids we have the cementation of iron by charcoal, the formation of
+  glass, and the crystallization together of isomorphous substances.</p>
+
+  <p>Although we have here spoken of the glasses as "solid solutions," it
+  should be mentioned that the term "solid" is used in its popular sense.
+  Strictly speaking, the glasses are to be regarded as supercooled liquids
+  (see also p. 53, <a href="#Nt96">footnote</a>).</p>
+
+  <p>In discussing the equilibria in systems containing a solid solution,
+  it is of essential importance to remember that a solid solution
+  constitutes only <i>one</i> phase, a phase of varying composition, as in
+  the case of liquid solutions.</p>
+
+  <p><b>Solution of Gases in Solids.</b>&mdash;Comparatively little work
+  has been done in this connection, the investigations being limited
+  chiefly to the phenomena of occlusion or adsorption of gases by
+  charcoal.<a name="NtA257" href="#Nt257"><sup>[257]</sup></a> We shall,
+  therefore, indicate only briefly <!-- Page 177 --><span
+  class="pagenum"><a name="page177"></a>{177}</span>and in a general
+  manner, the behaviour which the Phase Rule enables us to foresee.<a
+  name="NtA258" href="#Nt258"><sup>[258]</sup></a></p>
+
+  <p>In dealing with the systems formed by the two phases gas&mdash;solid,
+  three chief cases call for mention:&mdash;</p>
+
+  <p>I. <i>The gas is not absorbed by the solid, but when the pressure
+  reaches a certain value, combination of the two components can
+  result.</i></p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig046.png"><img style="width:100%" src="images/fig046.png"
+      alt="Fig. 46" title="Fig. 46" /></a>
+    Fig. 46.
+  </div>
+
+  <p>The graphic representation of such a system is shown in Fig. 46, the
+  ordinates being the pressures of the gas, and the abscissæ the
+  concentrations of the gaseous component in the solid phase. Since there
+  is no formation of a solid solution, the concentration of gas in the
+  solid phase remains zero until the pressure has increased to the point A.
+  At this point combination can take place. There will now be three phases
+  present, viz. solid component, compound, and vapour. The system is
+  therefore univariant, and if the temperature is maintained constant, the
+  vapour pressure will be constant, irrespective of the amount of compound
+  formed, <i>i.e.</i> irrespective of the relative amounts of gas and
+  solid. This is indicated by the line AB. When the solid component has
+  entirely disappeared, the system ceases to be univariant, and if no
+  absorption occurs, the pressure will increase again, as shown by BC. If a
+  second compound can be formed, then a second <i>pc</i>-line will be
+  obtained, similar to the preceding. To this group belong the salt
+  hydrates (Chap. VII.).</p>
+
+  <p>II. <i>The gas may be absorbed and may also form a compound.</i></p>
+
+  <p>If absorption of gas occurs with formation of a solid solution, then,
+  as the system consists of two phases, solution&mdash;vapour, it is
+  bivariant. At constant temperature, therefore, the pressure will still
+  vary with the concentration of the gaseous component in the solid phase.
+  This is represented by the curve AB in Fig. 47. When, however, the
+  pressure has reached a certain value, combination can take place; and
+  since there are now three phases present, the system is <!-- Page 178
+  --><span class="pagenum"><a name="page178"></a>{178}</span>univariant,
+  and at constant temperature the pressure is constant, as shown by the
+  line BC.</p>
+
+  <p>III. <i>Absorption of gas occurs, but at a certain concentration the
+  solid solution can separate into two immiscible solid solutions.</i></p>
+
+  <p>We have seen, in Chapter VI., that two liquids can form two immiscible
+  solutions, and the same has also been found true of solid solutions, as
+  we shall presently learn more fully. If, now, two immiscible solutions
+  are formed, then the system will become univariant, and at constant
+  temperature the <i>pc</i>-curve will be a straight line, as in the case
+  of the formation of a compound (cf. p.&nbsp;<a href="#page86">86</a>). The
+  behaviour of this system will, therefore, also be represented
+  diagrammatically by Fig. 47.</p>
+
+  <div class="figcenter" style="width:16%;">
+      <a href="images/fig047.png"><img style="width:100%" src="images/fig047.png"
+      alt="Fig. 47" title="Fig. 47" /></a>
+    <span class="sc">Fig.</span> 47.
+  </div>
+
+  <p><i>Palladium and Hydrogen.</i>&mdash;The phenomenon of the absorption
+  of hydrogen by palladium, to which Graham gave the name "occlusion," is
+  one that has claimed the attention of several investigators. Although
+  Graham was not of opinion that a compound is formed, but rather that the
+  gas undergoes very great condensation, acts as a quasi-metal (to which he
+  gave the name hydrogenium), and forms a homogeneous alloy with the
+  palladium, later investigations, especially those of Troost and
+  Hautefeuille,<a name="NtA259" href="#Nt259"><sup>[259]</sup></a> pointed
+  to the formation of a definite chemical compound, having the formula
+  Pd<sub>2</sub>H. This conclusion has, however, not been confirmed by
+  subsequent investigation.<a name="NtA260"
+  href="#Nt260"><sup>[260]</sup></a></p>
+
+  <p>Roozeboom and Hoitsema<a name="NtA261"
+  href="#Nt261"><sup>[261]</sup></a> sought to arrive at a final decision
+  as to the nature of the phenomenon by an investigation of the equilibrium
+  between hydrogen and palladium on the basis of the Phase Rule
+  classification given above. If a compound is formed, diminution of volume
+  would cause no increase of pressure, but only an increase in the amount
+  of the compound.</p>
+
+  <p>As this is the only case of gas absorption which has been <!-- Page
+  179 --><span class="pagenum"><a name="page179"></a>{179}</span>accurately
+  studied from this point of view, a brief account of the results obtained
+  will be given here, although these are not so clear and free from
+  ambiguity as one would desire.</p>
+
+  <p>The scientists just mentioned investigated the variation of the
+  pressure of hydrogen with the amount absorbed by the metal at different
+  temperatures, and a few of their results, typical of all, are represented
+  graphically in Fig. 48; the curves indicating the variation of the gas
+  pressure with the concentration of the hydrogen in the palladium at the
+  temperatures 120°, 170°, and 200°. As can be seen, the curve consists of
+  three parts, an ascending portion which passes gradually and continuously
+  into an almost horizontal but slightly ascending middle part, which in
+  turn passes without break into a second rapidly ascending curve. This, as
+  Fig. 48 indicates, is the general form of the curve; but the length of
+  the middle portion varies with the temperature, being shorter at higher
+  than at lower temperatures.</p>
+
+  <div class="figcenter" style="width:27%;">
+      <a href="images/fig048.png"><img style="width:100%" src="images/fig048.png"
+      alt="Fig. 48" title="Fig. 48" /></a>
+    <span class="sc">Fig. 48.</span>
+  </div>
+
+  <p>What is the interpretation to be put on these curves? With regard to
+  the two end portions, these represent bivariant, two-phase systems,
+  consisting of a solid solution and gas. They correspond, therefore, to
+  curve AB in Fig. 47. If the middle portion were horizontal, it would
+  indicate either the formation of a compound or of two immiscible solid
+  solutions. If a compound Pd<sub>2</sub>H were formed, then the middle
+  portion would at all temperatures end at the same value of the
+  concentration, viz. that corresponding to 0.5 atoms of hydrogen to 1 atom
+  of palladium. As the figure shows, however, this is not the case; the
+  higher the temperature, the lower is the concentration at which the
+  middle passes into the terminal portion of the curve. <!-- Page 180
+  --><span class="pagenum"><a name="page180"></a>{180}</span>Such a
+  behaviour would, however, agree with the assumption of the formation of
+  two solid solutions, the "miscibility" of which increases with the
+  temperature, as in the case of the liquid solutions of phenol and water
+  (p.&nbsp;<a href="#page97">97</a>). Nevertheless, although the assumption of
+  the formation of two solid solutions is more satisfactory than that of
+  the formation of a compound, it does not entirely explain the facts. If
+  two solid solutions are formed, the pressure curve should be horizontal,
+  but this is not the case; and the deviation from the horizontal does not
+  appear to be due to impurities either in the gas or in the metal, but is
+  apparently a peculiarity of the system. Further, the gradual instead of
+  abrupt passage of the three portions of the curve into one another
+  remains unexplained. Hoitsema has expressed the opinion that the
+  occlusion of hydrogen by palladium is a process of continuous absorption,
+  the peculiar form of the curve&mdash;the flat middle portion&mdash;being
+  possibly due to a condensation of the gas, even at temperatures far above
+  the critical temperature of liquid hydrogen.</p>
+
+  <p>While, therefore, the occlusion of hydrogen by palladium still
+  presents some unexplained phenomena, the behaviour found by Hoitsema
+  would appear to disprove conclusively the formation of a definite
+  chemical compound.<a name="NtA262" href="#Nt262"><sup>[262]</sup></a></p>
+
+<p class="cenhead"><span class="sc">Solution of Solids in Solids. Mixed Crystals.</span></p>
+
+  <p>The introduction by van't Hoff of the term "solid solution" resulted
+  from the discovery of a number of deviations from the Raoult-van't Hoff
+  law for the depression of the freezing point by dissolved substances. In
+  all cases, the depression was too small; in some instances, indeed, the
+  freezing point may be raised. To explain these irregularities, van't Hoff
+  assumed that the dissolved substance crystallized out along with the
+  solid solvent; and he showed how this would account for the <!-- Page 181
+  --><span class="pagenum"><a name="page181"></a>{181}</span>deviations
+  from the law of the depression of the freezing point, which had been
+  developed on the assumption that only the pure solvent crystallized out
+  from the solution.<a name="NtA263" href="#Nt263"><sup>[263]</sup></a></p>
+
+  <p>The "mixed crystals" which were thus obtained, and which van't Hoff
+  called dilute solid solutions, showed great resemblance in their
+  behaviour to ordinary liquid solutions, and obeyed the laws applicable to
+  these. These laws, however, can no longer be applied in the case of the
+  concentrated solid solutions formed by the crystallization together of
+  isomorphous substances, and known as isomorphous mixtures. Indeed, it has
+  been contended<a name="NtA264" href="#Nt264"><sup>[264]</sup></a> that
+  these isomorphous mixtures should not be considered as solid solutions at
+  all, although no sharp line of demarcation can be drawn between the two
+  classes. The differences, however, in the behaviour of the two groups are
+  of a quantitative rather than a qualitative nature; and since we are
+  concerned at present only with the qualitative behaviour, we shall make
+  no distinction between the crystalline solid solutions and the
+  isomorphous mixtures, but shall study the behaviour of the two classes
+  under the head of "mixed crystals."</p>
+
+  <p>Mixed crystals can be formed either by sublimation<a name="NtA265"
+  href="#Nt265"><sup>[265]</sup></a> or from a liquid phase; and in the
+  latter case the mixed crystals can be deposited either from solution in a
+  common solvent or from a mixture of the fused components. In this method
+  of formation, which alone will be discussed in the present chapter, we
+  are dealing with the fusion curves of two substances, where, however, the
+  liquid solution is in equilibrium not with one of the pure components,
+  but with a solid solution or mixed crystal. The simple scheme (Fig. 29,
+  p.&nbsp;<a href="#page117">117</a>) which was obtained in the case of two
+  components which crystallize out in the pure state, is no longer
+  sufficient in the case of the formation of mixed crystals. With the help
+  of the Phase Rule, however, the different possible systems can be
+  classified; and examples of the different cases predicted by the Phase
+  Rule have also been obtained by experiment.</p>
+
+<p><!-- Page 182 --><span class="pagenum"><a name="page182"></a>{182}</span></p>
+
+  <p>We shall now consider briefly the formation of mixed crystals by
+  isomorphous substances; the consideration of the formation of mixed
+  crystals of isodimorphous substances will, on account of the complexity
+  of the relationships, not be undertaken here.<a name="NtA266"
+  href="#Nt266"><sup>[266]</sup></a></p>
+
+<p class="cenhead"><i>Formation of Mixed Crystals of Isomorphous Substances.</i></p>
+
+  <p>For the purpose of representing the relationships found here we shall
+  employ a temperature-concentration diagram,<a name="NtA267"
+  href="#Nt267"><sup>[267]</sup></a> in which the ordinates represent the
+  temperature and the abscissæ the concentration of the components. Since
+  there are two solutions, the liquid and the solid, and since the
+  concentration of the components in these two phases is not, in general,
+  the same, two curves will be required for each system, one relating to
+  the liquid phase, the other relating to the solid. The temperature at
+  which solid begins to be deposited from the liquid solution will be
+  called the <i>freezing point</i> of the mixture, and the temperature at
+  which the solid solution just begins to liquefy will be called the
+  <i>melting point</i> of the solid solution. The temperature-concentration
+  curve for the liquid phase will therefore be the freezing-point curve;
+  that for the solid solution, the melting-point curve. The latter will be
+  represented by a dotted line.<a name="NtA268"
+  href="#Nt268"><sup>[268]</sup></a></p>
+
+<p><!-- Page 183 --><span class="pagenum"><a name="page183"></a>{183}</span></p>
+
+<p class="cenhead">I.&mdash;<span class="sc">The Two Components can form an Unbroken Series of Mixed Crystals.</span></p>
+
+  <p>Since, as has already been pointed out (p.&nbsp;<a
+  href="#page176">176</a>), a mixed crystal (solid solution) constitutes
+  only one phase, it is evident that if the two components are miscible
+  with one another in all proportions in the solid state, there can never
+  be more than one solid phase present, viz. the solid solution or mixed
+  crystal. If the components are completely miscible in the solid state,
+  they will also be completely miscible in the liquid state, and there can
+  therefore be only one liquid phase. The system can at no point become
+  invariant, because there can never be more than three phases present.
+  When, therefore, the two components form a continuous series of mixed
+  crystals, the equilibrium curve must also be continuous. Of these systems
+  three types are found.</p>
+
+  <div class="figcenter" style="width:17%;">
+      <a href="images/fig049.png"><img style="width:100%" src="images/fig049.png"
+      alt="Fig. 49" title="Fig. 49" /></a>
+    <span class="sc">Fig.</span> 49.
+  </div>
+
+  <p>(<i>a</i>) <i>The freezing points of all mixtures lie between the
+  freezing points of the pure components</i> (Curve I., Fig. 49).</p>
+
+  <p><b>Examples.</b>&mdash;This type of curve is represented by the mixed
+  crystals of naphthalene and <span class="grk">&beta;</span>-naphthol.<a
+  name="NtA269" href="#Nt269"><sup>[269]</sup></a> The addition of <span
+  class="grk">&beta;</span>-naphthol to naphthalene raises the freezing
+  point of the latter, and the rise is directly proportional to the amount
+  of naphthol added. The freezing point curve is therefore a straight line
+  joining the melting points of the two components. This behaviour,
+  however, is rather exceptional, the freezing-point curve lying generally
+  above, sometimes also below, the straight line joining the melting points
+  of the pure components. Thus the freezing-point curve of mixtures of
+  <span class="grk">&alpha;</span>-monochlorocinnamic aldehyde and <span
+  class="grk">&alpha;</span>-monobromocinnamic aldehyde<a name="NtA270"
+  href="#Nt270"><sup>[270]</sup></a> lies above the <!-- Page 184 --><span
+  class="pagenum"><a name="page184"></a>{184}</span>straight line joining
+  the melting points of the pure components (31.22° and 69.56°), as is
+  evident from the following table:&mdash;</p>
+
+<table class="nobctr" summary="Freezing points of chloro- and bromo-cinnamic aldehyde mixture" title="Freezing points of chloro- and bromo-cinnamic aldehyde mixture">
+<tr><td class="toribob" style="text-align:center"> Molecules of bromo-<br />cinnamic aldehyde in<br />100 mols. of mixture.
+</td><td class="toboleb" style="text-align:center"> Freezing point.
+</td><td class="toboleb" style="text-align:center"> Deviation from straight<br />line.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 0.00 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 31.22° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> &nbsp; &mdash;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 10.48 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 37.28° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 2.04°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 21.91 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 43.12° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 3.50°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 30.07 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 46.80° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 4.05°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 45.04 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 52.94° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 4.45°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 62.16 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 58.82° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 3.77°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 82.98 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 65.07° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 2.03°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:4em;"> 93.50 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 67.91° </td><td class="leb" style="text-align:left; padding-left:4em; padding-right:2em;"> 0.84°</td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:4em;"> 100.00 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 69.56° </td><td class="boleb" style="text-align:left; padding-left:4em; padding-right:2em;"> &nbsp; &mdash;</td></tr>
+
+</table>
+
+  <p><b>Melting-point Curve.</b>&mdash;This curve, like the freezing-point
+  curve, must also be continuous, and the melting points of the different
+  solid solutions will lie between the melting points of the pure
+  components. This is represented by the dotted line in Fig. 49, I. The
+  relative position of the two curves, which can be deduced with the help
+  of thermodynamics and also by experimental determination, is found in all
+  cases to be in accordance with the following rule: At any given
+  temperature, <i>the concentration of that component by the addition of
+  which the freezing point is depressed, is greater in the liquid than in
+  the solid phase</i>; or, conversely, <i>the concentration of that
+  component by the addition of which the freezing point is raised, is
+  greater in the solid than in the liquid phase</i>. An illustration of
+  this rule is afforded by the two substances chloro- and bromo-cinnamic
+  aldehyde already mentioned. As can be seen from the above table, the
+  addition of chlorocinnamic aldehyde lowers the melting point of the
+  bromo-compound. In accordance with the rule, therefore, the concentration
+  of the chloro-compound in the liquid phase must be greater than in the
+  solid phase; and this was found experimentally. At a temperature of
+  49.44°, the liquid contained 58.52 per cent., the solid only 52.57 per
+  cent. of the chlorocinnamic aldehyde.</p>
+
+  <p>From this it will also be clear that on cooling a fused mixture of two
+  substances capable of forming mixed crystals, <!-- Page 185 --><span
+  class="pagenum"><a name="page185"></a>{185}</span>the temperature of
+  solidification will not remain constant during the separation of the
+  solid; nor, on the other hand, will the temperature of liquefaction of
+  the solid solution be constant. Thus, for example, if a liquid solution
+  of two components, A and B, having the composition represented by the
+  point <i>x</i> (Fig. 50), is allowed to cool, the system will pass along
+  the line <i>xx&prime;</i>. At the temperature of the point <i>a</i>,
+  mixed crystals will be deposited, the composition of which will be that
+  represented by <i>b</i>. As the temperature continues to fall, more and
+  more solid will be deposited; and since the solid phase is relatively
+  rich in the component B, the liquid will become relatively poorer in
+  this. The composition of the liquid solution will therefore pass along
+  the curve <i>ad</i>, the composition of the solid solution at the same
+  time passing along the curve <i>bc</i>; at the point <i>c</i> the liquid
+  will solidify completely.<a name="NtA271"
+  href="#Nt271"><sup>[271]</sup></a></p>
+
+  <div class="figcenter" style="width:21%;">
+      <a href="images/fig050.png"><img style="width:100%" src="images/fig050.png"
+      alt="Fig. 50" title="Fig. 50" /></a>
+    <span class="sc">Fig.</span> 50.
+  </div>
+
+  <p>Conversely, if mixed crystals of the composition and at the
+  temperature <i>x&prime;</i> are heated, liquefaction will begin at the
+  temperature <i>c</i>, yielding a liquid of the composition <i>d</i>. On
+  continuing to add heat, the temperature of the mass will rise, more of
+  the solid will melt, and the composition of the two phases will change as
+  represented by the curves <i>da</i> and <i>cb</i>. When the temperature
+  has risen to <i>a</i>, complete liquefaction will have occurred. The
+  process of solidification or of liquefaction is therefore extended over a
+  temperature interval <i>ac</i>.</p>
+
+  <p>Even when the freezing-point curve is a straight line joining <!--
+  Page 186 --><span class="pagenum"><a name="page186"></a>{186}</span>the
+  melting points of the pure components, the melting-point curve will not
+  necessarily coincide with the freezing-point curve, although it may
+  approach very near to it; complete coincidence can take place only when
+  the melting points of the two components are identical. An example of
+  this will be given later (Chap. XII.).</p>
+
+  <p>(<i>b</i>) <i>The freezing-point curve passes through a maximum</i>
+  (Curve II., Fig. 49).</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig051.png"><img style="width:100%" src="images/fig051.png"
+      alt="Fig. 51" title="Fig. 51" /></a>
+    <span class="sc">Fig.</span> 51.
+  </div>
+
+  <p>This curve exhibits the greatest degree of contrast to the
+  freezing-point curve which is obtained when the pure components
+  crystallize out. For, since the curve passes through a maximum, it is
+  evident that the freezing point of each of the components must be
+  <i>raised</i> by the addition of the other component.</p>
+
+  <p><b>Example.</b>&mdash;Very few cases belonging to this type are known.
+  The best example is found in the freezing-point curve of mixtures of
+  <i>d</i>- and <i>l</i>-carvoxime<a name="NtA272"
+  href="#Nt272"><sup>[272]</sup></a> (C<sub>10</sub>H<sub>14</sub>N.OH).
+  The freezing points and melting points of the different mixtures of
+  <i>d</i>- and <i>l</i>-carvoxime are given in the following table, and
+  represented graphically in Fig. 51:&mdash;</p>
+
+<table class="nobctr" summary="Freezing and melting points of d- and l-carvoxime mixture" title="Freezing and melting points of d- and l-carvoxime mixture">
+<tr><td class="toribob" style="text-align:center"> Per cent. of<br /><i>d</i>-carvoxime.
+</td><td class="toboleb" style="text-align:center"> Per cent. of<br /><i>l</i>-carvoxime.
+</td><td class="toboleb" style="text-align:center"> Freezing point.
+</td><td class="toboleb" style="text-align:center"> Melting point.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 100 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 99 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 1 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 98 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 2 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 73.0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 95 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 5 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 75.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 73.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 90 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 10 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 79.0° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 75.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 80 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 20 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 84.6° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 80.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 70 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 30 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 88.2° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 85.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 60 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 40 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 90.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 50 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 50 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 91.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 91.4°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 25 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 75 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 86.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 82.0°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 8 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 92 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 77.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:2em;"> 1 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 99 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.4° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:2em;"> 0 </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 100 </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.0° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 72.0°</td></tr>
+
+</table>
+
+<p><!-- Page 187 --><span class="pagenum"><a name="page187"></a>{187}</span></p>
+
+  <p>In this figure, the melting-point curve, <i>i.e.</i> the
+  temperature-concentration curve for the mixed crystals, is represented by
+  the lower curve. Since the addition of the lævo-form to the dextro-form
+  raises the melting point of the latter, the concentration of the
+  lævo-form (on the right-hand branch of the curve) must, in accordance
+  with the rule given, be greater in the solid phase than in the liquid.
+  Similarly, since addition of the dextro-form raises the melting point of
+  the lævo-form, the solid phase (on the left-hand branch of the curve)
+  must be richer in dextro- than in lævo-carvoxime. At the maximum point,
+  the melting-point and freezing-point curves touch; at this point,
+  therefore, the composition of the solid and liquid phases must be
+  identical. It is evident, therefore, that at the maximum point the liquid
+  will solidify, or the solid will liquefy completely without change of
+  temperature; and, accordingly, mixed crystals of the composition
+  represented by the maximum point will exhibit a definite melting point,
+  and will in this respect behave like a simple substance.</p>
+
+  <p>(<i>c</i>) <i>The freezing-point curve passes through a minimum</i>
+  (Curve III., Fig. 49).</p>
+
+  <p>In this case, as in the case of those systems where the pure
+  components are deposited, a minimum freezing point is obtained. In the
+  latter case, however, there are two freezing-point curves which intersect
+  at a eutectic point; in the case where mixed crystals are formed there is
+  only one continuous curve. On one side of the minimum point the liquid
+  phase contains relatively more, on the other side relatively less, of the
+  one component than does the solid phase; while at the minimum point the
+  composition of the two phases is the same. At this point, therefore,
+  complete solidification or complete liquefaction will occur without
+  change of temperature, and the mixed crystals will accordingly exhibit a
+  definite melting point.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/fig052.png"><img style="width:100%" src="images/fig052.png"
+      alt="Fig. 52" title="Fig. 52" /></a>
+    <span class="sc">Fig.</span> 52.
+  </div>
+
+<p><!-- Page 188 --><span class="pagenum"><a name="page188"></a>{188}</span></p>
+
+  <p><b>Example.</b>&mdash;As an example of this there may be taken the
+  mixed crystals of mercuric bromide and iodide.<a name="NtA273"
+  href="#Nt273"><sup>[273]</sup></a> Mercuric bromide melts at 236.5°, and
+  mercuric iodide at 255.4°. The mixed crystal of definite constant melting
+  point (minimum point) contains 59 mols. per cent. of mercuric bromide,
+  the melting point being 216.1°.</p>
+
+  <p>The numerical data are contained in the following table, and
+  represented graphically in Fig. 52:&mdash;</p>
+
+<table class="nobctr" summary="Freezing and melting points of mercuric bromide and iodide mixture" title="Freezing and melting points of mercuric bromide and iodide mixture">
+<tr><td class="toribob" style="text-align:center"> Mols. per cent. of<br />HgBr<sub>2</sub>.
+</td><td class="toboleb" style="text-align:center"> Freezing point.
+</td><td class="toboleb" style="text-align:center"> Melting point.</td></tr>
+
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 100 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 236.5° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 236°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 90 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 228.8° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 226°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 80 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 222.2° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 219°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 70 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217.8° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 65 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216.6° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 60 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216.1° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 215.5°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 55 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216.3° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 50 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217.3° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 40 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 221.1° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 218°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 30 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 227.8° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 223°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 20 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 236.2° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 231°</td></tr>
+<tr><td class="rib" style="text-align:right; padding-right:3em;"> 10 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 245.5° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 242°</td></tr>
+
+<tr><td class="ribob" style="text-align:right; padding-right:3em;"> 0 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 255.4° </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 254°</td></tr>
+
+</table>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig053.png"><img style="width:100%" src="images/fig053.png"
+      alt="Fig. 53" title="Fig. 53" /></a>
+    <span class="sc">Fig.</span> 53.
+  </div>
+
+  <p><b>Fractional Crystallization of Mixed Crystals.</b>&mdash;With the
+  help of the diagrams already given it will be possible to predict what
+  will be the result of the fractional crystallization of a fused mixture
+  of two substances which can form mixed crystals. Suppose, for example, a
+  fused mixture of the composition <i>x</i> (Fig. 53) is cooled down; then,
+  as we have already seen, when the temperature has fallen to <i>a</i>,
+  mixed crystals of composition, <i>b</i>, are deposited. If the
+  temperature is allowed to fall <!-- Page 189 --><span class="pagenum"><a
+  name="page189"></a>{189}</span>to <i>x&prime;</i>, and the solid then
+  separated from the liquid, the mixed crystals so obtained will have the
+  composition represented by <i>e</i>. If, now, the mixed crystals <i>e</i>
+  are completely fused and the fused mass allowed to cool, separation of
+  solid will occur when the temperature has fallen to the point <i>f</i>.
+  The mixed crystals which are deposited have now the composition
+  represented by <i>g</i>, i.e. <i>they are richer in B than the original
+  mixed crystals</i>. By repeating this process, the composition of the
+  successive crops of mixed crystals which are obtained approximates more
+  and more to that of the pure component B, while, on the other hand, the
+  composition of the liquid phase produced tends to that of pure A. By a
+  systematic and methodical repetition of the process of fractional
+  crystallization, therefore, a <i>practically</i> complete separation of
+  the components can be effected; a perfect separation is theoretically
+  impossible.</p>
+
+  <p>From this it will be readily understood that in the case of substances
+  the freezing point of which passes through a maximum, fractional
+  crystallization will ultimately lead to mixed crystals having the
+  composition of the maximum point, while the liquid phase will more and
+  more assume the composition of either pure A or pure B, according as the
+  initial composition was on the A side or the B side of the maximum point.
+  In those cases, however, where the curves exhibit a minimum, the solid
+  phase which separates out will ultimately be one of the pure components,
+  while a liquid phase will finally be obtained which has the composition
+  of the minimum point.</p>
+
+<p class="cenhead">II.&mdash;<span class="sc">The Two Components do not form a Continuous Series of Mixed Crystals.</span></p>
+
+  <p>This case corresponds to that of the partial miscibility of liquids.
+  The solid component A can "dissolve" the component B until the
+  concentration of the latter in the mixed crystal has reached a certain
+  value. Addition of a further amount of B will not alter the composition
+  of the mixed crystal, but there will be formed a second solid phase
+  consisting <!-- Page 190 --><span class="pagenum"><a
+  name="page190"></a>{190}</span>of a solution of A in B. At this point the
+  four phases, mixed crystals containing excess of A, mixed crystals
+  containing excess of B, liquid solution, vapour, can coexist; this will
+  therefore be an invariant point. The temperature-concentration curves
+  will therefore no longer be continuous, but will exhibit a break or
+  discontinuity at the point at which the invariant system is formed.</p>
+
+  <p>(<i>a</i>) <i>The freezing-point curve exhibits a transition point</i>
+  (Curve I., Fig. 54).</p>
+
+  <p>As is evident from the figure, addition of B raises the melting point
+  of A, and, in accordance with the rule previously given, the
+  concentration of B in the mixed crystals will be greater than in the
+  solution. This is represented in the figure by the dotted curve AD. On
+  the other hand, addition of A lowers the melting point of B, and the two
+  curves BC and BE are obtained for the liquid and solid phases
+  respectively. At the temperature of the line CDE the liquid solution of
+  the composition represented by C is in equilibrium with the two different
+  mixed crystals represented by D and E. At this temperature, therefore,
+  the <i>tc</i>-curve for the solid phase exhibits a discontinuity; and,
+  since the solid phase undergoes change at this point, the freezing-point
+  curve must show a break (p.&nbsp;<a href="#page111">111</a>).</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig054.png"><img style="width:100%" src="images/fig054.png"
+      alt="Fig. 54" title="Fig. 54" /></a>
+    <span class="sc">Fig. 54.</span>
+  </div>
+
+  <p><b>Example.</b>&mdash;Curves of the form given in Fig. 54 I. have been
+  found experimentally in the case of silver nitrate and sodium nitrate.<a
+  name="NtA274" href="#Nt274"><sup>[274]</sup></a> The following table
+  contains the numerical data, which are also represented graphically in
+  Fig. 55:&mdash;</p>
+
+<p><!-- Page 191 --><span class="pagenum"><a name="page191"></a>{191}</span></p>
+
+<table class="nobctr" summary="Freezing and melting points of silver and sodium nitrate mixture" title="Freezing and melting points of silver and sodium nitrate mixture">
+<tr><td class="toribob" style="text-align:center"> Molecules NaNO<sub>3</sub><br />per cent.
+</td><td class="toboleb" style="text-align:center"> Freezing point.
+</td><td class="toboleb" style="text-align:center"> Melting point.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> &nbsp; 0 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 208.6° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 208.6°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> &nbsp; 8 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 211.4° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 210°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 15.06 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 215° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 212°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 19.46 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217.2° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 214.8°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 21.9 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 222° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 215°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 26 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 228.4° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 216.5°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 29.7 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 234.8° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217.5°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 36.2 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 244.4° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 217.5°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 47.3 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 259.4° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 237.6°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 58.9 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 272° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 257°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:3em;"> 72 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 284° </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 274°</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:3em;"> 100 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 308° </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 308°</td></tr>
+
+</table>
+
+  <p>The temperature of the transition point is 217.5°; at this point the
+  liquid contains 19.5, and the two conjugate solid solutions 26 and 38
+  molecules of sodium nitrate per cent. respectively.</p>
+
+  <div class="figcenter" style="width:17%;">
+      <a href="images/fig055.png"><img style="width:100%" src="images/fig055.png"
+      alt="Fig. 55" title="Fig. 55" /></a>
+    <span class="sc">Fig.</span> 55.
+  </div>
+
+  <div class="figcenter" style="width:27%;">
+      <a href="images/fig056.png"><img style="width:100%" src="images/fig056.png"
+      alt="Fig. 56" title="Fig. 56" /></a>
+    <span class="sc">Fig.</span> 56.
+  </div>
+
+  <p>(<i>b</i>) <i>The freezing-point curve exhibits a eutectic point</i>
+  (Curve II., Fig. 54). <!-- Page 192 --><span class="pagenum"><a
+  name="page192"></a>{192}</span></p>
+
+  <p>In this case the freezing point of each of the components is lowered
+  by the addition of the other, until at last a point is reached at which
+  the liquid solution solidifies to a mixture or conglomerate of two mixed
+  crystals.</p>
+
+  <p><b>Examples.</b>&mdash;Curves belonging to this class have been
+  obtained in the case of potassium and thallium nitrates<a name="NtA275"
+  href="#Nt275"><sup>[275]</sup></a> and of naphthalene and monochloracetic
+  acid.<a name="NtA276" href="#Nt276"><sup>[276]</sup></a> The data for the
+  latter are given in the following table and represented in Fig.
+  56:&mdash;</p>
+
+<table class="nobctr" summary="Solutions of naphthalene and monochloracetic acid" title="Solutions of naphthalene and monochloracetic acid">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Liquid solution.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Solid solution.</td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Per cent.<br />naphthalene.
+</td><td class="toboleb" style="text-align:center"> Per cent. acid.
+</td><td class="toboleb" style="text-align:center"> Per cent.<br />naphthalene.
+</td><td class="toboleb" style="text-align:center"> Per cent. acid.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 62° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 100&nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 100&nbsp; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 60° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 4.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 96.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 1.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 98.3</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 55° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 21.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 79.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 2.1 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 97.9</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 53.5° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 29.4 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 70.0 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 55° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 31.3 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 68.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 59.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 40.4</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 60° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 42.4 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 57.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 80.3 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 19.7</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 65° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 53.3 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 46.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 89.2 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 10.8</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 70° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 69.7 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 32.3 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 95.4 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 4.6</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 75° </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 84.4 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 15.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 96.6 </td><td class="leb" style="text-align:right; padding-left:2em; padding-right:2em;"> 3.4</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 79.9° </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 100&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp; </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> 100&nbsp; &nbsp; </td><td class="boleb" style="text-align:right; padding-left:2em; padding-right:2em;"> &mdash; &nbsp;</td></tr>
+
+</table>
+
+  <p>At the eutectic point the liquid solution is in equilibrium with two
+  different mixed crystals the composition of which is represented by D and
+  E respectively. If, therefore, a fused mixture containing the two
+  components A and B in the proportions represented by C is cooled down, it
+  will, when the temperature has reached the point C, solidify completely
+  to a <i>conglomerate</i> of mixed crystals, D and E.</p>
+
+  <div class="figright" style="width:23%;">
+      <a href="images/fig058.png"><img style="width:100%" src="images/fig058.png"
+      alt="Fig. 58" title="Fig. 58" /></a>
+    <span class="sc">Fig.</span> 58.
+  </div>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig057.png"><img style="width:100%" src="images/fig057.png"
+      alt="Fig. 57" title="Fig. 57" /></a>
+    <span class="sc">Fig.</span> 57.
+  </div>
+
+  <p><b>Changes in Mixed Crystals with the Temperature.</b>&mdash;In the
+  case of the different types of systems represented in Fig. 49, a
+  homogeneous liquid solution of the two components will exist at
+  temperatures above the freezing-point curve, a homogeneous mixed crystal
+  at temperatures below the melting-point curve, while at any point between
+  the freezing-point and melting-point <!-- Page 193 --><span
+  class="pagenum"><a name="page193"></a>{193}</span>curves the mixture will
+  separate into a solid phase and a liquid phase. In the case, however, of
+  the two types shown in Fig. 54 the relationships are somewhat more
+  complicated. As before, the area above the freezing-point curve gives the
+  conditions under which homogeneous liquid solutions can exist; but below
+  the melting-point curve two different mixed crystals can coexist. This
+  will be best understood from Figs. 57 and 58. D and E represent, as we
+  have seen, the composition of two mixed crystals which are in equilibrium
+  with the liquid solution at the temperature of the point C. These two
+  mixed crystals represent, in the one case, a saturated solution of B in A
+  (point D), and the other a saturated solution of A in B (point E). Just
+  as we saw that the mutual solubility of two liquids varied with the
+  temperature, so also in the case of two solids; as the temperature
+  alters, the solubility of the two solid components in one another will
+  change. This alteration is indicated diagrammatically in Figs. 57 and 58
+  by the dotted curve similar to the solubility curves for two mutually
+  soluble liquids (p.&nbsp;<a href="#page101">101</a>).</p>
+
+  <p>Suppose, now, that a mixed crystal of the composition <i>x</i> is
+  cooled down, it will remain unchanged until, when the temperature has
+  fallen to <i>t&prime;</i>, the homogeneous mixed crystal breaks up into a
+  conglomerate of two mixed crystals the composition of <!-- Page 194
+  --><span class="pagenum"><a name="page194"></a>{194}</span>which is
+  represented by <i>x&prime;</i> and <i>x&Prime;</i> respectively. From
+  this, then, it can be seen that in the case of substances which form two
+  solid solutions, the mixed crystals which are desposited from the liquid
+  fused mass need not remain unchanged in the solid state, but may at some
+  lower temperature lose their homogeneity. This fact is of considerable
+  importance for the formation of alloys.<a name="NtA277"
+  href="#Nt277"><sup>[277]</sup></a></p>
+
+  <p>A good example of this will soon be met with in the case of the iron
+  and carbon alloys. The alloys of copper and tin also furnish examples of
+  the great changes which may take place in the alloy between the
+  temperature at which it separates out from the fused mass and the
+  ordinary temperature. Thus, for example, one of the alloys of copper and
+  tin which separates out from the liquid as a solid solution breaks up, on
+  cooling, into the compound Cu<sub>3</sub>Sn and liquid:<a name="NtA278"
+  href="#Nt278"><sup>[278]</sup></a> a striking example of a solid
+  substance partially liquefying on being cooled.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 195 --><span class="pagenum"><a name="page195"></a>{195}</span></p>
+
+<h3>CHAPTER XI</h3>
+
+<p class="cenhead">EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES</p>
+
+  <p>It has long been known that certain substances, <i>e.g.</i>
+  acetoacetic ester, are capable when in solution or in the fused state, of
+  reacting as if they possessed two different constitutions; and in order
+  to explain this behaviour the view was advanced (by Laar) that in such
+  cases a hydrogen atom oscillated between two positions in the molecule,
+  being at one time attached to oxygen, at another time to carbon, as
+  represented by the formula&mdash;</p>
+
+  <div class="figcenter" style="width:12%;">
+      <a href="images/211.png"><img style="width:100%" src="images/211.png"
+      alt="acetoacetic ester" title="acetoacetic ester" /></a>
+  </div>
+  <p>When the hydrogen is in one position, the substance will act as an
+  hydroxy-compound; with hydrogen in the other position, as a ketone.
+  Substances possessing this double function are called
+  <i>tautomeric</i>.</p>
+
+  <p>Doubt, however, arose as to the validity of the above explanation, and
+  this doubt was confirmed by the isolation of the two isomerides in the
+  solid state, and also by the fact that the velocity of change of the one
+  isomeride into the other could in some cases be quantitatively measured.
+  These and other observations then led to the view, in harmony with the
+  laws of chemical dynamics, that tautomeric substances in the dissolved or
+  fused state represent a <i>mixture</i> of two isomeric forms, and that
+  equilibrium is established not by <i>intra</i>- but by
+  <i>inter</i>-molecular change, as expressed by the equation&mdash;</p>
+
+<p class="cenhead">CH<sub>3</sub>.CO.CH<sub>2</sub>.CO<sub>2</sub>C<sub>2</sub>H<sub>5</sub> <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CH<sub>3</sub>.C(OH):CH.CO<sub>2</sub>C<sub>2</sub>H<sub>5</sub></p>
+
+<p><!-- Page 196 --><span class="pagenum"><a name="page196"></a>{196}</span></p>
+
+  <p>In the solid state, the one or other of the isomerides represents the
+  stable form; but in the liquid state (solution or fusion) the stable
+  condition is an equilibrium between the two forms.</p>
+
+  <p>A similar behaviour is also found in the case of other isomeric
+  substances where the isomerism is due to difference of structure,
+  <i>i.e.</i> structure isomerism (<i>e.g.</i> in the case of the oximes <a
+  href="images/212a.png"><img src="images/212a.png" class="middle"
+  style="height:6ex" alt="beta form" /></a> and <a
+  href="images/212b.png"><img src="images/212b.png" class="middle"
+  style="height:6ex" alt="alpha form" /></a>, or to difference in
+  configuration, <i>i.e.</i> stereoisomerism (<i>e.g.</i> optically active
+  substances), or to polymerism (<i>e.g.</i> acetaldehyde and paraldehyde).
+  In all such cases, although the different solid forms correspond to a
+  single definite constitution, in the liquid state a condition of
+  equilibrium between the two modifications is established. As a general
+  name for these different classes of substances, the term "dynamic
+  isomerides" has been introduced; and the different kinds of isomerism are
+  classed together under the title "dynamic isomerism."<a name="NtA279"
+  href="#Nt279"><sup>[279]</sup></a></p>
+
+  <p>By reason of the importance of these phenomena in the study more
+  especially of Organic Chemistry, a brief account of the equilibrium
+  relations exhibited by systems composed of dynamic isomerides may be
+  given here.<a name="NtA280" href="#Nt280"><sup>[280]</sup></a></p>
+
+  <p>In studying the fusion and solidification of those substances which
+  exhibit the relationships of dynamic isomerism, the phenomena observed
+  will vary somewhat according as the reversible transformation of the one
+  form into the other takes place with measurable velocity at temperatures
+  in the neighbourhood of the melting points, or only at some higher
+  temperature. If the transformation is very rapid, the system will behave
+  like a one-component system, but if the isomeric change is comparatively
+  slow, the behaviour will be that of a two-component system.</p>
+
+  <p><b>Temperature-Concentration Diagram.</b>&mdash;The relationships
+  which are met with here will be most readily understood with <!-- Page
+  197 --><span class="pagenum"><a name="page197"></a>{197}</span>the help
+  of Fig. 59. Suppose, in the first instance, that isomeric transformation
+  does not take place at the temperature of the melting point, then the
+  freezing point curve will have the simple form ACB; the formation of
+  compounds being for the present excluded. This is the simplest type of
+  curve, and gives the composition of the solutions in equilibrium with the
+  one modification (<span class="grk">&alpha;</span> modification) at
+  different temperatures (curve AC); and of the solutions in equilibrium
+  with the other modification (<span class="grk">&beta;</span>
+  modification) at different temperatures (curve BC). C is the eutectic
+  point at which the two solid isomerides can exist side by side in contact
+  with the solution.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig059.png"><img style="width:100%" src="images/fig059.png"
+      alt="Fig. 59" title="Fig. 59" /></a>
+    <span class="sc">Fig</span>. 59.
+  </div>
+
+  <p>Now, suppose that isomeric transformation takes place with measurable
+  velocity. If the pure <span class="grk">&alpha;</span>-modification is
+  heated to a temperature <i>t&prime;</i> above its melting point, and the
+  liquid maintained at that temperature until equilibrium has been
+  established, a certain amount of the <span class="grk">&beta;</span>-form
+  will be present in the liquid, the composition of which will be
+  represented by the point <i>x&prime;</i>. The same condition of
+  equilibrium will also be reached by starting with pure <span
+  class="grk">&beta;</span>. Similarly, if the temperature of the liquid is
+  maintained at the temperature <i>t&Prime;</i>, equilibrium will be
+  reached, we shall suppose, when the solution has the composition
+  <i>x&Prime;</i>. The curve DE, therefore, which passes through all the
+  different values of <i>x</i> corresponding to different values of
+  <i>t</i>, will represent the change of equilibrium with the temperature.
+  It will slope to the right (as in the figure) if the transformation of
+  <span class="grk">&alpha;</span> into <span class="grk">&beta;</span> is
+  accompanied by absorption of heat; to the left if the transformation is
+  accompanied by evolution of heat, in accordance with van't Hoff's Law of
+  movable equilibrium. If transformation occurs without heat effect, the
+  equilibrium will be independent of the <!-- Page 198 --><span
+  class="pagenum"><a name="page198"></a>{198}</span>temperature, and the
+  equilibrium curve DE will therefore be perpendicular and parallel to the
+  temperature axis.</p>
+
+  <p>We must now find the meaning of the point D. Suppose the pure <span
+  class="grk">&alpha;</span>- or pure <span class="grk">&beta;</span>-form
+  heated to the temperature <i>t&prime;</i>, and the temperature maintained
+  constant until the liquid has the composition <i>x&prime;</i>
+  corresponding to the equilibrium at that temperature. If the temperature
+  is now allowed to fall sufficiently slowly so that the condition of
+  equilibrium is continually readjusted as the temperature changes, the
+  composition of the solution will gradually alter as represented by the
+  curve <i>x&prime;</i>D. Since D is on the freezing point curve of pure
+  <span class="grk">&alpha;</span>, this form will be deposited on cooling;
+  and since D is also on the equilibrium curve of the liquid, D is the only
+  point at which solid can exist in stable equilibrium with the liquid
+  phase. (The vapour phase may be omitted from consideration, as we shall
+  suppose the experiments carried out in open vessels.) All systems
+  consisting of the two hylotropic<a name="NtA281"
+  href="#Nt281"><sup>[281]</sup></a> isomeric substances <span
+  class="grk">&alpha;</span> and <span class="grk">&beta;</span> will,
+  therefore, ultimately freeze at the point D, which is called the
+  "natural" freezing point<a name="NtA282"
+  href="#Nt282"><sup>[282]</sup></a> of the system; provided, of course,
+  that sufficient time is allowed for equilibrium to be established. From
+  this it is apparent that <i>the stable modification at temperatures in
+  the neighbourhood of the melting point is that which is in equilibrium
+  with the liquid phase at the natural freezing point</i>.</p>
+
+  <p>From what has been said, it will be easy to predict what will be the
+  behaviour of the system under different conditions. If pure <span
+  class="grk">&alpha;</span> is heated, a temperature will be reached at
+  which it will melt, but this melting point will be sharp only if the
+  velocity of isomeric transformation is comparatively slow; <i>i.e.</i>
+  slow in comparison with the determination of the melting point. If the
+  substance be maintained in the fused condition for some time, a certain
+  amount of the <span class="grk">&beta;</span> modification will be
+  formed, and on lowering the temperature the pure <span
+  class="grk">&alpha;</span> form will be deposited, not at the temperature
+  of the melting point, but at some lower temperature depending on the
+  concentration of the <span class="grk">&beta;</span> modification in the
+  liquid phase. If isomeric transformation <!-- Page 199 --><span
+  class="pagenum"><a name="page199"></a>{199}</span>takes place slowly in
+  comparison with the rate at which deposition of the solid occurs, the
+  liquid will become increasingly rich in the <span
+  class="grk">&beta;</span> modification, and the freezing point will,
+  therefore, sink continuously. At the eutectic point, however, the <span
+  class="grk">&beta;</span> modification will also be deposited, and the
+  temperature will remain constant until all has become solid. If, on the
+  other hand, the velocity of transformation is sufficiently rapid, then as
+  quickly as the <span class="grk">&alpha;</span> modification is
+  deposited, the equilibrium between the two isomeric forms in the liquid
+  phase will continuously readjust itself, and the end-point of
+  solidification will be the natural freezing point.</p>
+
+  <p>Similarly, starting with the pure <span class="grk">&beta;</span>
+  modification, the freezing point after fusion will gradually fall owing
+  to the formation of the <span class="grk">&alpha;</span> modification;
+  and the composition of the liquid phase will pass along the curve BC. If,
+  now, the rate of cooling is not too great, or if the velocity of isomeric
+  transformation is sufficiently rapid, complete solidification will not
+  occur at the eutectic point; for at this temperature solid and liquid are
+  not in stable equilibrium with one another. On the contrary, a further
+  quantity of the <span class="grk">&beta;</span> modification will undergo
+  isomeric change, the liquid phase will become richer in the <span
+  class="grk">&alpha;</span> form, and the freezing point will <i>rise</i>;
+  the solid phase in contact with the liquid being now the <span
+  class="grk">&alpha;</span> modification. The freezing point will continue
+  to rise until the point D is reached, at which complete solidification
+  will take place without further change of temperature.</p>
+
+  <p>The diagram also allows us to predict what will be the result of
+  rapidly cooling a fused mixture of the two isomerides. Suppose that
+  either the <span class="grk">&alpha;</span> or the <span
+  class="grk">&beta;</span> modification has been maintained in the fused
+  state at the temperature <i>t&prime;</i> sufficiently long for
+  equilibrium to be established. The composition of the liquid phase will
+  be represented by <i>x&prime;</i>. If the liquid is now <i>rapidly</i>
+  cooled, the composition will remain unchanged as represented by the
+  dotted line <i>x&prime;</i>G. At the temperature of the point G solid
+  <span class="grk">&alpha;</span> modification will be deposited. If the
+  cooling is not carried below the point G, so as to cause complete
+  solidification, the freezing point will be found to rise with time, owing
+  to the conversion of some of the <span class="grk">&beta;</span> form
+  into the <span class="grk">&alpha;</span> form <!-- Page 200 --><span
+  class="pagenum"><a name="page200"></a>{200}</span>in the liquid phase;
+  and this will continue until the composition of the liquid has reached
+  the point D. From what has just been said, it can also be seen that if
+  the freezing point curves can be obtained by actual determination of the
+  freezing points of different synthetic mixtures of the two isomerides, it
+  will be possible to determine the condition of equilibrium in the fused
+  state at any given temperature without having recourse to analysis. All
+  that is necessary is to rapidly cool the fused mass, after equilibrium
+  has been established, and find the freezing point at which solid is
+  deposited; that is, find the point at which the line of constant
+  temperature cuts the freezing point curve. The composition corresponding
+  to this temperature gives the composition of the equilibrium mixture at
+  the given temperature.</p>
+
+  <p>It will be evident, from what has gone before, that the degree of
+  completeness with which the different curves can be realised will depend
+  on the velocity with which isomeric change takes place, and on the
+  rapidity with which the determinations of the freezing point can be
+  carried out. As the two extremes we have, on the one hand, practically
+  instantaneous transformation, and on the other, practically infinite
+  slowness of transformation. In the former case, only one melting and
+  freezing point will be found, viz. the natural freezing point; in the
+  latter case, the two isomerides will behave as two perfectly independent
+  components, and the equilibrium curve DE will not be realised.</p>
+
+  <p>The diagram which is obtained when isomeric transformation does not
+  occur within measurable time at the temperature of the melting point is
+  somewhat different from that already given in Fig. 59. In this case, the
+  two freezing point curves AC and BC (Fig. 60) can be readily realized, as
+  no isomeric change occurs in the liquid phase. Suppose, however, that at
+  a higher temperature, <i>t&prime;</i>, reversible isomeric transformation
+  can take place, the composition of the liquid phase will alter until at
+  the point <i>x&prime;</i> a condition of equilibrium is reached; and the
+  composition of the liquid at higher temperatures will be represented by
+  the curve <i>x&prime;</i>F. Below the temperature <i>t&prime;</i> the
+  position of the equilibrium curve is hypothetical; but as the temperature
+  <!-- Page 201 --><span class="pagenum"><a
+  name="page201"></a>{201}</span>falls the velocity of transformation
+  diminishes, and at last becomes <i>practically</i> zero. The equilibrium
+  curve can therefore be regarded as dividing into two branches
+  <i>x&prime;</i>G and <i>x&prime;</i>H. At temperatures between G and
+  <i>t&prime;</i> the <span class="grk">&alpha;</span> modification can
+  undergo isomeric change leading to a point on the curve G<i>x&prime;</i>;
+  and the <span class="grk">&beta;</span> modification can undergo change
+  leading to a point on the curve H<i>x&prime;</i>. The same condition of
+  equilibrium is therefore not reached from each side, and we are therefore
+  dealing not with true but with false equilibrium (p.&nbsp;<a
+  href="#page5">5</a>). Below the temperatures G and H, isomeric
+  transformation does not occur in measurable time. We shall not, however,
+  enter into a detailed discussion of the equilibria in such systems, more
+  especially as they are not systems in true equilibrium, and as the
+  temperature at which true equilibrium can be established with appreciable
+  velocity alters under the influence of catalytic agents.<a name="NtA283"
+  href="#Nt283"><sup>[283]</sup></a> Examples of such systems will no doubt
+  be found in the case of optically active substances, where both
+  isomerides are apparently quite stable at the melting point. In the case
+  of such substances, also, the action of catalytic agents in producing
+  isomeric transformation (racemisation) is well known.</p>
+
+  <div class="figcenter" style="width:22%;">
+      <a href="images/fig060.png"><img style="width:100%" src="images/fig060.png"
+      alt="Fig. 60" title="Fig. 60" /></a>
+    <span class="sc">Fig.</span> 60.
+  </div>
+
+  <p><b>Transformation of the Unstable into the Stable Form.</b>&mdash;As
+  has already been stated, the stable modification in the neighbourhood of
+  the melting point is that one which is in equilibrium with the liquid
+  phase at the natural freezing point. In the case of polymorphic
+  substances, we have seen (p.&nbsp;<a href="#page39">39</a>) that that form
+  which is stable in the neighbourhood of the melting point melts at the
+  higher temperature. That was a <!-- Page 202 --><span class="pagenum"><a
+  name="page202"></a>{202}</span>consequence of the fact that the two
+  polymorphic forms on melting gave identical liquid phases. In the present
+  case, however, the above rule does not apply, for the simple reason that
+  the liquid phase obtained by the fusion of the one modification is not
+  identical with that obtained by the fusion of the other. In the case of
+  isomeric substances, therefore, the form of lower melting point
+  <i>may</i> be the more stable; and where this behaviour is found it is a
+  sign that the two forms are isomeric (or polymeric) and not
+  polymorphic.<a name="NtA284" href="#Nt284"><sup>[284]</sup></a> An
+  example of this is found in the case of the isomeric benzaldoximes (p.&nbsp;<a
+  href="#page203">203</a>).</p>
+
+  <p>Since in Fig. 59 the <span class="grk">&alpha;</span> modification has
+  been represented as the stable form, the transformation of the <span
+  class="grk">&beta;</span> into the <span class="grk">&alpha;</span> form
+  will be possible at all temperatures down to the transition point. At
+  temperatures below the eutectic point, transformation will occur without
+  formation of a liquid phase; but at temperatures above the eutectic point
+  liquefaction can take place. This will be more readily understood by
+  drawing a line of constant temperature, HK, at some point between C and
+  B. Then if the <span class="grk">&beta;</span> modification is maintained
+  for a sufficiently long time at that temperature, a certain amount of the
+  <span class="grk">&alpha;</span> modification will be formed; and when
+  the composition of the mixture has reached the point H, fusion will
+  occur. If the temperature is maintained constant, isomeric transformation
+  will continue to take place in the liquid phase until the equilibrium
+  point for that temperature is reached. If this temperature is higher than
+  the natural melting point, the mixture will remain liquid all the time;
+  but if it is below the natural melting point, then the <span
+  class="grk">&alpha;</span> modification will be deposited when the system
+  reaches the condition represented by the point on the curve AC
+  corresponding to the particular temperature. As isomeric transformation
+  continues, the freezing point of the system will rise until it reaches
+  the natural freezing point D. Similarly, if the <span
+  class="grk">&alpha;</span> modification is maintained at a temperature
+  above that of the point D, liquefaction will ultimately occur, and the
+  system will again reach the final state represented by D.<a name="NtA285"
+  href="#Nt285"><sup>[285]</sup></a></p>
+
+<p><!-- Page 203 --><span class="pagenum"><a name="page203"></a>{203}</span></p>
+
+  <p><b>Examples.</b>&mdash;<i>Benzaldoximes.</i> The relationships which
+  have just been discussed from the theoretical point of view will be
+  rendered clearer by a brief description of cases which have been
+  experimentally investigated. The first we shall consider is that of the
+  two isomeric benzaldoximes:<a name="NtA286"
+  href="#Nt286"><sup>[286]</sup></a>&mdash;</p>
+
+<table class="nobctr" summary="benzaldoximes" title="benzaldoximes">
+<tr><td class="nob" style="text-align:center"> <a href="images/219a.png"><img src="images/219a.png" class="middle" style="height:6ex" alt="alpha form" /></a><br />
+Benzantialdoxime (<span class="grk">&alpha;</span>-modification).
+</td><td class="nob" style="text-align:center"> <a href="images/219b.png"><img src="images/219b.png" class="middle" style="height:6ex" alt="alpha form" /></a><br />
+Benzsynaldoxime (<span class="grk">&beta;</span>-modification).</td></tr>
+</table>
+
+  <p>Fig. 61 gives a graphic representation of the results obtained.</p>
+
+  <p>The melting point of the <span class="grk">&alpha;</span> modification
+  is 34-35°; the melting point of the unstable <span
+  class="grk">&beta;</span>-modification being 130°. The
+  freezing curves AC and BC were obtained by determining the freezing
+  points of different mixtures of known composition, and the numbers so
+  obtained are given in the following table.</p>
+
+<p><!-- Page 204 --><span class="pagenum"><a name="page204"></a>{204}</span></p>
+
+<table class="nobctr" summary="Freezing point of alpha and beta-benzaldoxime mixture" title="Freezing point of alpha and beta-benzaldoxime mixture">
+<tr><td class="toribob" style="text-align:center"> Grams of the <span class="grk">&alpha;</span> modification<br />in 100 gm. of mixture.
+</td><td class="toboleb" style="text-align:center"> Freezing point.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:4em;"> 26.2 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 101°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:4em;"> 49.2 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 79°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:4em;"> 73.7 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 46°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:4em;"> 91.7 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 26.2°</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:4em;"> 95.0 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 28.6°</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:4em;"> 96.0 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 30.0°</td></tr>
+
+</table>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig061.png"><img style="width:100%" src="images/fig061.png"
+      alt="Fig. 61" title="Fig. 61" /></a>
+    <span class="sc">Fig.</span> 61.
+  </div>
+
+  <p>The eutectic point C was found to lie at 25-26°, and the natural
+  freezing point D was found to be 27.7°. The equilibrium curve DE was
+  determined by heating the liquid mixtures at different temperatures until
+  equilibrium was attained, and then rapidly cooling the liquid. In all
+  cases the freezing point was practically that of the point D. From this
+  it is seen that the equilibrium curve must be a straight line parallel to
+  the temperature axis; and, therefore, isomeric transformation in the case
+  of the two benzaldoximes is not accompanied by any heat effect (p.&nbsp;<a
+  href="#page197">197</a>). This behaviour has also been found in the case
+  of acetaldoxime.<a name="NtA287" href="#Nt287"><sup>[287]</sup></a></p>
+
+  <p>The isomeric benzaldoximes are also of interest from the fact that the
+  stable modification has the <i>lower</i> melting point (<i>v.</i> p.
+  202).</p>
+
+  <p><i>Acetaldehyde and Paraldehyde.</i>&mdash;As a second example of the
+  equilibria between two isomerides, we shall take the two isomeric
+  (polymeric) forms of acetaldehyde, which have recently been exhaustively
+  studied.<a name="NtA288" href="#Nt288"><sup>[288]</sup></a></p>
+
+<p><!-- Page 205 --><span class="pagenum"><a name="page205"></a>{205}</span></p>
+
+  <p>In the case of these two substances the reaction</p>
+
+<p class="cenhead">3CH<sub>3</sub>.CHO <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> (CH<sub>3</sub>.CHO)<sub>3</sub></p>
+
+  <p>takes place at the ordinary temperature with very great slowness. For
+  this reason it is possible to determine the freezing point curves of
+  acetaldehyde and paraldehyde. The three chief points on these curves,
+  represented graphically in Fig. 62, are:&mdash;</p>
+
+<table class="nobctr" summary="Acetaldehyde and paraldehyde data" title="Acetaldehyde and paraldehyde data">
+<tr><td class="nob" style="text-align:left"> m.p. of acetaldehyde </td><td class="nob" style="text-align:left"> - 118.45°</td></tr>
+<tr><td class="nob" style="text-align:left"> m.p. of paraldehyde </td><td class="nob" style="text-align:left"> +&nbsp; 12.55°</td></tr>
+<tr><td class="nob" style="text-align:left"> eutectic point </td><td class="nob" style="text-align:left"> - 119.9°</td></tr>
+</table>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig062.png"><img style="width:100%" src="images/fig062.png"
+      alt="Fig. 62" title="Fig. 62" /></a>
+    <span class="sc">Fig.</span> 62.
+  </div>
+
+  <p>In order to determine the position of the natural melting point, it
+  was necessary, on account of the slowness of transformation, to employ a
+  catalytic agent in order to increase the velocity with which the
+  equilibrium was established. A drop of concentrated sulphuric acid served
+  the purpose. In presence of a trace of this substance, isomeric
+  transformation very speedily occurs, and leads to the condition of
+  equilibrium. Starting in the one case with fused paraldehyde, and in the
+  other case with acetaldehyde, the same freezing point, viz. 6.75°, was
+  obtained, the solid phase being paraldehyde. This temperature, 6.75°, is
+  therefore the natural freezing point, and paraldehyde, the solid in
+  equilibrium with the liquid phase at this point, is the stable form.</p>
+
+  <p>With regard to the change of equilibrium with the temperature, it was
+  found that whereas the liquid phase contained 11.7 molecules per cent. of
+  acetaldehyde at the natural freezing point, the liquid at the temperature
+  of 41.6° contains 46.6 molecules per cent. of acetaldehyde. As the
+  temperature <!-- Page 206 --><span class="pagenum"><a
+  name="page206"></a>{206}</span>rises, therefore, there is increased
+  formation of acetaldehyde, or a decreasing amount of polymerisation. This
+  is in harmony with the fact that the polymerisation of acetaldehyde is
+  accompanied by evolution of heat.</p>
+
+  <p>While speaking of these isomerides, it may be mentioned that at the
+  temperature 41.6° the equilibrium mixture has a vapour pressure equal to
+  the atmospheric pressure. At this temperature, therefore, the equilibrium
+  mixture (obtained quickly with the help of a trace of sulphuric acid)
+  boils.<a name="NtA289" href="#Nt289"><sup>[289]</sup></a></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 207 --><span class="pagenum"><a name="page207"></a>{207}</span></p>
+
+<h3>CHAPTER XII</h3>
+
+<p class="cenhead">SUMMARY.&mdash;APPLICATION OF THE PHASE RULE TO
+THE STUDY OF SYSTEMS OF TWO COMPONENTS</p>
+
+  <p>In this concluding chapter on two-component systems, it is proposed to
+  indicate briefly how the Phase Rule has been applied to the elucidation
+  of a number of problems connected with the equilibria between two
+  components, and how it has been employed for the interpretation of the
+  data obtained by experiment. It is hoped that the practical value of the
+  Phase Rule may thereby become more apparent, and its application to other
+  cases be rendered easier.</p>
+
+  <p>The interest and importance of investigations into the conditions of
+  equilibrium between two substances, lie in the determination not only of
+  the conditions for the stable existence of the participating substances,
+  but also of whether or not chemical action takes place between these two
+  components; and if combination occurs, in the determination of the nature
+  of the compounds formed and the range of their existence. In all such
+  investigations, the Phase Rule becomes of conspicuous value on account of
+  the fact that its principles afford, as it were, a touchstone by which
+  the character of the system can be determined, and that from the form of
+  the equilibrium curves obtained, conclusions can be drawn as to the
+  nature of the interaction between the two substances. In order to
+  exemplify the application of the principles of the Phase Rule more fully
+  than has already been done, illustrations will be drawn from
+  investigations on the interaction of organic compounds; on the equilibria
+  between optically active compounds; and on alloys. <!-- Page 208 --><span
+  class="pagenum"><a name="page208"></a>{208}</span></p>
+
+  <p><b>Summary of the Different Systems of Two
+  Components.</b>&mdash;Before passing to the consideration of the
+  application of the Phase Rule to the investigation of particular
+  problems, it will be well to collect together the different types of
+  equilibrium curves with which we are already acquainted; to compare them
+  with one another, in order that we may then employ these characteristic
+  curves for the interpretation of the curves obtained as the result of
+  experiment.</p>
+
+  <p>In investigating the equilibria between two components, three chief
+  classes of curves will be obtained according as&mdash;</p>
+
+  <p>I. No combination takes place between the two components.</p>
+
+  <p>II. The components can form definite compounds.</p>
+
+  <p>III. The components separate out in the form of mixed crystals.</p>
+
+  <p>The different types of curves which are obtained in these three cases
+  are represented in Figs. 63, 64, 65. These different diagrams represent
+  the whole series of equilibria, from the melting point of the one
+  component (A) to that of the other component (B). The curves represent,
+  in all cases, the composition of the solution, or phase of variable
+  composition; the temperature being measured along one axis, and the
+  composition along the other.</p>
+
+  <p>We shall now recapitulate very briefly the characteristics of the
+  different curves.</p>
+
+  <div class="figright" style="width:35%;">
+      <a href="images/fig063.png"><img style="width:100%" src="images/fig063.png"
+      alt="Fig. 63" title="Fig. 63" /></a>
+    <span class="sc">Fig.</span> 63.
+  </div>
+
+  <p>If no compound is formed between the two components, <!-- Page 209
+  --><span class="pagenum"><a name="page209"></a>{209}</span>the general
+  form of the equilibrium curve will be that of curve I. or II., Fig. 63.
+  Type I. is the simplest form of curve found, and consists, as the diagram
+  shows, of only two branches, AC and BC, meeting at the point C, <i>which
+  lies below the melting point of either component</i>. The solid phase
+  which is in equilibrium with the solutions AC is pure A; that in
+  equilibrium with BC, pure B. C is the eutectic point. Although at the
+  eutectic point the solution solidifies entirely without change of
+  temperature, the solid which is deposited is not a homogeneous solid
+  phase, but a mixture, or conglomerate of the two components. <i>The
+  eutectic point, therefore, represents the melting or freezing point, not
+  of a compound, but of a mixture</i> (p.&nbsp;<a href="#page119">119</a>).</p>
+
+  <p>Curve II., Fig. 63, is obtained when two liquid phases are formed. C
+  is an eutectic point, D and F are transition points at which there can
+  co-exist the four phases&mdash;solid, two liquid phases, vapour. DEF
+  represents the change in the composition of the two liquid phases with
+  rise of temperature; the curve might also have the reversed form with the
+  critical solution point below the transition points D and F.</p>
+
+<div style="clear: both"></div>
+  <div class="figright" style="width:22%;">
+      <a href="images/fig064.png"><img style="width:100%" src="images/fig064.png"
+      alt="Fig. 64" title="Fig. 64" /></a>
+    <span class="sc">Fig.</span> 64.
+  </div>
+
+  <p>In the second class of systems (Fig. 64), that in which combination
+  between the components occurs, there are again two types according as the
+  compound formed has a definite melting point (<i>i.e.</i> can exist in
+  equilibrium with a solution of the same composition), or undergoes only
+  partial fusion; that is, exhibits a transition point.</p>
+
+  <p>If a compound possessing a definite melting point is formed, the
+  equilibrium curve will have the general form shown by curve I., Fig. 64.
+  A, B, and D are the melting points of pure A, pure B, and of the compound
+  A<sub>x</sub>B<sub>y</sub> respectively. AC <!-- Page 210 --><span
+  class="pagenum"><a name="page210"></a>{210}</span>is the freezing point
+  curve of A in presence of B; BE that of B in presence of A; and DC and DE
+  the freezing point curves of the compound in presence of a solution
+  containing excess of one of the components. C and E are eutectic points
+  at which mixtures of A and A<sub>x</sub>B<sub>y</sub>, or B and
+  A<sub>x</sub>B<sub>y</sub> can co-exist in contact with solution. The
+  curve CDE may be large or small, and the melting point of the compound,
+  D, may lie above or below that of each of the components, or may have an
+  intermediate position. If more than one compound can be formed, a series
+  of curves similar to CDE will be obtained (<i>cf.</i> p.&nbsp;<a
+  href="#page152">152</a>).</p>
+
+  <p>On the other hand, if the compound undergoes transition to another
+  solid phase at a temperature below its melting point, a curve of the form
+  II., Fig. 64, will be found. This corresponds to the case where a
+  compound can exist only in contact with solutions containing excess of
+  one of the components. The metastable continuation of the equilibrium
+  curve for the compound is indicated by the dotted line, the summit of
+  which would be the melting point of the compound. Before this temperature
+  is reached, however, the solid compound ceases to be able to exist in
+  contact with solution, and transition to a different solid phase occurs
+  at the point E (<i>cf.</i> p.&nbsp;<a href="#page134">134</a>). This point,
+  therefore, represents the limit of the existence of the compound AB. If a
+  series of compounds can be formed none of which possess a definite
+  melting point, then a series of curves will be obtained which do not
+  exhibit a temperature-maximum, and there will be only one eutectic point.
+  The limits of existence of each compound will be marked by a break in the
+  curve (<i>cf.</i> p.&nbsp;<a href="#page143">143</a>).</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/fig065.png"><img style="width:100%" src="images/fig065.png"
+      alt="Fig. 65" title="Fig. 65" /></a>
+    <span class="sc">Fig.</span> 65.
+  </div>
+
+  <p>Turning, lastly, to the third class of systems, in which formation of
+  mixed crystals can occur, five different types of curves can be obtained,
+  as shown in Fig. 65. With regard to the first three types, curves I.,
+  II., and III., <!-- Page 211 --><span class="pagenum"><a
+  name="page211"></a>{211}</span>these differ entirely from those of the
+  previous classes, in that they are continuous; they exhibit no eutectic
+  point, and no transition point. Curve II. bears some resemblance to the
+  melting-point curve of a compound (<i>e.g.</i> CDE, Fig. 64, I.), but
+  differs markedly from it in not ending in eutectic points.</p>
+
+  <p>Further, in the case of the formation of a compound, the composition
+  of the solid phase remains unchanged throughout the whole curve between
+  the eutectic points; whereas, when mixed crystals are produced, the
+  composition of the solid phase varies with the composition of the liquid
+  solution. On passing through the maximum, the relative proportions of A
+  and B in the solid and the liquid phase undergo change; on the one side
+  of the maximum, the solid phase contains relatively more A, and on the
+  other side of the maximum, relatively more B than the liquid phase.
+  Lastly, when mixed crystals are formed, the temperature at which complete
+  solidification occurs changes as the composition of the solution changes,
+  whereas in the case of the formation of compounds, the temperature of
+  complete solidification for all solutions is a eutectic point.</p>
+
+  <p>The third type of curve, Fig. 65, can be distinguished in a similar
+  manner from the ordinary eutectic curve, Fig. 63, I., to which it bears a
+  certain resemblance. Whereas in the case of the latter, the eutectic
+  point is the temperature of complete solidification of all solutions, the
+  point of minimum temperature in the case of the formation of mixed
+  crystals, is the solidification point only of solutions having one
+  particular composition; that, namely, of the minimum point. For all other
+  solutions, the temperature of complete solidification is different.
+  Whereas, also, in the case of the simple eutectic curve, the solid which
+  separates out from the solutions represented by either curve remains the
+  same throughout the whole extent of that curve, the composition of the
+  mixed crystal varies with variation of the composition of the liquid
+  phase, and the relative proportions of the two components in the solid
+  and the liquid phase are reversed on passing through the minimum.<a
+  name="NtA290" href="#Nt290"><sup>[290]</sup></a></p>
+
+  <p>In a similar manner, type IV., Fig. 65, can be distinguished from type
+  II., Fig. 64, by the fact that it does not exhibit a <!-- Page 212
+  --><span class="pagenum"><a name="page212"></a>{212}</span>eutectic
+  point, and that the composition of the solid phase undergoes continuous
+  variation with variation of the liquid phase on either side of the
+  transition point. Lastly, type V., which does exhibit a eutectic point,
+  differs from the eutectic curve of Fig. 63, in that the eutectic point
+  does not constitute the point of complete solidification for all
+  solutions, and that the composition of the solid phase varies with the
+  composition of the liquid phase.</p>
+
+  <p>Such, then, are the chief general types of equilibrium curves for
+  two-components; they are the pattern curves with which other curves,
+  experimentally determined, can be compared; and from the comparison it
+  will be possible to draw conclusions as to the nature of the equilibria
+  between the two components under investigation.</p>
+
+<p class="cenhead">1. <i>Organic Compounds.</i></p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/fig066.png"><img style="width:100%" src="images/fig066.png"
+      alt="Fig. 66" title="Fig. 66" /></a>
+    <span class="sc">Fig.</span> 66.
+  </div>
+
+  <p>The principles of the Phase Rule have been applied to the
+  investigation of the equilibria between organic compounds, and Figs.
+  66-69 reproduce some of the results which have been obtained.<a
+  name="NtA291" href="#Nt291"><sup>[291]</sup></a></p>
+
+<p><!-- Page 213 --><span class="pagenum"><a name="page213"></a>{213}</span></p>
+
+  <p>Fig. 66, the freezing point curve (curve of equilibrium) for
+  <i>o</i>-nitrophenol and <i>p</i>-toluidine, shows a curve of the
+  simplest type<a name="NtA292" href="#Nt292"><sup>[292]</sup></a> (type
+  I., Fig. 63), in which two branches meet at an eutectic point. The solid
+  phase in equilibrium with solutions represented by the left-hand branch
+  of the curve was <i>o</i>-nitrophenol (m.p. 44.1°); that in equilibrium
+  with the solutions represented by the right-hand branch, was
+  <i>p</i>-toluidine (m.p. 43.3°). At the eutectic point (15.6°), these two
+  solid phases could co-exist with the liquid phase. This equilibrium
+  curve, therefore, shows that <i>o</i>-nitrophenol and <i>p</i>-toluidine
+  do not combine with one another.</p>
+
+  <p>In connection with this curve, attention may be called to the
+  interesting fact that although the solid produced by cooling the liquid
+  phase at the eutectic point has a composition approximating to that of a
+  compound of equimolecular proportions of the phenol and toluidine, and a
+  constant melting point, it is nevertheless a <i>mixture</i>. Although, as
+  a rule, the constituents of the eutectic mixture are not present in
+  simple molecular proportions, there is no reason why they should not be
+  so; and it is therefore necessary to beware of assuming the formation of
+  compounds in such cases.<a name="NtA293"
+  href="#Nt293"><sup>[293]</sup></a></p>
+
+  <p>Fig. 67, on the other hand, indicates with perfect certainty the
+  formation of a compound between phenol and <span
+  class="grk">&alpha;</span>-naphthylamine.<a name="NtA294"
+  href="#Nt294"><sup>[294]</sup></a> (<i>Cf.</i> curve I., Fig. 64.)</p>
+
+  <p>Phenol freezes at 40.4°, but the addition of <span
+  class="grk">&alpha;</span>-naphthylamine lowers the freezing point as
+  represented by the curve AC. At C (16.0°) the compound
+  C<sub>6</sub>H<sub>5</sub>OH,C<sub>10</sub>H<sub>7</sub>NH<sub>2</sub> is
+  formed, and the system becomes invariant. On increasing the amount of the
+  amine, the temperature of equilibrium rises, the solid phase now being
+  the compound. At D, the curve passes through a maximum (28.8°), at which
+  the solid and liquid phases have the same composition. This is the
+  melting point of the compound. Further addition of the amine lowers the
+  temperature of equilibrium, until at E solid <span
+  class="grk">&alpha;</span>-naphthylamine separates out, and a second
+  eutectic point (24.0°) is obtained. BE is the <!-- Page 214 --><span
+  class="pagenum"><a name="page214"></a>{214}</span>freezing-point curve of
+  <span class="grk">&alpha;</span>-naphthylamine in presence of phenol, the
+  freezing point of the pure amine being 48.3°.</p>
+
+  <p>On account of the great sluggishness with which the compound of phenol
+  and <span class="grk">&alpha;</span>-naphthylamine crystallizes, it was
+  found possible to follow the freezing point curves of phenol and the
+  amine to temperatures considerably below the eutectic points, as shown by
+  the curves CF and EG.</p>
+
+  <div class="figcenter" style="width:45%;">
+      <a href="images/fig067.png"><img style="width:100%" src="images/fig067.png"
+      alt="Fig. 67" title="Fig. 67" /></a>
+    <span class="sc">Fig.</span> 67.
+  </div>
+
+  <p>Phenol can also combine with <i>p</i>-toluidine in equimolecular
+  proportions; and this compound is of interest, from the fact that it
+  exists in two crystalline forms melting at 28.5° and 30°. Each of these
+  forms now must have its own equilibrium curve, and it was found that the
+  intermediate portion of the freezing point curve was duplicated, as shown
+  in Fig. 68.<a name="NtA295" href="#Nt295"><sup>[295]</sup></a></p>
+
+<p><!-- Page 215 --><span class="pagenum"><a name="page215"></a>{215}</span></p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/fig068.png"><img style="width:100%" src="images/fig068.png"
+      alt="Fig. 68" title="Fig. 68" /></a>
+    <span class="sc">Fig.</span> 68.
+  </div>
+
+  <div class="figcenter" style="width:45%;">
+      <a href="images/fig069.png"><img style="width:100%" src="images/fig069.png"
+      alt="Fig. 69" title="Fig. 69" /></a>
+    <span class="sc">Fig.</span> 69.
+  </div>
+
+<p><!-- Page 216 --><span class="pagenum"><a name="page216"></a>{216}</span></p>
+
+  <p>Lastly, a curve is given, Fig. 69,<a name="NtA296"
+  href="#Nt296"><sup>[296]</sup></a> which corresponds with curve II., Fig.
+  64. Picric acid and benzene can form a compound, which, however, can
+  exist only in contact with solutions <i>containing excess of benzene</i>.
+  When the temperature is raised, a point (K) is reached at which the
+  compound melts with separation of solid picric acid. The point, K, is,
+  therefore, a <i>transition point</i>; analysis, however, showed that the
+  composition of the solution at this point is very nearly that of the
+  compound
+  C<sub>6</sub>H<sub>2</sub>(NO<sub>2</sub>)<sub>3</sub>OH,C<sub>6</sub>H<sub>6</sub>,
+  so that the melting point of the compound can almost be reached. The
+  fusion of the compound of benzene and picric acid with separation of the
+  latter is analogous to the (partial) fusion of Glauber's salt with
+  separation of anhydrous sodium sulphate.</p>
+
+<p class="cenhead">2. <i>Optically Active Substances.</i></p>
+
+  <p>The question as to whether a resolvable inactive body is a mixture of
+  the two oppositely active constituents (a <i>dl</i>-mixture), or a
+  racemic compound, is one which has given rise to considerable discussion
+  during the past decade; and several investigators have endeavoured to
+  establish general rules by which the question could be decided. In the
+  case of inactive liquids it is a matter of great difficulty to arrive at
+  a certain conclusion as to whether one is dealing with a mixture or a
+  compound, for in this case the usual physical methods give but a dubious
+  answer; and although the existence of a racemate in the liquid state (in
+  the case of conine) has been asserted,<a name="NtA297"
+  href="#Nt297"><sup>[297]</sup></a> most chemists incline to the belief
+  that such a thing is improbable.</p>
+
+  <p>Even in the case of crystalline substances, where the differences
+  between the various forms is greater, it was not always easy to
+  discriminate between the <i>dl</i>-mixture and the racemic compound. The
+  occurrence of hemihedral faces was considered by Pasteur to be a
+  sufficient criterion for an optically active substance. It has, however,
+  been found that hemihedry in crystals, although a frequent accompaniment
+  of <!-- Page 217 --><span class="pagenum"><a
+  name="page217"></a>{217}</span>optical activity, is by no means a
+  necessary or constant expression of this property. Other rules, also,
+  which were given, although in some cases reliable, were in other cases
+  insufficient; and all were in so far unsatisfactory that they lacked a
+  theoretical basis.</p>
+
+  <p>With the help of the Phase Rule, however, it is possible from a study
+  of the solubility or fusion curves of the optically active and inactive
+  substances, to decide the nature of the inactive substance, at least
+  under certain conditions. On account of the interest and importance which
+  these compounds possess, a brief description of the application of the
+  Phase Rule to the study of such substances will be given here;<a
+  name="NtA298" href="#Nt298"><sup>[298]</sup></a> the two optical
+  antipodes being regarded as the two components.</p>
+
+  <p>In the present chapter we shall consider only the fusion curves, the
+  solubility curves being discussed in the next section on three-component
+  systems. The rules which are hereby obtained, have reference only to the
+  nature of the inactive substance in the neighbourhood of the melting
+  points.</p>
+
+  <p>I. <i>The inactive substance is a </i>dl<i>-mixture.</i></p>
+
+  <p>In this case the fusion curves will have the simple form shown in type
+  I, Fig. 63. A and B are the melting points of the two optical isomerides,
+  and C the eutectic point at which the inactive mixture consisting of
+  equal amounts of d- and l-form melts. Owing to the similar effect of the
+  one form on the freezing point of the other, the figure is symmetrical.
+  No example of this simple case has been investigated.</p>
+
+  <p>II. <i>The two components form a racemic compound.</i></p>
+
+  <p>In this case there will be three melting point curves as in Fig. 64,
+  type I. In this case also the figure must be symmetrical.</p>
+
+  <p><b>Examples.</b>&mdash;As examples of this, may be taken dimethyl
+  tartrate and mandelic acid, the freezing point curves of which are given
+  in Figs. 70 and 71.<a name="NtA299" href="#Nt299"><sup>[299]</sup></a> As
+  can be seen, the curve for the racemic tartrate occupies a large part of
+  the diagram, <!-- Page 218 --><span class="pagenum"><a
+  name="page218"></a>{218}</span>while that for racemic mandelic acid is
+  much smaller. In the case of dimethyldiacetyl tartrate, this middle
+  portion is still less.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/fig070.png"><img style="width:100%" src="images/fig070.png"
+      alt="Fig. 70" title="Fig. 70" /></a>
+    <span class="sc">Fig.</span> 70.
+  </div>
+
+  <div class="figleft" style="width:19%;">
+      <a href="images/fig071.png"><img style="width:100%" src="images/fig071.png"
+      alt="Fig. 71" title="Fig. 71" /></a>
+    <span class="sc">Fig.</span> 71.
+  </div>
+
+  <div class="figleft" style="width:43%;">
+      <a href="images/fig072.png"><img style="width:100%" src="images/fig072.png"
+      alt="Fig. 72" title="Fig. 72" /></a>
+    <span class="sc">Fig.</span> 72.
+  </div>
+
+<div style="clear: both"></div>
+  <p>Active dimethyl tartrate melts at 43.3°; racemic dimethyl tartrate at
+  89.4°. Active mandelic acid melts at 132.8°; the racemic acid at 118.0°.
+  In the one case, therefore, the racemic compound has a higher, in the
+  other a lower melting point than the active forms. <!-- Page 219 --><span
+  class="pagenum"><a name="page219"></a>{219}</span></p>
+
+  <p>In the case of partially racemic compounds (<i>i.e.</i> the compound
+  of a racemate with an optically active substance) the type of curve will
+  be the same, but the figure will no longer be symmetrical. Such a curve
+  has been found in the case of the l-menthyl esters of d- and l-mandelic
+  acid (Fig. 72).<a name="NtA300" href="#Nt300"><sup>[300]</sup></a> The
+  freezing point of l-menthyl d-mandelate is 97.2°, of l-menthyl
+  l-mandelate 77.6°, and of l-menthyl r-mandelate 83.7.° It will be
+  observed that the summit of the curve for the partially racemic mandelate
+  is very flat, indicating that the compound is largely dissociated into
+  its components at the temperature of fusion.</p>
+
+  <p>III. <i>The inactive substance is a pseudo-racemic mixed
+  crystal.</i></p>
+
+  <p>In cases where the active components can form mixed crystals, the
+  freezing-point curve will exhibit one of the forms given in Fig. 65. The
+  inactive mixed crystal containing 50 per cent. of the dextro and laevo
+  compound, is known as a pseudo-racemic mixed crystal.<a name="NtA301"
+  href="#Nt301"><sup>[301]</sup></a> So far, only curves of the types I.
+  and II. have been obtained.</p>
+
+  <p><b>Examples.</b>&mdash;The two active camphor oximes are of interest
+  from the fact that they form a continuous series of mixed crystals,
+  <i>all of which have the same melting point</i>. The curve which is
+  obtained in this case is, therefore, a straight line joining the melting
+  points of the pure active components; the melting point of the active
+  isomerides and of the whole series of mixed crystals being 118.8°.</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig073.png"><img style="width:100%" src="images/fig073.png"
+      alt="Fig. 73" title="Fig. 73" /></a>
+    <span class="sc">Fig.</span> 73.
+  </div>
+
+  <p>In the case of the carvoximes mixed crystals are also formed, but the
+  equilibrium curve in this case exhibits a maximum (Fig. 73). At this
+  maximum point the composition of the solid and of the liquid solution is
+  the same. Since the curve must be symmetrical, this maximum point must
+  occur in the case of the solution containing 50 per cent. <!-- Page 220
+  --><span class="pagenum"><a name="page220"></a>{220}</span>of each
+  component, which will therefore be inactive. Further, this inactive mixed
+  crystal will melt and solidify at the same temperature, and behave,
+  therefore, like a chemical compound (p.&nbsp;<a href="#page187">187</a>). The
+  melting point of the active compounds is 72°; that of the inactive
+  pseudo-racemic mixed crystal is 91.4°·</p>
+
+  <p><b>Transformations.</b>&mdash;As has already been remarked, the
+  conclusions which can be drawn from the fusion curves regarding the
+  nature of the inactive substances formed hold only for temperatures in
+  the neighbourhood of the melting points. At temperatures below the
+  melting point transformation may occur; <i>e.g.</i> a racemate may break
+  up into a <i>dl</i>-mixture, or a pseudo-racemic mixed crystal may form a
+  racemic compound. We shall at a later point meet with examples of a
+  racemic compound changing into a <i>dl</i>-mixture at a definite
+  transition point; and the pseudo-racemic mixed crystal of camphoroxime is
+  an example of the second transformation. Although at temperatures in the
+  neighbourhood of the melting point the two active camphoroximes form only
+  mixed crystals but no compound, a racemic compound is formed at
+  temperatures below 103°. At this temperature the inactive pseudo-racemic
+  mixed crystal changes into a racemic compound; and in the case of the
+  other mixed crystals transformation to racemate and (excess of) active
+  component also occurs, although at a lower temperature than in the case
+  of the inactive mixed crystal. Although this behaviour is one of
+  considerable importance, this brief reference to it must suffice here.<a
+  name="NtA302" href="#Nt302"><sup>[302]</sup></a></p>
+
+<p class="cenhead">3. <i>Alloys.</i></p>
+
+  <p>One of the most important classes of substances in the study of which
+  the Phase Rule has been of very considerable importance, is that formed
+  by the mixtures or compounds of metals with one another known as alloys.
+  Although in the investigation of the nature of these bodies various
+  methods are employed, one of the most important is the determination of
+  the character of the freezing-point curve; for from the form of this,
+  valuable information can, as we have already learned, be <!-- Page 221
+  --><span class="pagenum"><a name="page221"></a>{221}</span>obtained
+  regarding the nature of the solid substances which separate out from the
+  molten mixture.</p>
+
+  <p>Although it is impossible here to discuss fully the experimental
+  results and the oftentimes very complicated relationships which the study
+  of the alloys has brought to light, a brief reference to these bodies
+  will be advisable on account both of the scientific interest and of the
+  industrial importance attaching to them.<a name="NtA303"
+  href="#Nt303"><sup>[303]</sup></a></p>
+
+  <p>We have already seen that there are three chief types of
+  freezing-point curves in systems of two components, viz. those obtained
+  when (1) the pure components crystallize out from the molten mass; (2)
+  the components form one or more compounds; (3) the components form mixed
+  crystals. In the case of the metals, representatives of these three
+  classes are also found.</p>
+
+  <p>1. <i>The components separate out in the pure state.</i></p>
+
+  <p>In this case the freezing-point curve is of the simple type, Fig. 63,
+  I. Such curves have been obtained in the case of a number of pairs of
+  metals, <i>e.g.</i> zinc&mdash;cadmium, zinc&mdash;aluminium,
+  copper&mdash;silver (Heycock and Neville), tin&mdash;zinc,
+  bismuth&mdash;lead (Gautier), and in other cases. From molten mixtures
+  represented by one branch of the freezing-point curve one of the metals
+  will be deposited; while from mixtures represented by the other branch,
+  the other metal will separate out. At the eutectic point the molten mass
+  will solidify to a <i>heterogeneous mixture</i> of the two metals,
+  forming what is known as the <i>eutectic alloy</i>. Such an alloy,
+  therefore, will melt at a definite temperature lower than the melting
+  point of either of the pure metals.</p>
+
+<p><!-- Page 222 --><span class="pagenum"><a name="page222"></a>{222}</span></p>
+
+  <p>In the following table are given the temperature and the composition
+  of the liquid at the eutectic point, for three pairs of
+  metals:&mdash;</p>
+
+<table class="nobctr" summary="Eutectic mixtures of metals" title="Eutectic mixtures of metals">
+<tr><td class="torib" style="text-align:center">
+</td><td class="toboleb" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Composition of liquid.</td></tr>
+
+<tr><td class="nob" style="text-align:left;"> Zinc&mdash;cadmium </td><td class="rileb" style="text-align:left; padding-left:2em; padding-right:2em;"> 264.5° </td><td class="nob" style="text-align:left;"> 73.5 atoms per cent. of cadmium.</td></tr>
+<tr><td class="nob" style="text-align:left;"> Zinc&mdash;aluminium </td><td class="rileb" style="text-align:left; padding-left:2em; padding-right:2em;"> 380.5° </td><td class="nob" style="text-align:left;"> 11 &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; aluminium.</td></tr>
+
+<tr><td class="bob" style="text-align:left;"> Copper&mdash;silver </td><td class="riboleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 778° </td><td class="bob" style="text-align:left;"> 40 &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; copper.</td></tr>
+
+</table>
+
+  <p>The melting points of the pure metals are, zinc, 419°; cadmium, 322°;
+  silver, 960°; copper, 1081°; aluminium, 650°.</p>
+
+  <p>2. <i>The two metals can form one or more compounds.</i></p>
+
+  <p>In this case there will be obtained not only the freezing-point curves
+  of the pure metals, but each compound formed will have its own
+  freezing-point curve, exhibiting a point of maximum temperature, and
+  ending on either side in an eutectic point. The simplest curve of this
+  type will be obtained when only one compound is formed, as is the case
+  with mercury and thallium.<a name="NtA304"
+  href="#Nt304"><sup>[304]</sup></a> This curve is represented in Fig. 74,
+  where the summit of the intermediate curve corresponds with a composition
+  TlHg<sub>2</sub>. Similar curves are also given by nickel and tin, by
+  aluminium and silver, and by other metals, the formation of definite
+  compounds between these pairs of metals being thereby indicated.<a
+  name="NtA305" href="#Nt305"><sup>[305]</sup></a></p>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/fig074.png"><img style="width:100%" src="images/fig074.png"
+      alt="Fig. 74" title="Fig. 74" /></a>
+    <span class="sc">Fig.</span> 74.
+  </div>
+
+<p><!-- Page 223 --><span class="pagenum"><a name="page223"></a>{223}</span></p>
+
+  <p>A curve belonging to the same type, but more complicated, is obtained
+  with gold and aluminium;<a name="NtA306"
+  href="#Nt306"><sup>[306]</sup></a> in this case, several compounds are
+  formed, some of which have a definite melting point, while others exhibit
+  only a transition point. The chief compound is AuAl<sub>2</sub>, which
+  has practically the same melting point as pure gold.</p>
+
+  <p>3. <i>The two metals form mixed crystals (solid solutions).</i></p>
+
+  <p>The simplest case in which the metals crystallize out together is
+  found in silver and gold.<a name="NtA307"
+  href="#Nt307"><sup>[307]</sup></a> The freezing-point curve in this case
+  is an almost straight line joining the freezing points of the pure metals
+  (<i>cf.</i> curve I., Fig. 65, p.&nbsp;<a href="#page210">210</a>). These two
+  metals, therefore, can form an unbroken series of mixed crystals.</p>
+
+  <p>In some cases, however, the two metals do not form an unbroken series
+  of mixed crystals. In the case of zinc and silver,<a name="NtA308"
+  href="#Nt308"><sup>[308]</sup></a> for example, the addition of silver
+  <i>raises</i> the freezing point of the mixture, until a transition point
+  is reached. This corresponds with curve IV., Fig. 65. Silver and copper,
+  and gold and copper, on the other hand, do not form unbroken series of
+  mixed crystals, but the freezing-point curve exhibits an eutectic point,
+  as in curve V., Fig. 65.</p>
+
+  <p>Not only may there be these three different types of curves, but there
+  may also be combinations of these. Thus the two metals may not only form
+  compounds, but one of the metals may not separate out in the pure state
+  at all, but form mixed crystals. In this case the freezing point may rise
+  (as in the case of silver and zinc), and one of the eutectic points will
+  be absent.</p>
+
+  <p><b>Iron-Carbon Alloys.</b>&mdash;Of all the different binary alloys,
+  probably the most important are those formed by iron and carbon: alloys
+  consisting not of two metals, but of a metal and a non-metal. On account
+  of the importance of these alloys, an attempt will be made to describe in
+  brief some of the most important relationships met with.</p>
+
+  <p>Before proceeding to discuss the applications of the Phase Rule to the
+  study of the iron-carbon alloys, however, the main <!-- Page 224 --><span
+  class="pagenum"><a name="page224"></a>{224}</span>facts with which we
+  have to deal may be stated very briefly. With regard to the metal itself,
+  it is known to exist in three different allotropic modifications, called
+  <span class="grk">&alpha;</span>-, <span class="grk">&beta;</span>-, and
+  <span class="grk">&gamma;</span>-ferrite respectively. Like the two
+  modifications of sulphur and of tin, these different forms exhibit
+  transition points at which the relative stability of the forms changes.
+  Thus the transition point for <span class="grk">&alpha;</span>- and <span
+  class="grk">&beta;</span>-ferrite is about 780°; and below this
+  temperature the <span class="grk">&alpha;</span>- form, above it the
+  <span class="grk">&beta;</span>- form is stable. For <span
+  class="grk">&beta;</span>- and <span class="grk">&gamma;</span>-ferrite,
+  the transition point is about 870°, the <span class="grk">&gamma;</span>-
+  form being the stable modification above this temperature.</p>
+
+  <p>The different modifications of iron also possess different properties.
+  Thus, <span class="grk">&alpha;</span>-ferrite is magnetic, but does not
+  possess the power of dissolving carbon; <span
+  class="grk">&beta;</span>-ferrite is non-magnetic, and likewise does not
+  dissolve carbon; <span class="grk">&gamma;</span>-ferrite is also
+  non-magnetic, but possesses the power of dissolving carbon, and of thus
+  giving rise to solid solutions of carbon in iron.</p>
+
+  <p>Various alloys of iron and carbon, also, have to be distinguished.
+  First of all, there is <i>hard steel</i>, which contains varying amounts
+  of carbon up to 2 per cent. Microscopic examination shows that these
+  mixtures are all homogeneous; and they are therefore to be regarded as
+  solid solutions of carbon in iron (<span
+  class="grk">&gamma;</span>-ferrite). To these solutions the name
+  <i>martensite</i> has been given. <i>Pearlite</i> contains about 0.8 per
+  cent. of carbon, and, on microscopic examination, is found to be a
+  heterogeneous mixture. If heated above 670°, pearlite becomes
+  homogeneous, and forms martensite. Lastly, there is a definite compound
+  of iron and carbon, iron carbide or <i>cementite</i>, having the formula
+  Fe<sub>3</sub>C.</p>
+
+  <p>A short description may now be given of the application of the Phase
+  Rule to the two-component system iron&mdash;carbon; and of the diagram
+  showing how the different systems are related, and with the help of which
+  the behaviour of the different mixtures under given conditions can be
+  predicted. Although, with regard to the main features of this diagram,
+  the different areas to be mapped and the position of the frontier lines,
+  there is general agreement; a final decision has not yet been reached
+  with regard to the interpretation to be put on all the curves.</p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/fig075.png"><img style="width:100%" src="images/fig075.png"
+      alt="Fig. 75" title="Fig. 75" /></a>
+    <span class="sc">Fig.</span> 75.
+  </div>
+
+  <p>The chief relationships met with in the case of the <!-- Page 225
+  --><span class="pagenum"><a name="page225"></a>{225}</span>iron-carbon
+  alloys are represented graphically in Fig. 75.<a name="NtA309"
+  href="#Nt309"><sup>[309]</sup></a> The curve AC is the freezing-point
+  curve for iron,<a name="NtA310" href="#Nt310"><sup>[310]</sup></a> BC the
+  unknown freezing-point curve for graphite. C is an <span
+  class="correction" title="Original reads 'eutetic'.">eutectic</span>
+  point. Suppose, now, that we start with a mixture of iron and carbon,
+  represented by the point <i>x</i>. On lowering the temperature, a point,
+  <i>y</i>, will be reached at which solid begins to separate out. This
+  solid phase, however, is not pure iron, but a solid solution of carbon in
+  iron, having the composition represented by <i>y&prime;</i> (cf. p.&nbsp;<a
+  href="#page185">185</a>). As the temperature continues to fall, the <!--
+  Page 226 --><span class="pagenum"><a
+  name="page226"></a>{226}</span>composition of the liquid phase changes in
+  the direction of <i>y</i>C, while the composition of the solid which
+  separates out changes in the direction <i>y&prime;</i>D; and, finally,
+  when the composition of the molten mass is that of the point C (4.3 per
+  cent. of carbon), the whole mass solidifies to a heterogeneous mixture of
+  two solid solutions, one of which is represented by D (containing 2 per
+  cent. of carbon), while the other will consist practically of pure
+  graphite, and is not shown in the figure. The temperature of the eutectic
+  point is 1130°.</p>
+
+  <p>Even below the solidification point, however, changes can take place.
+  As has been said, the solid phase which finally separates out from the
+  molten mass is a solid solution represented by the point D; and the curve
+  DE represents the change in the composition of this solid solution with
+  the temperature. As indicated in the figure, DE forms a part of a curve
+  representing the mutual solubility of graphite in iron and iron in
+  graphite; the latter solutions, however, not being shown, as they would
+  lie far outside the diagram. As the temperature falls below 1130°, more
+  and more graphite separates out, until at E, when the temperature is
+  1000°, the solid solution contains only 1.8 per cent. of carbon. At this
+  temperature cementite also begins to be formed, so that as the
+  temperature continues to fall, separation of cementite (represented by
+  the line E&prime;F&prime;) occurs, and the composition of the solid
+  solution undergoes alteration, as represented by the curve EF. Below the
+  temperature of the point F (670°) the martensite becomes heterogeneous,
+  and forms pearlite.</p>
+
+  <p>From the above description, therefore, it follows that if we start
+  with a molten mixture of iron and carbon, the composition of which is
+  represented by any point between D and C (from 2 to 4.3 per cent. of
+  carbon), we shall obtain, on cooling the mass, first of all solid
+  solutions, the composition of which will be represented by points on the
+  line AD; that then, after the mass has completely solidified at 1130°,
+  further cooling will lead to a separation of graphite and a change in the
+  composition of the martensite (from 2 to 1.8 per cent. of carbon). On
+  cooling below 1000°, however, the martensite and graphite will give rise
+  to cementite and solid solutions <!-- Page 227 --><span
+  class="pagenum"><a name="page227"></a>{227}</span>containing less carbon
+  than before, until, at temperatures below 670°, we are left with a
+  mixture of pearlite and cementite.</p>
+
+  <p>We have already said that iron consists in three allotropic
+  modifications, the regions of stability of which are separated by
+  definite transition points. The transition point for <span
+  class="grk">&alpha;</span>- and <span class="grk">&beta;</span>-ferrite
+  (780°) is represented in Fig. 75 by the point H; and the transition point
+  for <span class="grk">&beta;</span>- and <span
+  class="grk">&gamma;</span>-ferrite (870°) by the point I. Since neither
+  the <span class="grk">&alpha;</span>- nor the <span
+  class="grk">&beta;</span>-ferrite dissolves carbon, the transition point
+  will be unaffected by addition of carbon, and we therefore obtain the
+  horizontal transition curve HG. In the case of the <span
+  class="grk">&beta;</span>- and <span class="grk">&gamma;</span>-ferrite,
+  however, the latter dissolves carbon, and the transition point is
+  consequently affected by the amount of carbon present. This is shown by
+  the line IG.</p>
+
+  <p>If a martensite containing less carbon than that represented by the
+  point G is cooled down from a temperature of, say, 900°, then when the
+  temperature has fallen to that, represented by a point on the curve IG,
+  <span class="grk">&beta;</span>-ferrite will separate out, and, as the
+  temperature falls, the composition of the solid solution will alter as
+  represented by IG. On passing below the temperature of HG, the <span
+  class="grk">&beta;</span>-ferrite will be converted into <span
+  class="grk">&alpha;</span>-ferrite, and, as the temperature falls, the
+  latter will separate out more and more, while the composition of the
+  solid solution alters in the direction GF. On passing to still lower
+  temperatures, the solid solution at F (0.8 per cent. of carbon) breaks up
+  into pearlite. If the percentage of carbon in the original solid solution
+  was between that represented by the points G and F, then, on cooling
+  down, no <span class="grk">&beta;</span>-ferrite, but only <span
+  class="grk">&alpha;</span>-ferrite would separate out.</p>
+
+  <p>We see, therefore, that when martensite is allowed to cool
+  <i>slowly</i>, it yields a heterogeneous mixture either of ferrite and
+  pearlite (when the original mixture contained up to 0.8 per cent. of
+  carbon), or pearlite and cementite (when the original mixture contained
+  between 0.8 and 2 per cent. of carbon). These heterogeneous mixtures
+  constitute soft steels, or, when the carbon content is low, wrought
+  iron.</p>
+
+  <p>The case, however, is different if the solid solution of carbon in
+  iron is <i>rapidly</i> cooled (quenched) from a temperature above the
+  curve IGFE to a temperature below this <!-- Page 228 --><span
+  class="pagenum"><a name="page228"></a>{228}</span>curve. In this case,
+  the rapid cooling does not allow time for the various changes which have
+  been described to take place; so that the homogeneous solid solution, on
+  being rapidly cooled, remains homogeneous. In this way hard steel is
+  obtained. By varying the rapidity of cooling, as is done in the tempering
+  of steel, varying degrees of hardness can be obtained.</p>
+
+  <p>The interpretation of the curves given above is that due essentially
+  to Roozeboom, who concluded from the experimental data that at
+  temperatures below 1000° the stable systems are martensite and cementite,
+  or ferrite and cementite, graphite being labile. It has, however, been
+  pointed out, more especially by E. Heyn,<a name="NtA311"
+  href="#Nt311"><sup>[311]</sup></a> that this is not in harmony with the
+  facts of metallurgy, which show that graphite is undoubtedly formed on
+  slow cooling, and more especially when small quantities of silicon are
+  present in the iron.<a name="NtA312" href="#Nt312"><sup>[312]</sup></a>
+  While, therefore, the relationships represented by Fig. 75 are obtained
+  under certain conditions (especially when manganese is present), Heyn
+  considers that all the curves in that figure, except ACB, represent
+  <i>metastable</i> systems&mdash;systems, therefore, akin to supercooled
+  liquids. Rapid cooling will favour the production of the metastable
+  systems containing cementite, and therefore give rise to relationships
+  represented by Fig. 75; whereas slow cooling will lead to the stable
+  system ferrite and graphite. Presence of silicon tends to prevent,
+  presence of manganese tends to assist, the production of the metastable
+  systems.</p>
+
+  <p><span class="correction" title="Original reads 'Athough'."
+  >Although</span> this view put forward by Heyn has not been conclusively
+  proved, it must be said that there is much evidence in its favour.
+  Further investigation is, however, required before a final decision as to
+  the interpretation of the curves can be reached.</p>
+
+  <p><b>Determination of the Composition of Compounds, without
+  Analysis.</b>&mdash;Since the equilibrium between a solid and a liquid
+  phase depends not only on the composition of the liquid (solution) but
+  also on that of the solid, it is necessary <!-- Page 229 --><span
+  class="pagenum"><a name="page229"></a>{229}</span>to determine the
+  composition of the latter. In some cases this is easily effected by
+  separating the solid from the liquid phase and analyzing it. In other
+  cases, however, this method is inapplicable, or is accompanied by
+  difficulties, due either to the fact that the solid phase undergoes
+  decomposition (<i>e.g.</i> when it contains a volatile constituent), or
+  to the difficulty of completely separating the mother liquor; as, for
+  example, in the case of alloys. In all such cases, therefore, recourse
+  must be had to other methods.</p>
+
+  <p>In the first place, synthetic methods may be employed.<a name="NtA313"
+  href="#Nt313"><sup>[313]</sup></a> In this case we start with a solution
+  of the two components, to which a third substance is added, which,
+  however, does not enter into the solid phase.<a name="NtA314"
+  href="#Nt314"><sup>[314]</sup></a> We will assume that the initial
+  solution contains <i>x</i> gm. of A and <i>y</i> gm. of B to 1 gm. of C.
+  After the solution has been cooled down to such a temperature that solid
+  substance separates out, a portion of the liquid phase is removed with a
+  pipette and analyzed. If, now, the composition of the solution is such
+  that there are <i>x&prime;</i> gm. of A and <i>y&prime;</i> gm. of B to 1
+  gm. of C., then the composition of the solid phase is <i>x</i> -
+  <i>x&prime;</i> gm. of A and <i>y</i> - <i>y&prime;</i> gm. of B. When
+  <i>x</i> = <i>x&prime;</i>, the solid phase is pure B; when <i>y</i> =
+  <i>y&prime;</i>, the solid phase is pure A.</p>
+
+  <p>We have assumed here that there is only one solid phase present,
+  containing A and B. To make sure that the solid phase is not a solid
+  solution in which A and B are present in the same ratio as in the liquid
+  solution, a second determination of the composition must be made, with
+  different initial and end concentrations. If the solid phase is a solid
+  solution, the composition will now be found different from that found
+  previously.</p>
+
+  <p>The composition of the solid phase can, however, be determined in
+  another manner, viz. by studying the fusion curve and the curve of
+  cooling. From the form of the fusion curve alone, it is possible to
+  decide whether the two components <!-- Page 230 --><span
+  class="pagenum"><a name="page230"></a>{230}</span>form a compound or not;
+  and if the compounds which may be formed have a definite melting point,
+  the position of the latter gives at once the composition of the compounds
+  (cf. p.&nbsp;<a href="#page231">231</a>).</p>
+
+  <p>This method, however, cannot be applied when the compounds undergo
+  decomposition before the melting point is reached. In such cases,
+  however, the form of the cooling curve enables one to decide the
+  composition of the solid phase.<a name="NtA315"
+  href="#Nt315"><sup>[315]</sup></a> If a solution is allowed to cool
+  slowly, and the temperature noted at definite times, the graphic
+  representation of the rate of cooling will give a continuous curve;
+  <i>e.g.</i> <i>ab</i> in Fig. 76. So soon, however, as a solid phase
+  begins to be formed, the rate of cooling alters abruptly, and the cooling
+  curve then exhibits a break, or change in direction (point <i>b</i>).
+  When the eutectic point is reached, the temperature remains constant,
+  until all the liquid has solidified. This is represented by the line
+  <i>cd</i>. When complete solidification has occurred, the fall of
+  temperature again becomes uniform (<i>de</i>).</p>
+
+  <div class="figcenter" style="width:16%;">
+      <a href="images/fig076.png"><img style="width:100%" src="images/fig076.png"
+      alt="Fig. 76" title="Fig. 76" /></a>
+    <span class="sc">Fig.</span> 76.
+  </div>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/fig078.png"><img style="width:100%" src="images/fig078.png"
+      alt="Fig. 78" title="Fig. 78" /></a>
+    <span class="sc">Fig.</span> 78.
+  </div>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/fig077.png"><img style="width:100%" src="images/fig077.png"
+      alt="Fig. 77" title="Fig. 77" /></a>
+    <span class="sc">Fig.</span> 77.
+  </div>
+
+  <p>The length of time during which the temperature remains constant at
+  the point <i>c</i>, depends, of course, on the eutectic solution. If,
+  therefore, we take equal amounts of solution having a different initial
+  composition, the period of constant temperature in the cooling curve will
+  evidently be greatest in the case of the solution having the composition
+  of the eutectic point; and the period will become less and less as we
+  increase the amount of one of the components. The relationship between
+  initial composition of solution and the duration of constant temperature
+  at the eutectic point is represented by the curve
+  <i>a&prime;c&prime;b&prime;</i> (Fig. 77). When a compound possessing a
+  definite melting point is formed, it behaves as a pure substance. If,
+  therefore, the initial composition of the <!-- Page 231 --><span
+  class="pagenum"><a name="page231"></a>{231}</span>solution is the same as
+  that of the compound, no eutectic solution will be obtained; and
+  therefore no line of constant temperature, such as <i>cd</i> (Fig. 76).
+  In such a case, if we represent graphically the relation between the
+  initial composition of the solution and the duration of constant
+  temperature, a diagram is obtained such as shown in Fig. 78. The two
+  maxima on the time-composition curve represent eutectic points, and the
+  minima, <i>a&prime;</i>, <i>b&prime;</i>, <i>e&prime;</i>, pure
+  substances. The position of <i>e&prime;</i> gives the composition of the
+  compound. When a series of compounds is formed, then for each compound a
+  minimum is found on the time-composition curve.</p>
+
+  <div class="figright" style="width:17%;">
+      <a href="images/fig079.png"><img style="width:100%" src="images/fig079.png"
+      alt="Fig. 79" title="Fig. 79" /></a>
+    <span class="sc">Fig.</span> 79.
+  </div>
+
+  <p>If the compound formed has no definite melting point, the diagram
+  obtained is like that shown in Fig. 79. If we start with a solution, the
+  composition of which is represented by a point between <i>d</i> and
+  <i>b</i>, then, on cooling, <i>b</i> will separate out first, and the
+  temperature will fall until the point <i>d</i> is reached. The
+  temperature then remains constant until the component <i>b</i>, which has
+  separated out, is converted into the compound. After this the temperature
+  again falls, until it again remains constant at the eutectic point
+  <i>c</i>. In the case of the first halt, the period of constant
+  temperature is greatest when the initial composition of the solution is
+  the same as that of the compound; and it becomes shorter and shorter with
+  <!-- Page 232 --><span class="pagenum"><a
+  name="page232"></a>{232}</span>increase in the amount of either
+  component. In this way we obtain the time-composition curve
+  <i>b&prime;e&Prime;d&prime;</i>, of which the maximum point
+  <i>e&Prime;</i> gives the composition of the compound.</p>
+
+  <p>On the other hand, the period of constant temperature for the eutectic
+  point <span class="correction" title="Original reads 'e'."
+  ><i>c</i></span> is greatest in the case of solutions having the same
+  initial <i>composition</i> as that corresponding with the eutectic point;
+  and it decreases the more the initial composition approaches that of the
+  pure component <i>a</i> or the component <i>e</i>. In this way we obtain
+  the time-composition curve <i>a&prime;c&prime;e&prime;</i>. Here also the
+  point <i>e&prime;</i> represents the composition of the compound. We see,
+  therefore, that from the graphic representation of the freezing-point
+  curve, and from the duration of the temperature-arrests on the cooling
+  curve, for solutions of different initial composition, it is possible,
+  without having recourse to analysis, to decide what solid phases are
+  formed, and what is their composition.</p>
+
+  <p><b>Formation of Minerals.</b>&mdash;Important and interesting as is
+  the application of the Phase Rule to the study of alloys, its application
+  to the study of the conditions regulating the formation of minerals is no
+  less so; and although we do not propose to consider different cases in
+  detail here, still attention must be drawn to certain points connected
+  with this interesting subject.</p>
+
+  <p>In the first place, it will be evident from what has already been
+  said, that that mineral which first crystallizes out from a molten magma
+  is not necessarily the one with the highest melting point. The
+  <i>composition</i> of the fused mass must be taken into account. When the
+  system consists of two components which do not form a compound, one or
+  other of these will separate out in a pure state, according as the
+  composition of the molten mass lies on one or other side of the eutectic
+  composition; and the separation of the one component will continue until
+  the composition of the eutectic point is reached. Further cooling will
+  then lead to the simultaneous separation of the two components.</p>
+
+  <p>If, however, the two components form a stable compound (<i>e.g.</i>
+  orthoclase, from a fused mixture of silica and potassium aluminate), then
+  the freezing-point curve will resemble that <!-- Page 233 --><span
+  class="pagenum"><a name="page233"></a>{233}</span>shown in Fig. 64;
+  <i>i.e.</i> there will be a middle curve possessing a dystectic point,
+  and ending on either side at a eutectic point. This curve would represent
+  the conditions under which orthoclase is in equilibrium with the molten
+  magma. If the initial composition of the magma is represented by a point
+  between the two eutectic points, orthoclase will separate first. The
+  composition of the magma will thereby change, and the mass will finally
+  solidify to a mixture of orthoclase and silica, or orthoclase and
+  potassium aluminate, according to the initial composition.</p>
+
+  <p>What has just been said holds, however, only for stable equilibria,
+  and it must not be forgotten that complications can arise owing to
+  suspended transformation (when, for example, the magma is rapidly cooled)
+  and the production of metastable equilibria. These conditions occur very
+  frequently in nature.</p>
+
+  <p>The study of the formation of minerals from the point of view of the
+  Phase Rule is still in its initial stages, but the results which have
+  already been obtained give promise of a rich harvest in the future.<a
+  name="NtA316" href="#Nt316"><sup>[316]</sup></a></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 234 --><span class="pagenum"><a name="page234"></a>{234}</span></p>
+
+<h3>CHAPTER XIII</h3>
+
+<p class="cenhead">SYSTEMS OF THREE COMPONENTS</p>
+
+  <p><b>General.</b>&mdash;It has already been made evident that an
+  increase in the number of the components from one to two gives rise to a
+  considerable increase in the possible number of systems, and introduces
+  not a few complications into the equilibrium relations of these. No less
+  is this the case when the number of components increases from two to
+  three; and although examples of all the possible types of systems of
+  three components have not been investigated, nor, indeed, any one type
+  fully, nevertheless, among the systems which have been studied
+  experimentally, cases occur which not only possess a high scientific
+  interest, but are also of great industrial importance. On account not
+  only of the number, but more especially of the complexity of the systems
+  constituted of three components, no attempt will be made to give a full
+  account, or, indeed, even a survey of all the cases which have been
+  subjected to a more or less complete experimental investigation; on the
+  contrary, only a few of the more important classes will be selected, and
+  the most important points in connection with the behaviour of these
+  described.</p>
+
+  <p>On applying the Phase Rule</p>
+
+<p class="cenhead">P + F = C + 2</p>
+
+  <p>to the systems of three components, we see that in order that the
+  system shall be invariant, no fewer than five phases must be present
+  together, and an invariant system will therefore exist at a
+  <i>quintuple</i> point. Since the number of liquid phases can never
+  exceed the number of the components, and since there can be only one
+  vapour phase, it is evident that in this case, <!-- Page 235 --><span
+  class="pagenum"><a name="page235"></a>{235}</span>as in others, there
+  must always be at least one solid phase present at the quintuple point.
+  As the number of phases diminishes, the variability of the system can
+  increase from one to four, so that in the last case the condition of the
+  system will not be completely defined until not only the temperature and
+  the total pressure of the system, but also the concentrations of two of
+  the components have been fixed. Or, instead of the concentrations, the
+  partial pressures of the components may also be taken as independent
+  variables.</p>
+
+  <p><b>Graphic Representation.</b>&mdash;Hitherto the concentrations of
+  the components have been represented by means of rectangular
+  co-ordinates, although the numerical relationships have been expressed in
+  two different ways. In the one case, the concentration of the one
+  component was expressed in terms of a fixed amount of the other
+  component. Thus, the solubility of a salt was expressed by the number of
+  grams of salt dissolved by 100 grams of water or other solvent; and the
+  numbers so obtained were measured along one of the co-ordinates. The
+  second co-ordinate was then employed to indicate the change of another
+  independent variable, <i>e.g.</i> temperature. In the other case, the
+  combined weights of the two components A and B were put equal to unity,
+  and the concentration of the one expressed as a fraction of the whole
+  amount. This method allows of the representation of the complete series
+  of concentrations, from pure A to pure B, and was employed, for example,
+  in the graphic representation of the freezing point curves.</p>
+
+  <p>Even in the case of three components rectangular co-ordinates can also
+  be employed, and, indeed, are the most convenient in those cases where
+  the behaviour of two of the components to one another is very different
+  from their behaviour to the third component; as, for example, in the case
+  of two salts and water. In these cases, the composition of the system can
+  be represented by measuring the amounts of each of the two components in
+  a given weight of the third, along two co-ordinates at right angles to
+  one another; and the change of the system with the temperature can then
+  be represented by a third axis at right angles to the first two. In those
+  cases, <!-- Page 236 --><span class="pagenum"><a
+  name="page236"></a>{236}</span>however, where the three components behave
+  in much the same manner towards one another, the rectangular co-ordinates
+  are not at all suitable, and instead of these a <i>triangular diagram</i>
+  is employed. Various methods have been proposed for the graphic
+  representation of systems of three components by means of a triangle, but
+  only two of these have been employed to any considerable extent; and a
+  short description of these two methods will therefore suffice.<a
+  name="NtA317" href="#Nt317"><sup>[317]</sup></a></p>
+
+  <div class="figcenter" style="width:39%;">
+      <a href="images/fig080.png"><img style="width:100%" src="images/fig080.png"
+      alt="Fig. 80" title="Fig. 80" /></a>
+    <span class="sc">Fig. 80.</span>
+  </div>
+
+  <p>In the method proposed by Gibbs an equilateral triangle of unit height
+  is used (Fig 80).<a name="NtA318" href="#Nt318"><sup>[318]</sup></a> The
+  quantities of the different components are expressed as fractional parts
+  of the whole, and the sum of their concentrations is therefore equal to
+  unity, and can be represented by the height of the triangle. The corners
+  <!-- Page 237 --><span class="pagenum"><a
+  name="page237"></a>{237}</span>of the triangle represent the pure
+  substances A, B, and C respectively. A point on one of the sides of the
+  triangle will give the composition of a mixture in which only two
+  components are present, while a point within the triangle will represent
+  the composition of a ternary mixture. Since every point within the
+  triangle has the property that the sum of the perpendiculars from that
+  point on the sides of the triangle is equal to unity (the height of the
+  triangle), it is evident that the composition of a ternary mixture can be
+  represented by fixing a point within the triangle such that the lengths
+  of the <i>perpendiculars</i> from the point to the sides of the triangle
+  are equal respectively to the fractional amounts of the three components
+  present; the fractional amount of A, B, or C being represented by the
+  perpendicular distance from the side of the triangle <i>opposite</i> the
+  corners A, B, and C respectively.</p>
+
+  <p>The location of this point is simplified by dividing the normals from
+  each of the corners on the opposite side into ten or one hundred parts,
+  and drawing through these divisions lines at right angles to the normal
+  and parallel to the side of the triangle. A network of rhombohedra is
+  thus obtained, and the position of any point can be read off in
+  practically the same manner as in the case of rectangular co-ordinates.
+  Thus the point P in Fig. 80 represents a ternary mixture of the
+  composition A = 0.5, B = 0.3, C = 0.2; the perpendiculars P<i>a</i>,
+  P<i>b</i>, and P<i>c</i> being equal respectively to 0.5, 0.2, and 0.3 of
+  the height of the triangle.</p>
+
+  <p>Another method of representation, due to Roozeboom, consists in
+  employing an equilateral triangle, the length of whose <i>side</i> is
+  made equal to unity, or one hundred; the sum of the fractional or
+  percentage amounts of the three components being represented therefore by
+  a side of the triangle. In this case the composition of a ternary mixture
+  is obtained by determining, not the <i>perpendicular</i> distance of a
+  point P from the three sides of the triangle, but the distance in a
+  direction <i>parallel</i> to the sides of the triangle (Fig. 81).
+  Conversely, in order to represent a mixture consisting of <i>a</i>,
+  <i>b</i>, and <i>c</i> parts of the components A, B, and C respectively,
+  one side of the triangle, say AB, is first of all divided into ten or one
+  <!-- Page 238 --><span class="pagenum"><a
+  name="page238"></a>{238}</span>hundred parts; a portion, B<i>x</i> =
+  <i>a</i>, is then measured off, and represents the amount of A present.
+  Similarly, a portion, A<i>x&prime;</i> = <i>b</i>, is measured off and
+  represents the fractional amount of B, while the remainder,
+  <i>xx&prime;</i> = <i>c</i>, represents the amount of C. From <i>x</i>
+  and <i>x&prime;</i> lines are drawn parallel to the sides of the
+  triangle, and the point of intersection, P, represents the composition of
+  the ternary mixture of given composition; for, as is evident from the
+  figure, the distance of the point P from the three sides of the triangle,
+  when measured in directions <i>parallel</i> to the sides, is equal to
+  <i>a</i>, <i>b</i>, and <i>c</i> respectively. From the division marks on
+  the side AB, it is seen that the point P in this figure also represents a
+  mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives
+  exactly the same result as the previous method. The employment of a
+  right-angled isosceles triangle has also been suggested,<a name="NtA319"
+  href="#Nt319"><sup>[319]</sup></a> but is not in general use.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig081.png"><img style="width:100%" src="images/fig081.png"
+      alt="Fig. 81" title="Fig. 81" /></a>
+    <span class="sc">Fig. 81.</span>
+  </div>
+
+  <p>In employing the triangular diagram, it will be of use to note a
+  property of the equilateral triangle. A line drawn from one corner of the
+  triangle to the opposite side, represents the composition of all mixtures
+  in which the <i>relative</i> amounts of two of the components remain
+  unchanged. Thus, as Fig. 82 shows, if the component C is added to a
+  mixture x, in which A and B are present in the proportions of <i>a</i> :
+  <i>b</i>, a mixture <i>x&prime;</i>, which is thereby obtained, also
+  contains A and B in the ratio <i>a</i> : <i>b</i>. For the two triangles
+  AC<i>x</i> and BC<i>x</i> are similar to the two triangles
+  HC<i>x&prime;</i> and KC<i>x&prime;</i>; and, <!-- Page 239 --><span
+  class="pagenum"><a name="page239"></a>{239}</span>therefore, A<i>x</i> :
+  B<i>x</i> = H<i>x&prime;</i> : K<i>x&prime;</i>. But A<i>x</i> =
+  D<i>x</i> and B<i>x</i> = E<i>x</i>; further H<i>x&prime;</i> =
+  F<i>x&prime;</i> and K<i>x&prime;</i> = G<i>x&prime;</i>. Therefore,
+  D<i>x</i> : E<i>x</i> = F<i>x&prime;</i> : G<i>x&prime;</i> = <i>b</i> :
+  <i>a</i>. At all points on the line C<i>x</i>, therefore, the ratio of A
+  to B is the same.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig082.png"><img style="width:100%" src="images/fig082.png"
+      alt="Fig. 82" title="Fig. 82" /></a>
+    <span class="sc">Fig. 82.</span>
+  </div>
+
+  <div class="figcenter" style="width:16%;">
+      <a href="images/fig083.png"><img style="width:100%" src="images/fig083.png"
+      alt="Fig. 83" title="Fig. 83" /></a>
+    <span class="sc">Fig. 83.</span>
+  </div>
+
+  <p>If it is desired to represent at the same time the change of another
+  independent variable, <i>e.g.</i> temperature, this can be done by
+  measuring the latter along axes drawn perpendicular to the corners of the
+  triangle. In this way a right prism (Fig. 83) is obtained, and each
+  section of this cut parallel to the base represents therefore an
+  <i>isothermal surface</i>.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 240 --><span class="pagenum"><a name="page240"></a>{240}</span></p>
+
+<h3>CHAPTER XIV</h3>
+
+<p class="cenhead">SOLUTIONS OF LIQUIDS IN LIQUIDS</p>
+
+  <p>We have already seen (p.&nbsp;<a href="#page95">95</a>) that when two
+  liquids are brought together, they may mix in all proportions and form
+  one homogeneous liquid phase; or, only partial miscibility may occur, and
+  two phases be formed consisting of two mutually saturated solutions. In
+  the latter case, the concentration of the components in either phase and
+  also the vapour pressure of the system had, at a given temperature,
+  perfectly definite values. In the case of three liquid components, a
+  similar behaviour may be found, although complete miscibility of three
+  components with the formation of only one liquid phase is of much rarer
+  occurrence than in the case of two components. When only partial
+  miscibility occurs, various cases are met with according as the three
+  components form one, two, or three pairs of partially miscible liquids.
+  Further, when two of the components are only partially miscible, the
+  addition of the third may cause either an increase or a diminution in the
+  mutual solubility of these. An increase in the mutual solubility is
+  generally found when the third component dissolves readily in each of the
+  other two; but when the third component dissolves only sparingly in the
+  other two, its addition diminishes the mutual solubility of the
+  latter.</p>
+
+  <p>We shall consider here only a few examples illustrating the three
+  chief cases which can occur, viz. (1) A and B, and also B and C are
+  miscible in all proportions, while A and C are only partially miscible.
+  (2) A and B are miscible in all proportions, but A and C and B and C are
+  only partially miscible. (3) A and B, B and C, and A and C are only
+  partially miscible. A, B, and C here represent the three components.</p>
+
+  <p>1.&mdash;<i>The three components form only one pair of partially
+  miscible liquids.</i> <!-- Page 241 --><span class="pagenum"><a
+  name="page241"></a>{241}</span></p>
+
+  <p>An example of this is found in the three substances: chloroform,
+  water, and acetic acid.<a name="NtA320"
+  href="#Nt320"><sup>[320]</sup></a> Chloroform and acetic acid, and water
+  and acetic acid, are miscible with one another in all proportions, but
+  chloroform and water are only partially miscible with one another. If,
+  therefore, chloroform is shaken with a larger quantity of water than it
+  can dissolve, two layers will be formed consisting one of a saturated
+  solution of water in chloroform, the other of a saturated solution of
+  chloroform in water. The composition of these two solutions at a
+  temperature of about 18°, will be represented by the points <i>a</i> and
+  <i>b</i> in Fig. 84; <i>a</i> representing a solution of the composition:
+  chloroform, 99 per cent.; water, 1 per cent.; and <i>b</i> a solution of
+  the composition: chloroform, 0.8 per cent.; water, 99.2 per cent. When
+  acetic acid is added, it distributes itself between the two liquid
+  layers, and two conjugate <i>ternary</i> solutions, consisting of
+  chloroform, water, and acetic acid are thereby produced which are in
+  equilibrium with one another, and the composition of which will be
+  represented by two points inside the triangle. In this way a series of
+  pairs of ternary solutions will be obtained by the addition of acetic
+  acid to the mixture of chloroform and water. By this addition, also, not
+  only do the two liquid phases become increasingly rich in acetic acid,
+  but the mutual solubility of the chloroform and water increases; so that
+  the layer <i>a</i> becomes relatively richer in water, and layer <i>b</i>
+  relatively richer in chloroform. This is seen from the following table,
+  which gives the percentage composition of different conjugate ternary
+  solutions at 18°.</p>
+
+<table class="nobctr" summary="Layers of chloroform-water-acetic acid" title="Layers of chloroform-water-acetic acid">
+<tr><td class="toribob" style="text-align:center" colspan="3"> Heavier layer.
+</td><td class="toboleb" style="text-align:center" colspan="3"> Lighter layer.</td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Chloroform.
+</td><td class="toboleb" style="text-align:center"> Water.
+</td><td class="toboleb" style="text-align:center"> Acetic acid.
+</td><td class="toboleb" style="text-align:center"> Chloroform.
+</td><td class="toboleb" style="text-align:center"> Water.
+</td><td class="toboleb" style="text-align:center"> Acetic acid.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 99.01 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 0.99 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 0 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 0.84 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 99.16 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 0</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 91.85 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 1.38 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 6.77 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 1.21 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 73.69 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 25.10</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 80.00 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 2.28 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 17.72 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> &nbsp; 7.30 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 48.58 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 44.12</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 70.13 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 4.12 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 25.75 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 15.11 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 34.71 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 50.18</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 67.15 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 5.20 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 27.65 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 18.33 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 31.11 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 50.56</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 59.99 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 7.93 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 32.08 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 25.20 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 25.39 </td><td class="leb" style="text-align:left; padding-left:2em; padding-right:2em;"> 49.41</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 55.81 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 9.58 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 34.61 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 28.85 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 23.28 </td><td class="boleb" style="text-align:left; padding-left:2em; padding-right:2em;"> 47.87</td></tr>
+
+</table>
+
+<p><!-- Page 242 --><span class="pagenum"><a name="page242"></a>{242}</span></p>
+
+  <p>By the continued addition of acetic acid, the composition of the
+  successive conjugate solutions in equilibrium with one another becomes,
+  as the table shows, more nearly the same, and a point is at length
+  reached at which the two solutions become identical. This will therefore
+  be a <i>critical point</i> (p.&nbsp;<a href="#page98">98</a>). Increased
+  addition of acetic acid beyond this point will lead to a single
+  homogeneous solution.</p>
+
+  <p>These relationships are represented graphically by the curve
+  <i>a</i>K<i>b</i>, Fig. 84. The points on the branch <i>a</i>K represent
+  the composition of the solutions relatively rich in chloroform (heavier
+  layer), those on the curve <i>b</i>K the composition of solutions
+  relatively rich in water (lighter layer); and the points on these two
+  branches representing conjugate solutions are joined together by
+  "tie-lines." Thus, the points <i>a&prime;b&prime;</i> represent conjugate
+  solutions, and the line <i>a&prime;b&prime;</i> is a tie-line.</p>
+
+  <div class="figcenter" style="width:32%;">
+      <a href="images/fig084.png"><img style="width:100%" src="images/fig084.png"
+      alt="Fig. 84" title="Fig. 84" /></a>
+    <span class="sc">Fig.</span> 84.
+  </div>
+
+  <p>Since, now, acetic acid when added to a heterogeneous mixture of
+  chloroform and water does not enter in equal amounts into the two layers,
+  but in amounts depending on its coefficient of distribution between
+  chloroform and water,<a name="NtA321" href="#Nt321"><sup>[321]</sup></a>
+  the <!-- Page 243 --><span class="pagenum"><a
+  name="page243"></a>{243}</span>tie-lines will not be parallel to AB, but
+  will be inclined at an angle. As the solutions become more nearly the
+  same, the tie-lines diminish in length, and at last, when the conjugate
+  solutions become identical, shrink to a point. For the reason that the
+  tie-lines are, in general, not parallel to the side of the triangle, the
+  critical point at which the tie-line vanishes will not be at the summit
+  of the curve, but somewhere below this, as represented by the point
+  K.</p>
+
+  <p>The curve <i>a</i>K<i>b</i>, further, forms the boundary between the
+  heterogeneous and homogeneous systems. A mixture of chloroform, water,
+  and acetic acid represented by any point outside the curve
+  <i>a</i>K<i>b</i>, will form only one homogeneous phase; while any
+  mixture represented by a point within the curve, will separate into two
+  layers having the composition represented by the ends of the tie-line
+  passing through that point. Thus, a mixture of the total composition
+  <i>x</i>, will separate into two layers having the composition
+  <i>a&prime;</i> and <i>b&prime;</i> respectively.</p>
+
+  <p>Since three components existing in three phases (two liquid and a
+  vapour phase) constitute a bivariant system, the final result,
+  <i>i.e.</i> the composition of the two layers and the total vapour
+  pressure, will not depend merely on the temperature, as in the case of
+  two-component systems (p.&nbsp;<a href="#page102">102</a>), but also on the
+  composition of the mixture with which we start. At constant temperature,
+  however, all mixtures, the composition of which is represented by a point
+  on one and the same tie-line, will separate into the same two liquid
+  phases, although the relative <i>amounts</i> of the two phases will vary.
+  If we omit the vapour phase, the condition of the system will depend on
+  the pressure as well as on the temperature and composition of the initial
+  mixture. By keeping the pressure constant, <i>e.g.</i> at atmospheric
+  pressure (by working with open vessels), the system again becomes
+  bivariant. We see, therefore, that the position of the curve
+  <i>a</i>K<i>b</i>, or, in other words, the composition of the different
+  conjugate ternary solutions, will vary with the temperature, and only
+  with the temperature, if we assume either constancy of pressure or the
+  presence of the vapour phase. Since at the critical point the condition
+  is imposed that the two liquid phases become identical, one degree of
+  freedom is thereby <!-- Page 244 --><span class="pagenum"><a
+  name="page244"></a>{244}</span>lost, and therefore only one degree of
+  freedom remains. The critical point, therefore, depends on the
+  temperature, and only on the temperature; always on the assumption, of
+  course, that the pressure is constant, or that a vapour phase is present.
+  Fig. 84, therefore, represents an isothermal (p.&nbsp;<a
+  href="#page239">239</a>).</p>
+
+  <p>It is of importance to note that the composition of the different
+  ternary solutions obtained by the addition of acetic acid to a
+  heterogeneous mixture of chloroform and water, will depend not only on
+  the amount of acetic acid added, but also on the relative amounts of
+  chloroform and water at the commencement. Suppose, for example, that we
+  start with chloroform and water in the proportions represented by the
+  point <i>c&prime;</i> (Fig. 84). On mixing these, two liquid layers
+  having the composition <i>a</i> and <i>b</i> respectively will be formed.
+  Since by the addition of acetic acid the relative amounts of these two
+  substances in the system as a whole cannot undergo alteration, the total
+  composition of the different ternary systems which will be obtained must
+  be represented by a point on the line C<i>c&prime;</i> (p.&nbsp;<a
+  href="#page238">238</a>). Thus, for example, by the addition of acetic
+  acid a system may be obtained, the total composition of which is
+  represented by the point <i>c&Prime;</i>. Such a system, however, will
+  separate into two conjugate ternary solutions, the composition of which
+  will be represented by the ends of the tie-line passing through the point
+  <i>c&Prime;</i>. So long as the total composition of the system lies
+  below the point S, <i>i.e.</i> the point of intersection of the line
+  C<i>c&prime;</i> with the boundary curve, two liquid layers will be
+  formed; while all systems having a total composition represented by a
+  point on the line C<i>c&prime;</i>, above S, will form only one
+  homogeneous solution.</p>
+
+  <p>From the figure, also, it is evident that as the amount of acetic acid
+  is increased, the relative amounts of the two liquid layers formed differ
+  more and more until at S a limiting position is reached, when the amount
+  of the one liquid layer dwindles to nought, and only one solution
+  remains.</p>
+
+  <p>The same reasoning can be carried through for different initial
+  amounts of chloroform and water, but it would be fruitless to discuss all
+  the different systems which can be obtained. The reason for the preceding
+  discussion was to show that <!-- Page 245 --><span class="pagenum"><a
+  name="page245"></a>{245}</span>although the addition of acetic acid to a
+  mixture of chloroform and water will, in all cases, lead ultimately to a
+  limiting system, beyond which homogeneity occurs, that point is not
+  necessarily the critical point. On the contrary, in order that addition
+  of acetic acid shall lead to the critical mixture, it is necessary to
+  start with a binary mixture of chloroform and water in the proportions
+  represented by the point <i>c&prime;</i>. In this case, addition of
+  acetic acid will give rise to a series of conjugate ternary solutions,
+  the composition of which will gradually approach to one another, and at
+  last become identical.</p>
+
+  <p>From the foregoing it will be evident that the amount of acetic acid
+  required to produce a homogenous solution, will depend on the relative
+  amounts of chloroform and water from which we start, and can be
+  ascertained by joining the corner C with the point on the line AB
+  representing the total composition of the initial binary system. The
+  point where this line intersects the boundary curve <i>a</i>K<i>b</i>
+  will indicate the minimum amount of acetic acid which, under these
+  particular conditions, is necessary to give one homogeneous solution.</p>
+
+  <p><b>Retrograde Solubility.</b>&mdash;As a consequence of the fact that
+  acetic acid distributes itself unequally between chloroform and water,
+  and the critical point K, therefore, does not lie at the summit of the
+  curve, it is possible to start with a homogeneous solution in which the
+  percentage amount of acetic acid is greater than at the critical point,
+  and to pass from this first to a heterogenous and then again to a
+  homogenous system merely by altering the relative amounts of chloroform
+  and water. This phenomenon, to which the term <i>retrograde
+  solubility</i> is applied, will be observed not only in the case of
+  chloroform, water, and acetic acid, but in all other systems in which the
+  critical point lies below the highest point of the boundary curve for
+  heterogeneous systems. This will be seen from the diagram, Fig. 85.
+  Starting with the homogeneous system represented by <i>x</i>, in which,
+  therefore, the concentration of C is greater than in the critical mixture
+  (K), if the relative amounts of A and B are altered in the direction
+  <i>xx&prime;</i>, while the amount of C is maintained constant, the
+  system will become heterogeneous when the composition reaches the point
+  <i>y</i>, and will remain <!-- Page 246 --><span class="pagenum"><a
+  name="page246"></a>{246}</span>heterogeneous with changing composition
+  until the point <i>y&prime;</i> is passed, when it will again become
+  homogeneous. If the relative concentration of C is increased above that
+  represented by the line SS, this phenomenon will, of course, no longer be
+  observed.</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig085.png"><img style="width:100%" src="images/fig085.png"
+      alt="Fig. 85" title="Fig. 85" /></a>
+    <span class="sc">Fig.</span> 85.
+  </div>
+
+  <p>Relationships similar to those described for chloroform, water, and
+  acetic acid are also found in the case of a number of other trios,
+  <i>e.g.</i> ether, water, and alcohol; chloroform, water, and alcohol.<a
+  name="NtA322" href="#Nt322"><sup>[322]</sup></a> They have also been
+  observed in the case of a considerable number of molten metals.<a
+  name="NtA323" href="#Nt323"><sup>[323]</sup></a> Thus, molten lead and
+  silver, as well as molten zinc and silver, mix in all proportions; but
+  molten lead and zinc are only partially miscible with one another. When
+  melted together, therefore, the last two metals will separate into two
+  liquid layers, one rich in lead, the other rich in zinc. If silver is now
+  added, and the temperature maintained above the freezing point of the
+  mixture, the silver passes for the most part, in accordance with the law
+  of distribution, into the upper layer, which is rich in zinc; silver
+  being more soluble in molten zinc than in molten lead. This is clearly
+  shown by the following figures:&mdash;<a name="NtA324"
+  href="#Nt324"><sup>[324]</sup></a></p>
+
+<p><!-- Page 247 --><span class="pagenum"><a name="page247"></a>{247}</span></p>
+
+<table class="nobctr" summary="Alloys of silver-lead-zinc" title="Alloys of silver-lead-zinc">
+<tr><td class="toribob" style="text-align:center" colspan="3"> Heavier alloy.
+</td><td class="toboleb" style="text-align:center" colspan="3"> Lighter alloy.</td></tr>
+
+<tr><td class="torib" style="text-align:center" colspan="3"> Percentage amount of
+</td><td class="toleb" style="text-align:center" colspan="3"> Percentage amount of</td></tr>
+
+<tr><td class="ribob" style="text-align:center"> Silver.
+</td><td class="boleb" style="text-align:center"> Lead.
+</td><td class="boleb" style="text-align:center"> Zinc.
+</td><td class="boleb" style="text-align:center"> Silver.
+</td><td class="boleb" style="text-align:center"> Lead.
+</td><td class="boleb" style="text-align:center"> Zinc.</td></tr>
+
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> 1.25 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 96.69 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 2.06 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 38.91 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 3.12 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 57.97</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> 1.71 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 96.43 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 1.86 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 45.01 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 3.37 </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> 51.62</td></tr>
+
+<tr><td class="ribob" style="text-align:center; padding-left:2em; padding-right:2em;"> 5.55 </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 93.16 </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 1.29 </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 54.93 </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 4.21 </td><td class="boleb" style="text-align:center; padding-left:2em; padding-right:2em;"> 40.86</td></tr>
+
+</table>
+
+  <p>The numbers in the same horizontal row give the composition of the
+  conjugate alloys, and it is evident that the upper layer consists almost
+  entirely of silver and zinc. On allowing the mixture to cool slightly,
+  the upper layer solidifies first, and can be separated from the still
+  molten lead layer. It is on this behaviour of silver towards a mixture of
+  molten lead and zinc that the Parkes's method for the desilverization of
+  lead depends.<a name="NtA325" href="#Nt325"><sup>[325]</sup></a> If
+  aluminium is also added, a still larger proportion of silver passes into
+  the lighter layer, and the desilverization of the lead is more
+  complete.<a name="NtA326" href="#Nt326"><sup>[326]</sup></a></p>
+
+  <div class="figright" style="width:16%;">
+      <a href="images/fig087.png"><img style="width:100%" src="images/fig087.png"
+      alt="Fig. 87" title="Fig. 87" /></a>
+    <span class="sc">Fig.</span> 87.
+  </div>
+
+  <div class="figright" style="width:14%;">
+      <a href="images/fig086.png"><img style="width:100%" src="images/fig086.png"
+      alt="Fig. 86" title="Fig. 86" /></a>
+    <span class="sc">Fig.</span> 86.
+  </div>
+
+  <p><b>The Influence of Temperature.</b>&mdash;As has already been said, a
+  ternary system existing in three phases possesses two degrees of freedom;
+  and the state of the system is therefore dependent not only on the
+  relative concentration of the components, but also on the temperature. As
+  the temperature changes, therefore, the boundary curve of the
+  heterogeneous system will also alter; and in order to represent this
+  alteration we shall make use of the right prism, in which the temperature
+  is measured upwards. In this way the boundary curve passes into a
+  boundary surface (called a dineric surface), as shown in Fig. 86. In this
+  figure the curve <i>akb</i> is the isothermal for the ternary system; the
+  curve <i>a</i>K<i>b</i> shows the change in the <i>binary</i> system AB
+  with the temperature, with <!-- Page 248 --><span class="pagenum"><a
+  name="page248"></a>{248}</span>a critical point at K. This curve has the
+  same meaning as those given in Chapter VI. The curve <i>k</i>K is a
+  critical curve joining together the critical points of the different
+  isothermals. In such a case as is shown in Fig. 86, there does not exist
+  any real critical temperature for the ternary system, for as the
+  temperature is raised, the amount of C in the "critical" solution becomes
+  less and less, and at K only two components, A and B, are present. In the
+  case, however, represented in Fig. 87, a real ternary critical point is
+  found. In this figure <i>ak&prime;b</i> is an isothermal,
+  <i>ak&Prime;</i> is the curve for the binary system, and K is the ternary
+  critical point. All points outside the helmet-shaped boundary surface
+  represent homogeneous ternary solutions, while all points within the
+  surface belong to heterogeneous systems. Above the temperature of the
+  point K, the three components are miscible in all proportions. An example
+  of a ternary system yielding such a boundary surface is that consisting
+  of phenol, water, and acetone.<a name="NtA327"
+  href="#Nt327"><sup>[327]</sup></a> In this case the critical temperature
+  K is 92°, and the composition at this ternary critical point
+  is&mdash;</p>
+
+<table class="nobctr" summary="Ternary critical composition, water-acetone-phenol" title="Ternary critical composition, water-acetone-phenol">
+<tr><td class="nob" style="text-align:left"> Water </td><td class="nob" style="text-align:left"> 59 per cent.</td></tr>
+<tr><td class="nob" style="text-align:left"> Acetone </td><td class="nob" style="text-align:left"> 12 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+<tr><td class="nob" style="text-align:left"> Phenol </td><td class="nob" style="text-align:left"> 29 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+</table>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig088.png"><img style="width:100%" src="images/fig088.png"
+      alt="Fig. 88" title="Fig. 88" /></a>
+    <span class="sc">Fig.</span> 88.
+  </div>
+
+  <p>The difference between the two classes of systems just mentioned, is
+  seen very clearly by a glance at the Figs. 88 and 89, which show the
+  projection of the isothermals on the base of the prism. In Fig. 88, the
+  projections yield paraboloid curves, the two branches of which are cut by
+  one side of the triangle; and the critical point is represented by a
+  point on <!-- Page 249 --><span class="pagenum"><a
+  name="page249"></a>{249}</span>this side. In the second case (Fig. 89),
+  however, the projections of the isothermals form ellipsoidal curves
+  surrounding the supreme critical point, which now lies <i>inside the
+  triangle</i>. At lower temperatures, these isothermal boundary curves are
+  cut by a side of the triangle; at the critical temperature,
+  <i>k&Prime;</i>, of the binary system AB, the boundary curve
+  <i>touches</i> the side AB, while at still higher temperatures the
+  boundary curve comes to lie entirely within the triangle. At any given
+  temperature, therefore, between the critical point of the binary system
+  (<i>k&Prime;</i>), and the supreme critical point of the ternary system
+  (K), each pair of the three components are miscible with one another in
+  all proportions; for the region of heterogeneous systems is now bounded
+  by a closed curve lying entirely within the triangle. Outside this curve
+  only homogeneous systems are found. Binary mixtures, therefore,
+  represented by any point on one of the sides of the triangle must be
+  homogeneous, for they all lie outside the boundary curve for
+  heterogeneous states.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig089.png"><img style="width:100%" src="images/fig089.png"
+      alt="Fig. 89" title="Fig. 89" /></a>
+    <span class="sc">Fig.</span> 89.
+  </div>
+
+  <p>2. <i>The three components can form two pairs of partially miscible
+  liquids.</i></p>
+
+  <p>In the case of the three components water, alcohol, and succinic
+  nitrile, water and alcohol are miscible in all proportions, but not so
+  water and succinic nitrile, or alcohol and succinic nitrile.</p>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/fig091.png"><img style="width:100%" src="images/fig091.png"
+      alt="Fig. 91" title="Fig. 91" /></a>
+    <span class="sc">Fig.</span> 91.
+  </div>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/fig090.png"><img style="width:100%" src="images/fig090.png"
+      alt="Fig. 90" title="Fig. 90" /></a>
+    <span class="sc">Fig.</span> 90.
+  </div>
+
+  <p>As we have already seen (p.&nbsp;<a href="#page122">122</a>), water and
+  succinic nitrile can form two liquid layers between the temperatures
+  18.5° and 55.5°; while alcohol and nitrile can form two liquid layers
+  <span class="correction" title="Original reads 'betwen'.">between</span>
+  13° and 31°. If, then, between these two temperature limits, alcohol is
+  added to a heterogeneous mixture of water and nitrile, or water is added
+  to a mixture of alcohol and nitrile, two heterogeneous ternary systems
+  will be formed, <!-- Page 250 --><span class="pagenum"><a
+  name="page250"></a>{250}</span>and two boundary curves will be obtained
+  in the triangular diagram, as shown in Fig. 90.<a name="NtA328"
+  href="#Nt328"><sup>[328]</sup></a> On changing the temperature, the
+  boundary curves will also undergo alteration, in a manner similar to that
+  just discussed. As the temperature falls, the two curves will spread out
+  more and more into the centre of the triangle, and might at last meet one
+  another; while at still lower temperatures we may imagine the curves
+  still further expanding so that the two heterogeneous regions flow into
+  one another and form a <i>band</i> on the triangular diagram (Fig. 91).
+  This, certainly, has not been realized in the case of the three
+  components mentioned, because at a temperature higher than that at which
+  the two heterogeneous regions could fuse together, solid separates
+  out.</p>
+
+  <div class="figcenter" style="width:16%;">
+      <a href="images/fig092.png"><img style="width:100%" src="images/fig092.png"
+      alt="Fig. 92" title="Fig. 92" /></a>
+    <span class="sc">Fig.</span> 92.
+  </div>
+
+  <p>The gradual expansion of a paraboloid into a band-like area of
+  heterogeneous ternary systems, has, however, been observed in the case of
+  water, phenol, and aniline.<a name="NtA329"
+  href="#Nt329"><sup>[329]</sup></a> In Fig. 92 are shown three
+  isothermals, viz. those for 148°, 95°, and 50°. At 148°, water and
+  aniline form two layers having the composition&mdash;</p>
+
+<table class="nobctr" summary="Layers of water and aniline at 148°" title="Layers of water and aniline at 148°">
+<tr><td class="nob" style="text-align:left"> Water, &nbsp;83.5 per cent.
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+</td><td class="nob" style="text-align:left" rowspan="2"> and
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left"> water, &nbsp;20 per cent.</td></tr>
+<tr><td class="nob" style="text-align:left"> Aniline, 16.5 &nbsp; &nbsp; &nbsp; ,, </td><td class="nob" style="text-align:left"> aniline, 80 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+</table>
+
+<p><!-- Page 251 --><span class="pagenum"><a name="page251"></a>{251}</span></p>
+
+  <p>and the critical point <i>k&prime;</i> has the composition&mdash;</p>
+
+<p class="cenhead">Water, 65; phenol, 13.2; aniline, 21.8 per cent.</p>
+
+  <p>At 95°, the composition of the two binary solutions is&mdash;</p>
+
+<table class="nobctr" summary="Layers of water and aniline at 95°" title="Layers of water and aniline at 95°">
+<tr><td class="nob" style="text-align:left"> Water, &nbsp;93 per cent.
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+</td><td class="nob" style="text-align:left" rowspan="2"> and
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left"> water, &nbsp; &nbsp;8 per cent.</td></tr>
+<tr><td class="nob" style="text-align:left"> Aniline, &nbsp; 7 &nbsp; &nbsp; &nbsp; ,, </td><td class="nob" style="text-align:left"> aniline, 92 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+</table>
+
+  <p>while the point <i>k&Prime;</i> has the composition</p>
+
+<p class="cenhead">Water, 69.9; phenol, 26.6; aniline, 3.5 per cent.</p>
+
+  <p>At 50°, the region of heterogeneous states now forms a band, and the
+  two layers formed by water and aniline have the composition&mdash;</p>
+
+<table class="nobctr" summary="Layers of water and aniline at 50°" title="Layers of water and aniline at 50°">
+<tr><td class="nob" style="text-align:left"> Water, &nbsp;96.5 per cent.
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+</td><td class="nob" style="text-align:left" rowspan="2"> and
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left"> water, &nbsp; &nbsp;5.5 per cent.</td></tr>
+<tr><td class="nob" style="text-align:left"> Aniline, &nbsp; 3.5 &nbsp; &nbsp; &nbsp; ,, </td><td class="nob" style="text-align:left"> aniline, 94.5 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+</table>
+
+  <p>while the two layers formed by water and phenol have the
+  composition&mdash;</p>
+
+<table class="nobctr" summary="Layers of water and phenol at 50°" title="Layers of water and phenol at 50°">
+<tr><td class="nob" style="text-align:left"> Water, &nbsp;89 per cent.
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+</td><td class="nob" style="text-align:left" rowspan="2"> and
+</td><td class="nob" style="text-align:left" rowspan="2"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left"> water, &nbsp;20 per cent.</td></tr>
+<tr><td class="nob" style="text-align:left"> Phenol, 11 &nbsp; &nbsp; &nbsp; ,, </td><td class="nob" style="text-align:left"> phenol, 80 &nbsp; &nbsp; &nbsp; ,,</td></tr>
+</table>
+
+  <p>All mixtures of water, phenol, and aniline, therefore, the composition
+  of which is represented by any point within the band <i>abcd</i>, will
+  form two ternary solutions; while if the composition is represented by a
+  point outside the band, only one homogeneous solution will be
+  produced.</p>
+
+  <p>3. <i>The three components form three pairs of partially miscible
+  liquids.</i></p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig093.png"><img style="width:100%" src="images/fig093.png"
+      alt="Fig. 93" title="Fig. 93" /></a>
+    <span class="sc">Fig.</span> 93.
+  </div>
+
+  <p>The third chief case which can occur is that no two of the components
+  are completely miscible with one another. In this case, therefore, we
+  shall obtain three paraboloid boundary curves, as shown in Fig. 93. If,
+  now, we imagine these three curves to expand in towards the centre of the
+  triangle, as might happen, for example, by lowering the temperature, a
+  point will <!-- Page 252 --><span class="pagenum"><a
+  name="page252"></a>{252}</span>be reached at which the curves partly
+  overlap, and we shall get the appearance shown in Fig. 94.</p>
+
+  <p>The points <i>a</i>, <i>b</i>, and <i>c</i> represent the points where
+  the three curves cut, and the triangle <i>abc</i> is a region where the
+  curves overlap. From this diagram we can see that any mixture having a
+  composition represented by a point in one of the clear spaces at the
+  corners of the larger triangle, will form a homogeneous solution; if the
+  composition corresponds to any point lying in one of the quadrilateral
+  regions <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub> or
+  <i>x</i><sub>3</sub>, two ternary solutions will be formed; while, if the
+  composition is represented by any point in the inner triangle, separation
+  into three layers will occur.</p>
+
+  <div class="figcenter" style="width:26%;">
+      <a href="images/fig094.png"><img style="width:100%" src="images/fig094.png"
+      alt="Fig. 94" title="Fig. 94" /></a>
+    <span class="sc">Fig.</span> 94.
+  </div>
+
+  <p>Since in the clear regions at the corners of the triangle we have
+  three components in two phases, liquid and vapour, the systems have three
+  degrees of freedom. At constant temperature, therefore, the condition of
+  the system is not defined until the concentrations of two of the
+  components are fixed. A system belonging to one of the quadrilateral
+  spaces has, as we have seen, two degrees of freedom; besides the
+  temperature, one concentration must be fixed. Lastly, a system the
+  composition of which falls within the inner triangle <i>abc</i>, will
+  form three layers, and will therefore possess only one degree of freedom.
+  If the temperature is fixed, the composition of the three layers is also
+  determined, viz. that of the points <i>a</i>, <i>b</i>, and <i>c</i>
+  respectively; and a change in the composition of the original mixture can
+  lead only to a difference in the relative amounts of the three layers,
+  not to a difference in their composition.</p>
+
+  <p>An example of a system which can form three liquid phases is found in
+  water, ether, and succinic nitrile.<a name="NtA330"
+  href="#Nt330"><sup>[330]</sup></a></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 253 --><span class="pagenum"><a name="page253"></a>{253}</span></p>
+
+<h3>CHAPTER XV</h3>
+
+<p class="cenhead">PRESENCE OF SOLID PHASES</p>
+
+  <p><b>A. The Ternary Eutectic Point.</b>&mdash;In passing to the
+  consideration of those ternary systems in which one or more solid phases
+  can exist together with one liquid phase, we shall first discuss not the
+  solubility curves, as in the case of two-component systems, but the
+  simpler relationships met with at the freezing point. That is, we shall
+  first of all examine the freezing point curves of ternary systems.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig095.png"><img style="width:100%" src="images/fig095.png"
+      alt="Fig. 95" title="Fig. 95" /></a>
+    <span class="sc">Fig.</span> 95.
+  </div>
+
+  <p>Since it is necessary to take into account not only the changing
+  composition of the liquid phase, but also the variation of the
+  temperature, we shall employ the right prism for the graphic
+  representation of the systems, as shown in Fig. 95. A, B, and C in this
+  figure, therefore, denote the melting points of the pure components. If
+  we start with the component A at its melting point, and add B, which is
+  capable of dissolving in liquid A, the freezing point of A will be
+  lowered; and, similarly, the freezing point of B by addition of A. In
+  this way we get the freezing point curve A<i>k</i><sub>1</sub>B for the
+  binary system; <i>k</i><sub>1</sub>; being an eutectic point. This curve
+  will of course lie in the plane formed by one face of the prism. In a
+  similar manner we obtain the freezing point curves A<i>k</i><sub>2</sub>C
+  and B<i>k</i><sub>3</sub>C. These curves give the composition of the
+  binary liquid phases in equilibrium <!-- Page 254 --><span
+  class="pagenum"><a name="page254"></a>{254}</span>with one of the pure
+  components, or at the eutectic points, with a mixture of two solid
+  components. If, now, to the system represented say by the point
+  <i>k</i><sub>1</sub>, a small quantity of the third component, C, is
+  added, the temperature at which the two solid phases A and B can exist in
+  equilibrium with the liquid phase is lowered; and this depression of the
+  eutectic point is all the greater the larger the addition of C. In this
+  way we obtain the curve <i>k</i><sub>1</sub>K, which slopes inwards and
+  downwards, and indicates the varying composition of the ternary liquid
+  phase with which a mixture of solid A and B are in equilibrium.
+  Similarly, the curves <i>k</i><sub>2</sub>K and <i>k</i><sub>3</sub>K are
+  the corresponding eutectic curves for A and C, and B and C in equilibrium
+  with ternary solutions. At the point K, the three solid components are in
+  equilibrium with the liquid phase; and this point, therefore, represents
+  <i>the lowest temperature attainable with the three components given</i>.
+  Each of the ternary eutectic curves, as they may be called, is produced
+  by the intersection of two surfaces, while at the ternary eutectic point,
+  three surfaces, viz. A<i>k</i><sub>1</sub>K<i>k</i><sub>2</sub>,
+  B<i>k</i><sub>1</sub>K<i>k</i><sub>3</sub>, and
+  C<i>k</i><sub>1</sub>K<i>k</i><sub>3</sub> intersect. Any point on one of
+  these surfaces represents a ternary solution in equilibrium with only one
+  component in the solid state; the lines or curves of intersection of
+  these represent equilibria with two solid phases, while at the point K,
+  the ternary eutectic point, there are three solid phases in equilibrium
+  with a liquid and a vapour phase. The surfaces just mentioned represent
+  bivariant systems. One component in the solid state can exist in
+  equilibrium with a ternary liquid phase under varying conditions of
+  temperature and concentration of the components in the solution; and
+  before the state of the system is defined, these two variables,
+  temperature and composition of the liquid phase, must be fixed. On the
+  other hand, the curves formed by the intersection of these planes
+  represent univariant systems; at a given temperature two solid phases can
+  exist in equilibrium with a ternary solution, only when the latter has a
+  definite composition. Lastly, the ternary eutectic point, K, represents
+  an invariant system; three solid phases can exist in equilibrium with a
+  ternary solution, only when the latter has one fixed composition and when
+  the temperature has a definite value. This eutectic point, therefore,
+  <!-- Page 255 --><span class="pagenum"><a
+  name="page255"></a>{255}</span>has a perfectly definite position,
+  depending only on the nature of the three components.</p>
+
+  <p>Instead of employing the prism, the change in the composition of the
+  ternary solutions can also be indicated by means of the
+  <i>projections</i> of the curves <i>k</i><sub>1</sub>K,
+  <i>k</i><sub>2</sub>K, and <i>k</i><sub>3</sub>K on the base of the
+  prism, the particular temperature being written beside the different
+  eutectic points and curves. This is shown in Fig. 96.</p>
+
+  <div class="figcenter" style="width:22%;">
+      <a href="images/fig096.png"><img style="width:100%" src="images/fig096.png"
+      alt="Fig. 96" title="Fig. 96" /></a>
+    <span class="sc">Fig.</span> 96.
+  </div>
+
+  <p>The numbers which are given in this diagram refer to the eutectic
+  points for the system bismuth&mdash;lead&mdash;tin, the data for which
+  are as follows:&mdash;<a name="NtA331"
+  href="#Nt331"><sup>[331]</sup></a></p>
+
+<table class="nobctr" summary="Mixtures of bismuth-lead-tin" title="Mixtures of bismuth-lead-tin">
+<tr><td class="toribob" style="text-align:center"> Melting point of<br />pure metal.
+</td><td class="toboleb" style="text-align:center" colspan="3"> Percentage composition of<br />binary eutectic mixture.
+</td><td class="toboleb" style="text-align:center"> Temperature of binary<br />eutectic point.</td></tr>
+
+<tr><td class="rib" style="text-align:left"> </td><td class="nob" style="text-align:center"> Bi </td><td class="nob" style="text-align:center"> Pb </td><td class="nob" style="text-align:center"> Sn </td><td class="leb" style="text-align:left"></td></tr>
+<tr><td class="rib" style="text-align:left"> Bismuth, 268° </td><td class="nob" style="text-align:center"> 55 </td><td class="nob" style="text-align:center"> 45 </td><td class="nob" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:left"> Bi&mdash;Pb, 127°</td></tr>
+<tr><td class="rib" style="text-align:left"> Lead, &nbsp; &nbsp; 325° </td><td class="nob" style="text-align:center"> 58 </td><td class="nob" style="text-align:center"> &mdash; </td><td class="nob" style="text-align:center"> 42 </td><td class="leb" style="text-align:left"> Bi&mdash;Sn, 133°</td></tr>
+
+<tr><td class="ribob" style="text-align:left"> Tin, &nbsp; &nbsp; &nbsp; &nbsp;232° </td><td class="bob" style="text-align:center"> &mdash; </td><td class="bob" style="text-align:center"> 37 </td><td class="bob" style="text-align:center"> 63 </td><td class="boleb" style="text-align:left"> Pb&mdash;Sn, 182°</td></tr>
+</table>
+
+  <p>&nbsp;</p>
+
+<table class="nobctr" summary="Ternary eutectic mixture of bismuth-lead-tin" title="Ternary eutectic mixture of bismuth-lead-tin">
+<tr><td class="toribob" style="text-align:center" colspan="3"> Percentage composition of<br />ternary eutectic mixture.
+</td><td class="toboleb" style="text-align:center"> Temperature of ternary<br />eutectic point.</td></tr>
+
+<tr><td class="nob" style="text-align:center"> Bi </td><td class="nob" style="text-align:center"> Pb </td><td class="nob" style="text-align:center"> Sn </td><td class="leb" style="text-align:center"></td></tr>
+<tr><td class="bob" style="text-align:center"> 52 </td><td class="bob" style="text-align:center"> 32 </td><td class="bob" style="text-align:center"> 16 </td><td class="boleb" style="text-align:center"> 96°</td></tr>
+</table>
+
+  <p><b>Formation of Compounds.</b>&mdash;In the case just discussed, the
+  components crystallized out from solution in the pure state. If, however,
+  combination can take place between two of the components, the
+  relationships will be somewhat different; the curves which are obtained
+  in such a case being represented in Fig. 97. From the figure, we see that
+  the two components B <!-- Page 256 --><span class="pagenum"><a
+  name="page256"></a>{256}</span>and C form a compound, and the freezing
+  point curve of the binary system has therefore the form shown in Fig. 64
+  (p.&nbsp;<a href="#page209">209</a>). Further, there are two <i>ternary</i>
+  eutectic points, K<sub>1</sub> and K<sub>2</sub>, the solid phases
+  present being A, B, and compound, and A, C, and compound
+  respectively.</p>
+
+  <div class="figcenter" style="width:17%;">
+      <a href="images/fig097.png"><img style="width:100%" src="images/fig097.png"
+      alt="Fig. 97" title="Fig. 97" /></a>
+    <span class="sc">Fig.</span> 97.
+  </div>
+
+  <p>The particular point, now, to which it is desired to draw attention is
+  this. Suppose the ternary eutectic curves projected on a plane parallel
+  to the face of the prism containing B and C, <i>i.e.</i> suppose the
+  concentrations of the two components B and C, between which interaction
+  can occur, expressed in terms of a constant amount of the third component
+  A,<a name="NtA332" href="#Nt332"><sup>[332]</sup></a> curves will then be
+  obtained which are in every respect analogous to the freezing point
+  curves of binary systems. Thus, suppose the eutectic curves
+  <i>k</i><sub>1</sub>K and <i>k</i><sub>2</sub>K in Fig. 95 projected on
+  the face BC of the prism, then evidently a curve will be obtained
+  consisting of two branches meeting in an eutectic point. On the other
+  hand, the projection of the ternary eutectic curves in Fig. 97 on the
+  face BC of the prism, will give a curve consisting of three portions, as
+  shown by the outline
+  <i>k</i><sub>1</sub>K<sub>1</sub>K<sub>2</sub><i>k</i><sub>2</sub> in
+  Fig. 97.</p>
+
+  <p>Various examples of this have been studied, and the following table
+  contains some of the data for the system ethylene bromide (A), picric
+  acid (B), and <span class="grk">&beta;</span>-naphthol (C), obtained by
+  Bruni.<a name="NtA333" href="#Nt333"><sup>[333]</sup></a></p>
+
+<p><!-- Page 257 --><span class="pagenum"><a name="page257"></a>{257}</span></p>
+
+<table class="nobctr" summary="Phases of ethylene bromide-picric acid-beta-naphthol system" title="Phases of ethylene bromide-picric acid-beta-naphthol system">
+<tr><td class="toribob" style="text-align:center">
+</td><td class="toboleb" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solid phases present.</td></tr>
+
+<tr><td class="rib" style="text-align:left"> Point <i>k</i><sub>1</sub> </td><td class="rileb" style="text-align:center"> 9.41° </td><td class="leb" style="text-align:left"> Ethylene bromide, picric acid.</td></tr>
+<tr><td class="rib" style="text-align:left"> Curve <i>k</i><sub>1</sub>K<sub>1</sub> </td><td class="rileb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:left"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left"> Point K<sub>1</sub> </td><td class="rileb" style="text-align:center"> 9.32° </td><td class="leb" style="text-align:left"> Ethylene bromide, picric acid, and <span class="grk">&beta;</span>-naphthol picrate.</td></tr>
+<tr><td class="rib" style="text-align:left"> Curve K<sub>1</sub>D&prime;K<sub>2</sub> </td><td class="rileb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:left"> Ethylene bromide, <span class="grk">&beta;</span>-<span class="correction" title="Original reads 'napthol'.">naphthol</span> picrate. </td></tr>
+<tr><td class="rib" style="text-align:left"> Point D&prime; </td><td class="rileb" style="text-align:center"> 9.75° </td><td class="leb" style="text-align:left"> &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,,</td></tr>
+<tr><td class="rib" style="text-align:left"> Point K<sub>2</sub> </td><td class="rileb" style="text-align:center"> 8.89° </td><td class="leb" style="text-align:left"> &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; <span class="grk">&beta;</span>-naphthol, and picrate.</td></tr>
+<tr><td class="rib" style="text-align:left"> Curve K<sub>2</sub><i>k</i><sub>2</sub> </td><td class="rileb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:left"> &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; <span class="grk">&beta;</span>-<span class="correction" title="Original reads 'napthol'.">naphthol</span>. </td></tr>
+<tr><td class="ribob" style="text-align:left"> Point <i>k</i><sub>2</sub> </td><td class="riboleb" style="text-align:center"> 9.04° </td><td class="boleb" style="text-align:left"> &nbsp; &nbsp; &nbsp;,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;,,</td></tr>
+
+</table>
+
+  <p>From what has been said, it will be apparent that if the ternary
+  eutectic curve of a three-component system (in which one of the
+  components is present in constant amount) is determined, it will be
+  possible to state, from the form of curve obtained, whether or not the
+  two components present in varying amount crystallize out pure or combine
+  with one another to form a compound. It may be left to the reader to work
+  out the curves for the other possible systems; but it will be apparent,
+  that the projections of the ternary eutectic curves in the manner given
+  will yield a series of curves alike in all points to the binary curves
+  given in Figs. 63-65, pp.&nbsp;<a href="#page208">208</a>-210.</p>
+
+  <p>Since, from the method of investigation, the temperatures of the
+  eutectic curves will depend on the melting point of the third component
+  (A), it is possible, by employing substances with widely differing
+  melting points, to investigate the interaction of the two components
+  (<i>e.g.</i> two optical antipodes) B and C over a range of temperature;
+  and thus determine the range of stability of the compound, if one is
+  formed. Since, in some cases, two substances which at one temperature
+  form mixed crystals combine at another temperature to form a definite
+  compound, the relationships which have just been described can be
+  employed, and indeed, have been employed, to determine the temperature at
+  which this change occurs.<a name="NtA334"
+  href="#Nt334"><sup>[334]</sup></a> By means of this method, Adriani found
+  that below 103° <i>i</i>-camphoroxime exists as a racemic compound, while
+  above <!-- Page 258 --><span class="pagenum"><a
+  name="page258"></a>{258}</span>that temperature it occurs as a racemic
+  mixed crystal<a name="NtA335" href="#Nt335"><sup>[335]</sup></a>
+  (<i>cf.</i> p.&nbsp;<a href="#page219">219</a>).</p>
+
+  <p><b>B. Equilibria at Higher Temperatures. Formation of Double
+  Salts.</b>&mdash;After having studied the relationships which are found
+  in the neighbourhood of the freezing points of the components, we now
+  pass to the discussion of the equilibria which are met with at higher
+  temperatures. In this connection we shall confine the discussion entirely
+  to the systems formed of two salts and water, dealing more particularly
+  with those cases in which the water is present in relatively large amount
+  and acts as solvent. Further, in studying these systems, one restriction
+  must be made, viz. that the single salts are salts either of the same
+  base or of the same acid; or are, in other words, capable of yielding a
+  common ion in solution. Such a restriction is necessary, because
+  otherwise the system would be one not of three but of four components.<a
+  name="NtA336" href="#Nt336"><sup>[336]</sup></a></p>
+
+  <p><b>Transition Point.</b>&mdash;As is very well known, there exist a
+  number of hydrated salts which, on being heated, undergo apparent partial
+  fusion; and in Chapter V. the behaviour of such hydrates was more fully
+  studied in the light of the Phase Rule. Glauber's salt, or sodium
+  sulphate decahydrate, for example, on being heated to a temperature of
+  about 32.5°, partially liquefies, owing to the fact that the water of
+  crystallization is split off and anhydrous sodium sulphate formed, as
+  shown by the equation&mdash;</p>
+
+<p class="cenhead">Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O = Na<sub>2</sub>SO<sub>4</sub> + 10H<sub>2</sub>O</p>
+
+  <p>The temperature of 32.5°, it was learned, constituted a <i>transition
+  point</i> for the decahydrate and anhydrous salt plus water;
+  decomposition of the hydrated salt occurring above this temperature,
+  combination of the anhydrous salt and water below it.</p>
+
+  <p>Analogous phenomena are met with in systems constituted of two salts
+  and water in which the formation of double salts can take place. Thus,
+  for example, if <i>d</i>-sodium potassium <!-- Page 259 --><span
+  class="pagenum"><a name="page259"></a>{259}</span>tartrate is heated to
+  above 55°, apparent partial fusion occurs, and the two single salts,
+  <i>d</i>-sodium tartrate and <i>d</i>-potassium tartrate, are deposited,
+  the change which occurs being represented by the equation&mdash;</p>
+
+<p class="cenhead">4NaKC<sub>4</sub>O<sub>6</sub>H<sub>4</sub>,4H<sub>2</sub>O = 2Na<sub>2</sub>C<sub>4</sub>O<sub>6</sub>H<sub>4</sub>,2H<sub>2</sub>O + 2K<sub>2</sub>C<sub>4</sub>O<sub>6</sub>H<sub>4</sub>,½H<sub>2</sub>O + 11H<sub>2</sub>O</p>
+
+  <p>On the other hand, if sodium and potassium tartrates are mixed with
+  water in the proportions shown on the right side of the equation, the
+  system will remain partially liquid so long as the temperature is
+  maintained above 55° (in a closed vessel to prevent loss of water), but
+  on allowing the temperature to fall below this point, complete
+  solidification will ensue, owing to the formation of the hydrated double
+  salt. Below 55°, therefore, the hydrated double salt is the stable
+  system, while above this temperature the two single salts plus saturated
+  solution are stable.<a name="NtA337"
+  href="#Nt337"><sup>[337]</sup></a></p>
+
+  <p>A similar behaviour is found in the case of the double salt copper
+  dipotassium chloride (CuCl<sub>2</sub>,2KCl,2H<sub>2</sub>O or
+  CuK<sub>2</sub>Cl<sub>4</sub>,2H<sub>2</sub>O).<a name="NtA338"
+  href="#Nt338"><sup>[338]</sup></a> When this salt is heated to 92°,
+  partial liquefaction occurs, and the original blue plate-shaped crystals
+  give place to brown crystalline needles and white cubes; while on
+  allowing the temperature to fall, re-formation of the blue double salt
+  ensues. The temperature 92° is, therefore, a transition point at which
+  the reversible reaction&mdash;</p>
+
+<p class="cenhead">CuK<sub>2</sub>Cl<sub>4</sub>,2H<sub>2</sub>O <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CuKCl<sub>3</sub> + KCl + 2H<sub>2</sub>O</p>
+
+  <p>takes place.</p>
+
+  <p>The decomposition of sodium potassium tartrate, or of copper
+  dipotassium chloride, differs in so far from that of Glauber's salt that
+  <i>two</i> new solid phases are formed; and in the case of copper
+  dipotassium chloride, one of the decomposition products is itself a
+  double salt.</p>
+
+  <p>In the two examples of double salt decomposition which have just been
+  mentioned, sufficient water was yielded to cause a partial liquefaction;
+  but other cases are known where this is not so. Thus, when copper calcium
+  acetate is heated to a <!-- Page 260 --><span class="pagenum"><a
+  name="page260"></a>{260}</span>temperature of 75°, although decomposition
+  of the double salt into the two single salts occurs as represented by the
+  equation<a name="NtA339" href="#Nt339"><sup>[339]</sup></a>&mdash;</p>
+
+<p class="cenhead">CuCa(C<sub>2</sub>H<sub>3</sub>O<sub>2</sub>)<sub>4</sub>,8H<sub>2</sub>O = Cu(C<sub>2</sub>H<sub>3</sub>O<sub>2</sub>)<sub>2</sub>,H<sub>2</sub>O + Ca(C<sub>2</sub>H<sub>3</sub>O<sub>2</sub>)<sub>2</sub>,H<sub>2</sub>O
+   + 6H<sub>2</sub>O</p>
+
+  <p>the amount of water split off is insufficient to give the appearance
+  of partial fusion, and, therefore, only a change in the crystals is
+  observed.</p>
+
+  <p>The preceding examples, in which decomposition of the double salt was
+  effected by a rise of temperature, were chosen for first consideration as
+  being more analogous to the case of Glauber's salt; but not a few
+  examples are known where the reverse change takes place, formation of the
+  double salt occurring <i>above</i> the transition point, and
+  decomposition into the constituent salts below it. Instances of this
+  behaviour are found in the case of the formation of astracanite from
+  sodium and magnesium sulphates, and of sodium ammonium racemate from the
+  two sodium ammonium tartrates, to which reference will be made later.
+  Between these various systems, however, there is no essential difference;
+  and whether decomposition or formation of the double salt occurs at
+  temperatures above the transition point, will of course depend on the
+  heat of change at that point. For, in accordance with van't Hoff's law of
+  movable equilibrium (p.&nbsp;<a href="#page58">58</a>), that change will take
+  place at the higher temperature which is accompanied by an absorption of
+  heat. If, therefore, the formation of the double salt from the single
+  salts is accompanied by an absorption of heat, the double salt will be
+  formed from the single salts on raising the temperature; but if the
+  reverse is the case, then the double salt on being heated will decompose
+  into the constituent salts.<a name="NtA340"
+  href="#Nt340"><sup>[340]</sup></a></p>
+
+  <p>In those cases, now, which have so far been studied, the change at the
+  transition point is accompanied by a taking up or a splitting off of
+  water; and <i>in such cases the general rule can be given, that if the
+  water of crystallization of the two constituent <!-- Page 261 --><span
+  class="pagenum"><a name="page261"></a>{261}</span>salts together is
+  greater than that of the double salt, the latter will be produced from
+  the former on raising the temperature</i> (<i>e.g.</i> astracanite from
+  sodium and magnesium sulphates); <i>but if the double salt contains more
+  water of crystallization than the two single salts, increase of
+  temperature will effect the decomposition of the double salt</i>. When we
+  seek for the connection between this rule and the law of van't Hoff, it
+  is found in the fact that the heat effect involved in the hydration or
+  dehydration of the salts is much greater than that of the other changes
+  which occur, and determines, therefore, the sign of the total heat
+  effect.<a name="NtA341" href="#Nt341"><sup>[341]</sup></a></p>
+
+  <p><b>Vapour Pressure. Quintuple Point.</b>&mdash;In the case of
+  Glauber's salt, we saw that at a certain temperature the vapour pressure
+  curve of the hydrated salt cut that of the saturated solution of
+  anhydrous sodium sulphate. That point, it will be remembered, was a
+  quadruple point at which the four phases sodium sulphate decahydrate,
+  anhydrous sodium sulphate, solution, and vapour, could co-exist; and was
+  also the point of intersection of the curves for four univariant systems.
+  In the case of the formation of double salts, similar relationships are
+  met with; and also certain differences, due to the fact that we are now
+  dealing with systems of three components. Two cases will be chosen here
+  for brief description, one in which formation, the other in which
+  decomposition of the double salt occurs with rise of temperature.</p>
+
+  <p>On heating a mixture of sodium sulphate decahydrate and magnesium
+  sulphate heptahydrate, it is found that at 22° partial liquefaction
+  occurs with formation of astracanite. At this temperature, therefore,
+  there can coexist the five phases&mdash;</p>
+
+<p class="cenhead">Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O; MgSO<sub>4</sub>,7H<sub>2</sub>O; Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O;
+solution; vapour.</p>
+
+  <p>This constitutes, therefore, a <i>quintuple point</i>; and since there
+  are three components present in five phases, the system is invariant.
+  This point, also, will be the point of intersection of curves for five
+  univariant systems, which, in this case, must each be composed of four
+  phases. These systems are&mdash;</p>
+
+<p><!-- Page 262 --><span class="pagenum"><a name="page262"></a>{262}</span></p>
+
+<blockquote class="b1n">
+
+  <p>I. Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O;
+  MgSO<sub>4</sub>,7H<sub>2</sub>O;
+  Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O; vapour.</p>
+
+  <p>II. Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O;
+  MgSO<sub>4</sub>,7H<sub>2</sub>O; solution; vapour.</p>
+
+  <p>III. MgSO<sub>4</sub>,7H<sub>2</sub>O;
+  Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O; solution;
+  vapour.</p>
+
+  <p>IV. Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O;
+  Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O; solution;
+  vapour.</p>
+
+  <p>V. Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O;
+  MgSO<sub>4</sub>,7H<sub>2</sub>O;
+  Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O;
+  solution.</p>
+
+</blockquote>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/fig098.png"><img style="width:100%" src="images/fig098.png"
+      alt="Fig. 98" title="Fig. 98" /></a>
+    <span class="sc">Fig.</span> 98.
+  </div>
+
+  <p>On representing the vapour pressures of these different systems
+  graphically, a diagram is obtained such as is shown in Fig. 98,<a
+  name="NtA342" href="#Nt342"><sup>[342]</sup></a> the curves being
+  numbered in accordance with the above list. When the system I. is heated,
+  the vapour pressure increases until at the quintuple point the liquid
+  phase (solution) is formed, and it will then depend on the relative
+  amounts of the different phases whether on further heating there is
+  formed system III., IV., or V. If either of the first two is produced, we
+  shall obtain the vapour pressure of the solutions saturated with respect
+  to both double salt and one of the single salts; while if the vapour
+  phase disappears, there will be obtained the pressure of the condensed
+  systems formed of double salt, two single salts and solution. This curve,
+  therefore, indicates the <i>change of the transition point with
+  pressure</i>; and since in the ordinary determinations of the transition
+  point in open vessels, we are in reality dealing with condensed systems
+  under the pressure of 1 atm., it will be evident that the transition
+  point does not accurately coincide with the quintuple point (at which the
+  system is under the pressure of its own vapour). As in the case of other
+  condensed systems, however, pressure has only a slight influence on the
+  temperature of the transition point. Whether or not pressure raises or
+  lowers the transition point will depend on whether transformation is
+  accompanied by an increase or <!-- Page 263 --><span class="pagenum"><a
+  name="page263"></a>{263}</span>diminution of volume (theorem of Le
+  Chatelier, p.&nbsp;<a href="#page58">58</a>). In the case of the formation of
+  astracanite, expansion occurs, and the transition point will therefore be
+  raised by increase of pressure. Although measurements have not been made
+  in the case of this system, the existence of such a curve has been
+  experimentally verified in the case of copper and calcium acetates and
+  water (v. <i>infra</i>).<a name="NtA343"
+  href="#Nt343"><sup>[343]</sup></a></p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig099.png"><img style="width:100%" src="images/fig099.png"
+      alt="Fig. 99" title="Fig. 99" /></a>
+    <span class="sc">Fig.</span> 99.
+  </div>
+
+  <p>The vapour pressure diagram in the case of copper calcium acetate and
+  water (Fig. 99), is almost the reverse of that already discussed. In this
+  case, the double salt decomposes on heating, and the decomposition is
+  accompanied by a contraction. Curve I. is the vapour pressure curve for
+  double salt, two single salts (p.&nbsp;<a href="#page260">260</a>), and
+  vapour; curves II. and III. give the vapour pressures of solutions
+  saturated with respect to double salt and one of the single salts; curve
+  IV. is the curve of pressures for the solutions saturated with respect to
+  the two single salts; while curve V. again represents the change of the
+  transition point with pressure. On examining this diagram, it is seen
+  that whereas <!-- Page 264 --><span class="pagenum"><a
+  name="page264"></a>{264}</span>astracanite could exist both above and
+  below the quintuple point, copper calcium acetate can exist only
+  <i>below</i> the quintuple point. This behaviour is found only in those
+  cases in which the double salt is decomposed by rise of temperature, and
+  where the decomposition is accompanied by a diminution of volume.<a
+  name="NtA344" href="#Nt344"><sup>[344]</sup></a></p>
+
+  <p>As already mentioned, the decomposition of copper calcium acetate into
+  the single salts and saturated solution is accompanied by a contraction,
+  and it was therefore to be expected that increase of pressure would
+  <i>lower</i> the transition point. This expectation of theory was
+  confirmed by experiment, for van't Hoff and Spring found that although
+  the transition point under atmospheric pressure is about 75°,
+  decomposition of the double salt took place even at the ordinary
+  temperature when the pressure was increased to 6000 atm.<a name="NtA345"
+  href="#Nt345"><sup>[345]</sup></a></p>
+
+  <p><b>Solubility Curves at the Transition Point.</b>&mdash;At the
+  transition point, as has already been shown, the double salt and the two
+  constituent salts can exist in equilibrium with the same solution. The
+  transition point, therefore, must be the point of intersection of two
+  solubility curves; the solubility curve of the double salt and the
+  solubility curve of the mixtures of the two constituent salts. It should
+  be noted here that we are not dealing with the solubility curves of the
+  single salts separately, for since the systems are composed of three
+  components, a single solid phase can, at a given temperature, be in
+  equilibrium with solutions of different composition, and two solid phases
+  in contact with solution (and vapour) are therefore necessary to give an
+  univariant system. The same applies, of course, to the solubility of the
+  double salt; for a double salt also constitutes a single phase, and can
+  therefore exist in equilibrium with solutions of varying composition. If,
+  however, we make the restriction (which we do for the present) that the
+  double salt is not decomposed by water, then the solution will contain
+  the constituent salts in the same relative proportions as they are
+  contained in the double salt, and the system may therefore be regarded as
+  one of <i>two</i> components, viz. double salt and water. In this case
+  one solid phase is sufficient, with solution and <!-- Page 265 --><span
+  class="pagenum"><a name="page265"></a>{265}</span>vapour, to give an
+  univariant system; and at a given temperature, therefore, the solubility
+  will have a perfectly definite value.</p>
+
+  <p>Since in almost all cases the solubility is determined in open
+  vessels, we shall in the following discussion consider that the vapour
+  phase is absent, and that the system is under a constant pressure, that
+  of the atmosphere. With this restriction, therefore, four phases will
+  constitute an invariant system, three phases an univariant, and two
+  phases a bivariant system.</p>
+
+  <p>It has already been learned that in the case of sodium sulphate and
+  water, the solubility curve of the salt undergoes a sudden change in
+  direction at the transition point, and that this is accompanied by a
+  change in the solid phase in equilibrium with the solution. The same
+  behaviour is also found in the case of double salts. To illustrate this,
+  we shall briefly discuss the solubility relations of a few double salts,
+  beginning with one of the simplest cases, that of the formation of
+  rubidium racemate from rubidium <i>d</i>- and <i>l</i>-tartrates. The
+  solubilities are represented diagrammatically in Fig. 100, the numerical
+  data being contained in the following table, in which the solubility is
+  expressed as the number of gram-molecules
+  Rb<sub>2</sub>C<sub>4</sub>H<sub>4</sub>O<sub>6</sub> in 100
+  gm.-molecules of water.<a name="NtA346"
+  href="#Nt346"><sup>[346]</sup></a></p>
+
+<table class="nobctr" summary="Solubility of rubidium tartrates" title="Solubility of rubidium tartrates">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Solubility of tartrate<br />mixture.
+</td><td class="toboleb" style="text-align:center"> Solubility of racemate.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 25° </td><td class="leb" style="text-align:center"> 13.03 </td><td class="leb" style="text-align:center"> 10.91</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 35° </td><td class="leb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:center"> 12.63</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 40.4° </td><td class="leb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:center"> 13.48</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 40.7° </td><td class="leb" style="text-align:center"> 13.46 </td><td class="leb" style="text-align:center"> &mdash;</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 54° </td><td class="boleb" style="text-align:center"> 13.83 </td><td class="boleb" style="text-align:center"> &mdash;</td></tr>
+
+</table>
+
+  <p>In Fig. 100 the curve AB represents the solubility of the racemate,
+  while A&prime;BC represents the solubility of the mixed tartrates. Below
+  the transition point, therefore, the solubility of the racemate is less
+  than that of the mixed tartrates. The solution, saturated with respect to
+  the latter, will be supersaturated with respect to the racemate; and if a
+  nucleus of this is present, racemate will be deposited, and the mixed
+  tartrates, if present in equimolecular amounts, will ultimately <!-- Page
+  266 --><span class="pagenum"><a name="page266"></a>{266}</span>entirely
+  disappear, and only racemate will be left as solid phase. The solution
+  will then have the composition represented by a point on the curve AB.
+  Conversely, above the transition point, the saturated solution of the
+  racemate would be supersaturated with respect to the two tartrates, and
+  transformation into the latter would ensue. If, therefore, a solution of
+  equimolecular proportions of rubidium <i>d</i>- and <i>l</i>-tartrates is
+  allowed to evaporate at a temperature above 40°, a mixture of the two
+  tartrates will be deposited; while at temperatures below 40° the racemate
+  will separate out.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig100.png"><img style="width:100%" src="images/fig100.png"
+      alt="Fig. 100" title="Fig. 100" /></a>
+    <span class="sc">Fig.</span> 100.
+  </div>
+
+  <p>Similar relationships are met with in the case of sodium ammonium
+  <i>d</i>- and <i>l</i>-tartrate and sodium ammonium racemate; but in this
+  case the racemate is the stable form in contact with solution above the
+  transition point (27°).<a name="NtA347"
+  href="#Nt347"><sup>[347]</sup></a> Below the transition point, therefore,
+  the solubility curve of the mixed tartrates will lie below the solubility
+  curve of the racemate. Below the transition point, therefore, sodium
+  ammonium racemate will break up in contact with solution into a mixture
+  of sodium ammonium <i>d</i>- and <i>l</i>-tartrates. At a higher
+  temperature, 35°, sodium ammonium racemate undergoes decomposition into
+  sodium racemate and ammonium racemate.<a name="NtA348"
+  href="#Nt348"><sup>[348]</sup></a></p>
+
+  <p>The behaviour of sodium ammonium racemate is of interest from the fact
+  that it was the first racemic substance to be resolved into its optically
+  active forms by a process of crystallization. On neutralizing a solution
+  of racemic tartaric acid, half with soda and half with ammonia, and
+  allowing the solution to evaporate, Pasteur<a name="NtA349"
+  href="#Nt349"><sup>[349]</sup></a> obtained a mixture of sodium ammonium
+  <!-- Page 267 --><span class="pagenum"><a
+  name="page267"></a>{267}</span><i>d</i>- and <i>l</i>-tartrates. Since
+  Pasteur was unaware of the existence of a transition point, the success
+  of his experiment was due to the happy chance that he allowed the
+  solution to evaporate at a temperature below 27°; for had he employed a
+  temperature above this, separation of the racemate into the two
+  enantiomorphous forms would not have occurred. For this reason the
+  attempt of Staedel to perform the same resolution met only with
+  failure.<a name="NtA350" href="#Nt350"><sup>[350]</sup></a></p>
+
+  <p><b>Decomposition of the Double Salt by Water.</b>&mdash;In the two
+  cases just described, the solubility relationships at the transition
+  point are of a simpler character than in the case of most double salts.
+  If, at a temperature above the transition point, a mixture of rubidium
+  <i>d</i>- and <i>l</i>-tartrates in equimolecular proportions is brought
+  in contact with water a solution will be obtained, which is saturated
+  with respect to both enantiomorphous forms; and since the solubility of
+  the two optical antipodes is identical, and the effect of one on the
+  solubility of the other also the same, the solution will contain
+  equimolecular amounts of the <i>d</i>- and <i>l</i>-salt. If, now, the
+  solution is cooled down in contact with the solid salts to just below the
+  transition point, it becomes supersaturated with respect to the racemate,
+  and this will be deposited. The solution thereby becomes unsaturated with
+  respect to the mixture of the active salts, and these must therefore pass
+  into solution. As the latter are equally soluble, equal amounts of each
+  will dissolve, and a further quantity of the racemate will be deposited.
+  These processes of solution and deposition will continue until the single
+  tartrates have completely disappeared, and only racemate is left as solid
+  phase. As a consequence of the identical solubility of the two tartrates,
+  therefore, no excess of either form will be left on passing through the
+  transition point. From this it will be evident that the racemate can
+  exist as single solid phase in contact with its saturated solution at the
+  transition point; or, in other words, the racemate is not decomposed by
+  water at the transition point. The same behaviour will evidently be
+  exhibited by sodium ammonium racemate at 27°, for the two enantiomorphous
+  sodium ammonium tartrates have also identical solubility.</p>
+
+<p><!-- Page 268 --><span class="pagenum"><a name="page268"></a>{268}</span></p>
+
+  <p>Very different, however, is the behaviour of, say, astracanite, or of
+  the majority of double salts; for the solubility of the constituent salts
+  is now no longer the same. If, for example, excess of a mixture of sodium
+  sulphate and magnesium sulphate, in equimolecular proportions, is brought
+  in contact with water below the transition point (22°), more magnesium
+  sulphate than sodium sulphate will dissolve, the solubility of these two
+  salts in a common solution being given by the following figures, which
+  express number of molecules of the salt in 100 molecules of water.<a
+  name="NtA351" href="#Nt351"><sup>[351]</sup></a></p>
+
+<p class="cenhead"><span class="sc">Composition of Solutions saturated with respect to</span>
+Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O <span class="sc">and</span> MgSO<sub>4</sub>,7H<sub>2</sub>O.</p>
+
+<table class="nobctr" summary="Solutions saturated with respect to sodium and magnesium sulphates" title="Solutions saturated with respect to sodium and magnesium sulphates">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Na<sub>2</sub>SO<sub>4</sub>.
+</td><td class="toboleb" style="text-align:center"> MgSO<sub>4</sub>.</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> 18.5° </td><td class="leb" style="text-align:center"> 2.16 </td><td class="leb" style="text-align:center"> 4.57</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> 24.5° </td><td class="boleb" style="text-align:center"> 3.43 </td><td class="boleb" style="text-align:center"> 4.68</td></tr>
+
+</table>
+
+  <p>At the transition point, then, it is evident that the solution
+  contains more magnesium sulphate than sodium sulphate: and this must
+  still be the case when astracanite, which contains sodium sulphate and
+  magnesium sulphate in equimolecular proportions, separates out. If,
+  therefore, the temperature is raised slightly above the transition point,
+  magnesium sulphate and sodium sulphate will pass into solution, the
+  former, however, in larger quantities than the latter, and astracanite
+  will be deposited; and this will go on until all the magnesium sulphate
+  has disappeared, and a mixture of astracanite and sodium sulphate
+  decahydrate is left as solid phases. Since there are now three phases
+  present, the system is univariant (by reason of the restriction
+  previously made that the vapour phase is absent), and at a given
+  temperature the solution will have a definite composition; as given in
+  the following table:&mdash;</p>
+
+<p class="cenhead"><span class="sc">Composition of Solutions saturated with respect to</span>
+Na<sub>2</sub>Mg(SO<sub>4</sub>)<sub>2</sub>,4H<sub>2</sub>O <span class="sc">and</span> Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O.</p>
+
+<table class="nobctr" summary="Solutions saturated with respect to sodium sulphate and astracanite" title="Solutions saturated with respect to sodium sulphate and astracanite">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Na<sub>2</sub>SO<sub>4</sub>.
+</td><td class="toboleb" style="text-align:center"> MgSO<sub>4</sub>.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 22° </td><td class="leb" style="text-align:center"> 2.95 </td><td class="leb" style="text-align:center"> 4.70</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 24.5° </td><td class="boleb" style="text-align:center"> 3.45 </td><td class="boleb" style="text-align:center"> 3.62</td></tr>
+</table>
+
+<p><!-- Page 269 --><span class="pagenum"><a name="page269"></a>{269}</span></p>
+
+  <p>From the above figures, therefore, it will be seen that at a
+  temperature just above the transition point a solution in contact with
+  the two solid phases, astracanite and Glauber's salt, contains a
+  relatively smaller amount of sodium sulphate than a pure solution of
+  astracanite would; for in this case there would be equal molecular
+  amounts of Na<sub>2</sub>SO<sub>4</sub> and MgSO<sub>4</sub>. A solution
+  which is saturated with respect to astracanite alone, will contain more
+  sodium sulphate than the solution saturated with respect to astracanite
+  plus Glauber's salt, and the latter will therefore be deposited. From
+  this, therefore, it is clear that if astracanite is brought in contact
+  with water at about the transition point, it will undergo decomposition
+  with separation of Glauber's salt (supersaturation being excluded).</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/fig101.png"><img style="width:100%" src="images/fig101.png"
+      alt="Fig. 101" title="Fig. 101" /></a>
+    <span class="sc">Fig.</span> 101.
+  </div>
+
+  <p>This will perhaps be made clearer by considering Fig. 101. In this
+  diagram the ordinates represent the ratio of sodium sulphate to magnesium
+  sulphate in the solutions, and the abscissæ represent the temperatures.
+  The line AB represents solutions saturated with respect to a mixture of
+  the single salts (p.&nbsp;<a href="#page268">268</a>); BC refers to solutions
+  in equilibrium with astracanite and magnesium sulphate; while BX
+  represents the composition of solutions in contact with the solid phases
+  astracanite and Glauber's salt. The values of the solubility are
+  contained in the following table, and in that on p.&nbsp;<a
+  href="#page268">268</a>, and are, as before, expressed in gm.-molecules
+  of salt in 100 gm.-molecules of water.<a name="NtA352"
+  href="#Nt352"><sup>[352]</sup></a></p>
+
+<p><!-- Page 270 --><span class="pagenum"><a name="page270"></a>{270}</span></p>
+
+<table class="nobctr" summary="Solutions of sodium and magnesium sulphate" title="Solutions of sodium and magnesium sulphate">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Astracanite + sodium sulphate.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Astracanite + magnesium sulphate.</td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Na<sub>2</sub>SO<sub>4</sub>.
+</td><td class="toboleb" style="text-align:center"> MgSO<sub>4</sub>.
+</td><td class="toboleb" style="text-align:center"> Na<sub>2</sub>SO<sub>4</sub>.
+</td><td class="toboleb" style="text-align:center"> MgSO<sub>4</sub>.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 18.5° </td><td class="leb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:center"> &mdash; </td><td class="leb" style="text-align:center"> 3.41 </td><td class="leb" style="text-align:center"> 4.27</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 22° </td><td class="leb" style="text-align:center"> 2.95 </td><td class="leb" style="text-align:center"> 4.70 </td><td class="leb" style="text-align:center"> 2.85 </td><td class="leb" style="text-align:center"> 4.63</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 24.5° </td><td class="leb" style="text-align:center"> 3.45 </td><td class="leb" style="text-align:center"> 3.62 </td><td class="leb" style="text-align:center"> 2.68 </td><td class="leb" style="text-align:center"> 4.76</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:2em;"> 30° </td><td class="leb" style="text-align:center"> 4.58 </td><td class="leb" style="text-align:center"> 2.91 </td><td class="leb" style="text-align:center"> 2.30 </td><td class="leb" style="text-align:center"> 5.31</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:2em;"> 35° </td><td class="boleb" style="text-align:center"> 4.30 </td><td class="boleb" style="text-align:center"> 2.76 </td><td class="boleb" style="text-align:center"> 1.73 </td><td class="boleb" style="text-align:center"> 5.88</td></tr>
+</table>
+
+  <p>At the transition point the ratio of sodium sulphate to magnesium
+  sulphate is approximately 1 : 1.6. In the case of solutions saturated
+  with respect to both astracanite and Glauber's salt, the relative amount
+  of sodium sulphate increases as the temperature rises, while in the
+  solutions saturated for astracanite and magnesium sulphate, the ratio of
+  sodium sulphate to magnesium sulphate decreases.</p>
+
+  <p>If, now, we consider only the temperatures above the transition point,
+  we see from the figure that solutions represented by points above the
+  line BX contain relatively more sodium sulphate than solutions in contact
+  with astracanite and Glauber's salt; and solutions lying below the line
+  BC contain relatively more magnesium sulphate than solutions saturated
+  with this salt and astracanite. These solutions will therefore not be
+  stable, but will deposit in the one case, astracanite and Glauber's salt,
+  and in the other case, astracanite and magnesium sulphate, until a point
+  on BX or BC is reached. All solutions, however, lying to the right of
+  CBX, will be <i>unsaturated</i> with respect to these two pairs of salts,
+  and only the solutions represented by the line XY (and which contain
+  equimolecular amounts of sodium and magnesium sulphates) will be
+  saturated with respect to the pure double salt.</p>
+
+  <p><b>Transition Interval.</b>&mdash;Fig. 101 will also render
+  intelligible a point of great importance in connection with astracanite,
+  and of double salts generally. At temperatures between those represented
+  by the points B and X, the double salt when brought in contact with water
+  will be decomposed with separation of sodium sulphate. Above the
+  temperature of the point <!-- Page 271 --><span class="pagenum"><a
+  name="page271"></a>{271}</span>X, however, the solution of the pure
+  double salt is stable, because it can still take up a little of either of
+  the components. At temperatures, then, above that at which the solution
+  in contact with the double salt and the less soluble single salt,
+  contains the single salts in the ratio in which they are present in the
+  double salt, solution of the latter will take place without
+  decomposition. <i>The range of temperature between that at which double
+  salt can begin to be formed (the transition point) and that at which it
+  ceases to be decomposed by water is called the transition interval.</i><a
+  name="NtA353" href="#Nt353"><sup>[353]</sup></a> If the two single salts
+  have identical solubility at the transition point, the transition
+  interval diminishes to nought.</p>
+
+  <p>In those cases where the double salt is the stable form below the
+  transition point, the transition interval will extend downwards to a
+  lower temperature. Fig. 101 will then have the reverse form.</p>
+
+  <p><b>Summary.</b>&mdash;With regard to double salts we have learned that
+  their formation from and their decomposition into the single salts, is
+  connected with a definite temperature, the <i>transition temperature</i>.
+  At this transition temperature two vapour pressure curves cut, viz. a
+  curve of dehydration of a mixture of the single salts and the solubility
+  curve of the double salt; or the dehydration curve of the double salt and
+  the solubility curve of the mixed single salts. The solubility curves,
+  also, of these two systems intersect at the transition point, but
+  although the formation of the double salt commences at the transition
+  point, complete stability in contact with water may not be attained till
+  some temperature above (or below) that point. <i>Only when the
+  temperature is beyond the transition interval, will a double salt
+  dissolve in water without decomposition (</i>e.g.<i> the alums).</i></p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 272 --><span class="pagenum"><a name="page272"></a>{272}</span></p>
+
+<h3>CHAPTER XVI</h3>
+
+<p class="cenhead">ISOTHERMAL CURVES AND THE SPACE MODEL</p>
+
+  <p>In the preceding chapter we considered the changes in the solubility
+  of double salts and of mixtures of their constituent salts with the
+  temperature; noting, more especially, the relationships between the two
+  systems at the transition point. It is now proposed to conclude the study
+  of the three-component systems by discussing very briefly the solubility
+  relations at constant temperature, or the isothermal solubility curves.
+  In this way fresh light will be thrown on the change in the solubility of
+  one component by the addition of another component, and also on the
+  conditions of formation and stable existence of double salts in solution.
+  With the help of these isothermal curves, also, the phenomena of
+  crystallization at constant temperature&mdash;phenomena which have not
+  only a scientific interest but also an important bearing on the
+  industrial preparation of double salts&mdash;will be more clearly
+  understood.<a name="NtA354" href="#Nt354"><sup>[354]</sup></a></p>
+
+  <p>A brief description will also be given of the method of representing
+  the variation of the concentration of the two salts in the solution with
+  the temperature.</p>
+
+  <p><b>Non-formation of Double Salts.</b>&mdash;In Fig. 102 are shown the
+  solubility curves of two salts, A and B, which at the given temperature
+  do not form a double salt.<a name="NtA355"
+  href="#Nt355"><sup>[355]</sup></a> The ordinates represent the amount of
+  A, the abscissæ the amount of B in a <i>constant amount</i> of the third
+  component, the solvent. The <!-- Page 273 --><span class="pagenum"><a
+  name="page273"></a>{273}</span>point A, therefore, represents the
+  solubility of the salt A at the given temperature; and similarly, point B
+  represents the solubility of B. Since we are dealing with a
+  three-component system, one solid phase in contact with solution will
+  constitute a bivariant system (in the absence of the vapour phase and
+  under a constant pressure). At any given temperature, therefore, the
+  concentration of the solution in equilibrium with the solid can undergo
+  change. If, now, to a pure solution of A a small quantity of B is added,
+  the solubility of A will in general be altered; as a rule it is
+  diminished, but sometimes it is increased.<a name="NtA356"
+  href="#Nt356"><sup>[356]</sup></a> The curve AC represents the varying
+  composition of the solution in equilibrium with the solid component A.
+  Similarly, the curve BC represents the composition of the solutions in
+  contact with pure B as solid phase. At the point, C, where these two
+  curves intersect, there are two solid phases, viz. pure A and pure B, in
+  equilibrium with solution, and the system becomes invariant. At this
+  point the solution is saturated with respect to both A and B, and at a
+  given temperature must have a perfectly definite composition. To take an
+  example, if we suppose A to represent sodium sulphate decahydrate, and B,
+  magnesium sulphate heptahydrate, and the temperature to be 18.5°
+  (<i>i.e.</i> below the transition point), the point C would represent a
+  solution containing 2.16 gm.-molecules Na<sub>2</sub>SO<sub>4</sub> and
+  4.57 gm.-molecules MgSO<sub>4</sub> per 100 gm.-molecules of water (p.&nbsp;<a
+  href="#page268">268</a>). The curve ACB is the boundary curve for
+  saturated solutions; solutions lying outside this curve are
+  supersaturated, those lying within the area ACBO, are unsaturated.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/fig102.png"><img style="width:100%" src="images/fig102.png"
+      alt="Fig. 102" title="Fig. 102" /></a>
+    <span class="sc">Fig.</span> 102.
+  </div>
+
+  <div class="figright" style="width:22%;">
+      <a href="images/fig104.png"><img style="width:100%" src="images/fig104.png"
+      alt="Fig. 104" title="Fig. 104" /></a>
+    <span class="sc">Fig.</span> 104.
+  </div>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig103.png"><img style="width:100%" src="images/fig103.png"
+      alt="Fig. 103" title="Fig. 103" /></a>
+    <span class="sc">Fig.</span> 103.
+  </div>
+
+  <p><b>Formation of Double Salt.</b>&mdash;We have already learned in the
+  preceding chapter that if the temperature is outside<a name="NtA357"
+  href="#Nt357"><sup>[357]</sup></a> the <!-- Page 274 --><span
+  class="pagenum"><a name="page274"></a>{274}</span>transition interval, it
+  is possible to prepare a pure saturated solution of the double salt. If,
+  now, we suppose the double salt to contain the two constituent salts in
+  equimolecular proportions, its saturated solution must be represented by
+  a point lying on the line which bisects the angle AOB; <i>e.g.</i> point
+  D, Fig. 103. But a double salt constitutes only a single phase, and can
+  exist, therefore, in contact with solutions of varying concentration, as
+  represented by EDF.</p>
+
+  <p>Let us compare, now, the relations between the solubility curve for
+  the double salt, and those for the two constituent salts. We shall
+  suppose that the double salt is formed from the single salts when the
+  temperature is raised above a certain point (as in the formation of
+  astracanite). At a temperature below the transition point, as we have
+  already seen, the solubility of the double salt is greater than that of a
+  mixture of the single salts. The curve EDF, therefore, must lie above the
+  point C, in the region representing solutions supersaturated with respect
+  to the single salts (Fig. 104). Such a solution, however, would be
+  metastable, and on being brought in contact with the single salts would
+  deposit these and yield a solution represented by the point C. At this
+  particular temperature, therefore, the isothermal solubility curve will
+  consist of only two branches.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/fig105.png"><img style="width:100%" src="images/fig105.png"
+      alt="Fig. 105" title="Fig. 105" /></a>
+    <span class="sc">Fig. 105.</span>
+  </div>
+
+  <p>Suppose, now, that the temperature is that of the transition point. At
+  this point, the double salt can exist together with the single salts in
+  contact with solution. The solubility curve <!-- Page 275 --><span
+  class="pagenum"><a name="page275"></a>{275}</span>of the double salt
+  must, therefore, pass through the point C, as shown in Fig. 105.</p>
+
+  <p>From this figure, now, it is seen that a solution saturated with
+  respect to double salt alone (point D), is supersaturated with respect to
+  the component A. If, then, at the temperature of the transition point,
+  excess of the double salt is brought in contact with water,<a
+  name="NtA358" href="#Nt358"><sup>[358]</sup></a> and if supersaturation
+  is excluded, <i>the double salt will undergo decomposition and the
+  component A will be deposited</i>. The relative concentration of the
+  component B in the solution will, therefore, increase, and the
+  composition of the solution will be thereby altered in the direction DC.
+  When the solution has the composition of C, the single salt ceases to be
+  deposited, for at this point the solution is saturated for both double
+  and single salt; and the system becomes invariant.</p>
+
+  <p>This diagram explains very clearly the phenomenon of the decomposition
+  of a double salt at the transition point. As is evident, this
+  decomposition will occur when the solution which is saturated at the
+  temperature of the transition point, with respect to the two single salts
+  (point C), does not contain these salts in the same ratio in which they
+  are present in the double salt. If point C lay on the dotted line
+  bisecting the right angle, then the pure saturated solution of the double
+  salt would not be supersaturated with respect to either of the single
+  salts, and the double salt would, therefore, not be decomposed by water.
+  As has already been mentioned, this behaviour is found in the case of
+  optically active isomerides, the solubilities of which are identical.</p>
+
+  <p>At the transition point, therefore, the isothermal curve also consists
+  of two branches; but the point of intersection of the two branches now
+  represents a solution which is saturated not <!-- Page 276 --><span
+  class="pagenum"><a name="page276"></a>{276}</span>only with respect to
+  the single salts, but also for the double salt in presence of the single
+  salts.</p>
+
+  <p>We have just seen that by a change of temperature the two solubility
+  curves, that for the two single salts and that for the double salt, were
+  made to approach one another (<i>cf.</i> Figs. 104 and 105). In the
+  previous chapter, however, we found that on passing the transition point
+  to the region of stability for the double salt, the solution which is
+  saturated for a mixture of the two constituent salts, is supersaturated
+  for the double salt. In this case, therefore, point C must lie above the
+  solubility curve of the pure double salt (Fig. 106), and a solution of
+  the composition C, if brought in contact with double salt, will deposit
+  the latter. If the single salts were also present, then as the double
+  salt separated out, the single salts would pass into solution, because so
+  long as the two single salts are present, the composition of the solution
+  must remain unaltered. If one of the single salts disappear before the
+  other, there will be left double salt plus A or double salt plus B,
+  according to which was in excess; and the composition of the solution
+  will be either that represented by D (saturated for double salt plus A),
+  or that of the point F (saturated for double salt plus B).</p>
+
+  <div class="figcenter" style="width:18%;">
+      <a href="images/fig106.png"><img style="width:100%" src="images/fig106.png"
+      alt="Fig. 106" title="Fig. 106" /></a>
+    <span class="sc">Fig. 106.</span>
+  </div>
+
+  <p>In connection with the isothermal represented in Fig. 106, it should
+  be noted that at this particular temperature a solution saturated with
+  respect to the pure double salt is no longer supersaturated for one of
+  the single salts (point D); so that at the temperature of this isothermal
+  the double salt is not decomposed by water. At this temperature, further,
+  the boundary curve consists of three branches AD, DF, and FB, which give
+  the composition of the solutions in equilibrium with pure A, double salt,
+  and pure B respectively; while the points D and F represent solutions
+  saturated for double salt plus A and double salt plus B.</p>
+
+  <p>On continuing to alter the temperature in the same direction <!-- Page
+  277 --><span class="pagenum"><a name="page277"></a>{277}</span>as before,
+  the relative shifting of the solubility curves becomes more marked, as
+  shown in Fig. 107. At the temperature of this isothermal, the solution
+  saturated for the double salt now lies in a region of distinct
+  unsaturation with respect to the single salts; and the double salt can
+  now exist as solid phase in contact with solutions containing both
+  relatively more of A (curve ED), and relatively more of B (curve DF),
+  than is contained in the double salt itself.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/fig107.png"><img style="width:100%" src="images/fig107.png"
+      alt="Fig. 107" title="Fig. 107" /></a>
+    <span class="sc">Fig. 107.</span>
+  </div>
+
+  <p><b>Transition Interval.</b>&mdash;From what has been said, and from an
+  examination of the isothermal diagrams, Figs. 104-107, it will be seen
+  that by a variation of the temperature we can pass from a condition where
+  the double salt is quite incapable of existing in contact with solution
+  (supersaturation being excluded), to a condition where the existence of
+  the double salt in presence of solution becomes possible; only in the
+  presence, however, of one of the single salts (<i>transition point</i>,
+  Fig. 105). A further change of temperature leads to a condition where the
+  stable existence of the pure double salt in contact with solution just
+  becomes possible (Fig. 106); and from this point onwards, pure saturated
+  solutions of the double salt can be obtained (Fig. 107). <i>At any
+  temperature, therefore, between that represented by Fig. 105, and that
+  represented by Fig. 106, the double salt undergoes partial decomposition,
+  with deposition of one of the constituent salts.</i> The temperature
+  range between the transition point and the temperature at which a stable
+  saturated solution of the pure double salt just begins to be possible, is
+  known as the <i>transition interval</i> (p.&nbsp;<a href="#page270">270</a>).
+  As the figures show, the transition interval is limited on the one side
+  by the transition temperature, and on the other by the temperature at
+  which the solution saturated for double salt and the less soluble of the
+  single salts, contains the component salts in the same ratio as they are
+  present in the double salt. The greater the difference in the solubility
+  of the single salts, the larger will be the transition interval. <!--
+  Page 278 --><span class="pagenum"><a name="page278"></a>{278}</span></p>
+
+  <p><b>Isothermal Evaporation.</b>&mdash;The isothermal solubility curves
+  are of great importance for obtaining an insight into the behaviour of a
+  solution when subjected to isothermal evaporation. To simplify the
+  discussion of the relationships found here, we shall still suppose that
+  the double salt contains the single salts in equimolecular proportions;
+  and we shall, in the first instance, suppose that the unsaturated
+  solution with which we commence, also contains the single salts in the
+  same ratio. The composition of the solution must, therefore, be
+  represented by some point lying on the line OD, the bisectrix of the
+  right angle.</p>
+
+  <p>From what has been said, it is evident that when the formation of a
+  double salt can occur, three temperature intervals can be distinguished,
+  viz. the single-salt interval, the transition interval, and the
+  double-salt interval.<a name="NtA359" href="#Nt359"><sup>[359]</sup></a>
+  When the temperature lies in the first interval, evaporation leads first
+  of all to the crystallization of one of the single salts, and then to the
+  separation of both the single salts together. In the second temperature
+  interval, evaporation again leads, in the first place, to the deposition
+  of one of the single salts, and afterwards to the crystallization of the
+  double salt. In the third temperature interval, only the double salt
+  crystallizes out. This will become clearer from what follows.</p>
+
+  <div class="figright" style="width:18%;">
+      <a href="images/fig109.png"><img style="width:100%" src="images/fig109.png"
+      alt="Fig. 109" title="Fig. 109" /></a>
+    <span class="sc">Fig. 109.</span>
+  </div>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig108.png"><img style="width:100%" src="images/fig108.png"
+      alt="Fig. 108" title="Fig. 108" /></a>
+    <span class="sc">Fig. 108.</span>
+  </div>
+
+  <p>If an unsaturated solution of the two single salts in equimolecular
+  proportion (<i>e.g.</i> point <i>x</i>, Fig. 108) is evaporated at a
+  temperature at which the formation of double salt is impossible, the
+  component A, the solubility curve of which is <!-- Page 279 --><span
+  class="pagenum"><a name="page279"></a>{279}</span>cut by the line OD,
+  will first separate out; the solution will thereby become richer in B. On
+  continued evaporation, more A will be deposited, and the composition of
+  the solution will change until it attains the composition represented by
+  the point C, when both A and B will be deposited, and the composition of
+  the solution will remain unchanged. The result of evaporation will
+  therefore be a mixture of the two components.</p>
+
+  <p>If the formation of double salt is possible, but if the temperature
+  lies within the transition interval, the relations will be represented by
+  a diagram like Fig. 109. Isothermal evaporation of the solution X will
+  lead to the deposition of the component A, and the composition of the
+  solution will alter in the direction DE; at the latter point the double
+  salt will be formed, and the composition of the solution will remain
+  unchanged so long as the two solid phases are present. As can be seen
+  from the diagram, however, the solution in E contains less of component A
+  than is contained in the double salt. Deposition of the double salt at E,
+  therefore, would lead to a relative decrease in the concentration of A in
+  the solution, and to counterbalance this, <i>the salt which separated out
+  at the commencement must redissolve</i>.</p>
+
+  <p>Since the salts were originally present in equimolecular proportions,
+  the final result of evaporation will be the pure double salt. If when the
+  solution has reached the point E the salt A which had separated out is
+  removed, double salt only will be left as solid phase. At a given
+  temperature, however, a single solid phase can exist in equilibrium with
+  solutions of different composition. If, therefore, isothermal evaporation
+  is continued after the removal of the salt A, double salt will be
+  deposited, and the composition of the solution will change in the
+  direction EF. At the point F the salt B will separate out, and on
+  evaporation both double salt and the salt B will be deposited. In the
+  former case (when the salt A disappears on evaporation) we are dealing
+  with an <i>incongruently saturated solution</i>; but in the latter case,
+  where both solid phases continue to be deposited, the solution is said to
+  be <i>congruently saturated</i>.<a name="NtA360"
+  href="#Nt360"><sup>[360]</sup></a></p>
+
+  <p>A "congruently saturated solution" is one from which the <!-- Page 280
+  --><span class="pagenum"><a name="page280"></a>{280}</span>solid phases
+  are continuously deposited during isothermal evaporation to dryness,
+  whereas in the case of "incongruently saturated solutions," at least one
+  of the solid phases disappears during the process of evaporation.</p>
+
+  <div class="figcenter" style="width:17%;">
+      <a href="images/fig110.png"><img style="width:100%" src="images/fig110.png"
+      alt="Fig. 110" title="Fig. 110" /></a>
+    <span class="sc">Fig.</span> 110.
+  </div>
+
+  <p>Lastly, if the temperature lies outside the transition interval,
+  isothermal evaporation of an unsaturated solution of the composition X
+  (Fig. 110) will lead to the deposition of pure double salt from beginning
+  to end. If a solution of the composition Y is evaporated, the component A
+  will first be deposited and the composition of the solution will alter in
+  the direction of E, at which point double salt will separate out. Since
+  the solution at this point contains relatively more of A than is present
+  in the double salt, both the double salt and the single salt A will be
+  deposited on continued evaporation, in order that the composition of the
+  solution shall remain unchanged. In the case of solution Z, first
+  component B and afterwards the double salt will be deposited. The result
+  will, therefore, be a mixture of double salt and the salt B (congruently
+  saturated solutions),</p>
+
+  <p>It may be stated here that the same relationships as have been
+  explained above for double salts are also found in the resolution of
+  racemic compounds by means of optically active substances (third method
+  of Pasteur). In this case the single salts are doubly active substances
+  (<i>e.g.</i> strychnine-<i>d</i>-tartrate and
+  strychnine-<i>l</i>-tartrate), and the double salt is a partially racemic
+  compound.<a name="NtA361" href="#Nt361"><sup>[361]</sup></a></p>
+
+  <p><b>Crystallization of Double Salt from Solutions containing Excess of
+  One Component.</b>&mdash;One more case of isothermal crystallization may
+  be discussed. It is well known that a double salt which is decomposed by
+  pure water can nevertheless be obtained pure by crystallization from a
+  solution containing excess of one of the single salts (<i>e.g.</i> in the
+  case of carnallite). Since the double salt is partially decomposed by
+  water, the temperature of the experiment must be within the transition
+  <!-- Page 281 --><span class="pagenum"><a
+  name="page281"></a>{281}</span>interval, and the relations will,
+  therefore, be represented by a diagram like Fig. 109. If, now, instead of
+  starting with an unsaturated solution containing the single salts in
+  equimolecular proportions, we commence with one in which excess of one of
+  the salts is present, as represented by the point Y, isothermal
+  evaporation will cause the composition to alter in the direction
+  YD&prime;, the relative amounts of the single salts remaining the same
+  throughout. When the composition of the solution reaches the point
+  D&prime;, pure double salt will be deposited. The separation of double
+  salt will, however, cause a relative decrease in the concentration of the
+  salt A, and the composition of the solution will, therefore, alter in the
+  direction D&prime;F. If the evaporation is discontinued before the
+  solution has attained the composition F, only double salt will have
+  separated out. Even within the transition interval, therefore, pure
+  double salt can be obtained by crystallization, provided the original
+  solution has a composition represented by a point lying between the two
+  lines OE and OF. Since, as already shown, the composition of the solution
+  alters on evaporation in the direction EF, it will be best to employ a
+  solution having a composition near to the line OE.</p>
+
+  <p><b>Formation of Mixed Crystals.</b>&mdash;If the two single salts A
+  and B do not crystallize out pure from solution, but form an unbroken
+  series of mixed crystals, it is evident that an invariant system cannot
+  be produced. The solubility curve will therefore be continuous from A to
+  B; the liquid solutions of varying composition being in equilibrium with
+  solid solutions also of varying composition. If, however, the series of
+  mixed crystals is not continuous, there will be a break in the solubility
+  curve at which two solid solutions of different composition will be in
+  equilibrium with liquid solution. This, of course, will constitute an
+  invariant system, and the point will correspond to the point C in Fig.
+  108. A full discussion of these systems would, however, lead us too far,
+  and the above indication of the behaviour must suffice.<a name="NtA362"
+  href="#Nt362"><sup>[362]</sup></a></p>
+
+<p><!-- Page 282 --><span class="pagenum"><a name="page282"></a>{282}</span></p>
+
+  <p><b>Application to the Characterization of Racemates.</b>&mdash;The
+  form of the isothermal solubility curves is also of great value for
+  determining whether an inactive substance is a racemic compound or a
+  conglomerate of equal proportions of the optical antipodes.<a
+  name="NtA363" href="#Nt363"><sup>[363]</sup></a></p>
+
+  <p>As has already been pointed out, the formation of racemic compounds
+  from the two enantiomorphous isomerides, is analogous to the formation of
+  double salts. The isothermal solubility curves, also, have a similar
+  form. In the case of the latter, indeed, the relationships are simplified
+  by the fact that the two enantiomorphous forms have identical solubility,
+  and the solubility curves are therefore symmetrical to the line bisecting
+  the angle of the co-ordinates. Further, with the exception of the
+  partially racemic compounds to be mentioned later, there is no transition
+  interval.</p>
+
+  <p>In Fig. 111, are given diagrammatically two isothermal solubility
+  curves for optically active substances. From what has been said in the
+  immediately preceding pages, the figure ought really to explain itself.
+  The upper isothermal <i>acb</i> represents the solubility relations when
+  the formation of a racemic compound is excluded, as, <i>e.g.</i> in the
+  case of rubidium <i>d</i>- and <i>l</i>-tartrates above the transition
+  point (p.&nbsp;<a href="#page265">265</a>). The solution at the point <i>c</i>
+  is, of course, inactive, and <i>is unaffected by addition of either the
+  </i>d<i>- or </i>l<i>- form</i>. The lower isothermal, on the other hand,
+  would be obtained at a temperature at which the racemic compound could be
+  formed. The curve <i>a&prime;e</i> is the solubility curve for the
+  <i>l</i>- form; <i>b&prime;f</i>, that for the <i>d</i>- form; and
+  <i>edf</i>, that for the racemic compound in presence of solutions of
+  varying concentration. The point <i>d</i> corresponds to saturation for
+  the pure racemic compound.</p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/fig111.png"><img style="width:100%" src="images/fig111.png"
+      alt="Fig. 111" title="Fig. 111" /></a>
+    <span class="sc">Fig.</span> 111.
+  </div>
+
+  <p>From these curves now, it will be evident that it will be possible, in
+  any given case, to decide whether or not an inactive body is a mixture or
+  a racemic compound. For this purpose, <!-- Page 283 --><span
+  class="pagenum"><a name="page283"></a>{283}</span>two solubility
+  determinations are made, first with the inactive material alone (in
+  excess), and then with the inactive material plus excess of one of the
+  optically active forms. If we are dealing with a mixture, the two
+  solutions thus obtained will be identical; both will have the composition
+  corresponding to the point <i>c</i>, and will be inactive. If, however,
+  the inactive material is a racemic compound, then two different solutions
+  will be obtained; namely, an inactive solution corresponding to the point
+  <i>d</i> (Fig. 111), and an <i>active</i> solution corresponding either
+  to <i>e</i> or to <i>f</i>, according to which enantiomorphous form was
+  added.</p>
+
+  <p><i>Partially racemic compounds.</i><a name="NtA364"
+  href="#Nt364"><sup>[364]</sup></a> In this case we are no longer dealing
+  with enantiomorphous forms, and the solubility of the two oppositely
+  active isomerides is no longer the same. The symmetry of the solubility
+  curves therefore disappears, and a figure is obtained which is identical
+  in its general form with that found in the case of ordinary double salts
+  (Fig. 112). In this case there is a transition interval.</p>
+
+  <div class="figcenter" style="width:19%;">
+      <a href="images/fig112.png"><img style="width:100%" src="images/fig112.png"
+      alt="Fig. 112" title="Fig. 112" /></a>
+    <span class="sc">Fig.</span> 112.
+  </div>
+
+  <p>The curves <i>acb</i> belong to a temperature at which the partially
+  racemic compound cannot be formed; <i>a&prime;dfb&prime;</i>, to the
+  temperature at which the compound just begins to be stable in contact
+  with water, and <i>a&Prime;ed&prime;f&prime;b&Prime;</i> belongs to a
+  temperature at which the partially racemic compound is quite stable in
+  contact with water. Suppose now solubility determinations, made in the
+  first case with the original material alone, and then with the original
+  body plus each of the two compounds, formed from the enantiomorphous
+  substances separately, then if the original body was a mixture, identical
+  solutions will be obtained in all three cases (point <i>c</i>); if it was
+  a partially racemic compound, three different solutions (<i>e</i>,
+  <i>d&prime;</i>, and <i>f&prime;</i>) will be obtained if the temperature
+  was outside the transition interval, and two solutions, <i>d</i> and
+  <i>f</i>, if the temperature belonged to the transition interval.</p>
+
+<p><!-- Page 284 --><span class="pagenum"><a name="page284"></a>{284}</span></p>
+
+<p class="cenhead"><i>Representation in Space.</i></p>
+
+  <p><b>Space Model for Carnallite.</b>&mdash;Interesting and important as
+  the isothermal solubility curves are, they are insufficient for the
+  purpose of obtaining a clear insight into the complete behaviour of the
+  systems of two salts and water. A short description will, therefore, be
+  given here of the representation in space of the solubility relations of
+  potassium and magnesium chlorides, and of the double salt which they
+  form, carnallite.<a name="NtA365" href="#Nt365"><sup>[365]</sup></a></p>
+
+  <div class="figcenter" style="width:47%;">
+      <a href="images/fig113.png"><img style="width:100%" src="images/fig113.png"
+      alt="Fig. 113" title="Fig. 113" /></a>
+    <span class="sc">Fig.</span> 113.
+  </div>
+
+  <p>Fig. 113 is a diagrammatic sketch of the model for carnallite looked
+  at sideways from above. Along the X-axis is measured the concentration of
+  magnesium chloride in the <!-- Page 285 --><span class="pagenum"><a
+  name="page285"></a>{285}</span>solution; along the Y-axis, the
+  concentration of potassium chloride; while along the T-axis is measured
+  the temperature. The three axes are at right angles to one another. The
+  XT-plane, therefore, contains the solubility curve of magnesium chloride;
+  the YT-plane, the solubility curve of potassium chloride, and in the
+  space between the two planes, there are represented the composition of
+  solutions containing both magnesium and potassium chlorides. Any
+  <i>surface</i> between the two planes will represent the various
+  solutions in equilibrium with only one solid phase, and will therefore
+  indicate the area or field of existence of bivariant ternary systems. A
+  <i>line</i> or <i>curve</i> formed by the intersection of two surfaces
+  will represent solutions in equilibrium with two solid phases (viz. those
+  belonging to the intersecting surfaces), and will show the conditions for
+  the existence of univariant systems. Lastly, <i>points</i> formed by the
+  intersection of three surfaces will represent invariant systems, in which
+  a solution can exist in equilibrium with three solid phases (viz. those
+  belonging to the three surfaces).</p>
+
+  <p>We shall first consider the solubility relations of the single salts.
+  The complete equilibrium curve for magnesium chloride and water is
+  represented in Fig. 113 by the series of curves ABF<sub>1</sub>
+  G<sub>1</sub> H<sub>1</sub> J<sub>1</sub> L<sub>1</sub> N<sub>1</sub>. AB
+  is the freezing-point curve of ice in contact with solutions containing
+  magnesium chloride, and B is the cryohydric point at which the solid
+  phases ice and MgCl<sub>2</sub>,12H<sub>2</sub>O can co-exist with
+  solution. BFG is the solubility curve of magnesium chloride
+  dodecahydrate. This curve shows a point of maximum temperature at
+  F<sub>1</sub>, and a retroflex portion F<sub>1</sub>G<sub>1</sub>. The
+  curve is therefore of the form exhibited by calcium chloride hexahydrate,
+  or the hydrates of ferric chloride (Chapter VIII.). G<sub>1</sub> is a
+  transition point at which the solid phase changes from dodecahydrate to
+  octahydrate, the solubility of which is represented by the curve
+  G<sub>1</sub>H<sub>1</sub>. At H<sub>1</sub> the octahydrate gives place
+  to the hexahydrate, which is the solid phase in equilibrium with the
+  solutions represented by the curve H<sub>1</sub>J<sub>1</sub>.
+  J<sub>1</sub> and L<sub>1</sub> are also transition points at which the
+  solid phase undergoes change, in the former case from hexahydrate to
+  tetrahydrate; and in the latter case, <!-- Page 286 --><span
+  class="pagenum"><a name="page286"></a>{286}</span>from tetrahydrate to
+  dihydrate. The complete curve of equilibrium for magnesium chloride and
+  water is, therefore, somewhat complicated, and is a good example of the
+  solubility curves obtained with salts capable of forming several
+  hydrates.</p>
+
+  <p>The solubility curve of potassium chloride is of the simplest form,
+  consisting only of the two branches AC, the freezing-point curve of ice,
+  and CO, the solubility curve of the salt. C is the cryohydric point. This
+  point and the two curves lie in the YT-plane.</p>
+
+  <p>On passing to the ternary systems, the composition of the solutions
+  must be represented by points or curves situated <i>between</i> the two
+  planes. We shall now turn to the consideration of these. BD and CD are
+  ternary eutectic curves (p.&nbsp;<a href="#page284">284</a>). They give the
+  composition of solutions in equilibrium with ice and magnesium chloride
+  dodecahydrate (BD), and with ice and potassium chloride (CD). D is a
+  <i>ternary cryohydric point</i>. If the temperature is raised and the ice
+  allowed to disappear, we shall pass to the solubility curve for
+  MgCl<sub>2</sub>,12H<sub>2</sub>O + KCl (curve DE). At E carnallite is
+  formed and the potassium chloride disappears; EFG is then the solubility
+  curve for MgCl<sub>2</sub>,12H<sub>2</sub>O + carnallite
+  (KMgCl<sub>3</sub>,6H<sub>2</sub>O). This curve also shows a point of
+  maximum temperature (F) and a retroflex portion. GH and HJ represent the
+  solubility curves of carnallite + MgCl<sub>2</sub>,8H<sub>2</sub>O and
+  carnallite + MgCl<sub>2</sub>,6H<sub>2</sub>O, G and H being transition
+  points. JK is the solubility curve for carnallite +
+  MgCl<sub>2</sub>,4H<sub>2</sub>O. At the point K we have the <i>highest
+  temperature at which carnallite can exist with magnesium chloride in
+  contact with solution</i>. Above this temperature decomposition takes
+  place and potassium chloride separates out.</p>
+
+  <p>If at the point E, at which the two single salts and the double salt
+  are present, excess of potassium chloride is added, the magnesium
+  chloride will all disappear owing to the formation of carnallite, and
+  there will be left carnallite and potassium chloride. The solubility
+  curve for a mixture of these two salts is represented by EMK; a simple
+  curve exhibiting, however, a temperature maximum at M. This maximum point
+  corresponds with the fact that dry carnallite melts at this temperature
+  with separation of potassium chloride. <i>At all temperatures <!-- Page
+  287 --><span class="pagenum"><a name="page287"></a>{287}</span>above this
+  point, the formation of double salt is impossible</i>. The retroflex
+  portion of the curve represents solutions in equilibrium with carnallite
+  and potassium chloride, but in which the ratio MgCl<sub>2</sub> : KCl is
+  greater than in the double salt.</p>
+
+  <p>Throughout its whole course, <i>the curve EMK represents solutions in
+  which the ratio of MgCl<sub>2</sub> : KCl is greater than in the double
+  salt</i>. As this is a point of some importance, it will be well,
+  perhaps, to make it clearer by giving one of the isothermal curves,
+  <i>e.g.</i> the curve for 10°, which is represented diagrammatically in
+  Fig. 114. E and F here represent solutions saturated for carnallite plus
+  magnesium chloride hydrate, and for carnallite plus potassium chloride.
+  As is evident, the point F lies above the line representing equimolecular
+  proportions of the salts (OD).</p>
+
+  <div class="figcenter" style="width:16%;">
+      <a href="images/fig114.png"><img style="width:100%" src="images/fig114.png"
+      alt="Fig. 114" title="Fig. 114" /></a>
+    <span class="sc">Fig.</span> 114.
+  </div>
+
+  <p><b>Summary and Numerical Data.</b>&mdash;We may now sum up the
+  different systems which can be formed, and give the numerical data from
+  which <span class="correction" title="Original reads 'he'.">the</span>
+  model is constructed.<a name="NtA366"
+  href="#Nt366"><sup>[366]</sup></a></p>
+
+<p class="cenhead">I. <i>Bivariant Systems.</i></p>
+
+<table class="nobctr" summary="Phases of potassium and magnesium chloride" title="Phases of potassium and magnesium chloride">
+<tr><td class="toribob" style="text-align:center"> Solid phase.
+</td><td class="toboleb" style="text-align:center"> Area of existence.</td></tr>
+
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> Ice </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> ABDC</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> KCl </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> CDEMKLNO</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> Carnallite </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> EFGHJKM</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> MgCl<sub>2</sub>,12H<sub>2</sub>O </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> BF<sub>1</sub>G<sub>1</sub>GFED</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> MgCl<sub>2</sub>,8H<sub>2</sub>O </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> G<sub>1</sub>H<sub>1</sub>HG</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> MgCl<sub>2</sub>,6H<sub>2</sub>O </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> H<sub>1</sub>I<sub>1</sub>IH</td></tr>
+<tr><td class="rib" style="text-align:center; padding-left:2em; padding-right:2em;"> MgCl<sub>2</sub>,4H<sub>2</sub>O </td><td class="leb" style="text-align:center; padding-left:2em; padding-right:2em;"> I<sub>1</sub>L<sub>1</sub>LKI</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> MgCl<sub>2</sub>,2H<sub>2</sub>O </td><td class="boleb" style="text-align:center"> L<sub>1</sub>N<sub>1</sub>NL</td></tr>
+
+</table>
+
+  <p>II. <i>Univariant Systems.</i>&mdash;The different univariant systems
+  have already been described. The course of the curves will be
+  sufficiently indicated if the temperature and composition of the
+  solutions for the different invariant systems are given.</p>
+
+<p><!-- Page 288 --><span class="pagenum"><a name="page288"></a>{288}</span></p>
+
+<p class="cenhead">III.&mdash;<i>Invariant Systems&mdash;Binary and Ternary.</i></p>
+
+<table class="nobctr" summary="Invariant Systems&mdash;Binary and Ternary" title="Invariant Systems&mdash;Binary and Ternary">
+<tr><td class="toribob" style="text-align:center"> Point.
+</td><td class="toboleb" style="text-align:center" colspan="3"> Solid Phases.
+</td><td class="toboleb" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Composition of solution. Gram-<br />molecules of salt per 1000 gram-<br />mol. water.</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> A </td><td class="rileb" style="text-align:center;" colspan="3"> Ice </td><td class="rileb" style="text-align:right; padding-right:1em;"> 0°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> B </td><td class="rileb" style="text-align:center;" colspan="3"> Ice; MgCl<sub>2</sub>,12H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> -33.6° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 49.2 MgCl<sub>2</sub></td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> C </td><td class="rileb" style="text-align:center;" colspan="3"> Ice; KCl </td><td class="rileb" style="text-align:right; padding-right:1em;"> -11.1° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 59.4 KCl</td></tr>
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> D </td><td class="rileb" style="text-align:center;" colspan="3"> Ice; MgCl<sub>2</sub>,12H<sub>2</sub>O; KCl </td><td class="rileb" style="text-align:right; padding-right:1em;"> -34.3° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 43 MgCl<sub>2</sub>; 3 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> E </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,12H<sub>2</sub>O; KCl;<br />carnallite</td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -21°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 66.1 MgCl<sub>2</sub>; 4.9 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> F<sub>1</sub> </td><td class="rileb" style="text-align:center;" colspan="3"> MgCl<sub>2</sub>,12H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> -16.4° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 83.33 MgCl<sub>2</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> F </td><td class="rileb" style="text-align:center;" colspan="3"> MgCl<sub>2</sub>,12H<sub>2</sub>O; carnallite </td><td class="rileb" style="text-align:right; padding-right:1em;"> -16.6° &nbsp; </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left; padding-left:0em; padding-right:0em;"> Almost same as F<sub>1</sub>; contains<br />small amount of KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> G<sub>1</sub> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,12H<sub>2</sub>O;<br />MgCl<sub>2</sub>,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -16.8° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 87.5 MgCl<sub>2</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> G </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,12H<sub>2</sub>O;<br />MgCl<sub>2</sub>,8H<sub>2</sub>O; carnallite </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -16.9° &nbsp; </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left; padding-left:0em; padding-right:0em;"> Almost same as G<sub>1</sub>, but<br />contains small quantity of KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> H<sub>1</sub> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,8H<sub>2</sub>O;<br />MgCl<sub>2</sub>,6H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -3.4° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 99 MgCl<sub>2</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> H </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,8H<sub>2</sub>O;<br />MgCl<sub>2</sub>,6H<sub>2</sub>O; carnallite </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> ca. -3.4° &nbsp; </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:left; padding-left:0em; padding-right:0em;"> Almost same as H<sub>1</sub>, but<br />contains small amount of KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> J<sub>1</sub> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,6H<sub>2</sub>O;<br />MgCl<sub>2</sub>,4H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 116.67° </td><td class="leb" style="text-align:center;" colspan="2"> 161.8 MgCl<sub>2</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> J </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,6H<sub>2</sub>O;<br />MgCl<sub>2</sub>,4H<sub>2</sub>O; carnallite </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 115.7° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 162 MgCl<sub>2</sub>; 4 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> K </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,4H<sub>2</sub>O; KCl;<br />carnallite </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 152.5° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 200 MgCl<sub>2</sub>; 24 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> L<sub>1</sub> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,4H<sub>2</sub>O;<br />MgCl<sub>2</sub>,2H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 181°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 238.1 MgCl<sub>2</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> L </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> MgCl<sub>2</sub>,4H<sub>2</sub>O;<br />MgCl<sub>2</sub>,2H<sub>2</sub>O; KCl </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 176°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 240 MgCl<sub>2</sub>; 41 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> M </td><td class="rileb" style="text-align:center;" colspan="3"> Carnallite; KCl </td><td class="rileb" style="text-align:right; padding-right:1em;"> 167.5° &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 166.7 MgCl<sub>2</sub>; 41.7 KCl</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> [N<sub>1</sub> </td><td class="rileb" style="text-align:center;" colspan="3"> MgCl<sub>2</sub>,2H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> 186°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> ca. 241 MgCl<sub>2</sub>]</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> N </td><td class="rileb" style="text-align:center;" colspan="3"> MgCl<sub>2</sub>,2H<sub>2</sub>O; KCl </td><td class="rileb" style="text-align:right; padding-right:1em;"> 186°&nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:center;" colspan="2"> 240 MgCl<sub>2</sub>; 63 KCl</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:1em;"> [O </td><td class="riboleb" style="text-align:center;" colspan="3"> KCl </td><td class="riboleb" style="text-align:right; padding-right:1em;"> 186°&nbsp; &nbsp; &nbsp; </td><td class="boleb" style="text-align:center;" colspan="2"> 195.6 KCl]</td></tr>
+</table>
+
+  <p>With the help of the data in the preceding table and of the solid
+  model it will be possible to state in any given case what will be the
+  behaviour of a system composed of magnesium chloride, potassium chloride
+  and water. One or two different cases will be very briefly described; and
+  the reader should have no difficulty in working out the behaviour under
+  other conditions with the help of the model and the numerical data just
+  given. <!-- Page 289 --><span class="pagenum"><a
+  name="page289"></a>{289}</span></p>
+
+  <p>In the first place it may be again noted that at a temperature above
+  167.5° (point M) carnallite cannot exist. If, therefore, a solution of
+  magnesium and potassium chlorides is evaporated at a temperature above
+  this point, the result will be a mixture of potassium chloride and either
+  magnesium chloride tetrahydrate or magnesium chloride dihydrate,
+  according as the temperature is below or above 176°. The isothermal curve
+  here consists of only two branches.</p>
+
+  <p>Further, reference has already been made to the fact that all points
+  of the carnallite area correspond to solutions in equilibrium with
+  carnallite, but in which the ratio of MgCl<sub>2</sub> to KCl is greater
+  than in the double salt. A solution which is saturated with respect to
+  double salt alone will be supersaturated with respect to potassium
+  chloride. At all temperatures, therefore, carnallite is decomposed by
+  water with separation of potassium chloride; hence all solutions obtained
+  by adding excess of carnallite to water will lie on the curve EM. <i>A
+  pure saturated solution of carnallite cannot be obtained.</i></p>
+
+  <p>If an unsaturated solution of the two salts in equimolecular amounts
+  is evaporated, potassium chloride will first be deposited, because the
+  plane bisecting the right angle formed by the X and Y axes cuts the area
+  for that salt. Deposition of potassium chloride will lead to a relative
+  increase in the concentration of magnesium chloride in the solution; and
+  on continued evaporation a point (on the curve EM) will be reached at
+  which carnallite will separate out. So long as the two solid phases are
+  present, the composition of the solution must remain unchanged. Since the
+  separation of carnallite causes a decrease in the relative concentration
+  of the potassium chloride in the solution, the portion of this salt which
+  was deposited at the commencement must <i>redissolve</i>, and carnallite
+  will be left on evaporating to dryness. (<i>Incongruently saturated
+  solution.</i>)</p>
+
+  <p>Although carnallite is decomposed by pure water, it will be possible
+  to crystallize it from a solution having a composition represented by any
+  point in the carnallite area. Since during the separation of the double
+  salt the relative amount of magnesium chloride increases, it is most
+  advantageous to <!-- Page 290 --><span class="pagenum"><a
+  name="page290"></a>{290}</span>commence with a solution the composition
+  of which is represented by a point lying just above the curve EM (cf.
+  p.&nbsp;<a href="#page281">281</a>).</p>
+
+  <p>From the above description of the behaviour of carnallite in solution,
+  the processes usually employed for obtaining potassium chloride will be
+  readily intelligible.<a name="NtA367"
+  href="#Nt367"><sup>[367]</sup></a></p>
+
+  <p><b>Ferric Chloride&mdash;Hydrogen Chloride&mdash;Water.</b>&mdash;In
+  the case of another system of three components which we shall now
+  describe, the relationships are considerably more complicated than in
+  those already discussed. They deserve discussion, however, on account of
+  the fact that they exhibit a number of new phenomena.</p>
+
+  <p>In the system formed by the three components, ferric chloride,
+  hydrogen chloride, and water, not only can various compounds of ferric
+  chloride and water (p.&nbsp;<a href="#page152">152</a>), and of hydrogen
+  chloride and water be formed, each of which possesses a definite melting
+  point, but various ternary compounds are also known. Thus we have the
+  following solid phases:&mdash;</p>
+
+<table class="nobctr" summary="Solid phases, ferric and hydrogen chlorides" title="Solid phases, ferric and hydrogen chlorides">
+<tr><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O </td><td class="nob" style="text-align:left"> HCl,3H<sub>2</sub>O </td><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O</td></tr>
+<tr><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,7H<sub>2</sub>O </td><td class="nob" style="text-align:left"> HCl,2H<sub>2</sub>O </td><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O</td></tr>
+<tr><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,5H<sub>2</sub>O </td><td class="nob" style="text-align:left"> HCl,H<sub>2</sub>O </td><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O</td></tr>
+<tr><td class="nob" style="text-align:left"> 2FeCl<sub>3</sub>,4H<sub>2</sub>O</td></tr>
+<tr><td class="nob" style="text-align:left"> FeCl<sub>3</sub></td></tr>
+</table>
+
+  <p>From this it will be readily understood that the complete study of the
+  conditions of temperature and concentration under which solutions can
+  exist, either with one solid phase or with two or three solid phases, are
+  exceedingly complicated; and, as a matter of fact, only a few of the
+  possible equilibria have been investigated. We shall attempt here only a
+  brief description of the most important of these.<a name="NtA368"
+  href="#Nt368"><sup>[368]</sup></a></p>
+
+  <p>If we again employ rectangular co-ordinates for the graphic <!-- Page
+  291 --><span class="pagenum"><a
+  name="page291"></a>{291}</span>representation of the results, we have the
+  two planes XOT and YOT (Fig. 115): the concentration of ferric chloride
+  being measured along the X-axis, the concentration of hydrogen chloride
+  along the Y-axis, and the temperature along the T-axis. The curve
+  ABCDEFGHJK is, therefore, the solubility curve of ferric chloride in
+  water (p.&nbsp;<a href="#page152">152</a>), and the curve
+  A&prime;B&prime;C&prime;D&prime;E&prime;F&prime; the solubility curve of
+  hydrogen chloride and its hydrates. B&prime; and D&prime; are the melting
+  points of the hydrates HCl,3H<sub>2</sub>O and HCl,2H<sub>2</sub>O. In
+  the space between these two planes are represented those systems in which
+  all three components are present. As already stated, only a few of the
+  possible ternary systems have been investigated, and these are
+  represented in Fig. 116. The figure shows the model resting on the
+  XOT-plane, so that the lower edge represents the solubility curve of
+  ferric chloride, the concentration increasing from right to left. The
+  concentration of hydrogen chloride is measured upwards, and the
+  temperature forwards. The further end of the model represents the
+  isothermal surface for -30°. The surface of the model on the left does
+  not correspond with the plane YOT in Fig. 115, but with a parallel plane
+  which cuts the concentration axis for ferric chloride at a point
+  representing 65 gm.-molecules FeCl<sub>3</sub> in 100 gm.-molecules of
+  water. The upper surface corresponds with a plane parallel to the axis
+  XOT, at a distance corresponding with the concentration of 50
+  gm.-molecules HCl in 100 gm.-molecules of water.</p>
+
+  <div class="figcenter" style="width:24%;">
+      <a href="images/fig115.png"><img style="width:100%" src="images/fig115.png"
+      alt="Fig. 115" title="Fig. 115" /></a>
+    <span class="sc">Fig.</span> 115.
+  </div>
+
+  <p><b>Ternary Systems.</b>&mdash;We pass over the binary system
+  FeCl<sub>3</sub>&mdash;H<sub>2</sub>O, which has already been discussed
+  (p.&nbsp;<a href="#page152">152</a>), and the similar system
+  HCl&mdash;H<sub>2</sub>O (see Fig. 115), and turn to the discussion of
+  some of the ternary systems represented by <!-- Page 292 --><span
+  class="pagenum"><a name="page292"></a>{292}</span>points on the surface
+  of the model between the planes XOT and YOT. As in the case of
+  carnallite, a plane represents the conditions of concentration of
+  solution and temperature under which a ternary solution can be in
+  equilibrium with a <i>single</i> solid phase (bivariant systems), a line
+  represents the conditions for the coexistence of a solution with two
+  solid phases (univariant systems), and a point the conditions for
+  equilibrium with three solid phases (invariant systems).</p>
+
+  <div class="figcenter" style="width:22%;">
+      <a href="images/fig116.jpg"><img style="width:100%" src="images/fig116.jpg"
+      alt="Fig. 116" title="Fig. 116" /></a>
+    <span class="sc">Fig.</span> 116.
+  </div>
+
+  <p>In the case of a binary system, in which
+  2FeCl<sub>3</sub>,12H<sub>2</sub>O is in equilibrium with a solution of
+  the same composition, addition of hydrogen chloride must evidently lower
+  the temperature at which equilibrium can exist; and the same holds, of
+  course, <!-- Page 293 --><span class="pagenum"><a
+  name="page293"></a>{293}</span>for all other binary solutions in
+  equilibrium with this solid phase. In this way we obtain the surface I.,
+  which represents the temperatures and concentrations of solutions in
+  which 2FeCl<sub>3</sub>,12H<sub>2</sub>O can be in equilibrium with a
+  ternary solution containing ferric chloride, hydrogen chloride, and
+  water. This surface is analogous to the curved surface
+  K<sub>1</sub>K<sub>2</sub><i>k</i><sub>4</sub><i>k</i><sub>3</sub> in
+  Fig. 97 (p.&nbsp;<a href="#page256">256</a>). Similarly, the surfaces II.,
+  III., IV., and V. represent the conditions for equilibrium between the
+  solid phases 2FeCl<sub>3</sub>,7H<sub>2</sub>O;
+  2FeCl<sub>3</sub>,5H<sub>2</sub>O; 2FeCl<sub>3</sub>,4H<sub>2</sub>O;
+  FeCl<sub>3</sub> and ternary solutions respectively. The lines CL, EM,
+  GN, and IO on the model represent univariant systems in which a ternary
+  solution is in equilibrium with two solid phases, viz. with those
+  represented by the adjoining fields. These lines correspond with the
+  ternary eutectic curves <i>k</i><sub>3</sub>K<sub>1</sub> and
+  <i>k</i><sub>4</sub>K<sub>2</sub> in Fig. 97. Besides the surfaces
+  already mentioned, there are still three others, VI., VII., and VIII.,
+  which also represent the conditions for equilibrium between one solid
+  phase and a ternary solution; but in these cases, the solid phase is not
+  a binary compound or an anhydrous salt, but a ternary compound containing
+  all three components. The solid phases which are in equilibrium with the
+  ternary solutions represented by the surfaces VI., VII., and VIII., are
+  2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O;
+  2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O; and
+  2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O respectively.</p>
+
+  <p>The model for FeCl<sub>3</sub>&mdash;HCl&mdash;H<sub>2</sub>O exhibits
+  certain other peculiarities not found in the case of
+  MgCl<sub>2</sub>&mdash;KCl&mdash;H<sub>2</sub>O. On examining the model
+  more closely, it is found that the field of the ternary compound
+  2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O (VII.) resembles the surface of a
+  sugar cone, and has a projecting point, the end of which corresponds with
+  a higher temperature than does any other point of the surface. At the
+  point of maximum temperature the composition of the liquid phase is the
+  same as that of the solid. This point, therefore, represents the melting
+  point of the double salt of the above composition.</p>
+
+  <p>The curves representing univariant systems are of two kinds. In the
+  one case, the two solid phases present are both binary compounds; or one
+  is a binary compound and the other is one of the components. In the other
+  case, either one or both solid phases are ternary compounds. Curves
+  belonging <!-- Page 294 --><span class="pagenum"><a
+  name="page294"></a>{294}</span>to the former class (so-called <i>border
+  curves</i>) start from binary eutectic points, and their course is always
+  towards lower temperatures, <i>e.g.</i> CL, EM, GN, IO. Curves belonging
+  to the latter class (so-called <i>medial curves</i>) would, in a
+  triangular diagram, lie entirely within the triangle. Such curves are YV,
+  WV, VL, LM, MV, NS, ST, SO, OZ. These curves do not always run from
+  higher to lower temperatures, but may even exhibit a point of maximum
+  temperature. Such maxima are found, for example, at U (Fig. 116), and
+  also on the curves ST and LV.</p>
+
+  <p>Finally, whereas all the other ternary univariant curves run in
+  valleys between the adjoining surfaces, we find at the point X a similar
+  appearance to that found in the case of carnallite, as the univariant
+  curve here rises above the surrounding surface. The point X, therefore,
+  does not correspond with a eutectic point, but with a transition point.
+  At this point the ternary compound
+  2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O melts with separation of
+  2FeCl<sub>3</sub>,12H<sub>2</sub>O, just as carnallite melts at 168° with
+  separation of potassium chloride.</p>
+
+  <p><b>The Isothermal Curves.</b>&mdash;A deeper insight into the
+  behaviour of the system FeCl<sub>3</sub>&mdash;HCl&mdash;H<sub>2</sub>O
+  is obtained from a study of the isothermal curves, the complete series of
+  which, so far as they have been studied, is given in Fig. 117.<a
+  name="NtA369" href="#Nt369"><sup>[369]</sup></a> In this figure the
+  lightly drawn curves represent isothermal solubility curves, the
+  particular temperature being printed beside the curve.<a name="NtA370"
+  href="#Nt370"><sup>[370]</sup></a> The dark lines give the composition of
+  the univariant systems at different temperatures. The point of
+  intersection of a dark with a light curve gives the composition of the
+  univariant solution at the temperature represented by the light curve;
+  and the point of intersection of two dark lines gives the composition of
+  the invariant solution in equilibrium with three solid phases. The dotted
+  lines represent metastable systems, and the points P, Q, and R represent
+  solutions of <!-- Page 295 --><span class="pagenum"><a
+  name="page295"></a>{295}</span>the composition of the ternary salts,
+  2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O;
+  2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O; and
+  2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O.</p>
+
+  <div class="figcenter" style="width:90%;">
+      <a href="images/fig117.png"><img style="width:100%" src="images/fig117.png"
+      alt="Fig. 117" title="Fig. 117" /></a>
+    <span class="sc">Fig.</span> 117.
+  </div>
+
+  <p>The farther end of the model (Fig. 116) corresponds, as already
+  mentioned, to the temperature -30°, so that the outline evidently
+  represents the isothermal curve for that temperature. Fig. 117 does not
+  show this. We can, however, follow the isothermal for -20°, which is the
+  extreme curve on the right in Fig. 117. Point A represents the solubility
+  of 2FeCl<sub>3</sub>,12H<sub>2</sub>O in water. If hydrogen chloride is
+  added, the concentration of ferric chloride in the solution first
+  decreases and then increases, until at point 34 the ternary double salt
+  2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O is formed. If the addition of
+  hydrogen chloride is continued, the ferric chloride disappears
+  ultimately, and only the ternary double salt remains. This salt can
+  coexist with solutions of the composition represented by the curve which
+  passes through the points 173, 174, 175. At the last-mentioned point, the
+  ternary salt with 8H<sub>2</sub>O is formed. The composition of the
+  solutions with which this salt is in equilibrium at -20° is represented
+  by the curve which passes through a point of maximal concentration with
+  respect to HCl, and cuts the curve SN at the point 112, at which the
+  solution is in equilibrium with the two solid phases
+  2FeCl<sub>3</sub>,4H<sub>2</sub>O and
+  2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O. The succeeding portion of the
+  isotherm represents the solubility curve at -20° of
+  2FeCl<sub>3</sub>,4H<sub>2</sub>O, which cuts the dark line OS at point
+  113, at which the solution is in equilibrium with the two solid phases
+  2FeCl<sub>3</sub>,4H<sub>2</sub>O and
+  2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O. Thereafter comes the solubility
+  curve of the latter compound.</p>
+
+  <p>The other isothermal curves can be followed in a similar manner. If
+  the temperature is raised, the region of existence of the ternary double
+  salts becomes smaller and smaller, and at temperatures above 30° the
+  ternary salts with 12H<sub>2</sub>O and 8H<sub>2</sub>O are no longer
+  capable of existing. If the temperature is raised above 46°, only the
+  binary compounds of ferric chloride and water and the anhydrous salt can
+  exist as solid phases. The isothermal curve for 0° represents the
+  solubility curve for 2FeCl<sub>3</sub>,12H<sub>2</sub>O;
+  2FeCl<sub>3</sub>,7H<sub>2</sub>O; 2FeCl<sub>3</sub>,5H<sub>2</sub>O; and
+  2FeCl<sub>3</sub>,4H<sub>2</sub>O. <!-- Page 296 --><span
+  class="pagenum"><a name="page296"></a>{296}</span></p>
+
+  <p>Finally, in the case of the system
+  FeCl<sub>3</sub>&mdash;HCl&mdash;H<sub>2</sub>O, we find <i>closed</i>
+  isothermal curves. Since, as already stated, the salt
+  2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O has a definite melting point, the
+  temperature of which is therefore higher than that at which this compound
+  is in equilibrium with solutions of other composition, it follows that
+  the line of intersection of an isothermal plane corresponding with a
+  temperature immediately below the melting point of the salt with the
+  cone-shaped surface of its region of existence, will form a closed curve.
+  This is shown by the isotherm for -4.5°, which surrounds the point Q, the
+  melting point of the ternary salt.</p>
+
+  <p>The following table gives some of the numerical data from which the
+  curves and the model have been constructed:&mdash;</p>
+
+<table class="nobctr" summary="Data for ferric and hydrogen chloride solutions" title="Data for ferric and hydrogen chloride solutions">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Point.
+</td><td class="toboleb" style="text-align:center" rowspan="2" colspan="3"> Solid Phases.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Composition of the<br />solution in gm.-mols. salt<br />to 100 gm.-mols. water.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> HCl
+</td><td class="toboleb" style="text-align:center"> FeCl<sub>3</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> A </td><td class="rileb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> -20°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> &mdash;&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 6.56</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> C </td><td class="rileb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O; 2FeCl<sub>3</sub>,7H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> 27.4° </td><td class="rileb" style="text-align:right; padding-right:1em;"> &mdash;&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 24.30</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> E </td><td class="rileb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,7H<sub>2</sub>O; 2FeCl<sub>3</sub>,5H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> 30°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> &mdash;&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 30.24</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> G </td><td class="rileb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,5H<sub>2</sub>O; 2FeCl<sub>3</sub>,4H<sub>2</sub>O </td><td class="rileb" style="text-align:right; padding-right:1em;"> 55°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> &mdash;&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 40.64</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> J </td><td class="rileb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,4H<sub>2</sub>O; FeCl<sub>3</sub> </td><td class="rileb" style="text-align:right; padding-right:1em;"> 66°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> &mdash;&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 58.40</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> L </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O; 2FeCl<sub>3</sub>,7H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -7.5° </td><td class="rileb" style="text-align:right; padding-right:1em;"> 19.22 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 23.72</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> M </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,7H<sub>2</sub>O; 2FeCl<sub>3</sub>,5H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -7.3° </td><td class="rileb" style="text-align:right; padding-right:1em;"> 23.08 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 28.55</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> N </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,5H<sub>2</sub>O; 2FeCl<sub>3</sub>,4H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -16°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 28.40 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 31.89</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> S </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:8ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,4H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:8ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -27.5° </td><td class="rileb" style="text-align:right; padding-right:1em;"> 32.33 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 34.21</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,4H<sub>2</sub>O; FeCl<sub>3</sub>;<br />2FeCl<sub>3</sub>,2HCl,4H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> 29°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 33.71 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 49.84</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> U </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,7H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -4.5° </td><td class="rileb" style="text-align:right; padding-right:1em;"> 20.66 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 25.74</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> V </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:8ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:8ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -13°&nbsp; &nbsp; </td><td class="rileb" style="text-align:right; padding-right:1em;"> 22.40 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 18.00</td></tr>
+
+<tr><td class="rib" style="text-align:left; padding-left:1em;"> X </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> 2FeCl<sub>3</sub>,12H<sub>2</sub>O;<br />2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex; width:0.7em" alt="brace" /></a>
+   </td><td class="rileb" style="text-align:right; padding-right:1em;"> -12.5° </td><td class="rileb" style="text-align:right; padding-right:1em;"> 22.14 </td><td class="rileb" style="text-align:right; padding-right:1em;"> 16.69</td></tr>
+
+<tr><td class="ribob" style="text-align:left; padding-left:1em;"> Q </td><td class="riboleb" style="text-align:center;" colspan="3"> 2FeCl<sub>3</sub>,2HCl,8H<sub>2</sub>O </td><td class="boleb" style="text-align:left;" colspan="3"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; -3° (melting point)</td></tr>
+
+</table>
+
+  <p><b>Basic Salts.</b>&mdash;Another class of systems in the study of
+  <!-- Page 297 --><span class="pagenum"><a
+  name="page297"></a>{297}</span>which the Phase Rule has performed
+  exceptional service, is that of the basic salts. In many cases it is
+  impossible, by the ordinary methods of analysis, to decide whether one is
+  dealing with a definite chemical individual or with a mixture. The
+  question whether a solid phase is a chemical individual can, however, be
+  answered, in most cases, with the help of the principles which we have
+  already learnt. Let us consider, for example, the formation of basic
+  salts from bismuth nitrate, and water. In this case we can choose as
+  components Bi<sub>2</sub>O<sub>3</sub>, N<sub>2</sub>O<sub>5</sub>, and
+  H<sub>2</sub>O; since all the systems consist of these in varying
+  amounts. If we are dealing with a condition of equilibrium at constant
+  temperature between liquid and solid phases, three cases can be
+  distinguished,<a name="NtA371" href="#Nt371"><sup>[371]</sup></a>
+  viz.&mdash;</p>
+
+  <p>1. The solutions in different experiments have the same composition,
+  but the composition of the precipitate alters. In this case there must be
+  two solid phases.</p>
+
+  <p>2. The solutions in different experiments can have varying
+  composition, while the composition of the precipitate remains unchanged.
+  In this case only one solid phase exists, a definite compound.</p>
+
+  <p>3. The composition both of the solution and of the precipitate varies.
+  In this case the solid phase is a solid solution or a mixed crystal.</p>
+
+  <p>In order, therefore, to decide what is the nature of a precipitate
+  produced by the hydrolysis of a normal salt, it is only necessary to
+  ascertain whether and how the composition of the precipitate alters with
+  alteration in the composition of the solution. If the composition of the
+  solution is represented by abscissæ, and the composition of the
+  precipitate by ordinates, the form of the curves obtained would enable us
+  to answer our question; for vertical lines would indicate the presence of
+  two solid phases (1st case), horizontal lines the presence of only one
+  solid phase (2nd case), and slanting lines the presence of mixed crystals
+  (3rd case). This method of representation cannot, however, be carried out
+  in most cases. It is, however, <!-- Page 298 --><span class="pagenum"><a
+  name="page298"></a>{298}</span>generally possible to find one pair or
+  several pairs of components, the <i>relative amounts</i> of which in the
+  solution or in the precipitate undergo change when, and only when, the
+  composition of the solution or of the precipitate changes. Thus, in the
+  case of bismuth, nitrate, and water, we can represent the ratio of
+  Bi<sub>2</sub>O<sub>3</sub> : N<sub>2</sub>O<sub>5</sub> in the
+  precipitate as ordinates, and N<sub>2</sub>O<sub>5</sub> : H<sub>2</sub>O
+  in the solution as abscissæ. A horizontal line then indicates a single
+  solid phase, and a vertical line two solid phases. An example of this is
+  given in Fig. 118.<a name="NtA372" href="#Nt372"><sup>[372]</sup></a></p>
+
+  <div class="figcenter" style="width:31%;">
+      <a href="images/fig118.png"><img style="width:100%" src="images/fig118.png"
+      alt="Fig. 118" title="Fig. 118" /></a>
+    <span class="sc">Fig.</span> 118.
+  </div>
+
+  <p><b>Bi<sub>2</sub>O<sub>3</sub>&mdash;N<sub>2</sub>O<sub>5</sub>&mdash;H<sub>2</sub>O.</b>&mdash;Although
+  various systems have been studied in which there is formation of basic
+  salts,<a name="NtA373" href="#Nt373"><sup>[373]</sup></a> we shall
+  content ourselves here with the description of some of the conditions for
+  the formation of basic salts of bismuth nitrate, and for their
+  equilibrium in contact with solutions.<a name="NtA374"
+  href="#Nt374"><sup>[374]</sup></a></p>
+
+  <p>Three normal salts of bismuth oxide and nitric acid are known, viz.
+  Bi<sub>2</sub>O<sub>3</sub>,3N<sub>2</sub>O<sub>5</sub>,10H<sub>2</sub>O(S<sub>10</sub>);
+  Bi<sub>2</sub>O<sub>3</sub>,3N<sub>2</sub>O<sub>5</sub>,4H<sub>2</sub>O(S<sub>4</sub>);
+  and
+  Bi<sub>2</sub>O<sub>3</sub>,3N<sub>2</sub>O<sub>5</sub>,3H<sub>2</sub>O(S<sub>3</sub>).
+  Besides these normal salts, there are the following basic
+  salts:&mdash;</p>
+
+<p><!-- Page 299 --><span class="pagenum"><a name="page299"></a>{299}</span></p>
+
+<table class="nobctr" summary="Basic bismuth nitrates" title="Basic bismuth nitrates">
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; Bi<sub>2</sub>O<sub>3</sub>,N<sub>2</sub>O<sub>5</sub>,2H<sub>2</sub>O </td><td class="nob" style="text-align:left"> (represented by B<sub>1-1-2</sub>)</td></tr>
+<tr><td class="nob" style="text-align:left"> &nbsp; &nbsp; Bi<sub>2</sub>O<sub>3</sub>,N<sub>2</sub>O<sub>5</sub>,H<sub>2</sub>O </td><td class="nob" style="text-align:left"> (&nbsp; &nbsp; &nbsp; &nbsp; ,,&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp;B<sub>1-1-1</sub>)</td></tr>
+<tr><td class="nob" style="text-align:left"> 6Bi<sub>2</sub>O<sub>3</sub>,5N<sub>2</sub>O<sub>5</sub>,9H<sub>2</sub>O </td><td class="nob" style="text-align:left"> (&nbsp; &nbsp; &nbsp; &nbsp; ,,&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp;B<sub>6-5-9</sub>)</td></tr>
+<tr><td class="nob" style="text-align:left"> 2Bi<sub>2</sub>O<sub>3</sub>,N<sub>2</sub>O<sub>5</sub>,H<sub>2</sub>O </td><td class="nob" style="text-align:left"> (&nbsp; &nbsp; &nbsp; &nbsp; ,,&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ,, &nbsp;B<sub>2-1-1</sub>)</td></tr>
+</table>
+
+  <p>Probably some others also exist. The problem now is to find the
+  conditions under which these different normal and basic salts can be in
+  equilibrium with solutions of varying concentration of the three
+  components. Having determined the equilibrium conditions for the
+  different salts, it is then possible to construct a model similar to that
+  for MgCl<sub>2</sub>&mdash;KCl&mdash;H<sub>2</sub>O or for
+  FeCl<sub>3</sub>&mdash;HCl&mdash;H<sub>2</sub>O, from which it will be
+  possible to determine the limits of stability of the different salts, and
+  to predict what will occur when we bring the salts in contact with
+  solutions of nitric acid of different concentrations and at different
+  temperatures.</p>
+
+  <p>For our present purpose it is sufficient to pick out only some of the
+  equilibria which have been studied, and which are represented in the
+  model (Fig. 119). In this case use has been made of the triangular method
+  of representation, so that the surface of the model lies within the
+  prism.</p>
+
+  <div class="figcenter" style="width:34%;">
+      <a href="images/fig119.png"><img style="width:100%" src="images/fig119.png"
+      alt="Fig. 119" title="Fig. 119" /></a>
+    <span class="sc">Fig.</span> 119.
+  </div>
+
+  <p>This model shows the three surfaces, A, B, and C, which represent the
+  conditions for the stable existence of the salts B<sub>1-1-1</sub>,
+  S<sub>10</sub>, and S<sub>3</sub> in contact with solution at different
+  <!-- Page 300 --><span class="pagenum"><a
+  name="page300"></a>{300}</span>temperatures. The front surface of the
+  model represents the temperature 9°, and the farther end the temperature
+  75.5°. The dotted curve represents the isotherm for 20°. The prominences
+  between the surfaces represent, of course, solutions which are saturated
+  in respect of two solid phases. Thus, for example, <i>pabc</i> represents
+  solutions in equilibrium with B<sub>1-1-1</sub> and S<sub>10</sub>; and
+  the ridge <i>qdc</i>, solutions in equilibrium with S<sub>10</sub> and
+  S<sub>3</sub>. The point <i>b</i>, which lies at 75.5°, is the point of
+  maximum temperature for S<sub>10</sub>. If the temperature is raised
+  above this point, S<sub>10</sub> decomposes into the basic salt
+  B<sub>1-1-1</sub> and solution. This point is therefore analogous to the
+  point M in the carnallite model, at which this salt decomposes into
+  potassium chloride and solution (p.&nbsp;<a href="#page284">284</a>); or to
+  the point at which the salt 2FeCl<sub>3</sub>,2HCl,12H<sub>2</sub>O
+  decomposes into 2FeCl<sub>3</sub>,12H<sub>2</sub>O and solution (p.&nbsp;<a
+  href="#page294">294</a>). The curve <i>pab</i> has been followed to the
+  temperature of 72° (point <i>c</i>). The end of the model is incomplete,
+  but it is probable that in the neighbourhood of the point <i>c</i> there
+  exists a quintuple point at which the basic salt B<sub>1-2-2</sub>
+  appears. In the neighbourhood of <i>e</i> also there probably exists
+  another quintuple point at which S<sub>4</sub> is formed. These systems
+  have, however, not been studied.</p>
+
+  <p>The following tables give some of the numerical data:&mdash;</p>
+
+<p class="cenhead"><span class="sc">Isotherm for 20°.</span></p>
+
+<table class="nobctr" summary="Isotherm for 20°" title="Isotherm for 20°">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Solid Phase.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Composition of the solution. Gram-mols.<br />in 1000 gm.-mols. of water.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> Bi<sub>2</sub>O<sub>3</sub>
+</td><td class="toboleb" style="text-align:center"> N<sub>2</sub>O<sub>5</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left;"> B<sub>1-1-1</sub> </td><td class="leb" style="text-align:right; padding-right:3em;"> 10.50 </td><td class="leb" style="text-align:right; padding-right:3em;"> 38.65</td></tr>
+<tr><td class="rib" style="text-align:left;"> &mdash; </td><td class="leb" style="text-align:right; padding-right:3em;"> 27.20 </td><td class="leb" style="text-align:right; padding-right:3em;"> 83.84</td></tr>
+<tr><td class="rib" style="text-align:left;"> B<sub>1-1-1</sub>; S<sub>10</sub>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 30.15 </td><td class="leb" style="text-align:right; padding-right:3em;"> 97.97</td></tr>
+<tr><td class="rib" style="text-align:left;"> S<sub>10</sub> </td><td class="leb" style="text-align:right; padding-right:3em;"> 29.70 </td><td class="leb" style="text-align:right; padding-right:3em;"> 96.57</td></tr>
+<tr><td class="rib" style="text-align:left;"> &mdash; </td><td class="leb" style="text-align:right; padding-right:3em;"> 19.65 </td><td class="leb" style="text-align:right; padding-right:3em;"> 98.76</td></tr>
+<tr><td class="rib" style="text-align:left;"> &mdash; </td><td class="leb" style="text-align:right; padding-right:3em;"> 10.51 </td><td class="leb" style="text-align:right; padding-right:3em;"> 162.58</td></tr>
+<tr><td class="rib" style="text-align:left;"> &mdash; </td><td class="leb" style="text-align:right; padding-right:3em;"> 33.51 </td><td class="leb" style="text-align:right; padding-right:3em;"> 355.87</td></tr>
+<tr><td class="rib" style="text-align:left;"> S<sub>10</sub>; S<sub>3</sub> </td><td class="leb" style="text-align:right; padding-right:3em;"> 51.00 </td><td class="leb" style="text-align:right; padding-right:3em;"> 403.0 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> S<sub>3</sub> </td><td class="leb" style="text-align:right; padding-right:3em;"> 14.35 </td><td class="leb" style="text-align:right; padding-right:3em;"> 492.0 &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:left;"> &mdash; </td><td class="boleb" style="text-align:right; padding-right:3em;"> 7.45 </td><td class="boleb" style="text-align:right; padding-right:3em;"> 592.9 &nbsp;</td></tr>
+
+</table>
+
+<p><!-- Page 301 --><span class="pagenum"><a name="page301"></a>{301}</span></p>
+
+<p class="cenhead"><span class="sc">Systems in Equilibrium with B<sub>1-1-1</sub> and S<sub>10</sub> (Curve</span> <i>pabc</i>).</p>
+
+<table class="nobctr" summary="Systems in Equilibrium with B1-1-1 and S10" title="Systems in Equilibrium with B1-1-1 and S10">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Solid Phase.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Composition of the solution. Gram-mols.<br />in 1000 gm.-mols. of water.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> Bi<sub>2</sub>O<sub>3</sub>
+</td><td class="toboleb" style="text-align:center"> N<sub>2</sub>O<sub>5</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left;"> 9° </td><td class="leb" style="text-align:right; padding-right:3em;"> 26.7 &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 88.2 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> 20° (point <i>a</i>) </td><td class="leb" style="text-align:right; padding-right:3em;"> 30.15 </td><td class="leb" style="text-align:right; padding-right:3em;"> 97.97</td></tr>
+<tr><td class="rib" style="text-align:left;"> 30° </td><td class="leb" style="text-align:right; padding-right:3em;"> 33.6 &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 112.3 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> 50° </td><td class="leb" style="text-align:right; padding-right:3em;"> 41.8 &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 148.4 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> 65° </td><td class="leb" style="text-align:right; padding-right:3em;"> 57.21 </td><td class="leb" style="text-align:right; padding-right:3em;"> 190.8 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> 75.5° (point <i>b</i>)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 87.9 &nbsp; </td><td class="leb" style="text-align:right; padding-right:3em;"> 288.4 &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:left;"> 72° (point <i>c</i>) </td><td class="boleb" style="text-align:right; padding-right:3em;"> 96.0 &nbsp; </td><td class="boleb" style="text-align:right; padding-right:3em;"> 327.0 &nbsp;</td></tr>
+
+</table>
+
+<p class="cenhead"><span class="sc">Systems in Equilibrium with S<sub>10</sub> and S<sub>3</sub> (Curve</span> <i>qde</i>).</p>
+
+<table class="nobctr" summary="Systems in Equilibrium with S10 and S3" title="Systems in Equilibrium with S10 and S3">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Solid Phase.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Composition of the solution. Gram-mols.<br />in 1000 gm.-mols. of water.</td></tr>
+
+<tr><td class="toboleb" style="text-align:center"> Bi<sub>2</sub>O<sub>3</sub>
+</td><td class="toboleb" style="text-align:center"> N<sub>2</sub>O<sub>5</sub></td></tr>
+
+<tr><td class="rib" style="text-align:left;"> 11.5° </td><td class="leb" style="text-align:right; padding-right:3em;"> 44.5 </td><td class="leb" style="text-align:right; padding-right:3em;"> 396&nbsp; &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:left;"> 20° </td><td class="leb" style="text-align:right; padding-right:3em;"> 51.0 </td><td class="leb" style="text-align:right; padding-right:3em;"> 405.4</td></tr>
+<tr><td class="rib" style="text-align:left;"> 50° </td><td class="leb" style="text-align:right; padding-right:3em;"> 66.5 </td><td class="leb" style="text-align:right; padding-right:3em;"> 444.2</td></tr>
+
+<tr><td class="ribob" style="text-align:left;"> 65° </td><td class="boleb" style="text-align:right; padding-right:3em;"> 80.0 </td><td class="boleb" style="text-align:right; padding-right:3em;"> 454.4</td></tr>
+
+</table>
+
+  <p><b>Basic Mercury Salts.</b>&mdash;The Phase Rule has also been applied
+  by A. J. Cox<a name="NtA375" href="#Nt375"><sup>[375]</sup></a> in an
+  investigation of the basic salts of mercury, the result of which has been
+  to show that, of the salts mentioned in text-books, quite a number are
+  incorrectly stated to be chemical compounds or chemical individuals
+  (p.&nbsp;<a href="#page92">92</a>). The investigation, which was carried out
+  essentially in the manner described above, included the salts mentioned
+  in the following table; and of the basic salts said to be derived from
+  them, only those mentioned really exist. In the following table, the
+  numbers in the second column give the minimum values of the concentration
+  of the acid, expressed in equivalent normality, necessary for the
+  existence of the <!-- Page 302 --><span class="pagenum"><a
+  name="page302"></a>{302}</span>corresponding salts in contact with
+  solution at the temperature given in the third column:&mdash;</p>
+
+<table class="nobctr" summary="Salts of mercury" title="Salts of mercury">
+<tr><td class="toribob" style="text-align:center"> Salt.
+</td><td class="toboleb" style="text-align:center"> Normality of<br />acid.
+</td><td class="toboleb" style="text-align:center"> Temperature.</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> HgCrO<sub>4</sub> </td><td class="leb" style="text-align:center;"> 1.41 </td><td class="leb" style="text-align:center;"> 50°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 3HgO.CrO<sub>3</sub> </td><td class="leb" style="text-align:center;"> 2.6 × 10<sup>-4</sup> </td><td class="leb" style="text-align:center;"> 50°</td></tr>
+<tr><td class="rib" style="text-align:center;"> </td><td class="leb" style="text-align:center;"> </td><td class="leb" style="text-align:center;"></td></tr>
+<tr><td class="rib" style="text-align:center;"> Hg(NO<sub>3</sub>)<sub>2</sub>.H<sub>2</sub>O </td><td class="leb" style="text-align:center;"> 18.72 &nbsp; </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 3HgO.N<sub>2</sub>O<sub>5</sub> </td><td class="leb" style="text-align:center;"> &nbsp; 0.159 </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> </td><td class="leb" style="text-align:center;"> </td><td class="leb" style="text-align:center;"></td></tr>
+<tr><td class="rib" style="text-align:center;"> HgSO<sub>4</sub> </td><td class="leb" style="text-align:center;"> 6.87 </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 3HgO.SO<sub>3</sub> </td><td class="leb" style="text-align:center;"> 1.3 × 10<sup>-3</sup> </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> </td><td class="leb" style="text-align:center;"> </td><td class="leb" style="text-align:center;"></td></tr>
+<tr><td class="rib" style="text-align:center;"> HgF<sub>2</sub> </td><td class="leb" style="text-align:center;"> 1.14 </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> </td><td class="leb" style="text-align:center;"> </td><td class="leb" style="text-align:center;"></td></tr>
+<tr><td class="rib" style="text-align:center;"> HgNO<sub>3</sub>.H<sub>2</sub>O </td><td class="leb" style="text-align:center;"> 2.95 </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 5Hg<sub>2</sub>O.3N<sub>2</sub>O<sub>5</sub>.2H<sub>2</sub>O </td><td class="leb" style="text-align:center;"> ca. 0.293 &nbsp; &nbsp; </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 2Hg<sub>2</sub>O.N<sub>2</sub>O<sub>5</sub>(?) </td><td class="leb" style="text-align:center;"> &nbsp; 0.110 </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> 3Hg<sub>2</sub>O.N<sub>2</sub>O<sub>5</sub>.2H<sub>2</sub>O(?) </td><td class="leb" style="text-align:center;"> 1.7 × 10<sup>-3</sup> </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+<tr><td class="rib" style="text-align:center;"> </td><td class="leb" style="text-align:center;"> </td><td class="leb" style="text-align:center;"></td></tr>
+<tr><td class="rib" style="text-align:center;"> Hg<sub>2</sub>SO<sub>4</sub> </td><td class="leb" style="text-align:center;"> 4.2 × 10<sup>-3</sup> </td><td class="leb" style="text-align:center;"> 25°</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> 2Hg<sub>2</sub>O.SO<sub>3</sub>.H<sub>2</sub>O </td><td class="boleb" style="text-align:center;"> 5.6 × 10<sup>-4</sup> </td><td class="boleb" style="text-align:center;"> 25°</td></tr>
+
+</table>
+
+  <p>Mercuric fluoride does not form any basic salt.</p>
+
+  <p>Since two succeeding members of a series can coexist only in contact
+  with a solution of definite concentration, we can prepare acid solutions
+  of definite concentration by bringing an excess of two such salts in
+  contact with water.</p>
+
+  <p><b>Indirect Determination of the Composition of the Solid
+  Phase.</b>&mdash;It has already been shown (p.&nbsp;<a
+  href="#page228">228</a>) how the composition of the solid phase in a
+  system of two components can be determined without analysis, and we shall
+  now describe how this can be done in a system of three components.<a
+  name="NtA376" href="#Nt376"><sup>[376]</sup></a></p>
+
+  <p>We shall assume that we are dealing with the aqueous solution of two
+  salts which can give rise to a double salt, in which case we can
+  represent the solubility relations in a system of rectangular
+  co-ordinates. In this case we should obtain, as before (Fig. 120), the
+  isotherm <i>adcb</i>, if we express the <!-- Page 303 --><span
+  class="pagenum"><a name="page303"></a>{303}</span>composition of the
+  solution in gram-molecules of A or of B to 100 gram-molecules of
+  water.</p>
+
+  <div class="figcenter" style="width:23%;">
+      <a href="images/fig120.png"><img style="width:100%" src="images/fig120.png"
+      alt="Fig. 120" title="Fig. 120" /></a>
+    <span class="sc">Fig. 120.</span>
+  </div>
+
+  <p>Let us suppose, now, that the double salt is in equilibrium with the
+  solution at a definite temperature, and that the composition of the
+  solution is represented by the point <i>e</i>. The greater part of the
+  solution is now separated from the solid phase, and the latter,
+  <i>together with the adhering mother liquor</i>, is analyzed. The
+  composition (expressed, as before, in gram-molecules of A and B to 100
+  gram molecules of water) will be represented by a point (<i>e.g.</i>
+  <i>f</i>) on the line <i>e</i>S, where S represents the composition of
+  the double salt. That this is so will be evident when one considers that
+  the composition of the whole mass must lie between the composition of the
+  solution and that of the double salt, no matter what the relative amounts
+  of the solid phase and the mother liquor.</p>
+
+  <p>If, in a similar manner, we analyze a solution of a different
+  composition in equilibrium with the same double salt (not necessarily at
+  the same temperature as before), and also the mixture of solid phase and
+  solution, we shall obtain two other points, as, for example, <i>g</i> and
+  <i>h</i>, and the line joining these must likewise pass through S. The
+  method of finding the <!-- Page 304 --><span class="pagenum"><a
+  name="page304"></a>{304}</span>composition of an unknown double salt
+  consists, therefore, in finding, in the manner just described, the
+  position of two lines such as <i>ef</i> and <i>gh</i>. The point of
+  intersection of these lines then gives the composition of the double
+  salt.</p>
+
+  <p>If the double salt is anhydrous, the point S lies at infinity, and the
+  lines <i>ef</i> and <i>gh</i> are parallel to each other.</p>
+
+  <p>The same result is arrived at by means of the triangular method of
+  representation.<a name="NtA377" href="#Nt377"><sup>[377]</sup></a> If we
+  start with the three components in known amounts, and represent the
+  initial composition of the whole by a point in the triangle, and then
+  ascertain the final composition of the solution in equilibrium with the
+  solid phase at a definite temperature, the line joining the points
+  representing the initial and end concentration passes through the point
+  representing the composition of the solid phase. If two determinations
+  are made with solutions having different initial and final concentrations
+  in equilibrium with the same solid phase, then the point of intersection
+  of the two lines so obtained gives the composition of the solid
+  phase.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 305 --><span class="pagenum"><a name="page305"></a>{305}</span></p>
+
+<h3>CHAPTER XVII</h3>
+
+<p class="cenhead">ABSENCE OF A LIQUID PHASE</p>
+
+  <p>In the preceding chapters dealing with equilibria in three-component
+  systems, our attention was directed only to those cases in which liquid
+  solutions formed one or more phases. Mention must, however, be made of
+  certain systems which contain no liquid phase, and in which only solids
+  and gases are in equilibrium. Since, in all cases, there can be but one
+  gas phase, four solid phases will be necessary in order to form an
+  invariant system. When only three solid phases are present, the system is
+  univariant; and when only two solid phases coexist with gas, it is
+  bivariant. If, however, we make the restriction that the gas pressure is
+  constant, we diminish the variability by one.</p>
+
+  <p>On account of their great industrial importance, we shall describe
+  briefly some of the systems belonging to this class.</p>
+
+  <p><b>Iron, Carbon Monoxide, Carbon Dioxide.</b>&mdash;Some of the most
+  important systems of three components in which equilibrium exists between
+  solid and gas phases are those formed by the three components&mdash;iron,
+  carbon monoxide, and carbon dioxide&mdash;and they are of importance
+  especially for the study of the processes occurring in the blast
+  furnace.</p>
+
+  <p>If carbon monoxide is passed over reduced iron powder at a temperature
+  of about 600°, the iron is oxidized and the carbon monoxide reduced with
+  separation of carbon in accordance with the equation</p>
+
+<p class="cenhead">Fe + CO = FeO + C</p>
+
+  <p>This reaction is succeeded by the two reactions</p>
+
+<p class="cenhead">FeO + CO = Fe + CO<sub>2</sub><br />
+CO<sub>2</sub> + C = 2CO</p>
+
+<p><!-- Page 306 --><span class="pagenum"><a name="page306"></a>{306}</span></p>
+
+  <div class="figcenter" style="width:44%;">
+      <a href="images/fig121.png"><img style="width:100%" src="images/fig121.png"
+      alt="Fig. 121" title="Fig. 121" /></a>
+    <span class="sc">Fig. 121.</span>
+  </div>
+
+  <p>The former of these reactions is not complete, but leads to a definite
+  equilibrium. The result of the different reactions is therefore an
+  equilibrium between the three solid phases, carbon, iron, and ferrous
+  oxide, and the gas phase consisting of carbon monoxide and dioxide. We
+  have here four phases; and if the total pressure is maintained constant,
+  equilibrium can occur only at a definite temperature.</p>
+
+  <p>Since, under certain conditions, we can also have the reaction</p>
+
+<p class="cenhead">Fe<sub>3</sub>O<sub>4</sub> + CO = 3FeO + CO<sub>2</sub></p>
+
+<p><!-- Page 307 --><span class="pagenum"><a name="page307"></a>{307}</span></p>
+
+  <p>a second series of equilibria can be obtained of a character similar
+  to the former. These various equilibria have been investigated by Baur
+  and Glaessner,<a name="NtA378" href="#Nt378"><sup>[378]</sup></a> and the
+  following is a short account of the results of their work.</p>
+
+  <p>Mixtures of the solid phases in equilibrium with carbon monoxide and
+  dioxide were heated in a porcelain tube at a definite temperature until
+  equilibrium was produced, and the gas was then pumped off and analyzed.
+  The results which were obtained are given in the following tables, and
+  represented graphically in Fig. 121.</p>
+
+<p class="cenhead"><span class="sc">Solid Phases</span>: Fe<sub>3</sub>O<sub>4</sub>; FeO.</p>
+
+<table class="nobctr" summary="Gases over iron oxides" title="Gases over iron oxides">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> No.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Tube filled<br />with
+</td><td class="toribob" style="text-align:center" rowspan="2"> Duration of<br />the experiment<br />in hours.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toleb" style="text-align:center" colspan="2"> Percentage of</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> CO<sub>2</sub>
+</td><td class="boleb" style="text-align:center"> CO</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> 1 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 14 </td><td class="leb" style="text-align:left; padding-left:2em;"> 600° </td><td class="leb" style="text-align:left; padding-left:2em;"> 59.3 </td><td class="leb" style="text-align:left; padding-left:2em;"> 40.7</td></tr>
+<tr><td class="rib" style="text-align:right;"> 2 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 15 </td><td class="leb" style="text-align:left; padding-left:2em;"> 590° </td><td class="leb" style="text-align:left; padding-left:2em;"> 54.7 </td><td class="leb" style="text-align:left; padding-left:2em;"> 45.3</td></tr>
+<tr><td class="rib" style="text-align:right;"> 3 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 16 </td><td class="leb" style="text-align:left; padding-left:2em;"> 590° </td><td class="leb" style="text-align:left; padding-left:2em;"> 64.6 </td><td class="leb" style="text-align:left; padding-left:2em;"> 35.4</td></tr>
+<tr><td class="rib" style="text-align:right;"> 4 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> 590° </td><td class="leb" style="text-align:left; padding-left:2em;"> 58.4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 41.6</td></tr>
+<tr><td class="rib" style="text-align:right;"> 5 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 22 </td><td class="leb" style="text-align:left; padding-left:2em;"> 730° </td><td class="leb" style="text-align:left; padding-left:2em;"> 67.7 </td><td class="leb" style="text-align:left; padding-left:2em;"> 32.3</td></tr>
+<tr><td class="rib" style="text-align:right;"> 6 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 22 </td><td class="leb" style="text-align:left; padding-left:2em;"> 730° </td><td class="leb" style="text-align:left; padding-left:2em;"> 86.1 </td><td class="leb" style="text-align:left; padding-left:2em;"> 31.9</td></tr>
+<tr><td class="rib" style="text-align:right;"> 7 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 22 </td><td class="leb" style="text-align:left; padding-left:2em;"> 750° </td><td class="leb" style="text-align:left; padding-left:2em;"> 68.4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 31.6</td></tr>
+<tr><td class="rib" style="text-align:right;"> 8 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 22 </td><td class="leb" style="text-align:left; padding-left:2em;"> 610° </td><td class="leb" style="text-align:left; padding-left:2em;"> 64.9 </td><td class="leb" style="text-align:left; padding-left:2em;"> 35.1</td></tr>
+<tr><td class="rib" style="text-align:right;"> 9 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 23 </td><td class="leb" style="text-align:left; padding-left:2em;"> 420° </td><td class="leb" style="text-align:left; padding-left:2em;"> 56.0 </td><td class="leb" style="text-align:left; padding-left:2em;"> 44.0</td></tr>
+<tr><td class="rib" style="text-align:right;"> 10 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 47 </td><td class="leb" style="text-align:left; padding-left:2em;"> 350° </td><td class="leb" style="text-align:left; padding-left:2em;"> 65.6 </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.4</td></tr>
+<tr><td class="rib" style="text-align:right;"> 11 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 46 </td><td class="leb" style="text-align:left; padding-left:2em;"> 350° </td><td class="leb" style="text-align:left; padding-left:2em;"> 72.8 </td><td class="leb" style="text-align:left; padding-left:2em;"> 27.2</td></tr>
+<tr><td class="rib" style="text-align:right;"> 12 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 53 </td><td class="leb" style="text-align:left; padding-left:2em;"> 350° </td><td class="leb" style="text-align:left; padding-left:2em;"> 64.0 </td><td class="leb" style="text-align:left; padding-left:2em;"> 36.0</td></tr>
+<tr><td class="rib" style="text-align:right;"> 13 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 570° </td><td class="leb" style="text-align:left; padding-left:2em;"> 53.4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 46.6</td></tr>
+<tr><td class="rib" style="text-align:right;"> 14 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 19 </td><td class="leb" style="text-align:left; padding-left:2em;"> 680° </td><td class="leb" style="text-align:left; padding-left:2em;"> 60.5 </td><td class="leb" style="text-align:left; padding-left:2em;"> 39.5</td></tr>
+<tr><td class="rib" style="text-align:right;"> 15 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> 540° </td><td class="leb" style="text-align:left; padding-left:2em;"> 55.5 </td><td class="leb" style="text-align:left; padding-left:2em;"> 44.5</td></tr>
+<tr><td class="rib" style="text-align:right;"> 16 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 21 </td><td class="leb" style="text-align:left; padding-left:2em;"> 630° </td><td class="leb" style="text-align:left; padding-left:2em;"> 57.5 </td><td class="leb" style="text-align:left; padding-left:2em;"> 42.5</td></tr>
+<tr><td class="rib" style="text-align:right;"> 17 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 17 </td><td class="leb" style="text-align:left; padding-left:2em;"> 690° </td><td class="leb" style="text-align:left; padding-left:2em;"> 65.5 </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.5</td></tr>
+<tr><td class="rib" style="text-align:right;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 17 </td><td class="leb" style="text-align:left; padding-left:2em;"> 670° </td><td class="leb" style="text-align:left; padding-left:2em;"> 67.0 </td><td class="leb" style="text-align:left; padding-left:2em;"> 33.0</td></tr>
+<tr><td class="rib" style="text-align:right;"> 19 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> 410° </td><td class="leb" style="text-align:left; padding-left:2em;"> 58.5 </td><td class="leb" style="text-align:left; padding-left:2em;"> 41.5</td></tr>
+<tr><td class="rib" style="text-align:right;"> 20 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> 490° </td><td class="leb" style="text-align:left; padding-left:2em;"> 51.7 </td><td class="leb" style="text-align:left; padding-left:2em;"> 48.8</td></tr>
+<tr><td class="rib" style="text-align:right;"> 21 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 23 </td><td class="leb" style="text-align:left; padding-left:2em;"> 590° </td><td class="leb" style="text-align:left; padding-left:2em;"> 54.4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 45.6</td></tr>
+<tr><td class="rib" style="text-align:right;"> 22 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> &nbsp; 4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 950° </td><td class="leb" style="text-align:left; padding-left:2em;"> 77.0 </td><td class="leb" style="text-align:left; padding-left:2em;"> 23.0</td></tr>
+<tr><td class="rib" style="text-align:right;"> 23 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 15 </td><td class="leb" style="text-align:left; padding-left:2em;"> 850° </td><td class="leb" style="text-align:left; padding-left:2em;"> 73.4 </td><td class="leb" style="text-align:left; padding-left:2em;"> 26.6</td></tr>
+<tr><td class="rib" style="text-align:right;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> &nbsp; 8 </td><td class="leb" style="text-align:left; padding-left:2em;"> 800° </td><td class="leb" style="text-align:left; padding-left:2em;"> 71.2 </td><td class="leb" style="text-align:left; padding-left:2em;"> 28.8</td></tr>
+
+<tr><td class="ribob" style="text-align:right;"> 25 </td><td class="boleb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="boleb" style="text-align:left; padding-left:2em;"> 24 </td><td class="boleb" style="text-align:left; padding-left:2em;"> 540° </td><td class="boleb" style="text-align:left; padding-left:2em;"> 56.7 </td><td class="boleb" style="text-align:left; padding-left:2em;"> 43.3</td></tr>
+</table>
+
+<p><!-- Page 308 --><span class="pagenum"><a name="page308"></a>{308}</span></p>
+
+<p class="cenhead"><span class="sc">Solid Phases</span>: FeO; Fe.</p>
+
+<table class="nobctr" summary="Gases over iron and oxide" title="Gases over iron and oxide">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> No.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Tube filled<br />with
+</td><td class="toribob" style="text-align:center" rowspan="2"> Duration of<br />the experiment<br />in hours.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Temperature.
+</td><td class="toleb" style="text-align:center" colspan="2"> Percentage of</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> CO<sub>2</sub>
+</td><td class="boleb" style="text-align:center"> CO</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> I. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 15 </td><td class="leb" style="text-align:left; padding-left:2em;"> 800° </td><td class="leb" style="text-align:left; padding-left:2em;"> 35.2 </td><td class="leb" style="text-align:left; padding-left:2em;"> 64.8</td></tr>
+<tr><td class="rib" style="text-align:center;"> II. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 530° </td><td class="leb" style="text-align:left; padding-left:2em;"> 29.1 </td><td class="leb" style="text-align:left; padding-left:2em;"> 70.9</td></tr>
+<tr><td class="rib" style="text-align:center;"> III. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 13 </td><td class="leb" style="text-align:left; padding-left:2em;"> 880° </td><td class="leb" style="text-align:left; padding-left:2em;"> 30.2 </td><td class="leb" style="text-align:left; padding-left:2em;"> 69.6</td></tr>
+<tr><td class="rib" style="text-align:center;"> IV. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 24 </td><td class="leb" style="text-align:left; padding-left:2em;"> 870° </td><td class="leb" style="text-align:left; padding-left:2em;"> 32.3 </td><td class="leb" style="text-align:left; padding-left:2em;"> 67.7</td></tr>
+<tr><td class="rib" style="text-align:center;"> V. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 760° </td><td class="leb" style="text-align:left; padding-left:2em;"> 36.9 </td><td class="leb" style="text-align:left; padding-left:2em;"> 63.1</td></tr>
+<tr><td class="rib" style="text-align:center;"> VI. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 16 </td><td class="leb" style="text-align:left; padding-left:2em;"> 820° </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.7 </td><td class="leb" style="text-align:left; padding-left:2em;"> 65.3</td></tr>
+<tr><td class="rib" style="text-align:center;"> VII. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 730° </td><td class="leb" style="text-align:left; padding-left:2em;"> 41.1 </td><td class="leb" style="text-align:left; padding-left:2em;"> 58.9</td></tr>
+<tr><td class="rib" style="text-align:center;"> VIII. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 630° </td><td class="leb" style="text-align:left; padding-left:2em;"> 34.9 </td><td class="leb" style="text-align:left; padding-left:2em;"> 65.1</td></tr>
+<tr><td class="rib" style="text-align:center;"> IX. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="leb" style="text-align:left; padding-left:2em;"> 17 </td><td class="leb" style="text-align:left; padding-left:2em;"> 630° </td><td class="leb" style="text-align:left; padding-left:2em;"> 61.6 </td><td class="leb" style="text-align:left; padding-left:2em;"> 58.4</td></tr>
+<tr><td class="rib" style="text-align:center;"> X. </td><td class="leb" style="text-align:left; padding-left:2em;"> CO </td><td class="leb" style="text-align:left; padding-left:2em;"> 18 </td><td class="leb" style="text-align:left; padding-left:2em;"> 540° </td><td class="leb" style="text-align:left; padding-left:2em;"> 25.0 </td><td class="leb" style="text-align:left; padding-left:2em;"> 75.0</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> XI. </td><td class="boleb" style="text-align:left; padding-left:2em;"> CO<sub>2</sub> </td><td class="boleb" style="text-align:left; padding-left:2em;"> 25 </td><td class="boleb" style="text-align:left; padding-left:2em;"> 540° </td><td class="boleb" style="text-align:left; padding-left:2em;"> 36.5 </td><td class="boleb" style="text-align:left; padding-left:2em;"> 63.5</td></tr>
+
+</table>
+
+  <p>As is evident from the above tables and from the curves in Fig. 121,
+  the curve of equilibrium in the case of the reaction</p>
+
+<p class="cenhead">Fe<sub>3</sub>O<sub>4</sub> + CO = 3FeO + CO<sub>2</sub></p>
+
+  <p>exhibits a maximum for the ratio CO : CO<sub>2</sub>, at 490°, while,
+  for the reaction</p>
+
+<p class="cenhead">FeO + CO = Fe + CO<sub>2</sub></p>
+
+  <p>this ratio has a minimum value at 680°. From these curves can be
+  derived the conditions under which the different solid phases can exist
+  in contact with gas. Thus, for example, at a temperature of 690°, FeO and
+  Fe<sub>3</sub>O<sub>4</sub> can coexist with a mixture of 65.5 per cent.
+  of CO<sub>2</sub> and 34.5 per cent. of CO. If the partial pressure of
+  CO<sub>2</sub> is increased, there occurs the reaction</p>
+
+<p class="cenhead">3FeO + CO<sub>2</sub> = Fe<sub>3</sub>O<sub>4</sub> + CO</p>
+
+  <p>and if carbon dioxide is added in sufficient amount, the ferrous oxide
+  finally disappears completely. If, on the other hand, the partial
+  pressure of CO is increased, there occurs the reaction</p>
+
+<p class="cenhead">Fe<sub>3</sub>O<sub>4</sub> + CO = 3FeO + CO<sub>2</sub></p>
+
+  <p>and all the ferric oxide can be made to disappear. We see, therefore,
+  that Fe<sub>3</sub>O<sub>4</sub> can only exist at temperatures and in
+  <!-- Page 309 --><span class="pagenum"><a
+  name="page309"></a>{309}</span>contact with mixtures of carbon monoxide
+  and dioxide, represented by the area which lies below the under curve in
+  Fig. 121. Similarly, the region of existence of FeO is that represented
+  by the area between the two curves; while metallic iron can exist under
+  the conditions of temperature and composition of gas phase represented by
+  the area above the upper curve in Fig. 121. If, therefore, ferric oxide
+  or metallic iron is heated for a sufficiently long time at temperatures
+  above 700° (to the right of the dotted line; <i>vide infra</i>), complete
+  transformation to ferrous oxide finally occurs.</p>
+
+  <p>In another series of equilibria which can be obtained, carbon is one
+  of the solid phases. In Fig. 121 the equilibria between carbon, carbon
+  monoxide, and carbon dioxide under pressures of one and of a quarter
+  atmosphere, are represented by dotted lines.<a name="NtA379"
+  href="#Nt379"><sup>[379]</sup></a></p>
+
+  <p>If we consider only the dotted line on the right, representing the
+  equilibria under atmospheric pressure, we see that the points in which
+  the dotted line cuts the other two curves must represent systems in which
+  carbon monoxide and carbon dioxide are in equilibrium with FeO +
+  Fe<sub>3</sub>O<sub>4</sub> + C, on the one hand, and with Fe + FeO + C
+  on the other. These systems can only exist at one definite temperature,
+  if we make the restriction that the pressure is maintained constant
+  (atmospheric pressure). Starting, therefore, with the equilibrium FeO +
+  Fe<sub>3</sub>O<sub>4</sub> + CO + CO<sub>2</sub> at a temperature of
+  about 670°, and then add carbon to the system, the reaction</p>
+
+<p class="cenhead">C + CO<sub>2</sub> = 2CO</p>
+
+  <p>will occur, because the concentration of CO<sub>2</sub> is greater
+  than what corresponds with the system FeO + Fe<sub>3</sub>O<sub>4</sub> +
+  C in equilibrium with carbon monoxide and dioxide. In consequence of this
+  reaction, the equilibrium between FeO + Fe<sub>3</sub>O<sub>4</sub> and
+  the gas phase is disturbed, and the change in the composition of the gas
+  phase is opposed by the reaction Fe<sub>3</sub>O<sub>4</sub> + CO = 3FeO
+  + CO<sub>2</sub>, which continues until either all the carbon <!-- Page
+  310 --><span class="pagenum"><a name="page310"></a>{310}</span>or all the
+  ferric oxide is used up. If the ferric oxide first disappears, the
+  equilibrium corresponds with a point on the dotted line in the middle
+  area of Fig. 121, which represents equilibria between FeO + C as solid
+  phases, and a mixture of carbon monoxide and dioxide as gas phase. If the
+  temperature is higher than 685°, at which temperature the curve for
+  C&mdash;CO&mdash;CO<sub>2</sub> cuts that for
+  Fe&mdash;FeO&mdash;CO&mdash;CO<sub>2</sub>; then, when all the ferric
+  oxide has disappeared, the concentration of CO<sub>2</sub> is still too
+  great for the coexistence of FeO and C. Consequently, there occurs the
+  reaction C + CO<sub>2</sub> = 2CO, and the composition of the gas phase
+  alters until a point on the upper curve is reached. A further increase in
+  the concentration of CO is opposed by the reaction FeO + CO = Fe +
+  CO<sub>2</sub>, and the pressure remains constant until all the ferrous
+  oxide is reduced and only iron and carbon remain in equilibrium with gas.
+  If the quantities of the substances have been rightly chosen, we
+  ultimately reach a point on the dotted curve in the upper part of Fig.
+  121.</p>
+
+  <p>Fig. 121 shows us, also, what are the conditions under which the
+  reduction of ferric to ferrous oxide by carbon can occur. Let us suppose,
+  for example, that we start with a mixture of carbon monoxide and dioxide
+  at about 600° (the lowest point on the dotted line), and maintain the
+  total pressure constant and equal to one atmosphere. If the temperature
+  is increased, the concentration of the carbon dioxide will diminish,
+  owing to the reaction C + CO<sub>2</sub> = 2CO, but the ferric oxide will
+  undergo no change until the temperature reaches 647°, the point of
+  intersection of the dotted curve with the curve for FeO and
+  Fe<sub>3</sub>O<sub>4</sub>. At this point further increase in the
+  concentration of carbon monoxide is opposed by the reduction of ferric
+  oxide in accordance with the equation Fe<sub>3</sub>O<sub>4</sub> + CO =
+  3FeO + CO<sub>2</sub>. The pressure, therefore, remains constant until
+  all the ferric oxide has disappeared. If the temperature is still further
+  raised, we again obtain a univariant system, FeO + C, in equilibrium with
+  gas (univariant because the total pressure is constant); and if the
+  temperature is raised the composition of the gas must undergo change.
+  This is effected by the reaction C + CO<sub>2</sub> = 2CO. When the <!--
+  Page 311 --><span class="pagenum"><a
+  name="page311"></a>{311}</span>temperature rises to 685°, at which the
+  dotted curve cuts the curve for Fe&mdash;FeO, further change is prevented
+  by the reaction FeO + CO = Fe + CO<sub>2</sub>. When all the ferrous
+  oxide is used up, we obtain the system Fe + C in equilibrium with gas. If
+  the temperature is now raised, the composition of the gas undergoes
+  change, as shown by the dotted line. The two temperatures, 647° and 685°,
+  give, evidently, the limits within which ferric or ferrous oxide can be
+  reduced directly by carbon.</p>
+
+  <p>It is further evident that at any temperature to the right of the
+  dotted line, carbon is unstable in presence of iron or its oxides; while
+  at temperatures lower than those represented by the dotted line, it is
+  stable. In the blast furnace, therefore, separation of carbon can occur
+  only at lower temperatures, and the carbon must disappear on raising the
+  temperature.</p>
+
+  <p>Finally, it may be remarked that the equilibrium curves show that
+  ferrous oxide is most easily reduced at 680°, since the concentration of
+  the carbon monoxide required at this temperature is a minimum. On the
+  other hand, ferric oxide is reduced with greatest difficulty at 490°,
+  since at this temperature the requisite concentration of carbon monoxide
+  is a maximum.</p>
+
+  <p>Other equilibria between solid and gas phases are: Equilibrium between
+  iron, ferric oxide, water vapour, and hydrogen,<a name="NtA380"
+  href="#Nt380"><sup>[380]</sup></a> and the equilibria between carbon,
+  carbon monoxide, carbon dioxide, water vapour, and hydrogen,<a
+  name="NtA381" href="#Nt381"><sup>[381]</sup></a> which is of importance
+  for the manufacture of water gas.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 312 --><span class="pagenum"><a name="page312"></a>{312}</span></p>
+
+<h3>CHAPTER XVIII</h3>
+
+<p class="cenhead">SYSTEMS OF FOUR COMPONENTS</p>
+
+  <p>In the systems which have so far been studied, we have met with cases
+  where two or three components could enter into combination; but in no
+  case did we find double decomposition occurring. The reason of this is
+  that in the systems previously studied, in which double decomposition
+  might have been possible, namely in those systems in which two salts
+  acted as components, the restriction was imposed that either the basic or
+  the acid constituent of these salts must be the same; a restriction
+  imposed, indeed, for the very purpose of excluding double decomposition.
+  Now, however, we shall allow this restriction to fall, thereby extending
+  the range of study.</p>
+
+  <p>Hitherto, in connection with four-component systems, the attention has
+  been directed solely to the study of aqueous solutions of salts, and more
+  especially of the salts which occur in sea-water, <i>i.e.</i> chiefly,
+  the sulphates and chlorides of magnesium, potassium, and sodium. The
+  importance of these investigations will be recognized when one recollects
+  that by the evaporation of sea-water there have been formed the enormous
+  salt-beds at Stassfurt, which constitute at present the chief source of
+  the sulphates and chlorides of magnesium and potassium. The
+  investigations, therefore, are not only of great geological interest as
+  tending to elucidate the conditions under which these salt-beds have been
+  formed, but are of no less importance for the industrial working of the
+  deposits.</p>
+
+  <p>It is, however, not the intention to enter here into any detailed
+  description of the different systems which have so far been studied, and
+  of the sometimes very complex relationships <!-- Page 313 --><span
+  class="pagenum"><a name="page313"></a>{313}</span>met with, but merely to
+  refer briefly to some points of more general import in connection with
+  these systems.<a name="NtA382" href="#Nt382"><sup>[382]</sup></a></p>
+
+  <p><b>Reciprocal Salt-Pairs. Choice of Components.</b>&mdash;When two
+  salts undergo double decomposition, the interaction can be expressed by
+  an equation such as</p>
+
+<p class="cenhead">NH<sub>4</sub>Cl + NaNO<sub>3</sub> = NaCl + NH<sub>4</sub>NO<sub>3</sub></p>
+
+  <p>Since one pair of salts&mdash;NaCl +
+  NH<sub>4</sub>NO<sub>3</sub>&mdash;is formed from the other
+  pair&mdash;NH<sub>4</sub>Cl + NaNO<sub>3</sub>&mdash;by double
+  decomposition, the two pairs of salts are known as <i>reciprocal
+  salt-pairs</i>.<a name="NtA383" href="#Nt383"><sup>[383]</sup></a> It is
+  with systems in which the component salts form reciprocal salt-pairs that
+  we have to deal here.</p>
+
+  <p>It must be noted, however, that the four salts formed by two
+  reciprocal salt-pairs do not constitute a system of four, but only of
+  <i>three</i> components. This will be understood if it is recalled that
+  only so many constituents are taken as components as are necessary to
+  <i>express</i> the composition of all the phases present (p.&nbsp;<a
+  href="#page12">12</a>). It will be seen, now, that the composition of
+  each of the four salts which can be present together can be expressed in
+  terms of three of them. Thus, for example, in the case of
+  NH<sub>4</sub>Cl, NaNO<sub>3</sub>, NH<sub>4</sub>NO<sub>3</sub>, NaCl,
+  we can express the composition of NH<sub>4</sub>Cl by
+  NH<sub>4</sub>NO<sub>3</sub> + NaCl - NaNO<sub>3</sub>; or of
+  NaNO<sub>3</sub> by NH<sub>4</sub>NO<sub>3</sub> + NaCl -
+  NH<sub>4</sub>Cl. In all these cases it will be seen that negative
+  quantities of one of the components must be employed; but that we have
+  seen to be quite permissible (p.&nbsp;<a href="#page12">12</a>). The number of
+  components is, therefore, three; but any three of the four salts can be
+  chosen.</p>
+
+  <p>Since, then, two reciprocal salt-pairs constitute only three <!-- Page
+  314 --><span class="pagenum"><a name="page314"></a>{314}</span>components
+  or independently variable constituents, another component is necessary in
+  order to obtain a four-component system. As such, we shall choose
+  water.</p>
+
+  <p><b>Transition Point.</b>&mdash;In the case of the formation of double
+  salts from two single salts, we saw that there was a point&mdash;the
+  <i>quintuple point</i>&mdash;at which five phases could coexist. This
+  point we also saw to be a transition point, on one side of which the
+  double salt, on the other side the two single salts in contact with
+  solution, were found to be the stable system. A similar behaviour is
+  found in the case of reciprocal salt-pairs. The four-component system,
+  two reciprocal salt-pairs and water, can give rise to an invariant system
+  in which the six phases, four salts, solution, vapour, can coexist; the
+  temperature at which this is possible constitutes a <i>sextuple
+  point</i>. Now, this sextuple point is also a transition point, on the
+  one side of which the one salt-pair, on the other side the reciprocal
+  salt-pair, is stable in contact with solution.</p>
+
+  <p>The sextuple point is the point of intersection of the curves of six
+  univariant systems, viz. four solubility curves with three solid phases
+  each, a vapour-pressure curve for the system: two reciprocal
+  salt-pairs&mdash;vapour; and a transition curve for the condensed system:
+  two reciprocal salt-pairs&mdash;solution. If we omit the vapour phase and
+  work under atmospheric pressure (in open vessels), we find that the
+  transition point is the point of intersection of four solubility
+  curves.</p>
+
+  <p>Just as in the case of three-component systems we saw that the
+  presence of one of the single salts along with the double salt was
+  necessary in order to give a univariant system, so in the four-component
+  systems the presence of a third salt is necessary as solid phase along
+  with one of the salt-pairs. In the case of the reciprocal salt-pairs
+  mentioned above, the transition point would be the point of intersection
+  of the solubility curves of the systems with the following groups of
+  salts as solid phases: Below the transition point: NH<sub>4</sub>Cl +
+  NaNO<sub>3</sub> + NaCl; NH<sub>4</sub>Cl + NaNO<sub>3</sub> +
+  NH<sub>4</sub>NO<sub>3</sub>; above the transition point: NaCl +
+  NH<sub>4</sub>NO<sub>3</sub> + NaNO<sub>3</sub>; NaCl +
+  NH<sub>4</sub>NO<sub>3</sub> + NH<sub>4</sub>Cl. From this we see that
+  the two salts NH<sub>4</sub>Cl and NaNO<sub>3</sub> would be able to
+  exist together with solution below the transition point, but not above
+  it. This transition point has not been determined. <!-- Page 315 --><span
+  class="pagenum"><a name="page315"></a>{315}</span></p>
+
+  <p><b>Formation of Double Salts.</b>&mdash;In all cases of four-component
+  systems so far studied, the transition points have not been points at
+  which one salt-pair passed into its reciprocal, but at which a double
+  salt was formed. Thus, at 4.4° Glauber's salt and potassium chloride form
+  glaserite and sodium chloride, according to the equation</p>
+
+<p class="cenhead">2Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O + 3KCl = K<sub>3</sub>Na(SO<sub>4</sub>)<sub>2</sub> + 3NaCl + 20H<sub>2</sub>O</p>
+
+  <p>Above the transition point, therefore, there would be
+  K<sub>3</sub>Na(SO<sub>4</sub>)<sub>2</sub>, NaCl and KCl; and it may be
+  considered that at a higher temperature the double salt would interact
+  with the potassium chloride according to the equation</p>
+
+<p class="cenhead">K<sub>3</sub>Na(SO<sub>4</sub>)<sub>2</sub> + KCl = 2K<sub>2</sub>SO<sub>4</sub> + NaCl</p>
+
+  <p>thus giving the reciprocal of the original salt-pair. This point has,
+  however, not been experimentally realized.<a name="NtA384"
+  href="#Nt384"><sup>[384]</sup></a></p>
+
+  <p><b>Transition Interval.</b>&mdash;A double salt, we learned (p.&nbsp;<a
+  href="#page277">277</a>), when brought in contact with water at the
+  transition point undergoes partial decomposition with separation of one
+  of the constituent salts; and only after a certain range of temperature
+  (transition interval) has been passed, can a pure saturated solution be
+  obtained. A similar behaviour is also found in the case of reciprocal
+  salt-pairs. If one of the salt-pairs is brought in contact with water at
+  the transition point, interaction will occur and one of the salts of the
+  reciprocal salt-pair will be deposited; and this will be the case
+  throughout a certain range of temperature, after which it will be
+  possible to prepare a solution saturated only for the one salt-pair. In
+  the case of ammonium chloride and sodium nitrate the lower limit of the
+  transition interval is 5.5°, so that above this temperature and up to
+  that of the transition point (unknown), ammonium chloride and sodium
+  nitrate in contact with water would give rise to a third salt by double
+  decomposition, in this case to sodium chloride.<a name="NtA385"
+  href="#Nt385"><sup>[385]</sup></a></p>
+
+<p><!-- Page 316 --><span class="pagenum"><a name="page316"></a>{316}</span></p>
+
+  <p><b>Graphic Representation.</b>&mdash;For the graphic representation of
+  systems of four components, four axes may be chosen intersecting at a
+  point like the edges of a regular octahedron (Fig. 122).<a name="NtA386"
+  href="#Nt386"><sup>[386]</sup></a> Along these different axes the
+  equivalent molecular amounts of the different salts are measured.</p>
+
+  <div class="figright" style="width:19%;">
+      <a href="images/fig123.png"><img style="width:100%" src="images/fig123.png"
+      alt="Fig. 123" title="Fig. 123" /></a>
+    <span class="sc">Fig.</span> 123.
+  </div>
+
+  <div class="figright" style="width:21%;">
+      <a href="images/fig122.png"><img style="width:100%" src="images/fig122.png"
+      alt="Fig. 122" title="Fig. 122" /></a>
+    <span class="sc">Fig.</span> 122.
+  </div>
+
+  <p>To represent a given system consisting of <i>x</i>B, <i>y</i>C, and
+  <i>z</i>D in a given amount of water (where B, C, and D represent
+  equivalent molecular amounts of the salts), measure off on OB and OC
+  lengths equal to <i>x</i> and <i>y</i> respectively. The point of
+  intersection <i>a</i> (Fig. 122) represents a solution containing
+  <i>x</i>B and <i>y</i>C (<i>ab</i> = <i>x</i>; <i>ac</i> = <i>y</i>).
+  From <i>a</i> a line <i>a</i>P is drawn parallel to OD and equal to
+  <i>z</i>. P then represents the solution of the above composition.</p>
+
+  <p>It is usual, however, not to employ the three-dimensional figure, but
+  its horizontal and vertical projections. Fig. 122, if projected on the
+  base of the octahedron, would yield a diagram such as is shown in Fig.
+  123. The projection of the edges of the octahedron form two axes at right
+  angles and give rise to four quadrants similar to those employed for the
+  representation of ternary solutions (p.&nbsp;<a href="#page273">273</a>).
+  Here, the point <i>a</i> represents a ternary solution saturated with
+  respect to B and C; and <i>a</i>P, quaternary solutions in equilibrium
+  with the same two salts as solid phases. Such a diagram represents the
+  conditions of equilibrium only for one definite temperature, and
+  corresponds, therefore, to the isothermal diagrams for ternary systems
+  (p.&nbsp;<a href="#page273">273</a>). In such a diagram, since the temperature
+  and <!-- Page 317 --><span class="pagenum"><a
+  name="page317"></a>{317}</span>pressure are constant (vessels open to the
+  air), a surface will represent a solution in equilibrium with only one
+  solid phase; a line, a solution with two solid phases, and a point, one
+  in equilibrium with three solid phases.</p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/fig124.png"><img style="width:100%" src="images/fig124.png"
+      alt="Fig. 124" title="Fig. 124" /></a>
+    <span class="sc">Fig.</span> 124.
+  </div>
+
+  <p><b>Example.</b>&mdash;As an example of the complete isothermal
+  diagram, there may be given one representing the equilibria in the system
+  composed of water and the reciprocal salt-pair sodium
+  sulphate&mdash;potassium chloride for the temperature 0° (Fig. 124).<a
+  name="NtA387" href="#Nt387"><sup>[387]</sup></a> The amounts of the
+  different salts are measured along the four axes, and the composition of
+  the solution is <!-- Page 318 --><span class="pagenum"><a
+  name="page318"></a>{318}</span>expressed in equivalent gram-molecules per
+  1000 gram-molecules of water.<a name="NtA388"
+  href="#Nt388"><sup>[388]</sup></a></p>
+
+  <p>The outline of this figure represents four ternary solutions in which
+  the component salts have a common acid or basic constituent; viz. sodium
+  chloride&mdash;sodium sulphate, sodium sulphate&mdash;potassium sulphate,
+  potassium sulphate&mdash;potassium chloride, potassium
+  chloride&mdash;sodium chloride. These four sets of curves are therefore
+  similar to those discussed in the previous chapter. In the case of sodium
+  and potassium sulphate, a double salt, <i>glaserite</i>
+  [K<sub>3</sub>Na(SO<sub>4</sub>)<sub>2</sub>] is formed. Whether
+  glaserite is really a definite compound or not is still a matter of
+  doubt, since isomorphic mixtures of Na<sub>2</sub>SO<sub>4</sub> and
+  K<sub>2</sub>SO<sub>4</sub> have been obtained. According to van't Hoff
+  and Barscholl,<a name="NtA389" href="#Nt389"><sup>[389]</sup></a>
+  glaserite is an isomorphous mixture; but Gossner<a name="NtA390"
+  href="#Nt390"><sup>[390]</sup></a> considers it to be a definite compound
+  having the formula K<sub>3</sub>Na(SO<sub>4</sub>)<sub>2</sub>. Points
+  VIII. and IX. represent solutions saturated with respect to glaserite and
+  sodium sulphate, and glaserite and potassium sulphate respectively.</p>
+
+  <p>The lines which pass inwards from these boundary curves represent
+  solutions containing three salts, but in contact with only two solid
+  phases; and the points where three lines meet, or where three fields
+  meet, represent solutions in equilibrium with three solid phases; with
+  the phases, namely, belonging to the three concurrent fields.</p>
+
+  <p>If it is desired to represent a solution containing the salts say in
+  the proportions, 51Na<sub>2</sub>Cl<sub>2</sub>,
+  9.5K<sub>2</sub>Cl<sub>2</sub>, 3.5K<sub>2</sub>SO<sub>4</sub>, the
+  difficulty is met with that two of the salts, sodium chloride and
+  potassium sulphate, lie on opposite axes. To overcome this difficulty the
+  difference 51 - 3.5 = 47.5 is taken and measured off along the sodium
+  chloride axis; and the solution is therefore represented by the point
+  47.5Na<sub>2</sub>Cl<sub>2</sub>, 9.5K<sub>2</sub>Cl<sub>2</sub>. In
+  order, therefore, to find the amount of potassium sulphate present <!--
+  Page 319 --><span class="pagenum"><a name="page319"></a>{319}</span>from
+  such a diagram, it is necessary to know the total number of salt
+  molecules in the solution. When this is known, it is only necessary to
+  subtract from it the sum of the molecules of sodium and potassium
+  chloride, and the result is equal to twice the number of potassium
+  sulphate molecules. Thus, in the above example, the total number of salt
+  molecules is 64. The number of molecules of sodium and potassium chloride
+  is 57; 64 - 57 = 7, and therefore the number of potassium sulphate
+  molecules is 3.5.</p>
+
+  <p>Another method of representation employed is to indicate the amounts
+  of only two of the salts in a plane diagram, and to measure off the total
+  number of molecules along a vertical axis. In this way a solid model is
+  obtained.</p>
+
+  <p>The numerical data from which Fig. 124 was constructed are contained
+  in the following table, which gives the composition of the different
+  solutions at 0°:&mdash;<a name="NtA391"
+  href="#Nt391"><sup>[391]</sup></a></p>
+
+<table class="nobctr" summary="Solutions in Na-K-Cl-SO4 system" title="Solutions in Na-K-Cl-SO4 system">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Point.
+</td><td class="toribob" style="text-align:center" rowspan="2" colspan="3"> Solid phases.
+</td><td class="toboleb" style="text-align:center" colspan="4"> Composition of solution in gram-mols.<br />per 1000 gram-mols. water.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> Total<br />number<br />of salt<br />molecules.</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> Na<sub>2</sub>Cl<sub>2</sub>.
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>Cl<sub>2</sub>.
+</td><td class="boleb" style="text-align:center"> Na<sub>2</sub>SO<sub>4</sub>.
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>SO<sub>4</sub>.</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> I. </td><td class="rileb" style="text-align:center;" colspan="3"> NaCl </td><td class="leb" style="text-align:left; padding-left:1em;"> 55 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 55</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> II. </td><td class="rileb" style="text-align:center;" colspan="3"> KCl </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 34.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 34.5</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> III. </td><td class="rileb" style="text-align:center;" colspan="3"> Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 6 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 6</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> IV. </td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>SO<sub>4</sub> </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 9 </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 9</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> V. </td><td class="rileb" style="text-align:center;" colspan="3"> NaCl; KCl </td><td class="leb" style="text-align:left; padding-left:1em;"> 46.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> 12.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 59</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> VI. </td><td class="rileb" style="text-align:center;" colspan="3"> NaCl; Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O </td><td class="leb" style="text-align:left; padding-left:1em;"> 47.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 8 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 55.5</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> VII. </td><td class="rileb" style="text-align:center;" colspan="3"> KCl; K<sub>2</sub>SO<sub>4</sub> </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 34.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 1 </td><td class="leb" style="text-align:left; padding-left:1em;"> 35.5</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> VIII. </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> Glaserite;<br />Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O</td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> 10 </td><td class="leb" style="text-align:left; padding-left:1em;"> 10 </td><td class="leb" style="text-align:left; padding-left:1em;"> 20</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> IX. </td><td class="rileb" style="text-align:center;" colspan="3"> Glaserite; K<sub>2</sub>SO<sub>4</sub> </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 7.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> 10 </td><td class="leb" style="text-align:left; padding-left:1em;"> 17.5</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> X. </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O; KCl;<br />NaCl </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="leb" style="text-align:left; padding-left:1em;"> 51 </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 9.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 3.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> 64</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> XI. </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> Na<sub>2</sub>SO<sub>4</sub>,10H<sub>2</sub>O; KCl;<br />glaserite </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a>
+   </td><td class="leb" style="text-align:left; padding-left:1em;"> 40.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> 13 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 3.5 </td><td class="leb" style="text-align:left; padding-left:1em;"> 57</td></tr>
+
+<tr><td class="rib" style="text-align:right;"> XII. </td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>SO<sub>4</sub>; KCl; glaserite </td><td class="leb" style="text-align:left; padding-left:1em;"> 18 </td><td class="leb" style="text-align:left; padding-left:1em;"> 23 </td><td class="leb" style="text-align:left; padding-left:1em;"> &mdash; </td><td class="leb" style="text-align:left; padding-left:1em;"> &nbsp; 3 </td><td class="leb" style="text-align:left; padding-left:1em;"> 44</td></tr>
+
+</table>
+
+  <p>From the aspect of these diagrams the conditions under which the salts
+  can coexist can be read at a glance. Thus, <!-- Page 320 --><span
+  class="pagenum"><a name="page320"></a>{320}</span>for example, Fig. 124
+  shows that at 0° Glauber's salt and potassium chloride can exist together
+  with solution; namely, in contact with solutions having the composition
+  X&mdash;XI. This temperature must therefore be below the transition point
+  of this salt-pair (p.&nbsp;<a href="#page314">314</a>). On raising the
+  temperature to 4.4°, it is found that the curve VIII.&mdash;XI. moves so
+  that the point XI. coincides with point X. At this point, therefore,
+  there will be <i>four</i> concurrent fields, viz. Glauber's salt,
+  potassium chloride, glaserite, and sodium chloride. But these four salts
+  can coexist with solution only at the transition point; so that 4.4° is
+  the transition temperature of the salt-pair: Glauber's
+  salt&mdash;potassium chloride. At higher temperatures the line
+  VIII.&mdash;XI. moves still further to the left, so that the field for
+  Glauber's salt becomes entirely separated from the field for potassium
+  chloride. This shows that at temperatures above the transition point the
+  salt-pair Glauber's salt&mdash;potassium chloride cannot coexist in
+  presence of solution.</p>
+
+  <div class="figcenter" style="width:21%;">
+      <a href="images/fig125.png"><img style="width:100%" src="images/fig125.png"
+      alt="Fig. 125" title="Fig. 125" /></a>
+    <span class="sc">Fig.</span> 125.
+  </div>
+
+  <p>If it is only desired to indicate the mutual relationships of the
+  different components and the conditions for their coexistence
+  (<i>paragenesis</i>), a simpler diagram than Fig. 124 can be employed.
+  Thus if the boundary curves of Fig. 124 are so drawn that they cut one
+  another at right angles, a figure such as Fig. 125 is obtained, the Roman
+  numerals here corresponding with those in Fig. 124.</p>
+
+  <p><b>Ammonia-Soda Process.</b>&mdash;One of the most important
+  applications of the Phase Rule to systems of four components with
+  reciprocal salt-pairs has recently been made by Fedotieff<a name="NtA392"
+  href="#Nt392"><sup>[392]</sup></a> in his investigations of the
+  conditions for the formation of sodium carbonate by the so-called
+  ammonia-soda (Solvay) <!-- Page 321 --><span class="pagenum"><a
+  name="page321"></a>{321}</span>process.<a name="NtA393"
+  href="#Nt393"><sup>[393]</sup></a> This process consists, as is well
+  known, in passing carbon dioxide through a solution of common salt
+  saturated with ammonia.</p>
+
+  <p>Whatever differences of detail there may be in the process as carried
+  out in different manufactories, the reaction which forms the basis of the
+  process is that represented by the equation</p>
+
+<p class="cenhead">NaCl + NH<sub>4</sub>HCO<sub>3</sub> = NaHCO<sub>3</sub> + NH<sub>4</sub>Cl</p>
+
+  <p>We are dealing here, therefore, with reciprocal salt-pairs, the
+  behaviour of which has just been discussed in the preceding pages. The
+  present case is, however, simpler than that of the salt-pair
+  Na<sub>2</sub>SO<sub>4</sub>.10H<sub>2</sub>O + KCl, inasmuch as under
+  the conditions of experiment neither hydrates nor double salts are
+  formed. Since the study of the reaction is rendered more difficult on
+  account of the fact that ammonium bicarbonate in solution, when under
+  atmospheric pressure, undergoes decomposition at temperatures above 15°,
+  this temperature was the one chosen for the detailed investigation of the
+  conditions of equilibrium. Since, further, it has been shown by
+  Bodländer<a name="NtA394" href="#Nt394"><sup>[394]</sup></a> that the
+  bicarbonates possess a definite solubility only when the pressure of
+  carbon dioxide in the solution has a definite value, the measurements
+  were carried out in solutions saturated with this gas. This, however,
+  does not constitute another component, because we have made the
+  restriction that the sum of the partial pressures of carbon dioxide and
+  water vapour is equal to 1 atmosphere. The concentration of the carbon
+  dioxide is, therefore, not independently variable (p.&nbsp;<a
+  href="#page10">10</a>).</p>
+
+  <div class="figcenter" style="width:43%;">
+      <a href="images/fig126.png"><img style="width:100%" src="images/fig126.png"
+      alt="Fig. 126" title="Fig. 126" /></a>
+    <span class="sc">Fig.</span> 126.
+  </div>
+
+  <p>In order to obtain the data necessary for a discussion of the
+  conditions of soda formation by the ammonia-soda process, solubility
+  determinations with the four salts, NaCl, NH<sub>4</sub>Cl,
+  NH<sub>4</sub>HCO<sub>3</sub>, and NaHCO<sub>3</sub> were made, first
+  with the single salts and then <!-- Page 322 --><span class="pagenum"><a
+  name="page322"></a>{322}</span>with the salts in pairs. The results
+  obtained are represented graphically in Fig. 126, which is an isothermal
+  diagram similar to that given by Fig. 124. The points I., II., III., IV.,
+  represent the composition of solutions in equilibrium with two solid
+  salts. We have, however, seen (p.&nbsp;<a href="#page314">314</a>) that the
+  transition point, when the experiment is carried out under constant
+  pressure (atmospheric pressure), is the point of intersection of four
+  solubility curves, each of which represents the composition of solutions
+  in equilibrium with three salts, viz. one of the reciprocal salt-pairs
+  along with a third salt. Since, now, it was found that the stable
+  salt-pair at temperatures between 0° and 30° is sodium bicarbonate and
+  ammonium chloride, determinations were made of the composition of
+  solutions in equilibrium with NaHCO<sub>3</sub> + NH<sub>4</sub>Cl +
+  NH<sub>4</sub>HCO<sub>3</sub> and with NaHCO<sub>3</sub> +
+  NH<sub>4</sub>Cl + NaCl as solid phases. Under the <!-- Page 323 --><span
+  class="pagenum"><a name="page323"></a>{323}</span>conditions of
+  experiment (temperature = 15°) sodium chloride and ammonium bicarbonate
+  cannot coexist in contact with solution. These determinations gave the
+  data necessary for the construction of the complete isothermal diagram
+  (Fig. 127). The most important of these data are given in the following
+  table (temperature, 15°):&mdash;</p>
+
+<table class="nobctr" summary="Solutions in ammonia-soda process" title="Solutions in ammonia-soda process">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Point.
+</td><td class="toribob" style="text-align:center" rowspan="2"> Solid phases.
+</td><td class="toboleb" style="text-align:center" colspan="4"> Composition of solution in gram-mols.<br />per 1000 gram-mols. water.</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> NaHCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> NaCl
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub>HCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub>Cl</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> NaHCO<sub>3</sub> </td><td class="leb" style="text-align:center;"> 1.08 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> NaCl </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 6.12 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> NH<sub>4</sub>HCO<sub>3</sub> </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 2.36 </td><td class="leb" style="text-align:center;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> NH<sub>4</sub>Cl </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 6.64</td></tr>
+<tr><td class="rib" style="text-align:center;"> I. </td><td class="leb" style="text-align:center;"> NaHCO<sub>3</sub>; NaCl </td><td class="leb" style="text-align:center;"> 0.12 </td><td class="leb" style="text-align:center;"> 6.06 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:center;"> II. </td><td class="leb" style="text-align:center;"> NaCl; NH<sub>4</sub>Cl </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 4.55 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 3.72</td></tr>
+<tr><td class="rib" style="text-align:center;"> III. </td><td class="leb" style="text-align:center;"> NH<sub>4</sub>Cl; NH<sub>4</sub>HCO<sub>3</sub> </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 0.81 </td><td class="leb" style="text-align:center;"> 6.40</td></tr>
+<tr><td class="rib" style="text-align:center;"> IV. </td><td class="leb" style="text-align:center;"> NaHCO<sub>3</sub>; NH<sub>4</sub>HCO<sub>3</sub> </td><td class="leb" style="text-align:center;"> 0.71 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 2.16 </td><td class="leb" style="text-align:center;"> &mdash;</td></tr>
+<tr><td class="rib" style="text-align:center;"> P<sub>1</sub> </td><td class="leb" style="text-align:center;"> NaHCO<sub>3</sub>; NH<sub>4</sub>HCO<sub>3</sub>; NH<sub>4</sub>Cl </td><td class="leb" style="text-align:center;"> 0.93 </td><td class="leb" style="text-align:center;"> 0.51 </td><td class="leb" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 6.28</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> P<sub>2</sub> </td><td class="boleb" style="text-align:center;"> NaHCO<sub>3</sub>; NaCl; NH<sub>4</sub>Cl </td><td class="boleb" style="text-align:center;"> 0.18 </td><td class="boleb" style="text-align:center;"> 4.44 </td><td class="boleb" style="text-align:center;"> &mdash; </td><td class="boleb" style="text-align:center;"> 3.73</td></tr>
+
+</table>
+
+  <p>With reference to the solution represented by the point P<sub>1</sub>,
+  it may be remarked that it is an incongruently saturated solution (p.&nbsp;<a
+  href="#page279">279</a>). If sodium chloride is added to this solution,
+  the composition of the latter undergoes change; and if a sufficient
+  amount of the salt is added, the solution P<sub>2</sub> is obtained.</p>
+
+  <p>Turning now to the practical application of the data so obtained,
+  consider first what is the influence of concentration on the yield of
+  soda. Since the reaction consists essentially in a double decomposition
+  between sodium chloride and ammonium bicarbonate, then, after the
+  deposition of the sodium bicarbonate, we obtain a solution containing
+  sodium chloride, ammonium chloride, and sodium bicarbonate. In order to
+  ascertain to what extent the sodium chloride has been converted into
+  solid sodium bicarbonate, it is necessary to examine the composition of
+  the solution which is obtained <!-- Page 324 --><span class="pagenum"><a
+  name="page324"></a>{324}</span>with definite amounts of sodium chloride
+  and ammonium bicarbonate.</p>
+
+  <div class="figcenter" style="width:43%;">
+      <a href="images/fig127.png"><img style="width:100%" src="images/fig127.png"
+      alt="Fig. 127" title="Fig. 127" /></a>
+    <span class="sc">Fig.</span> 127.
+  </div>
+
+  <p>Consider, in the first place, the solutions represented by the curve
+  P<sub>2</sub>P<sub>1</sub>. With the help of this curve we can state the
+  conditions under which a solution, saturated for ammonium chloride, is
+  obtained, after deposition of sodium bicarbonate. In the following table
+  the composition of the solutions is given which are obtained with
+  different initial amounts of sodium chloride and ammonium bicarbonate.
+  The last two columns give the percentage amount of the sodium used, which
+  is deposited as solid sodium bicarbonate (U<sub>Na</sub>); and likewise
+  the percentage amount of ammonium bicarbonate which is usefully converted
+  into sodium bicarbonate, that is to say, the amount of the radical
+  HCO<sub>3</sub> deposited (U<sub>NH<sub>4</sub></sub>):&mdash; <!-- Page
+  325 --><span class="pagenum"><a name="page325"></a>{325}</span></p>
+
+<table class="nobctr" summary="Solutions for curve P2P1" title="Solutions for curve P2P1">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Point.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Initial composition<br />of the solutions:<br />grams of salt to 1000<br />grams of water.
+</td><td class="toboleb" style="text-align:center" colspan="4"> Composition of solutions obtained:<br />gram-equivalents per 1000 grams<br />of water.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> U<sub>Na</sub><br />per cent.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> U<sub>NH<sub>4</sub></sub><br />per cent.</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> NaCl
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub>HCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> HCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> Cl
+</td><td class="boleb" style="text-align:center"> Na
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub></td></tr>
+
+<tr><td class="rib" style="text-align:center;"> P<sub>2</sub> </td><td class="leb" style="text-align:center;"> 479 </td><td class="leb" style="text-align:center;"> 295 </td><td class="leb" style="text-align:center;"> 0.18 </td><td class="leb" style="text-align:center;"> 8.17 </td><td class="leb" style="text-align:center;"> 4.62 </td><td class="leb" style="text-align:center;"> 3.73 </td><td class="leb" style="text-align:center;"> 43.4 </td><td class="leb" style="text-align:center;"> 95.1</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 448 </td><td class="leb" style="text-align:center;"> 360 </td><td class="leb" style="text-align:center;"> 0.31 </td><td class="leb" style="text-align:center;"> 7.65 </td><td class="leb" style="text-align:center;"> 3.39 </td><td class="leb" style="text-align:center;"> 4.56 </td><td class="leb" style="text-align:center;"> 55.7 </td><td class="leb" style="text-align:center;"> 93.4</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 417 </td><td class="leb" style="text-align:center;"> 431 </td><td class="leb" style="text-align:center;"> 0.51 </td><td class="leb" style="text-align:center;"> 7.13 </td><td class="leb" style="text-align:center;"> 2.19 </td><td class="leb" style="text-align:center;"> 5.45 </td><td class="leb" style="text-align:center;"> 69.2 </td><td class="leb" style="text-align:center;"> 90.5</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> P<sub>1</sub> </td><td class="boleb" style="text-align:center;"> 397 </td><td class="boleb" style="text-align:center;"> 496 </td><td class="boleb" style="text-align:center;"> 0.92 </td><td class="boleb" style="text-align:center;"> 6.79 </td><td class="boleb" style="text-align:center;"> 1.44 </td><td class="boleb" style="text-align:center;"> 6.28 </td><td class="boleb" style="text-align:center;"> 78.8 </td><td class="boleb" style="text-align:center;"> 85.1</td></tr>
+
+</table>
+
+  <p>This table shows that the greater the excess of sodium chloride, the
+  greater is the percentage utilization of ammonia (Point P<sub>2</sub>);
+  and the more the amount of sodium chloride decreases, the greater is the
+  percentage amount of sodium chloride converted into bicarbonate. In the
+  latter case, however, the percentage utilization of the ammonium
+  bicarbonate decreases; that is to say, less sodium bicarbonate is
+  deposited, or more of it remains in solution.</p>
+
+  <p>Consider, in the same manner, the relations for solutions represented
+  by the curve P<sub>2</sub>IV, which gives the composition of solutions
+  saturated with respect to sodium bicarbonate and ammonium bicarbonate. In
+  this case we obtain the following results:&mdash;</p>
+
+<table class="nobctr" summary="Solutions for curve P2IV" title="Solutions for curve P2IV">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Point.
+</td><td class="toboleb" style="text-align:center" colspan="2"> Initial composition<br />of the solutions:<br />grams of salt to 1000<br />grams of water.
+</td><td class="toboleb" style="text-align:center" colspan="4"> Composition of solutions obtained:<br />gram-equivalents per 1000 grams<br />of water.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> U<sub>Na</sub><br />per cent.
+</td><td class="toboleb" style="text-align:center" rowspan="2"> U<sub>NH<sub>4</sub></sub><br />per cent.</td></tr>
+
+<tr><td class="boleb" style="text-align:center"> NaCl
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub>HCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> HCO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> Cl
+</td><td class="boleb" style="text-align:center"> Na
+</td><td class="boleb" style="text-align:center"> NH<sub>4</sub></td></tr>
+
+<tr><td class="rib" style="text-align:center;"> P<sub>1</sub> </td><td class="leb" style="text-align:center;"> 397 </td><td class="leb" style="text-align:center;"> 496 </td><td class="leb" style="text-align:center;"> 0.92 </td><td class="leb" style="text-align:center;"> 6.79 </td><td class="leb" style="text-align:center;"> 1.44 </td><td class="leb" style="text-align:center;"> 6.28 </td><td class="leb" style="text-align:center;"> 78.8 </td><td class="leb" style="text-align:center;"> 85.1</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 351 </td><td class="leb" style="text-align:center;"> 446 </td><td class="leb" style="text-align:center;"> 0.99 </td><td class="leb" style="text-align:center;"> 6.00 </td><td class="leb" style="text-align:center;"> 1.34 </td><td class="leb" style="text-align:center;"> 5.65 </td><td class="leb" style="text-align:center;"> 77.7 </td><td class="leb" style="text-align:center;"> 82.5</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 316 </td><td class="leb" style="text-align:center;"> 412 </td><td class="leb" style="text-align:center;"> 1.07 </td><td class="leb" style="text-align:center;"> 5.41 </td><td class="leb" style="text-align:center;"> 1.27 </td><td class="leb" style="text-align:center;"> 5.21 </td><td class="leb" style="text-align:center;"> 76.4 </td><td class="leb" style="text-align:center;"> 79.5</td></tr>
+<tr><td class="rib" style="text-align:center;"> &mdash; </td><td class="leb" style="text-align:center;"> 294 </td><td class="leb" style="text-align:center;"> 389 </td><td class="leb" style="text-align:center;"> 1.12 </td><td class="leb" style="text-align:center;"> 5.03 </td><td class="leb" style="text-align:center;"> 1.23 </td><td class="leb" style="text-align:center;"> 4.92 </td><td class="leb" style="text-align:center;"> 75.5 </td><td class="leb" style="text-align:center;"> 75.1</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> &mdash; </td><td class="boleb" style="text-align:center;"> 234 </td><td class="boleb" style="text-align:center;"> 327 </td><td class="boleb" style="text-align:center;"> 1.30 </td><td class="boleb" style="text-align:center;"> 4.00 </td><td class="boleb" style="text-align:center;"> 1.16 </td><td class="boleb" style="text-align:center;"> 4.14 </td><td class="boleb" style="text-align:center;"> 71.0 </td><td class="boleb" style="text-align:center;"> 68.6</td></tr>
+</table>
+
+  <p>As is evident from this table, diminution in the relative amount of
+  sodium chloride exercises only a slight influence <!-- Page 326 --><span
+  class="pagenum"><a name="page326"></a>{326}</span>on the utilization of
+  this salt, but is accompanied by a rapid diminution of the effective
+  transformation of the ammonium bicarbonate. So far as the efficient
+  conversion of the sodium is concerned, we see that it reaches its maximum
+  at the point P<sub>1</sub>, and that it decreases both with increase and
+  with decrease of the relative amount of sodium chloride employed; and
+  faster, indeed, in the former than in the latter case. On the other hand,
+  the effective transformation of the ammonium bicarbonate reaches its
+  maximum at the point P<sub>2</sub>, and diminishes with increase in the
+  relative amount of ammonium bicarbonate employed. Since sodium chloride
+  is, in comparison with ammonia&mdash;even when this is
+  regenerated&mdash;a cheap material, it is evidently more advantageous to
+  work with solutions which are relatively rich in sodium chloride
+  (solutions represented by the curve P<sub>1</sub>P<sub>2</sub>). This
+  fact has also been established empirically.</p>
+
+  <p>When, as is the case in industrial practice, we are dealing with
+  solutions which are saturated not for two salts but only for sodium
+  bicarbonate, it is evident that we have then to do with solutions the
+  composition of which is represented by points in the area
+  P<sub>1</sub>P<sub>2</sub>I,IV. Since in the commercial manufacture, the
+  aim must be to obtain as complete a utilization of the materials as
+  possible, the solutions employed industrially must lie in the
+  neighbourhood of the curves P<sub>2</sub>P<sub>1</sub>IV, as is indicated
+  by the shaded portion in Fig. 127. The best results, from the
+  manufacturer's standpoint, will be obtained, as already stated, when the
+  composition of the solutions approaches that given by a point on the
+  curve P<sub>2</sub>P<sub>1</sub>. Considered from the chemical
+  standpoint, the results of the experiments lead to the conclusion that
+  the Solvay process, <i>i.e.</i> passage of carbon dioxide through a
+  solution of sodium chloride saturated with ammonia, is not so good as the
+  newer method of Schlösing, which consists in bringing together sodium
+  chloride and ammonium bicarbonate with water.<a name="NtA395"
+  href="#Nt395"><sup>[395]</sup></a></p>
+
+<p><!-- Page 327 --><span class="pagenum"><a name="page327"></a>{327}</span></p>
+
+  <p><b>Preparation of Barium Nitrite.</b>&mdash;Mention may also be made
+  here of the preparation of barium nitrite by double decomposition of
+  barium chloride and sodium nitrite.<a name="NtA396"
+  href="#Nt396"><sup>[396]</sup></a></p>
+
+  <p>The reaction with which we are dealing here is represented by the
+  equation</p>
+
+<p class="cenhead">BaCl<sub>2</sub> + 2NaNO<sub>2</sub> = 2NaCl + Ba(NO<sub>2</sub>)<sub>2</sub></p>
+
+  <p>It was found that at the ordinary temperature NaCl and
+  Ba(NO<sub>2</sub>)<sub>2</sub> form the stable salt-pair. If, therefore,
+  barium chloride and sodium nitrite are brought together with an amount of
+  water insufficient for complete solution, transformation to the stable
+  salt-pair occurs, and sodium chloride and barium nitrite are deposited.
+  When, however, a stable salt-pair is in its transition interval (p.&nbsp;<a
+  href="#page315">315</a>), a third salt&mdash;in this case barium
+  chloride&mdash;will be deposited, as we have already learned. On bringing
+  barium chloride and sodium nitrite together with water, therefore, three
+  solid phases are obtained, viz. BaCl<sub>2</sub>, NaCl,
+  Ba(NO<sub>2</sub>)<sub>2</sub>. These three phases, together with
+  solution and vapour, constitute a univariant system, so that at each
+  temperature the composition of the solution must be constant.</p>
+
+  <p>Witt and Ludwig found that the presence of solid barium chloride can
+  be prevented by adding an excess of sodium nitrite, as can be readily
+  foreseen from what has been said. Since the solution in presence of the
+  three solid phases must have a definite composition at a definite
+  temperature, the addition of sodium nitrite to the solution must have, as
+  its consequence, the solution of an equivalent amount of barium chloride,
+  and the deposition of an equivalent amount of sodium chloride and barium
+  nitrite. By sufficient addition of sodium nitrite, the complete
+  disappearance of the solid barium chloride can be effected, and there
+  will remain only the stable salt-pair sodium chloride and barium nitrite.
+  As was pointed out by Meyerhoffer, however, the disappearance of the
+  barium chloride is effected, not by a change in the <!-- Page 328
+  --><span class="pagenum"><a name="page328"></a>{328}</span>composition of
+  the solution, but by the necessity for the composition of the solution
+  remaining constant.</p>
+
+  <div class="figcenter" style="width:36%;">
+      <a href="images/fig128.png"><img style="width:100%" src="images/fig128.png"
+      alt="Fig. 128" title="Fig. 128" /></a>
+    <span class="sc">Fig.</span> 128.
+  </div>
+
+  <p><b>Barium Carbonate and Potassium Sulphate.</b>&mdash;As has been
+  found by Meyerhoffer,<a name="NtA397" href="#Nt397"><sup>[397]</sup></a>
+  these two salts form the stable pair, not only at the ordinary
+  temperature, but also at the melting point. For the ordinary temperatures
+  this was proved in the following manner: A solution with the solid phases
+  K<sub>2</sub>SO<sub>4</sub> and
+  K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O in excess can only coexist in
+  contact either with BaCO<sub>3</sub> or with BaSO<sub>4</sub>, since,
+  evidently, in one of the two groups the stable system must be present.
+  Two solutions were prepared, each with excess of
+  K<sub>2</sub>SO<sub>4</sub> +
+  K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O, <!-- Page 329 --><span
+  class="pagenum"><a name="page329"></a>{329}</span>and to one was added
+  BaCO<sub>3</sub> and to the other BaSO<sub>4</sub>. After stirring for a
+  few days, the barium sulphate was completely transformed to
+  BaCO<sub>3</sub>, whereas the barium carbonate remained unchanged.
+  Consequently, BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub> +
+  K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O is stable, and, therefore, so
+  also is BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub>. That
+  BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub> is the stable pair also at
+  the melting point was proved by a special analytical method which allows
+  of the detection of K<sub>2</sub>CO<sub>3</sub> in a mixture of the four
+  solid salts. This analysis showed that a mixture of BaCO<sub>3</sub> +
+  K<sub>2</sub>SO<sub>4</sub>, after being fused and allowed to solidify,
+  contains only small amounts of K<sub>2</sub>CO<sub>3</sub>; and this is
+  due entirely to the fact that BaCO<sub>3</sub> +
+  K<sub>2</sub>SO<sub>4</sub> on fusion deposits a little BaSO<sub>4</sub>,
+  thereby giving rise at the same time to the separation of an equivalent
+  amount of K<sub>2</sub>CO<sub>3</sub>.</p>
+
+  <p>The different solubilities are shown in Fig. 128. In this diagram the
+  solubility of the two barium salts has been neglected. A is the
+  solubility of K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O; addition of
+  BaCO<sub>3</sub> does not alter this. B is the solubility of
+  K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O + K<sub>2</sub>SO<sub>4</sub>
+  + BaCO<sub>3</sub>. A and B almost coincide, since the potassium sulphate
+  is very slightly soluble in the concentrated solution of potassium
+  carbonate. D gives the concentration of the solution in equilibrium with
+  K<sub>2</sub>SO<sub>4</sub> + BaSO<sub>4</sub>. The most interesting
+  point is C. This solution is obtained by adding a small quantity of water
+  to BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub>, whereupon, being in
+  the transition interval, BaSO<sub>4</sub> separates out and an equivalent
+  amount of K<sub>2</sub>CO<sub>3</sub> goes into solution. C is the end
+  point of the curve CO, which is called the Guldberg-Waage curve, because
+  these investigators determined several points on it.</p>
+
+  <p>In their experiments, Guldberg and Waage found the ratio
+  K<sub>2</sub>CO<sub>3</sub> : K<sub>2</sub>SO<sub>4</sub> in solution to
+  be constant and equal to 4. This result is, however, not exact, for the
+  curve CO is not a straight line, as it should be if the above ratio were
+  constant; but it is concave to the abscissa axis, and more so at lower
+  than at higher temperatures.</p>
+
+  <p>The following table refers to the temperature of 25°. The Roman
+  numbers in the first column refer to the points in Fig. 128. The numbers
+  in the column <span class="grk">&Sigma;</span><i>k</i><sub>2</sub> give
+  the amount, <!-- Page 330 --><span class="pagenum"><a
+  name="page330"></a>{330}</span>in gram-molecules, of
+  K<sub>2</sub>CO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub> contained in
+  1000 gram-molecules of water:&mdash;</p>
+
+<p class="cenhead"><span class="sc">Solubility Determinations at 25°.</span></p>
+<table class="nobctr" summary="Solubility Determinations at 25°" title="Solubility Determinations at 25°">
+<tr><td class="toribob" style="text-align:center" rowspan="2" colspan="2"> No.
+</td><td class="toboleb" style="text-align:center" colspan="3" rowspan="2"> Solid phases.
+</td><td class="toleb" style="text-align:center" colspan="3"> 100 gms. of the<br />solution contain,<br />in grams,
+</td><td class="toleb" style="text-align:center" colspan="2"> 1000 moles<br />of water contain,<br />in moles,
+</td><td class="toboleb" style="text-align:center" rowspan="2"> <span class="grk">&Sigma;</span><i>k</i><sub>2</sub>
+</td><td class="toboleb" style="text-align:center" rowspan="2"> K<sub>2</sub>CO<sub>3</sub><br /><span class="over">K<sub>2</sub>SO<sub>4</sub></span></td></tr>
+
+<tr><td class="boleb" style="text-align:center" colspan="2"> K<sub>2</sub>CO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>SO<sub>4</sub>
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>CO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>SO<sub>4</sub></td></tr>
+
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> I. </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"></td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O + BaCO<sub>3</sub> </td><td class="leb" style="text-align:right;" colspan="2"> 53.2 &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> 147.9 &nbsp; &nbsp;</td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;</td></tr>
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> II. </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"></td><td class="leb" style="text-align:right; padding-left:0em; padding-right:0em;"><a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:center; padding-left:0em; padding-right:0em;"> K<sub>2</sub>CO<sub>3</sub>.2H<sub>2</sub>O + K<sub>2</sub>SO<sub>4</sub><br />+ BaCO<sub>3</sub> </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> <a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="leb" style="text-align:right;" colspan="2"> 53.0 &nbsp; </td><td class="leb" style="text-align:right;"> 0.023 </td><td class="leb" style="text-align:right;"> 147.8 &nbsp; &nbsp;</td><td class="leb" style="text-align:right;"> 0.051 </td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> III.<br />IV.</td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>SO<sub>4</sub> + BaCO<sub>3</sub> </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:right; padding-left:0em;"> 28.5 &nbsp;<br />22.1 &nbsp; </td><td class="leb" style="text-align:right;"> 0.886<br />1.72 &nbsp; </td><td class="leb" style="text-align:right;"> 52.58 &nbsp;<br />37.79 &nbsp;</td><td class="leb" style="text-align:right;"> 1.296<br />2.333 </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;<br />&mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;<br />&mdash; &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> V. </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> </td><td class="rileb" style="text-align:center;" colspan="3"> BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub> + BaSO<sub>4</sub> </td><td class="leb" style="text-align:right;" colspan="2"> 17.81 </td><td class="leb" style="text-align:right;"> 2.485 </td><td class="leb" style="text-align:right;"> 29.11 &nbsp;</td><td class="leb" style="text-align:right;"> 3.220 </td><td class="leb" style="text-align:right;"> 32.32 </td><td class="leb" style="text-align:right;"> 9.03</td></tr>
+
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> VI.<br />VII. </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>SO<sub>4</sub> + BaSO<sub>4</sub> </td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="nob" style="text-align:right; padding-left:0em;"> 12.6 &nbsp;<br />5.85 </td><td class="leb" style="text-align:right;"> 3.92 &nbsp;<br />6.76 &nbsp; </td><td class="leb" style="text-align:right;"> 19.66 &nbsp;<br />8.724</td><td class="leb" style="text-align:right;"> 4.853<br />7.995 </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;<br />&mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;<br />&mdash; &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"> VIII.</td><td class="nob" style="text-align:right; padding-left:0em; padding-right:0em;"></td><td class="rileb" style="text-align:center;" colspan="3"> K<sub>2</sub>SO<sub>4</sub> </td><td class="leb" style="text-align:right;" colspan="2"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> 10.76 &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> 12.47 &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp; </td><td class="leb" style="text-align:right;"> &mdash; &nbsp;</td></tr>
+
+<tr><td class="bob" style="text-align:right; padding-left:0em; padding-right:0em;"> IX.<br />X. </td><td class="bob" style="text-align:right; padding-left:0em; padding-right:0em;"><a href="images/$rbrace.png"><img src="images/$rbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a></td><td class="riboleb" style="text-align:center;" colspan="3"> BaCO<sub>3</sub> + BaSO<sub>4</sub> </td><td class="bob" style="text-align:right; padding-left:0em; padding-right:0em;"> <a href="images/$lbrace.png"><img src="images/$lbrace.png" class="middle" style="height:5.5ex" alt="brace" /></a> </td><td class="bob" style="text-align:right; padding-left:0em;"> 7.35<br />2.85 </td><td class="boleb" style="text-align:right;"> 0.602<br />0.173 </td><td class="boleb" style="text-align:right;"> 10.43 &nbsp;<br />3.828</td><td class="boleb" style="text-align:right;"> 0.676<br />0.184 </td><td class="boleb" style="text-align:right;"> 11.11<br />4.0 &nbsp; </td><td class="boleb" style="text-align:right;"> 15.0 &nbsp;<br />21.0 &nbsp;</td></tr>
+
+</table>
+
+  <p>The Guldberg-Waage curve at 100° was also determined, and it was found
+  that the ratio K<sub>2</sub>CO<sub>3</sub>: K<sub>2</sub>SO<sub>4</sub>
+  is also not constant, although the variations are not so great as at
+  25°.</p>
+
+<p class="cenhead"><span class="sc">Guldberg-Waage Curve at 100°.</span></p>
+<table class="nobctr" summary="Guldberg-Waage Curve at 100°" title="Guldberg-Waage Curve at 100°">
+<tr><td class="toribob" style="text-align:center" rowspan="2"> Solid phases.
+</td><td class="toleb" style="text-align:center" colspan="2"> 100 moles of water<br />contain, in moles,
+</td><td class="toboleb" style="text-align:center" rowspan="2"> <span class="grk">&Sigma;</span><i>k</i><sub>2</sub>
+</td><td class="toboleb" style="text-align:center" rowspan="2"> K<sub>2</sub>CO<sub>3</sub><br /><span class="over">K<sub>2</sub>SO<sub>4</sub></span></td></tr>
+
+<tr><td class="boleb" style="text-align:center"> K<sub>2</sub>CO<sub>3</sub>
+</td><td class="boleb" style="text-align:center"> K<sub>2</sub>SO<sub>4</sub></td></tr>
+
+<tr><td class="rib" style="text-align:center;"> BaCO<sub>3</sub> + K<sub>2</sub>SO<sub>4</sub> + BaSO<sub>4</sub></td><td class="leb" style="text-align:right;"> 23.9 &nbsp; </td><td class="leb" style="text-align:right;"> 12.65 &nbsp; </td><td class="leb" style="text-align:right;"> 35.65 &nbsp; </td><td class="leb" style="text-align:right;"> 1.82</td></tr>
+<tr><td class="rib" style="text-align:center;"> BaCO<sub>3</sub> + BaSO<sub>4</sub> </td><td class="leb" style="text-align:right;"> 6.28 </td><td class="leb" style="text-align:right;"> 2.02 &nbsp; </td><td class="leb" style="text-align:right;"> 8.3 &nbsp; &nbsp; </td><td class="leb" style="text-align:right;"> 3.1 &nbsp;</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> " &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; " </td><td class="boleb" style="text-align:right;"> 3.17 </td><td class="boleb" style="text-align:right;"> 0.851 </td><td class="boleb" style="text-align:right;"> 4.025 </td><td class="boleb" style="text-align:right;"> 3.7 &nbsp;</td></tr>
+
+</table>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 331 --><span class="pagenum"><a name="page331"></a>{331}</span></p>
+
+<h3>APPENDIX</h3>
+
+<p class="cenhead">EXPERIMENTAL DETERMINATION OF THE
+TRANSITION POINT</p>
+
+  <p>For the purpose of determining the transition temperature, a number of
+  methods have been employed, and the most important of these will be
+  briefly described here. In any given case it is sometimes possible to
+  employ more than one method, but all are not equally suitable, and the
+  values of the transition point obtained by the different methods are not
+  always identical. Indeed, a difference of several degrees in the value
+  found may quite well occur.<a name="NtA398"
+  href="#Nt398"><sup>[398]</sup></a> In each case, therefore, some care
+  must be taken to select the method most suitable for the purpose.</p>
+
+  <p><b>I. The Dilatometric Method.</b>&mdash;Since, in the majority of
+  cases, transformation at the transition point is accompanied by an
+  appreciable change of volume, it is only necessary to ascertain the
+  temperature at which this change of volume occurs, in order to determine
+  the transition point. For this purpose the <i>dilatometer</i> is
+  employed, an apparatus which consists of a bulb with capillary tube
+  attached, and which constitutes a sort of large thermometer (Fig. 129).
+  Some of the substance to be examined is passed into the bulb A through
+  the tube B, which is then sealed off. The rest of the bulb and a small
+  portion of the capillary tube is then filled with some liquid, which, of
+  course, must be without chemical action on the substance under
+  investigation. A liquid, however, may be employed which dissolves the
+  substance, for, as we have seen (p.&nbsp;<a href="#page70">70</a>), the
+  transformation at the transition point is, as a rule, accelerated by the
+  presence of a solvent. On the other hand, the liquid must not dissolve in
+  the substance under examination, for the temperature of transformation
+  would be thereby altered.</p>
+
+<p><!-- Page 332 --><span class="pagenum"><a name="page332"></a>{332}</span></p>
+
+  <p>In using the dilatometer, two methods of procedure may be followed.
+  According to the first method, the dilatometer containing the form stable
+  at lower temperatures is placed in a thermostat, maintained at a constant
+  temperature, until it has taken the temperature of the bath. The height
+  of the meniscus is then read on a millimetre scale attached to the
+  capillary. The temperature of the thermostat is then raised degree by
+  degree, and the height of the meniscus at each point ascertained. If,
+  now, no change takes place in the solid, the expansion will be
+  practically uniform, or the rise in the level of the meniscus per degree
+  of temperature will be practically the same at the different
+  temperatures, as represented diagrammatically by the line AB in Fig. 130.
+  On passing through the transition point, however, there will be a more or
+  less sudden increase in the rise of the meniscus per degree (line BC) if
+  the specific volume of the form stable at higher temperatures is greater
+  than that of the original modification; thereafter, the expansion will
+  again be uniform (line CD). Similarly, on cooling, contraction will at
+  first be uniform and then at the transition point there will be a
+  relatively large diminution of volume.</p>
+
+  <div class="figcenter" style="width:6%;">
+      <a href="images/fig129.png"><img style="width:100%" src="images/fig129.png"
+      alt="Fig. 129" title="Fig. 129" /></a>
+    <span class="sc">Fig.</span> 129.
+  </div>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/fig130.png"><img style="width:100%" src="images/fig130.png"
+      alt="Fig. 130" title="Fig. 130" /></a>
+    <span class="sc">Fig.</span> 130.
+  </div>
+
+  <p>If, now, transformation occurred immediately the transition point was
+  reached, the sudden expansion and contraction would take place at the
+  same temperature. It is, however, generally found that there is a lag,
+  and that with rising temperature the relatively large expansion does not
+  take place until a temperature somewhat higher than the transition point;
+  and with falling temperature the contraction occurs at a temperature
+  somewhat below the transition point. This is represented in Fig. 130 by
+  the lines BC and EF. The amount of lag will vary from case to case, and
+  will <!-- Page 333 --><span class="pagenum"><a
+  name="page333"></a>{333}</span>also depend on the length of time during
+  which the dilatometer is maintained at constant temperature.</p>
+
+  <p>As an example, there may be given the results obtained in the
+  determination of the transition point at which sodium sulphate and
+  magnesium sulphate form astracanite (p.&nbsp;<a href="#page268">268</a>).<a
+  name="NtA399" href="#Nt399"><sup>[399]</sup></a> The dilatometer was
+  charged with a mixture of the two sulphates.</p>
+
+<table class="nobctr" summary="Dilatometer readings for formation of astracanite" title="Dilatometer readings for formation of astracanite">
+<tr><td class="toribob" style="text-align:center"> Temperature.
+</td><td class="toboleb" style="text-align:center"> Level of oil in capillary.
+</td><td class="toboleb" style="text-align:center"> Rise per 1°.</td></tr>
+
+<tr><td class="ribob" style="text-align:center;"> 15.6°<br />16.6°<br />17.6°<br />18.6°<br />19.6°<br />20.6°<br />21.6°<br />22.6°<br />23.6°<br />24.6°
+</td><td class="boleb" style="text-align:center;"> 134<br />141<br />148<br />154<br />161<br />168<br />241<br />243<br />251<br />259
+</td><td class="boleb" style="text-align:center;"> 7<br />7<br />6<br />7<br />7<br />73 &nbsp;<br />2<br />8<br />8</td></tr>
+</table>
+
+  <p>The transition point, therefore, lies about 21.6° (p.&nbsp;<a
+  href="#page268">268</a>).</p>
+
+  <p>The second method of manipulation depends on the fact that, while
+  above or below the transition point transformation of one form into the
+  other can take place, at the transition point the two forms undergo no
+  change. The bulb of the dilatometer is, therefore, charged with a mixture
+  of the stable and metastable forms and a suitable liquid, and is then
+  immersed in a bath at constant temperature. After the temperature of the
+  bath has been acquired, readings of the height of the meniscus are made
+  from time to time to ascertain whether expansion or contraction occurs.
+  If expansion is found, the temperature of the thermostat is altered until
+  a temperature is obtained at which a gradual contraction takes place. The
+  transition point must then lie between these two temperatures; and by
+  repeating the determinations it will be possible to reduce the difference
+  between the temperatures at which expansion and contraction take place
+  to, say, 1°, and to fix the temperature of the transition point,
+  therefore, to within half a degree. By this method the transition point,
+  for example, of sulphur was found to be 95.6° under a pressure of 4
+  atm.<a name="NtA400" href="#Nt400"><sup>[400]</sup></a> The following are
+  the figures obtained by Reicher, who used a mixture <!-- Page 334
+  --><span class="pagenum"><a name="page334"></a>{334}</span>of 1 part of
+  carbon disulphide (solvent for sulphur) and 5 parts of turpentine as the
+  measuring liquid.</p>
+
+<table class="nobctr" summary="Dilatometer readings for sulphur" title="Dilatometer readings for sulphur">
+<tr><td class="nob" style="text-align:center; padding-top:1em; padding-bottom:0.5em" colspan="2"> <span class="sc">Temperature 95.1°.</span></td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Time in minutes.
+</td><td class="toboleb" style="text-align:center"> Level of liquid.</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> &nbsp; 5 </td><td class="leb" style="text-align:center;"> 343.5 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:center;"> 30 </td><td class="leb" style="text-align:center;"> 340.5 &nbsp;</td></tr>
+<tr><td class="rib" style="text-align:center;"> 55 </td><td class="leb" style="text-align:center;"> 335.75</td></tr>
+<tr><td class="ribob" style="text-align:center;"> 65 </td><td class="boleb" style="text-align:center;"> 333&nbsp; &nbsp; &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em; padding-bottom:0.5em" colspan="2"> <span class="sc">Temperature 96.1°.</span></td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Time in minutes.
+</td><td class="toboleb" style="text-align:center"> Level of liquid.</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> &nbsp; 5 </td><td class="leb" style="text-align:center;"> 342.75</td></tr>
+<tr><td class="rib" style="text-align:center;"> 30 </td><td class="leb" style="text-align:center;"> 354.75</td></tr>
+<tr><td class="rib" style="text-align:center;"> 55 </td><td class="leb" style="text-align:center;"> 360.5 &nbsp;</td></tr>
+<tr><td class="ribob" style="text-align:center;"> 60 </td><td class="boleb" style="text-align:center;"> 361.5 &nbsp;</td></tr>
+
+<tr><td class="nob" style="text-align:center; padding-top:1em; padding-bottom:0.5em" colspan="2"> <span class="sc">Temperature 95.6°.</span></td></tr>
+
+<tr><td class="toribob" style="text-align:center"> Time in minutes.
+</td><td class="toboleb" style="text-align:center"> Level of liquid.</td></tr>
+
+<tr><td class="rib" style="text-align:center;"> &nbsp; 5 </td><td class="leb" style="text-align:center;"> 368.75</td></tr>
+<tr><td class="rib" style="text-align:center;"> 100 &nbsp; </td><td class="leb" style="text-align:center;"> 368&nbsp; &nbsp; &nbsp;</td></tr>
+<tr><td class="ribob" style="text-align:center;"> 110 &nbsp; </td><td class="boleb" style="text-align:center;"> 368.75</td></tr>
+
+</table>
+
+  <p>At a temperature of 95.1° there is a contraction, <i>i.e.</i>
+  monoclinic sulphur passes into the rhombic, the specific volume of the
+  former being greater than that of the latter. At 96.1°, however, there is
+  expansion, showing that at this temperature rhombic sulphur passes into
+  monoclinic; while at 95.6° there is neither expansion nor contraction.
+  This is, therefore, the transition temperature; and since the dilatometer
+  was sealed up to prevent evaporation of the liquid, the pressure within
+  it was 4 atm.</p>
+
+  <p><b>II. Measurement of the Vapour Pressure.</b>&mdash;In the preceding
+  pages it has been seen repeatedly that the vapour pressures of the two
+  systems undergoing reciprocal transformation become identical at the
+  transition point (more strictly, at the triple or <!-- Page 335 --><span
+  class="pagenum"><a name="page335"></a>{335}</span>multiple point), and
+  the latter can therefore be determined by ascertaining the temperature at
+  which this identity of vapour pressure is established. The apparatus
+  usually employed for this purpose is the Bremer-Frowein tensimeter (p.&nbsp;<a
+  href="#page91">91</a>).</p>
+
+  <p>Although this method has not as yet been applied to systems of one
+  component, it has been used to a considerable extent in the case of
+  systems containing water or other volatile component. An example of this
+  has already been given in Glauber's salt (p.&nbsp;<a
+  href="#page139">139</a>).</p>
+
+  <p><b>III. Solubility Measurements.</b>&mdash;The temperature of the
+  transition point can also be fixed by means of solubility measurements,
+  for at that point the solubility of the two systems becomes identical.
+  Reference has already been made to several cases in which this method was
+  employed, <i>e.g.</i> ammonium nitrate (p.&nbsp;<a href="#page112">112</a>),
+  Glauber's salt (p.&nbsp;<a href="#page134">134</a>), astracanite and sodium
+  and magnesium sulphates (p.&nbsp;<a href="#page268">268</a>).</p>
+
+  <p>The determinations of the solubility can be carried out in various
+  ways. One of the simplest methods, which also gives sufficiently accurate
+  results when the temperature is not high or when the solvent is not very
+  volatile, can be carried out in the following manner. The solid substance
+  is finely powdered (in order to accelerate the process of solution), and
+  placed in sufficient quantity along with the solvent in a tube carefully
+  closed by a glass stopper; the latter is protected by a rubber cap, such
+  as a rubber finger-stall. The tube is then rotated in a thermostat, the
+  temperature of which does not vary more than one or two tenths of a
+  degree, until saturation is produced. The solution is withdrawn by means
+  of a pipette to which a small glass tube, filled with cotton wool to act
+  as a filter, is attached. The solution is then run into a weighing
+  bottle, and weighed; after which the amount of solid in solution is
+  determined in a suitable manner.</p>
+
+  <p>For more accurate determinations of the solubility, especially when
+  the solvent is appreciably volatile at the temperature of experiment,
+  other methods are preferable. In Fig. 131 is shown the apparatus employed
+  by H. Goldschmidt,<a name="NtA401" href="#Nt401"><sup>[401]</sup></a> and
+  used to a considerable extent in the laboratory of van't Hoff. This
+  consists essentially of three parts: <i>a</i>, a tube in which the
+  solvent and salt are placed; this is closed at the foot by an
+  india-rubber stopper. Through this stopper there passes the bent tube
+  <i>cb</i>, which connects the tube <i>a</i> with the weighing-tube
+  <i>d</i>. At <i>c</i> there is a plug of cotton wool. Tube <i>e</i> is
+  open to the air. The wider portion of the tube <i>cb</i>, which passes
+  through the rubber stopper in <i>a</i>, can be closed by a plug <!-- Page
+  336 --><span class="pagenum"><a name="page336"></a>{336}</span>attached
+  to a glass rod <i>ff</i>, which passes up through a hollow Witt stirrer,
+  <i>g</i>. After being fitted together, the whole apparatus is immersed in
+  the thermostat. After the solution has become saturated, the stopper of
+  the bent tube is raised by means of the rod <i>ff</i> and a suction-pump
+  attached to the end of <i>e</i>. The solution is thereby drawn into the
+  weighing-tube <i>d</i>, the undissolved salt being retained by the plug
+  at <i>c</i>. The apparatus is then removed from the thermostat, tube
+  <i>d</i> detached and immediately closed by a ground stopper. It is then
+  carefully dried and weighed.</p>
+
+  <div class="figcenter" style="width:18%;">
+      <a href="images/fig131.png"><img style="width:100%" src="images/fig131.png"
+      alt="Fig. 131" title="Fig. 131" /></a>
+    <span class="sc">Fig.</span> 131.
+  </div>
+
+  <p>Another form of solubility vessel, due to Meyerhoffer and Saunders, is
+  shown in Fig. 132.<a name="NtA402" href="#Nt402"><sup>[402]</sup></a>
+  This consists of a single tube, and the stirring is effected by means of
+  a glass screw.</p>
+
+  <div class="figcenter" style="width:21%;">
+      <a href="images/fig132.png"><img style="width:100%" src="images/fig132.png"
+      alt="Fig. 132" title="Fig. 132" /></a>
+    <span class="sc">Fig.</span> 132.
+  </div>
+
+  <p>The progress of the solution towards saturation can be very well
+  tested by determining the density of the solution from time to <!-- Page
+  337 --><span class="pagenum"><a name="page337"></a>{337}</span>time. This
+  is most conveniently carried out by means of the pipette shown in Fig.
+  133.<a name="NtA403" href="#Nt403"><sup>[403]</sup></a> With this pipette
+  the solution can not only be removed for weighing, but the volume can be
+  determined at the same time. It consists of the wide tube <i>a</i>, to
+  which the graduated capillary <i>b</i>, furnished with a cap <i>c</i>, is
+  attached. To the lower end of the pipette the tube <i>e</i>, with plug of
+  cotton wool, can be fixed. After the pipette has been filled by sucking
+  at the end of <i>b</i>, the stop-cock <i>d</i> is closed and the cap
+  <i>c</i> placed on the capillary. The apparatus can then be weighed, and
+  the volume of the solution be ascertained by means of the
+  graduations.</p>
+
+  <p>As has already been insisted, particular care must be paid to the
+  characterization of the solid in contact with the solution.</p>
+
+  <div class="figcenter" style="width:10%;">
+      <a href="images/fig133.png"><img style="width:100%" src="images/fig133.png"
+      alt="Fig. 133" title="Fig. 133" /></a>
+    <span class="sc">Fig.</span> 133.
+  </div>
+
+  <p><b>IV. Thermometric Method.</b>&mdash;If a substance is heated, its
+  temperature will gradually rise until the melting point is reached, and
+  the temperature will then remain constant until all the solid has passed
+  into liquid. Similarly, if a substance which can undergo transformation
+  is heated, the temperature will rise until the transition point is
+  reached, and will then remain constant until complete transformation has
+  taken place.</p>
+
+  <p>This method, it will be remembered, was employed by Richards for the
+  determination of the transition point of sodium sulphate decahydrate
+  (p.&nbsp;<a href="#page136">136</a>). The following figures give the results
+  obtained by Meyerhoffer in the case of the transformation:&mdash;</p>
+
+<p class="cenhead">CuK<sub>2</sub>Cl<sub>4</sub>,2H<sub>2</sub>O <a href="images/doublearrow.png"><img src="images/doublearrow.png" class="middle" style="height:2.5ex" alt="reversible arrow" /></a> CuKCl<sub>3</sub> + KCl + 2H<sub>2</sub>O</p>
+
+  <p>the temperature being noted from minute to minute: 95°, 93°, 91.8°,
+  91.7°, 92°, 92.3°, 92.4°, 92.2°, 92.2°, 92°, 90.5°, 89°, and then a rapid
+  fall in the temperature. From this we see that the transition point is
+  about 92.2°. It is also evident that a slight supercooling took place
+  (91.7°), owing to a delay in the transformation, but that then the
+  temperature rose to the transition point. This is analogous to the
+  supercooling of a liquid.</p>
+
+  <p>A similar halt in the temperature would be observed on passing from
+  lower to higher temperatures; but owing to a lag in the transformation,
+  the same temperature is not always obtained.</p>
+
+<p><!-- Page 338 --><span class="pagenum"><a name="page338"></a>{338}</span></p>
+
+  <p><b>V. Optical Method.</b>&mdash;The transition point can sometimes be
+  determined by noting the temperature at which some alteration in the
+  appearance of the substance occurs, such as a change of colour or of the
+  crystalline form. Thus mercuric iodide changes colour from red to yellow,
+  and the blue quadratic crystals of copper calcium acetate change, on
+  passing the transition point, into green rhombs of copper acetate and
+  white needles of calcium acetate (p.&nbsp;<a href="#page260">260</a>). Or
+  again, changes in the double refraction of the crystals may be also
+  employed to ascertain the temperature of the transition point. These
+  changes are best observed by means of a microscope.</p>
+
+  <p>For the purpose of regulating the temperature of the substance a small
+  copper air-bath is employed.<a name="NtA404"
+  href="#Nt404"><sup>[404]</sup></a></p>
+
+  <p><b>VI. Electrical Methods.</b>&mdash;Electrical methods for the
+  determination of the transition point are of two kinds, based on
+  measurements of conductivity or of electromotive force. Both methods are
+  restricted in their application, but where applicable give very exact
+  results.</p>
+
+  <p>The former method, which has been employed in several cases, need not
+  be described here. The second method, however, is of considerable
+  interest and importance, and calls for special reference.<a name="NtA405"
+  href="#Nt405"><sup>[405]</sup></a></p>
+
+  <p>If two pieces, say, of zinc, connected together by a conducting wire,
+  are placed in a solution of a zinc salt, <i>e.g.</i> zinc sulphate, the
+  potential of the two electrodes will be the same, and no current will be
+  produced in the connecting wire. If, however, the zinc electrodes are
+  immersed in two solutions of <i>different</i> concentration contained in
+  separate vessels, but placed in connection with one another by means of a
+  bent tube filled with a conducting solution, the potentials at the
+  electrodes will no longer be the same, and a current will now flow
+  through the connecting wire. The direction of this current <i>in the
+  cell</i> will be from the weaker to the more concentrated solution.</p>
+
+  <p>The greater the difference in the concentration of the solutions with
+  respect to zinc, the greater will be the difference of the potential at
+  the two electrodes, or the greater will be the E.M.F. of the cell. When
+  the concentration of the two solutions becomes the same, the E.M.F. will
+  become zero, and no current will pass.</p>
+
+  <p>It will be understood now how this method can be made use of <!-- Page
+  339 --><span class="pagenum"><a name="page339"></a>{339}</span>for
+  determining the transition point of a salt, when we bear in mind that at
+  the transition point the solubility of the two forms becomes identical.
+  Thus, for example, the transition point of zinc sulphate heptahydrate
+  into hexahydrate could be determined in the following manner. Tube A
+  (Fig. 134) contains, say, a saturated solution of the heptahydrate along
+  with some of the solid salt; tube B, a saturated solution of the
+  hexahydrate along with the solid salt. The tube C is a connecting tube
+  bent downwards so as to prevent the mixing of the solutions by convection
+  currents. ZZ are two zinc electrodes immersed in the solution; the cell
+  is placed in a thermostat and the zinc electrodes connected with a
+  galvanometer. Since, now, at temperatures below the transition point the
+  solubility of the hexahydrate (the metastable form) is greater than that
+  of the heptahydrate, a current will be produced, flowing in the cell from
+  heptahydrate to hexahydrate. As the temperature is raised towards the
+  transition point, the solubilities of the two hydrates also approach, and
+  the current produced will therefore become weaker, because the E.M.F. of
+  the cell becomes less; and when the transition point is attained, the
+  E.M.F. becomes zero, and the current ceases. If the temperature is raised
+  above this, the solubility of the heptahydrate becomes greater than that
+  of the hexahydrate, and a current will again be produced, but in the
+  opposite direction. By noting the temperature, therefore, at which the
+  current ceases, or the E.M.F. becomes zero, the transition temperature
+  can be ascertained.<a name="NtA406"
+  href="#Nt406"><sup>[406]</sup></a></p>
+
+  <div class="figcenter" style="width:17%;">
+      <a href="images/fig134.png"><img style="width:100%" src="images/fig134.png"
+      alt="Fig. 134" title="Fig. 134" /></a>
+    <span class="sc">Fig.</span> 134.
+  </div>
+
+  <p>In the case just described, the electrodes consisted of the same metal
+  as was contained in the salt. But in some cases, <i>e.g.</i> sodium
+  sulphate, electrodes of the metal contained in the salt cannot be
+  employed. Nevertheless, the above electrical method can be used <!-- Page
+  340 --><span class="pagenum"><a name="page340"></a>{340}</span>even in
+  those cases, if a suitable non-polarizable mercury electrode is
+  employed.<a name="NtA407" href="#Nt407"><sup>[407]</sup></a></p>
+
+  <p>Although, as we saw, no current was produced when two pieces of zinc
+  were immersed in the same solution of zinc salt, a current will be
+  obtained if two different metals, or even two different modifications of
+  the same metal, are employed. Thus an E.M.F. will be established when
+  electrodes of grey and of white tin are immersed in the same solution of
+  zinc salt, but at the transition point this E.M.F. will become zero. By
+  this method Cohen determined the transition point of grey and white tin
+  (p.&nbsp;<a href="#page42">42</a>).</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 341 --><span class="pagenum"><a name="page341"></a>{341}</span></p>
+
+<h3>NAME INDEX</h3>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="i8">A</p>
+      <p>Abegg, <a href="#page52">52</a></p>
+      <p>Adriani, <a href="#page186">186</a>, <a href="#page217">217</a>, <a href="#page220">220</a></p>
+      <p>Alexejeff, <a href="#page97">97</a>, <a href="#page125">125</a></p>
+      <p>Allan, <a href="#page298">298</a></p>
+      <p>Allen, L. E., <a href="#page109">109</a></p>
+      <p>Allen, R. W., <a href="#page63">63</a></p>
+      <p>Ampolla, <a href="#page213">213</a></p>
+      <p>Andreä, <a href="#page109">109</a></p>
+      <p>Aristotle, <a href="#page41">41</a></p>
+      <p>Armstrong, E. F., <a href="#page313">313</a></p>
+      <p>Armstrong, H. E., <a href="#page196">196</a></p>
+      <p>Arzruni, <a href="#page33">33</a></p>
+      <p>Aten, <a href="#page147">147</a>, <a href="#page163">163</a></p>
+      <p>Auerbach, <a href="#page326">326</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">B</p>
+      <p>Babo, <a href="#page126">126</a></p>
+      <p>Bancroft, <a href="#page102">102</a>, <a href="#page104">104</a>, <a href="#page161">161</a>, <a href="#page176">176</a>, <a href="#page196">196</a>, <a href="#page202">202</a>, <a href="#page229">229</a>, <a href="#page246">246</a>, <a href="#page260">260</a>, <a href="#page261">261</a>, <a href="#page272">272</a>, <a href="#page281">281</a>, <a href="#page302">302</a></p>
+      <p>Barnes, <a href="#page331">331</a>, <a href="#page339">339</a></p>
+      <p>Barschall, <a href="#page318">318</a></p>
+      <p>Barus, <a href="#page67">67</a></p>
+      <p>Battelli, <a href="#page23">23</a></p>
+      <p>Baur, <a href="#page233">233</a>, <a href="#page307">307</a></p>
+      <p>Beckmann, <a href="#page49">49</a></p>
+      <p>Bell, <a href="#page229">229</a></p>
+      <p>Berthollet, <a href="#page7">7</a></p>
+      <p>Bodländer, <a href="#page181">181</a>, <a href="#page247">247</a>, <a href="#page311">311</a>, <a href="#page321">321</a></p>
+      <p>Bogojawlenski, <a href="#page72">72</a></p>
+      <p>Boudouard, <a href="#page309">309</a>, <a href="#page311">311</a></p>
+      <p>Braun, <a href="#page107">107</a></p>
+      <p>Brauns, <a href="#page40">40</a>, <a href="#page51">51</a>, <a href="#page74">74</a></p>
+      <p>Bredig, <a href="#page52">52</a></p>
+      <p>Bremer, <a href="#page91">91</a></p>
+      <p>Brodie, <a href="#page34">34</a>, <a href="#page47">47</a></p>
+      <p>Bruner, <a href="#page126">126</a></p>
+      <p>Bruni, <a href="#page181">181</a>, <a href="#page182">182</a>, <a href="#page256">256</a>, <a href="#page257">257</a></p>
+      <p>Bunsen, <a href="#page67">67</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">C</p>
+      <p>Cady, <a href="#page192">192</a></p>
+      <p>Calvert, <a href="#page130">130</a></p>
+      <p>Cameron, <a href="#page203">203</a></p>
+      <p>Carnelley, <a href="#page47">47</a></p>
+      <p>Carpenter, <a href="#page225">225</a></p>
+      <p>Carveth, <a href="#page204">204</a>, <a href="#page255">255</a></p>
+      <p>Centnerszwer, <a href="#page158">158</a></p>
+      <p>Chapman, <a href="#page47">47</a></p>
+      <p>Chappuis, <a href="#page51">51</a>, <a href="#page176">176</a></p>
+      <p>Charpy, <a href="#page255">255</a></p>
+      <p>Churchill, <a href="#page140">140</a></p>
+      <p>Coehn, <a href="#page52">52</a></p>
+      <p>Cohen, <a href="#page41">41</a>, <a href="#page72">72</a>, <a href="#page136">136</a>, <a href="#page139">139</a>, <a href="#page140">140</a></p>
+      <p>Cooke, <a href="#page331">331</a>, <a href="#page339">339</a></p>
+      <p>Cox, <a href="#page301">301</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">D</p>
+      <p>Dawson, <a href="#page263">263</a></p>
+      <p>Debray, <a href="#page74">74</a>, <a href="#page81">81</a>, <a href="#page139">139</a></p>
+      <p>Deville, <a href="#page49">49</a>, <a href="#page74">74</a></p>
+      <p>Dewar, <a href="#page26">26</a>, <a href="#page51">51</a>, <a href="#page178">178</a></p>
+      <p>Dietz, <a href="#page157">157</a></p>
+<!-- Page 342 --><span class="pagenum"><a name="page342"></a>{342}</span>
+      <p>Doelter, <a href="#page233">233</a></p>
+      <p>Donnan, <a href="#page8">8</a>, <a href="#page18">18</a></p>
+      <p>Dreyer, <a href="#page73">73</a></p>
+      <p>Duhem, <a href="#page56">56</a>, <a href="#page151">151</a></p>
+      <p>Dutoit, <a href="#page204">204</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">E</p>
+      <p>Etard, <a href="#page115">115</a>, <a href="#page135">135</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">F</p>
+      <p>Fahrenheit, <a href="#page30">30</a></p>
+      <p>Faraday, <a href="#page82">82</a>, <a href="#page89">89</a></p>
+      <p>Fath, <a href="#page204">204</a></p>
+      <p>Fedotieff, <a href="#page315">315</a>, <a href="#page320">320</a></p>
+      <p>Findlay, <a href="#page111">111</a>, <a href="#page204">204</a>, <a href="#page206">206</a>, <a href="#page219">219</a></p>
+      <p>Foote, <a href="#page69">69</a></p>
+      <p>Friedländer, <a href="#page72">72</a></p>
+      <p>Fritsche, <a href="#page41">41</a></p>
+      <p>Frowein, <a href="#page91">91</a></p>
+      <p>Füchtbauer, <a href="#page75">75</a></p>
+      <p>Fyffe, <a href="#page143">143</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">G</p>
+      <p>Gattermann, <a href="#page51">51</a>, <a href="#page52">52</a></p>
+      <p>Gautier, <a href="#page222">222</a>, <a href="#page223">223</a></p>
+      <p>Gay-Lussac, <a href="#page135">135</a></p>
+      <p>Gernez, <a href="#page72">72</a></p>
+      <p>Gibbs, <a href="#page7">7</a>, <a href="#page8">8</a>, <a href="#page151">151</a>, <a href="#page236">236</a></p>
+      <p>Glaessner, <a href="#page307">307</a></p>
+      <p>Goldschmidt, E., <a href="#page41">41</a></p>
+      <p>Goldschmidt, H., <a href="#page335">335</a></p>
+      <p>Goldschmidt, V., <a href="#page32">32</a></p>
+      <p>Goossens, <a href="#page26">26</a></p>
+      <p>Gossner, <a href="#page318">318</a></p>
+      <p>Graham, <a href="#page178">178</a></p>
+      <p>Guertler, <a href="#page73">73</a></p>
+      <p>Guldberg, <a href="#page7">7</a></p>
+      <p>Guthrie, <a href="#page97">97</a>, <a href="#page104">104</a>, <a href="#page117">117</a>, <a href="#page118">118</a>, <a href="#page119">119</a>, <a href="#page233">233</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">H</p>
+      <p>Haber, <a href="#page311">311</a></p>
+      <p>Hahn, <a href="#page309">309</a>, <a href="#page311">311</a></p>
+      <p>Hallock, <a href="#page35">35</a></p>
+      <p>Hammerl, <a href="#page145">145</a></p>
+      <p>Hautefeuille, <a href="#page46">46</a>, <a href="#page49">49</a>, <a href="#page50">50</a>, <a href="#page51">51</a>, <a href="#page178">178</a></p>
+      <p>Heller, <a href="#page311">311</a></p>
+      <p>Henry, <a href="#page94">94</a></p>
+      <p>Herold, <a href="#page321">321</a></p>
+      <p>Hertz, <a href="#page49">49</a></p>
+      <p>Heycock, <a href="#page194">194</a>, <a href="#page221">221</a>, <a href="#page223">223</a></p>
+      <p>Heyn, <a href="#page225">225</a>, <a href="#page228">228</a></p>
+      <p>Hickmans, <a href="#page219">219</a></p>
+      <p>Hiorns, <a href="#page228">228</a></p>
+      <p>Hissink, <a href="#page115">115</a>, <a href="#page190">190</a></p>
+      <p>Hoitsema, <a href="#page14">14</a>, <a href="#page90">90</a>, <a href="#page177">177</a>, <a href="#page178">178</a>, <a href="#page298">298</a></p>
+      <p>Hollmann, <a href="#page204">204</a></p>
+      <p>Holsboer, <a href="#page110">110</a></p>
+      <p>Horstmann, <a href="#page8">8</a>, <a href="#page83">83</a>, <a href="#page89">89</a></p>
+      <p>Hudson, <a href="#page102">102</a></p>
+      <p>Hulett, <a href="#page10">10</a>, <a href="#page48">48</a>, <a href="#page52">52</a>, <a href="#page54">54</a>, <a href="#page67">67</a>, <a href="#page109">109</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">I</p>
+      <p>Isaac, <a href="#page114">114</a></p>
+      <p>Isambert, <a href="#page80">80</a>, <a href="#page82">82</a>, <a href="#page84">84</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">J</p>
+      <p>Jaffé, <a href="#page74">74</a>, <a href="#page114">114</a></p>
+      <p>Joulin, <a href="#page176">176</a></p>
+      <p>Juhlin, <a href="#page23">23</a>, <a href="#page24">24</a>, <a href="#page30">30</a></p>
+      <p>von Jüptner, <a href="#page225">225</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">K</p>
+      <p>Kastle, <a href="#page71">71</a></p>
+      <p>Kaufler, <a href="#page49">49</a></p>
+      <p>Kaufmann, <a href="#page112">112</a></p>
+      <p>Kayser, <a href="#page176">176</a></p>
+      <p>Keeling, <a href="#page225">225</a></p>
+      <p>Kelvin, <a href="#page25">25</a></p>
+      <p>Kenrick, <a href="#page263">263</a>, <a href="#page297">297</a></p>
+      <p>Kipping, <a href="#page219">219</a></p>
+      <p>Kirchhoff, <a href="#page32">32</a></p>
+      <p>Knorr, <a href="#page203">203</a></p>
+      <p>de Kock, <a href="#page53">53</a>, <a href="#page182">182</a>, <a href="#page194">194</a></p>
+      <p>Konowaloff, <a href="#page102">102</a>, <a href="#page103">103</a>, <a href="#page104">104</a></p>
+      <p>Krasnicki, <a href="#page144">144</a></p>
+      <p>Kremann, <a href="#page147">147</a>, <a href="#page212">212</a></p>
+      <p>Kuenen, <a href="#page105">105</a></p>
+      <p>Kultascheff, <a href="#page233">233</a></p>
+<!-- Page 343 --><span class="pagenum"><a name="page343"></a>{343}</span>
+      <p>Kuriloff, <a href="#page216">216</a></p>
+      <p>Kurnakoff, <a href="#page221">221</a>, <a href="#page222">222</a>, <a href="#page230">230</a></p>
+      <p>Küster, <a href="#page72">72</a>, <a href="#page181">181</a>, <a href="#page183">183</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">L</p>
+      <p>Laar, <a href="#page195">195</a></p>
+      <p>Labenburg, <a href="#page216">216</a></p>
+      <p>Lattey, <a href="#page101">101</a></p>
+      <p>Le Chatelier, <a href="#page58">58</a>, <a href="#page81">81</a>, <a href="#page233">233</a></p>
+      <p>Lehfeldt, <a href="#page338">338</a>, <a href="#page340">340</a></p>
+      <p>Lehmann, <a href="#page33">33</a>, <a href="#page52">52</a>, <a href="#page53">53</a></p>
+      <p>Lidbury, <a href="#page147">147</a></p>
+      <p>Loewel, <a href="#page134">134</a>, <a href="#page135">135</a></p>
+      <p>Loewenherz, <a href="#page134">134</a>, <a href="#page316">316</a></p>
+      <p>Lowry, <a href="#page196">196</a>, <a href="#page198">198</a></p>
+      <p>Ludwig, <a href="#page327">327</a></p>
+      <p>Lumsden, <a href="#page80">80</a>, <a href="#page109">109</a>, <a href="#page110">110</a></p>
+      <p>Lussana, <a href="#page68">68</a></p>
+      <p>Luther, <a href="#page22">22</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">M</p>
+      <p>Mack, <a href="#page67">67</a></p>
+      <p>Magnus, <a href="#page22">22</a></p>
+      <p>Mathews, <a href="#page221">221</a></p>
+      <p>Mellor, <a href="#page80">80</a></p>
+      <p>Meusser, <a href="#page142">142</a></p>
+      <p>Meyer, J., <a href="#page71">71</a></p>
+      <p>Meyer, V., <a href="#page47">47</a></p>
+      <p>Meyerhoffer, <a href="#page158">158</a>, <a href="#page233">233</a>, <a href="#page259">259</a>, <a href="#page268">268</a>, <a href="#page271">271</a>, <a href="#page278">278</a>, <a href="#page279">279</a>, <a href="#page280">280</a>, <a href="#page284">284</a>, <a href="#page313">313</a>, <a href="#page315">315</a>, <a href="#page317">317</a>, <a href="#page319">319</a>, <a href="#page327">327</a>, <a href="#page328">328</a>, <a href="#page336">336</a>, <a href="#page337">337</a></p>
+      <p>Middelberg, <a href="#page116">116</a></p>
+      <p>Miers, <a href="#page114">114</a></p>
+      <p>Miller, <a href="#page297">297</a></p>
+      <p>Mitscherlich, <a href="#page33">33</a>, <a href="#page49">49</a></p>
+      <p>Mond, <a href="#page178">178</a></p>
+      <p>Moore, <a href="#page72">72</a></p>
+      <p>Moss, <a href="#page66">66</a></p>
+      <p>Müller, <a href="#page112">112</a>, <a href="#page265">265</a></p>
+      <p>Mylius, <a href="#page109">109</a>, <a href="#page142">142</a>, <a href="#page157">157</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">N</p>
+      <p>Naumann, <a href="#page49">49</a></p>
+      <p>Neville, <a href="#page194">194</a>, <a href="#page221">221</a>, <a href="#page223">223</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">O</p>
+      <p>Offer, <a href="#page119">119</a></p>
+      <p>Ostwald, <a href="#page8">8</a>, <a href="#page10">10</a>, <a href="#page13">13</a>, <a href="#page16">16</a>, <a href="#page22">22</a>, <a href="#page44">44</a>, <a href="#page58">58</a>, <a href="#page68">68</a>, <a href="#page70">70</a>, <a href="#page74">74</a>, <a href="#page85">85</a>, <a href="#page88">88</a>, <a href="#page92">92</a>, <a href="#page102">102</a>, <a href="#page110">110</a>, <a href="#page117">117</a>, <a href="#page125">125</a>, <a href="#page127">127</a>, <a href="#page130">130</a>, <a href="#page141">141</a>, <a href="#page198">198</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">P</p>
+      <p>Padoa, <a href="#page73">73</a>, <a href="#page181">181</a></p>
+      <p>Parsons, <a href="#page298">298</a></p>
+      <p>Pasteur, <a href="#page266">266</a></p>
+      <p>Paternò, <a href="#page213">213</a></p>
+      <p>Payen, <a href="#page74">74</a></p>
+      <p>Pedler, <a href="#page47">47</a></p>
+      <p>Pfaundler, <a href="#page119">119</a></p>
+      <p>Philip, <a href="#page213">213</a>, <a href="#page214">214</a></p>
+      <p>von Pickardt, <a href="#page73">73</a></p>
+      <p>Planck, <a href="#page68">68</a></p>
+      <p>Pope, <a href="#page219">219</a></p>
+      <p>Poynting, <a href="#page68">68</a></p>
+      <p>Preuner, <a href="#page311">311</a></p>
+      <p>Puschin, <a href="#page222">222</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">Q</p>
+      <p>Quincke, <a href="#page52">52</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">R</p>
+      <p>Rabe, <a href="#page113">113</a></p>
+      <p>Ramsay, <a href="#page3">3</a>, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page24">24</a>, <a href="#page30">30</a>, <a href="#page32">32</a>, <a href="#page63">63</a>, <a href="#page64">64</a>, <a href="#page66">66</a>, <a href="#page79">79</a>, <a href="#page90">90</a>, <a href="#page165">165</a>, <a href="#page178">178</a></p>
+      <p>Raoult, <a href="#page180">180</a></p>
+      <p>Reed, <a href="#page71">71</a></p>
+      <p>Regnault, <a href="#page22">22</a></p>
+      <p>Reicher, <a href="#page36">36</a>, <a href="#page37">37</a>, <a href="#page110">110</a>, <a href="#page260">260</a>, <a href="#page333">333</a></p>
+      <p>Reinders, <a href="#page71">71</a>, <a href="#page185">185</a>, <a href="#page188">188</a></p>
+      <p>Reinitzer, <a href="#page51">51</a>, <a href="#page52">52</a></p>
+      <p>Richards, <a href="#page136">136</a>, <a href="#page140">140</a></p>
+      <p>Riddle, <a href="#page47">47</a></p>
+      <p>Riecke, <a href="#page48">48</a>, <a href="#page55">55</a></p>
+      <p>Roberts-Austen, <a href="#page63">63</a>, <a href="#page194">194</a>, <a href="#page221">221</a>, <a href="#page223">223</a>, <a href="#page225">225</a></p>
+      <p>Roloff, <a href="#page117">117</a></p>
+      <p>Roozeboom, <a href="#page10">10</a>, <a href="#page38">38</a>, <a href="#page45">45</a>, <a href="#page47">47</a>, <a href="#page49">49</a>, <a href="#page50">50</a>, <a href="#page51">51</a>, <a href="#page54">54</a>, <a href="#page56">56</a>, <a href="#page57">57</a>, <a href="#page62">62</a>, <a href="#page63">63</a>, <a href="#page68">68</a>, <a href="#page88">88</a>, <a href="#page103">103</a>, <a href="#page126">126</a>, <a href="#page145">145</a>, <a href="#page147">147</a>, <a href="#page150">150</a>, <a href="#page151">151</a>, <a href="#page157">157</a>, <a href="#page162">162</a>, <a href="#page170">170</a>, <a href="#page174">174</a>, <a href="#page178">178</a>, <a href="#page182">182</a>, <a href="#page196">196</a>, <a href="#page201">201</a>, <a href="#page211">211</a>, <a href="#page217">217</a>, <a href="#page220">220</a>, <a href="#page225">225</a>, <a href="#page236">236</a>, <a href="#page238">238</a>, <a href="#page262">262</a>, <a href="#page264">264</a>, <a href="#page269">269</a>, <a href="#page272">272</a>, <a href="#page273">273</a>, <a href="#page281">281</a>, <a href="#page282">282</a>, <a href="#page290">290</a>, <a href="#page331">331</a></p>
+<!-- Page 344 --><span class="pagenum"><a name="page344"></a>{344}</span>
+      <p>Rose, <a href="#page223">223</a></p>
+      <p>Rotarski, <a href="#page52">52</a></p>
+      <p>Rothmund, <a href="#page97">97</a>, <a href="#page98">98</a>, <a href="#page100">100</a></p>
+      <p>Rutten, <a href="#page298">298</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">S</p>
+      <p>Saposchnikoff, <a href="#page212">212</a></p>
+      <p>Saunders, <a href="#page313">313</a>, <a href="#page317">317</a>, <a href="#page319">319</a>, <a href="#page336">336</a>, <a href="#page337">337</a></p>
+      <p>Saurel, <a href="#page151">151</a></p>
+      <p>Schaum, <a href="#page49">49</a>, <a href="#page75">75</a></p>
+      <p>Scheel, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page30">30</a></p>
+      <p>Schenck, <a href="#page49">49</a>, <a href="#page52">52</a>, <a href="#page54">54</a>, <a href="#page311">311</a></p>
+      <p>Schneider, <a href="#page52">52</a></p>
+      <p>Schönbeck, <a href="#page75">75</a></p>
+      <p>Schreinemakers, <a href="#page122">122</a>, <a href="#page126">126</a>, <a href="#page246">246</a>, <a href="#page248">248</a>, <a href="#page250">250</a>, <a href="#page252">252</a>, <a href="#page290">290</a>, <a href="#page302">302</a></p>
+      <p>Schrötter, <a href="#page46">46</a></p>
+      <p>Schukowsky, <a href="#page52">52</a></p>
+      <p>Schwarz, <a href="#page331">331</a></p>
+      <p>Seitz, <a href="#page52">52</a></p>
+      <p>Shenstone, <a href="#page109">109</a>, <a href="#page115">115</a>, <a href="#page135">135</a></p>
+      <p>Shepherd, <a href="#page221">221</a>, <a href="#page255">255</a></p>
+      <p>Shields, <a href="#page178">178</a></p>
+      <p>Skirrow, <a href="#page130">130</a></p>
+      <p>Spring, <a href="#page63">63</a></p>
+      <p>von Stackelberg, <a href="#page107">107</a>, <a href="#page110">110</a></p>
+      <p>Staedel, <a href="#page267">267</a></p>
+      <p>Stansfield, <a href="#page194">194</a>, <a href="#page221">221</a></p>
+      <p>Stokes, <a href="#page236">236</a></p>
+      <p>Stortenbeker, <a href="#page44">44</a>, <a href="#page147">147</a>, <a href="#page161">161</a>, <a href="#page164">164</a>, <a href="#page281">281</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">T</p>
+      <p>Taber, <a href="#page229">229</a></p>
+      <p>Tammann, <a href="#page26">26</a>, <a href="#page32">32</a>, <a href="#page33">33</a>, <a href="#page37">37</a>, <a href="#page38">38</a>, <a href="#page39">39</a>, <a href="#page48">48</a>, <a href="#page52">52</a>, <a href="#page65">65</a>, <a href="#page67">67</a>, <a href="#page68">68</a>, <a href="#page72">72</a>, <a href="#page73">73</a>, <a href="#page140">140</a>, <a href="#page151">151</a>, <a href="#page176">176</a>, <a href="#page221">221</a>, <a href="#page230">230</a></p>
+      <p>Thiesen, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page30">30</a></p>
+      <p>Thomson, J., <a href="#page25">25</a>, <a href="#page28">28</a>, <a href="#page32">32</a></p>
+      <p>Thomson, W., <a href="#page25">25</a></p>
+      <p>Tilden, <a href="#page109">109</a>, <a href="#page115">115</a>, <a href="#page135">135</a></p>
+      <p>Trevor, <a href="#page16">16</a></p>
+      <p>Troost, <a href="#page46">46</a>, <a href="#page49">49</a>, <a href="#page50">50</a>, <a href="#page51">51</a></p>
+      <p>Tumlirz, <a href="#page72">72</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">V</p>
+      <p>Van Bemmelen, <a href="#page180">180</a></p>
+      <p>Van Deventer, <a href="#page110">110</a>, <a href="#page139">139</a>, <a href="#page266">266</a>, <a href="#page267">267</a>, <a href="#page333">333</a></p>
+      <p>Van Eyk, <a href="#page41">41</a>, <a href="#page63">63</a>, <a href="#page192">192</a>, <a href="#page338">338</a></p>
+      <p>Van't Hoff, <a href="#page36">36</a>, <a href="#page38">38</a>, <a href="#page58">58</a>, <a href="#page70">70</a>, <a href="#page90">90</a>, <a href="#page92">92</a>, <a href="#page108">108</a>, <a href="#page127">127</a>, <a href="#page139">139</a>, <a href="#page140">140</a>, <a href="#page165">165</a>, <a href="#page175">175</a>, <a href="#page225">225</a>, <a href="#page258">258</a>, <a href="#page260">260</a>, <a href="#page263">263</a>, <a href="#page265">265</a>, <a href="#page266">266</a>, <a href="#page267">267</a>, <a href="#page272">272</a>, <a href="#page284">284</a>, <a href="#page290">290</a>, <a href="#page313">313</a>, <a href="#page318">318</a>, <a href="#page333">333</a>, <a href="#page340">340</a></p>
+      <p>Van Leeuwen, <a href="#page259">259</a></p>
+      <p>Van Wyk, <a href="#page185">185</a></p>
+      <p>Vogt, <a href="#page5">5</a>, <a href="#page233">233</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">W</p>
+      <p>Waage, <a href="#page7">7</a></p>
+      <p>Wald, <a href="#page92">92</a></p>
+      <p>Walden, <a href="#page158">158</a></p>
+      <p>Walker, <a href="#page80">80</a>, <a href="#page105">105</a>, <a href="#page122">122</a>, <a href="#page126">126</a>, <a href="#page143">143</a></p>
+      <p>Wegscheider, <a href="#page10">10</a>, <a href="#page49">49</a>, <a href="#page202">202</a></p>
+      <p>Wells, <a href="#page136">136</a></p>
+      <p>Wenzel, <a href="#page7">7</a></p>
+      <p>Wiebe, <a href="#page22">22</a></p>
+      <p>Witt, <a href="#page327">327</a></p>
+      <p>Wright, <a href="#page241">241</a>, <a href="#page246">246</a>, <a href="#page247">247</a></p>
+      <p>von Wrochem, <a href="#page109">109</a>, <a href="#page142">142</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">Y</p>
+      <p>Young, <a href="#page3">3</a>, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page24">24</a>, <a href="#page30">30</a>, <a href="#page32">32</a>, <a href="#page63">63</a>, <a href="#page64">64</a>, <a href="#page66">66</a>, <a href="#page79">79</a>, <a href="#page105">105</a>, <a href="#page165">165</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">Z</p>
+      <p>Zacharias, <a href="#page180">180</a></p>
+      <p>Zawidski, <a href="#page63">63</a></p>
+      <p>Zenghelis, <a href="#page35">35</a></p>
+      <p>Zimmermann, <a href="#page311">311</a></p>
+      <p>Zincke, <a href="#page44">44</a></p>
+      <p>Ziz, <a href="#page141">141</a></p>
+    </div>
+  </div>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<p><!-- Page 345 --><span class="pagenum"><a name="page345"></a>{345}</span></p>
+
+<h3>SUBJECT INDEX</h3>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="i8">A</p>
+      <p>Acetaldehyde and paraldehyde, <a href="#page204">204</a></p>
+      <p>Acetic acid, chloroform, water, <a href="#page241">241</a></p>
+      <p>Acetone, phenol, water, <a href="#page248">248</a></p>
+      <p>Adsorption, <a href="#page176">176</a></p>
+      <p>Alcohol, chloroform, water, <a href="#page246">246</a></p>
+      <p class="i2">&mdash;&mdash;, ether, water, <a href="#page246">246</a></p>
+      <p>Alloys, equilibrium curves of, <a href="#page221">221</a></p>
+      <p class="i2">&mdash;&mdash; of copper and tin, liquefaction of, by cooling, <a href="#page194">194</a></p>
+      <p class="i2">&mdash;&mdash; of iron and carbon, <a href="#page223">223</a></p>
+      <p class="i2">&mdash;&mdash; of thallium and mercury, <a href="#page222">222</a></p>
+      <p class="i2">&mdash;&mdash;, ternary, <a href="#page246">246</a></p>
+      <p>Ammonia compounds of metal chlorides, <a href="#page82">82</a></p>
+      <p>Ammonia silver chlorides, <a href="#page82">82</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; &mdash;&mdash;, dissociation pressures of, <a href="#page84">84</a></p>
+      <p>Ammonia-soda process, <a href="#page320">320</a></p>
+      <p>Ammonium chloride, dissociation of, <a href="#page3">3</a>, <a href="#page79">79</a></p>
+      <p class="i2">&mdash;&mdash; cyanide, dissociation of, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; hydrosulphide, dissociation of, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; nitrate, solubility of, <a href="#page113">113</a></p>
+      <p>Aniline, phenol, water, <a href="#page250">250</a></p>
+      <p>Astracanite, <a href="#page260">260</a>, <a href="#page261">261</a>, <a href="#page268">268</a>, <a href="#page274">274</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">B</p>
+      <p>Babo, law of, <a href="#page126">126</a></p>
+      <p>Barium acetate, solubility of, <a href="#page143">143</a></p>
+      <p>Barium carbonate and potassium sulphate, <a href="#page328">328</a></p>
+      <p class="i2">&mdash;&mdash; nitrite, preparation of, <a href="#page327">327</a></p>
+      <p>Basic salts, <a href="#page296">296</a></p>
+      <p>Benzaldoximes, <a href="#page203">203</a></p>
+      <p>Benzene and picric acid, <a href="#page216">216</a></p>
+      <p>Bismuth, effect of pressure on the melting point of, <a href="#page67">67</a></p>
+      <p class="i2">&mdash;&mdash;, lead, tin, <a href="#page255">255</a></p>
+      <p class="i2">&mdash;&mdash; nitrates, basic, <a href="#page298">298</a></p>
+      <p>Bivariant systems, <a href="#page16">16</a></p>
+      <p>Bromocinnamic aldehyde and chlorocinnamic aldehyde, <a href="#page183">183</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">C</p>
+      <p>Calcium carbonate, dissociation of, <a href="#page3">3</a>, <a href="#page11">11</a>, <a href="#page81">81</a></p>
+      <p class="i2">&mdash;&mdash; chloride hexahydrate, solubility of, <a href="#page146">146</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, solubility of hydrates of, <a href="#page148">148</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, vapour-pressure of hydrates of, <a href="#page88">88</a></p>
+      <p>Camphor oximes, <a href="#page219">219</a>, <a href="#page257">257</a></p>
+      <p>Carnallite, <a href="#page284">284</a></p>
+      <p>Carvoximes, <a href="#page186">186</a>, <a href="#page219">219</a></p>
+      <p>Cementite, <a href="#page224">224</a></p>
+      <p>Chlorine and iodine, <a href="#page161">161</a></p>
+      <p>Chlorocinnamic aldehyde and bromocinnamic aldehyde, <a href="#page183">183</a></p>
+      <p>Chloroform, acetic acid, water, <a href="#page241">241</a></p>
+      <p class="i2">&mdash;&mdash;, alcohol, water, <a href="#page246">246</a></p>
+<!-- Page 346 --><span class="pagenum"><a name="page346"></a>{346}</span>
+      <p>Classification of systems, <a href="#page17">17</a></p>
+      <p>Component, <a href="#page8">8</a>, <a href="#page10">10</a>, <a href="#page12">12</a></p>
+      <p class="i2">&mdash;&mdash;, systems of one, <a href="#page21">21</a>, <a href="#page55">55</a></p>
+      <p>Components, choice of, <a href="#page12">12</a>, <a href="#page13">13</a>, <a href="#page14">14</a>, <a href="#page76">76</a>, <a href="#page313">313</a></p>
+      <p class="i2">&mdash;&mdash;, determination of number of, <a href="#page13">13</a></p>
+      <p class="i2">&mdash;&mdash;, systems of four, <a href="#page312">312</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; of three, <a href="#page234">234</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; of two, <a href="#page76">76</a>, <a href="#page207">207</a></p>
+      <p class="i2">&mdash;&mdash;, variation in number of, <a href="#page11">11</a>, <a href="#page14">14</a></p>
+      <p>Composition, determination of, without analysis, <a href="#page228">228</a>, <a href="#page302">302</a></p>
+      <p>Concentration-temperature curve for two liquids, <a href="#page101">101</a></p>
+      <p>Condensed systems, <a href="#page36">36</a></p>
+      <p>Constituent, <a href="#page10">10</a></p>
+      <p>Cooling curve, <a href="#page230">230</a></p>
+      <p>Copper calcium acetate, <a href="#page260">260</a></p>
+      <p class="i2">&mdash;&mdash; chloride, heat of solution of, <a href="#page110">110</a></p>
+      <p class="i2">&mdash;&mdash; dipotassium chloride, <a href="#page259">259</a></p>
+      <p class="i2">&mdash;&mdash; sulphate, <a href="#page85">85</a></p>
+      <p>Critical concentration, <a href="#page98">98</a>, <a href="#page242">242</a></p>
+      <p class="i2">&mdash;&mdash; pressure of water, <a href="#page23">23</a></p>
+      <p class="i2">&mdash;&mdash; solution temperature, <a href="#page98">98</a></p>
+      <p class="i2">&mdash;&mdash; temperature of water, <a href="#page23">23</a></p>
+      <p>Cryohydrates, <a href="#page117">117</a>, <a href="#page118">118</a></p>
+      <p>Cryohydric point, <a href="#page117">117</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, changes at the, <a href="#page119">119</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; for silver nitrate and ice, <a href="#page116">116</a></p>
+      <p>Crystals, liquid, <a href="#page51">51</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, equilibria of, <a href="#page53">53</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, list of, <a href="#page54">54</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, nature of, <a href="#page52">52</a></p>
+      <p class="i2">&mdash;&mdash;, mixed, <a href="#page180">180</a></p>
+      <p>Crystallization, velocity of, <a href="#page72">72</a>, <a href="#page74">74</a></p>
+      <p class="i2">&mdash;&mdash;, spontaneous, <a href="#page114">114</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">D</p>
+      <p>Deliquescence, <a href="#page130">130</a></p>
+      <p>Devitrification, <a href="#page73">73</a></p>
+      <p>Diethylamine and water, solubility of, <a href="#page101">101</a></p>
+      <p>Dilatometer, determination of transition points by, <a href="#page331">331</a></p>
+      <p>Dineric surface, <a href="#page247">247</a></p>
+      <p>Dissociation equilibrium, effect of addition of dissociation products on, <a href="#page4">4</a></p>
+      <p class="i2">&mdash;&mdash; of ammonia compounds of metal chlorides, <a href="#page82">82</a>, <a href="#page84">84</a></p>
+      <p class="i2">&mdash;&mdash; of ammonium chloride, <a href="#page3">3</a>, <a href="#page79">79</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; cyanide, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; hydrosulphide, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; of calcium carbonate, <a href="#page3">3</a>, <a href="#page81">81</a></p>
+      <p class="i2">&mdash;&mdash; of compounds, degree of, <a href="#page147">147</a></p>
+      <p class="i2">&mdash;&mdash; of phosphonium bromide, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; of salt hydrates, <a href="#page85">85</a></p>
+      <p class="i2">&mdash;&mdash;, phenomena of, <a href="#page79">79</a></p>
+      <p>Dissociation pressure, <a href="#page81">81</a></p>
+      <p>Distillation of supercooled liquid to solid, <a href="#page32">32</a>, <a href="#page50">50</a></p>
+      <p>Double salt interval, <a href="#page278">278</a></p>
+      <p class="i2">&mdash;&mdash; salts, crystallization from solution, <a href="#page280">280</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, decomposition by water, <a href="#page267">267</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, formation of, <a href="#page258">258</a>, <a href="#page273">273</a>, <a href="#page315">315</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">E</p>
+      <p>Efflorescence, <a href="#page86">86</a></p>
+      <p>Electrical methods of determining transition points, <a href="#page338">338</a></p>
+      <p>Enantiotropy, <a href="#page44">44</a>, <a href="#page51">51</a></p>
+      <p>Equilibria, Gibbs's theory of, <a href="#page8">8</a></p>
+      <p class="i2">&mdash;&mdash;, metastable, <a href="#page69">69</a></p>
+      <p>Equilibrium apparent (false), <a href="#page5">5</a>, <a href="#page6">6</a></p>
+      <p class="i2">&mdash;&mdash; between ice and solution, <a href="#page116">116</a></p>
+      <p class="i2">&mdash;&mdash; between ice and water, <a href="#page25">25</a></p>
+      <p class="i2">&mdash;&mdash; between ice, water, vapour, <a href="#page27">27</a></p>
+      <p class="i2">&mdash;&mdash; between water and vapour, <a href="#page21">21</a></p>
+      <p class="i2">&mdash;&mdash;, chemical, <a href="#page3">3</a>, <a href="#page16">16</a></p>
+      <p class="i2">&mdash;&mdash;, heterogeneous, <a href="#page5">5</a></p>
+      <p class="i2">&mdash;&mdash;, homogeneous, <a href="#page5">5</a></p>
+      <p class="i2">&mdash;&mdash;, independence of, on amounts of phases, <a href="#page9">9</a></p>
+      <p class="i2">&mdash;&mdash;, law of movable, <a href="#page58">58</a></p>
+<!-- Page 347 --><span class="pagenum"><a name="page347"></a>{347}</span>
+      <p class="i2">&mdash;&mdash;, physical, <a href="#page3">3</a>, <a href="#page16">16</a></p>
+      <p class="i2">&mdash;&mdash; real (true), <a href="#page5">5</a>, <a href="#page6">6</a></p>
+      <p>Ether, alcohol, water, <a href="#page246">246</a></p>
+      <p class="i2">&mdash;&mdash;, succinic nitrile, water, <a href="#page252">252</a></p>
+      <p>Ethylene bromide, picric acid, <span class="grk">&beta;</span>-naphthol, <a href="#page256">256</a></p>
+      <p>Eutectic mixtures, <a href="#page117">117</a>, <a href="#page191">191</a>, <a href="#page209">209</a>, <a href="#page255">255</a>, <a href="#page257">257</a></p>
+      <p class="i2">&mdash;&mdash; point, <a href="#page117">117</a>, <a href="#page209">209</a>, <a href="#page213">213</a>, <a href="#page253">253</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">F</p>
+      <p>Ferric chloride, evaporation of solutions of, <a href="#page155">155</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, hydrates of, <a href="#page151">151</a>, <a href="#page153">153</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, hydrogen chloride and water, systems of, <a href="#page290">290</a></p>
+      <p>Ferrite, modifications of, <a href="#page224">224</a></p>
+      <p>Freedom, degree of, <a href="#page14">14</a></p>
+      <p>Freezing mixtures, <a href="#page120">120</a></p>
+      <p class="i2">&mdash;&mdash; point, natural, <a href="#page198">198</a></p>
+      <p>Fusion curve, <a href="#page66">66</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of ice, <a href="#page25">25</a></p>
+      <p class="i2">&mdash;&mdash; of ice, influence of pressure on, <a href="#page26">26</a></p>
+      <p class="i2">&mdash;&mdash;, partial, <a href="#page139">139</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">G</p>
+      <p>Glaserite, <a href="#page315">315</a>, <a href="#page317">317</a></p>
+      <p>Glasses, <a href="#page176">176</a></p>
+      <p>Glauber's salt, <a href="#page13">13</a>, <a href="#page134">134</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, transition curve of, <a href="#page68">68</a>, <a href="#page140">140</a></p>
+      <p>Graphic representation in space, <a href="#page77">77</a>, <a href="#page284">284</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">H</p>
+      <p>Hydrates, range of existence of, <a href="#page89">89</a></p>
+      <p class="i2">&mdash;&mdash; chloride and water, <a href="#page174">174</a></p>
+      <p>Hydrogen bromide and water, <a href="#page174">174</a></p>
+      <p>Hylotropic substances, <a href="#page198">198</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">I</p>
+      <p>Ice I., <a href="#page32">32</a></p>
+      <p class="i2">&mdash;&mdash; II., <a href="#page32">32</a></p>
+      <p class="i2">&mdash;&mdash; III., <a href="#page32">32</a></p>
+      <p class="i2">&mdash;&mdash;, equilibrium between water and, <a href="#page25">25</a></p>
+      <p class="i2">&mdash;&mdash;, influence of pressure on melting point of, <a href="#page25">25</a>, <a href="#page26">26</a></p>
+      <p class="i2">&mdash;&mdash;, sublimation curve of, <a href="#page24">24</a></p>
+      <p class="i2">&mdash;&mdash;, vapour pressure of, <a href="#page25">25</a>, <a href="#page31">31</a></p>
+      <p>Indifferent point, <a href="#page150">150</a></p>
+      <p>Individual, chemical, <a href="#page92">92</a></p>
+      <p>Inversion temperature, <a href="#page36">36</a></p>
+      <p>Iodine and chlorine, <a href="#page161">161</a></p>
+      <p>Iron&mdash;carbon alloys, <a href="#page223">223</a></p>
+      <p class="i2">&mdash;&mdash;, carbon monoxide and carbon dioxide, <a href="#page305">305</a></p>
+      <p>Isomerides, dynamic, <a href="#page195">195</a>, <a href="#page196">196</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, equilibrium between, <a href="#page195">195</a>, <a href="#page196">196</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, equilibrium point of, <a href="#page198">198</a></p>
+      <p class="i2">&mdash;&mdash;, transformation of unstable into stable, <a href="#page201">201</a></p>
+      <p>Isomerism, dynamic, <a href="#page196">196</a></p>
+      <p>Isothermal evaporation, <a href="#page278">278</a></p>
+      <p class="i2">&mdash;&mdash; solubility curves, <a href="#page272">272</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">L</p>
+      <p>Lead, bismuth, tin, <a href="#page255">255</a></p>
+      <p class="i2">&mdash;&mdash;, desilverization of, <a href="#page247">247</a></p>
+      <p class="i2">&mdash;&mdash;, silver, zinc, <a href="#page246">246</a></p>
+      <p>Le Chatelier, theorem of, <a href="#page57">57</a></p>
+      <p>Lime, burning of, <a href="#page3">3</a></p>
+      <p>Liquidus curve, <a href="#page182">182</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">M</p>
+      <p>Mandelic acid, <a href="#page217">217</a></p>
+      <p>Martensite, <a href="#page224">224</a></p>
+      <p>Mass action, law of, <a href="#page7">7</a></p>
+      <p>Melting point, influence of pressure on, <a href="#page66">66</a></p>
+<!-- Page 348 --><span class="pagenum"><a name="page348"></a>{348}</span>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, congruent, <a href="#page146">146</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, incongruent, <a href="#page139">139</a></p>
+      <p class="i2">&mdash;&mdash; under the solvent, <a href="#page122">122</a></p>
+      <p>Menthyl mandelates, <a href="#page219">219</a></p>
+      <p>Mercuric bromide and iodide, <a href="#page188">188</a></p>
+      <p>Mercury salts, basic, <a href="#page301">301</a></p>
+      <p>Metastable equilibria, <a href="#page69">69</a></p>
+      <p class="i2">&mdash;&mdash; region, <a href="#page30">30</a></p>
+      <p class="i2">&mdash;&mdash; state, <a href="#page30">30</a></p>
+      <p>Methylethyl ketone and water, <a href="#page100">100</a></p>
+      <p>Minerals, formation of, <a href="#page232">232</a></p>
+      <p>Miscibility of liquids, complete, <a href="#page95">95</a>, <a href="#page104">104</a>, <a href="#page114">114</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, partial, <a href="#page95">95</a>, <a href="#page96">96</a>, <a href="#page121">121</a></p>
+      <p>Mixed crystals, <a href="#page180">180</a>, <a href="#page281">281</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, changes in, with temperature, <a href="#page192">192</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, examples of, <a href="#page183">183</a>, <a href="#page186">186</a>, <a href="#page187">187</a>, <a href="#page190">190</a>, <a href="#page192">192</a>, <a href="#page219">219</a>, <a href="#page223">223</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, formation of, <a href="#page181">181</a>, <a href="#page182">182</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, fractional crystallization of, <a href="#page188">188</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, freezing points of, <a href="#page182">182</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, melting points of, <a href="#page182">182</a>, <a href="#page184">184</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, pseudoracemic, <a href="#page219">219</a></p>
+      <p>Mixtures, isomorphous, <a href="#page181">181</a></p>
+      <p class="i2">&mdash;&mdash; of constant boiling point, <a href="#page105">105</a></p>
+      <p class="i2">&mdash;&mdash; of constant melting point, <a href="#page117">117</a>, <a href="#page186">186</a>, <a href="#page187">187</a>, <a href="#page192">192</a>, <a href="#page209">209</a>, <a href="#page255">255</a>, <a href="#page257">257</a></p>
+      <p>Monotropy, <a href="#page44">44</a>, <a href="#page51">51</a></p>
+      <p>Multivariant systems, <a href="#page16">16</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">N</p>
+      <p>Naphthalene and monochloracetic acid, <a href="#page192">192</a></p>
+      <p class="i2">&mdash;&mdash; and <span class="grk">&beta;</span>-naphthol, mixed crystals of, <a href="#page183">183</a></p>
+      <p><span class="grk">&beta;</span>-Naphthol, ethylene bromide, picric acid, <a href="#page256">256</a></p>
+      <p><span class="grk">&alpha;</span>-Naphthylamine and phenol, <a href="#page213">213</a></p>
+      <p>Nickel iodate, solubility of, <a href="#page142">142</a></p>
+      <p><i>o</i>-Nitrophenol and <i>p</i>-toluidine, <a href="#page213">213</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">O</p>
+      <p>Occlusion of gases, <a href="#page176">176</a></p>
+      <p>Optical method of determining transition points, <a href="#page338">338</a></p>
+      <p>Optically active substances, freezing-point curves of, <a href="#page216">216</a></p>
+      <p>Order of a system, <a href="#page13">13</a></p>
+      <p>Organic compounds, application of Phase Rule to, <a href="#page212">212</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">P</p>
+      <p>Palladium and hydrogen, <a href="#page90">90</a>, <a href="#page178">178</a></p>
+      <p>Paragenesis, <a href="#page320">320</a></p>
+      <p>Paraldehyde and acetaldehyde, <a href="#page204">204</a></p>
+      <p>Partial pressures of two components, <a href="#page102">102</a></p>
+      <p>Pearlite, <a href="#page224">224</a></p>
+      <p>Phase, <a href="#page8">8</a></p>
+      <p class="i2">&mdash;&mdash; Rule, <a href="#page8">8</a>, <a href="#page16">16</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, deduction of, <a href="#page18">18</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, scope of, <a href="#page1">1</a></p>
+      <p>Phases, formation of new, <a href="#page69">69</a></p>
+      <p class="i2">&mdash;&mdash;, number of, <a href="#page9">9</a></p>
+      <p>Phenol, acetone, water, <a href="#page248">248</a></p>
+      <p class="i2">&mdash;&mdash;, aniline, water, <a href="#page250">250</a></p>
+      <p class="i2">&mdash;&mdash; and <span class="grk">&alpha;</span>-naphthylamine, <a href="#page213">213</a></p>
+      <p class="i2">&mdash;&mdash; and <i>p</i>-toluidine, <a href="#page214">214</a></p>
+      <p class="i2">&mdash;&mdash; and water, solubility of, <a href="#page97">97</a></p>
+      <p>Phosphonium bromide, dissociation of, <a href="#page80">80</a></p>
+      <p class="i2">&mdash;&mdash; chloride, <a href="#page65">65</a></p>
+      <p>Phosphorus, <a href="#page46">46</a></p>
+      <p class="i2">&mdash;&mdash;, distillation of white to red, <a href="#page50">50</a></p>
+      <p class="i2">&mdash;&mdash;, melting point of red, <a href="#page47">47</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; &mdash;&mdash; of white, <a href="#page48">48</a></p>
+      <p class="i2">&mdash;&mdash;, solubility of white and red, <a href="#page47">47</a></p>
+      <p class="i2">&mdash;&mdash;, vapour pressure of white and red, <a href="#page46">46</a></p>
+      <p>Picric acid and benzene, <a href="#page216">216</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, ethylene bromide, and <span class="grk">&beta;</span>-naphthol, <a href="#page256">256</a></p>
+      <p>Polymorphic forms, solubility of, <a href="#page112">112</a></p>
+<!-- Page 349 --><span class="pagenum"><a name="page349"></a>{349}</span>
+      <p class="i2">&mdash;&mdash; substances, list of, <a href="#page63">63</a></p>
+      <p>Polymorphism, <a href="#page33">33</a></p>
+      <p>Potassium nitrate and thallium nitrate, <a href="#page192">192</a></p>
+      <p>Potential, chemical, <a href="#page19">19</a></p>
+      <p>Pressure-concentration diagram for two liquids, <a href="#page102">102</a></p>
+      <p>Pressure-temperature diagram for solutions, <a href="#page126">126</a></p>
+      <p>Pseudomonotropy, <a href="#page45">45</a></p>
+      <p>Pseudo-racemic mixed crystals, <a href="#page21">21</a></p>
+      <p>Pyridine and methyl iodide, <a href="#page147">147</a></p>
+      <p>Pyrometer, registering, <a href="#page230">230</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">Q</p>
+      <p>Quadruple point, <a href="#page116">116</a></p>
+      <p>Quintuple point, <a href="#page234">234</a>, <a href="#page261">261</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">R</p>
+      <p>Racemates, characterization of, <a href="#page217">217</a>, <a href="#page282">282</a></p>
+      <p>Reactions, law of successive, <a href="#page73">73</a></p>
+      <p>Reciprocal salt-pairs, <a href="#page313">313</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, transition point of, <a href="#page314">314</a></p>
+      <p>Rubidium tartrates, <a href="#page265">265</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">S</p>
+      <p>Salt hydrates, <a href="#page85">85</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, indefiniteness of vapour pressure of, <a href="#page87">87</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; with definite melting point, <a href="#page145">145</a></p>
+      <p>Separation of salt on evaporation, <a href="#page130">130</a></p>
+      <p>Silicates, hydrated, <a href="#page176">176</a></p>
+      <p>Silver, lead, zinc, <a href="#page246">246</a></p>
+      <p>Silver nitrate, solubility of, <a href="#page114">114</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; and sodium nitrate, <a href="#page190">190</a></p>
+      <p>Single salt interval, <a href="#page278">278</a></p>
+      <p>Sodium ammonium tartrates, <a href="#page266">266</a></p>
+      <p class="i2">&mdash;&mdash; nitrate and silver nitrate, <a href="#page190">190</a></p>
+      <p class="i2">&mdash;&mdash; sulphate and water, equilibria between, <a href="#page134">134</a></p>
+      <p>Sodium sulphate and water, vapour pressures of, <a href="#page138">138</a>, <a href="#page140">140</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, anhydrous, dehydration by, <a href="#page138">138</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, solubility of, <a href="#page135">135</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; decahydrate, solubility of, <a href="#page134">134</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; &mdash;&mdash;, transition point of, <a href="#page136">136</a>, <a href="#page139">139</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; heptahydrate, solubility of, <a href="#page136">136</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; &mdash;&mdash;, transition point of, <a href="#page137">137</a></p>
+      <p>Solidus curve, <a href="#page182">182</a></p>
+      <p>Solubility curve at higher temperatures, <a href="#page114">114</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, form of, <a href="#page108">108</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of anhydrous salts, <a href="#page111">111</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, retroflex, <a href="#page146">146</a>, <a href="#page151">151</a>, <a href="#page162">162</a></p>
+      <p class="i2">&mdash;&mdash; curves, interpolation and extrapolation of, <a href="#page111">111</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of three component systems, <a href="#page264">264</a></p>
+      <p class="i2">&mdash;&mdash;, determination of transition points by, <a href="#page335">335</a></p>
+      <p class="i2">&mdash;&mdash;, influence of pressure on, <a href="#page107">107</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; of subdivision on, <a href="#page10">10</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; of temperature on, <a href="#page109">109</a></p>
+      <p class="i2">&mdash;&mdash; of metastable forms, <a href="#page47">47</a>, <a href="#page112">112</a>, <a href="#page137">137</a></p>
+      <p>Solubility of polymorphic forms, <a href="#page112">112</a></p>
+      <p class="i2">&mdash;&mdash; of salt hydrates, <a href="#page133">133</a>, <a href="#page145">145</a></p>
+      <p class="i2">&mdash;&mdash; of supercooled liquids, <a href="#page125">125</a></p>
+      <p class="i2">&mdash;&mdash;, retrograde, <a href="#page245">245</a></p>
+      <p>Solute, <a href="#page93">93</a></p>
+      <p>Solution, definition of, <a href="#page92">92</a></p>
+      <p class="i2">&mdash;&mdash;, heat of, <a href="#page109">109</a>, <a href="#page110">110</a></p>
+      <p class="i2">&mdash;&mdash;, saturated, <a href="#page106">106</a>, <a href="#page108">108</a></p>
+      <p class="i2">&mdash;&mdash;, supersaturated, <a href="#page108">108</a></p>
+      <p class="i2">&mdash;&mdash; temperature, critical, <a href="#page98">98</a></p>
+      <p class="i2">&mdash;&mdash;, unsaturated, <a href="#page108">108</a></p>
+      <p>Solutions, bivariant systems, <a href="#page129">129</a></p>
+      <p class="i2">&mdash;&mdash;, congruently saturated, <a href="#page279">279</a></p>
+      <p class="i2">&mdash;&mdash; conjugate, <a href="#page97">97</a>, <a href="#page241">241</a></p>
+<!-- Page 350 --><span class="pagenum"><a name="page350"></a>{350}</span>
+      <p class="i2">&mdash;&mdash;, incongruently saturated, <a href="#page279">279</a>, <a href="#page289">289</a></p>
+      <p class="i2">&mdash;&mdash;, inevaporable, <a href="#page157">157</a></p>
+      <p class="i2">&mdash;&mdash; of gases in liquids, <a href="#page93">93</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; in solids, <a href="#page176">176</a></p>
+      <p class="i2">&mdash;&mdash; of liquids in liquids (binary), <a href="#page95">95</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; &mdash;&mdash; (ternary), <a href="#page240">240</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, influence of temperature on, <a href="#page247">247</a></p>
+      <p class="i2">&mdash;&mdash; of solids in liquids, <a href="#page106">106</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; in solids, <a href="#page180">180</a></p>
+      <p class="i2">&mdash;&mdash;, solid, <a href="#page175">175</a>, <a href="#page180">180</a></p>
+      <p class="i2">&mdash;&mdash;, univariant systems, <a href="#page127">127</a></p>
+      <p>Space model for carnallite, <a href="#page284">284</a></p>
+      <p>Stability limit, <a href="#page202">202</a></p>
+      <p>Steel, formation of, <a href="#page223">223</a></p>
+      <p>Sublimation curve, <a href="#page63">63</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of ice, <a href="#page24">24</a></p>
+      <p class="i2">&mdash;&mdash; without fusion, <a href="#page65">65</a></p>
+      <p>Succinic nitrile and water, <a href="#page122">122</a></p>
+      <p class="i2">&mdash;&mdash; ether, water, <a href="#page252">252</a></p>
+      <p>Sulphur, <a href="#page33">33</a>, <a href="#page34">34</a></p>
+      <p class="i2">&mdash;&mdash; dioxide and water, <a href="#page169">169</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; and potassium iodide, <a href="#page158">158</a></p>
+      <p class="i2">&mdash;&mdash;, transition point of rhombic and monoclinic, <a href="#page36">36</a></p>
+      <p>Supersaturation, <a href="#page113">113</a>, <a href="#page114">114</a>, <a href="#page124">124</a></p>
+      <p class="i2">&mdash;&mdash;, limits of, <a href="#page114">114</a></p>
+      <p>Systems, condensed, <a href="#page36">36</a></p>
+      <p class="i2">&mdash;&mdash; of one component, <a href="#page21">21</a></p>
+      <p class="i2">&mdash;&mdash; of two components, <a href="#page76">76</a>, <a href="#page77">77</a>, <a href="#page207">207</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">T</p>
+      <p>Tachydrite, influence of pressure on the transition point of, <a href="#page263">263</a></p>
+      <p>Tartrate, dimethyl, <a href="#page217">217</a></p>
+      <p class="i2">&mdash;&mdash;, sodium potassium, <a href="#page259">259</a></p>
+      <p>Tautomeric substances, <a href="#page195">195</a></p>
+      <p>Tensimeter, <a href="#page91">91</a></p>
+      <p>Thallium nitrate and potassium nitrate, <a href="#page192">192</a></p>
+      <p>Theorem of van't Hoff and Le Chatelier, <a href="#page57">57</a></p>
+      <p>Thermometric  determination of transition point, <a href="#page337">337</a></p>
+      <p>Tin, <a href="#page41">41</a></p>
+      <p class="i2">&mdash;&mdash;, lead, bismuth, <a href="#page255">255</a></p>
+      <p class="i2">&mdash;&mdash; plague, <a href="#page43">43</a></p>
+      <p class="i2">&mdash;&mdash;, transition point of white and grey, <a href="#page41">41</a></p>
+      <p><i>p</i>-Toluidine and <i>o</i>-nitrophenol, <a href="#page213">213</a></p>
+      <p class="i2">&mdash;&mdash; and phenol, <a href="#page214">214</a></p>
+      <p>Transformation of optically active substances, <a href="#page220">220</a></p>
+      <p class="i2">&mdash;&mdash;, suspended, <a href="#page37">37</a>, <a href="#page69">69</a>, <a href="#page89">89</a>, <a href="#page113">113</a>, <a href="#page137">137</a>, <a href="#page155">155</a></p>
+      <p class="i2">&mdash;&mdash;, velocity of, <a href="#page70">70</a></p>
+      <p>Transition curve, <a href="#page66">66</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of Glauber's salt, <a href="#page68">68</a>, <a href="#page140">140</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of rhombic and monoclinic sulphur, <a href="#page37">37</a></p>
+      <p class="i2">&mdash;&mdash; interval, <a href="#page270">270</a>, <a href="#page277">277</a>, <a href="#page315">315</a></p>
+      <p class="i2">&mdash;&mdash; point, <a href="#page34">34</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; for double salts, <a href="#page258">258</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, influence of pressure on the, <a href="#page68">68</a></p>
+      <p class="i2">&mdash;&mdash; points, as fixed points in thermometry, <a href="#page140">140</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, methods of determining, <a href="#page331">331</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of polymorphic substances, <a href="#page63">63</a></p>
+      <p>Triangle, graphic representation by, <a href="#page235">235</a></p>
+      <p>Triethylamine and water, <a href="#page101">101</a></p>
+      <p>Triple point, <a href="#page27">27</a>, <a href="#page55">55</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, arrangement of curves round, <a href="#page56">56</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, changes at, <a href="#page58">58</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, ice, water, vapour, <a href="#page27">27</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, ice II., ice III., and water, <a href="#page33">33</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, metastable, <a href="#page38">38</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, monoclinic sulphur, liquid, vapour, <a href="#page38">38</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, monoclinic and rhombic sulphur, liquid, <a href="#page38">38</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, monoclinic and rhombic sulphur, vapour, <a href="#page34">34</a></p>
+<!-- Page 351 --><span class="pagenum"><a name="page351"></a>{351}</span>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, red phosphorus, liquid, vapour, <a href="#page47">47</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, rhombic sulphur, liquid, vapour, <a href="#page38">38</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; solid, solid, vapour, <a href="#page62">62</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, white phosphorus, liquid, vapour, <a href="#page48">48</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">U</p>
+      <p>Univariant systems, <a href="#page16">16</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">V</p>
+      <p>Van't Hoff, theorem of, <a href="#page57">57</a></p>
+      <p>Vaporization curve, <a href="#page63">63</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, interpolation and extrapolation of, <a href="#page66">66</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of water, <a href="#page21">21</a>, <a href="#page23">23</a></p>
+      <p>Vapour pressure, constancy of, and formation of compounds, <a href="#page90">90</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, dependence of, on solid phase, <a href="#page88">88</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, influence of surface tension on, <a href="#page2">2</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; in three-component systems, <a href="#page261">261</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash;, measurement of, <a href="#page91">91</a>, <a href="#page334">334</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of calcium chloride solutions, <a href="#page150">150</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of ice, <a href="#page25">25</a>, <a href="#page31">31</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of small drops, <a href="#page10">10</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of sodium sulphate and water, <a href="#page138">138</a></p>
+      <p>Vapour pressure of solid, solution, vapour, <a href="#page126">126</a></p>
+      <p class="i2">&mdash;&mdash; &mdash;&mdash; of water, <a href="#page21">21</a>, <a href="#page31">31</a></p>
+      <p>Variability of a system, <a href="#page14">14</a>, <a href="#page16">16</a></p>
+      <p>Variance of a system, <a href="#page16">16</a></p>
+      <p>Volatile components, two, <a href="#page161">161</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">W</p>
+      <p>Water, <a href="#page21">21</a></p>
+      <p class="i2">&mdash;&mdash;, acetic acid, chloroform, <a href="#page241">241</a></p>
+      <p class="i2">&mdash;&mdash;, acetone, phenol, <a href="#page248">248</a></p>
+      <p class="i2">&mdash;&mdash;, alcohol, ether, <a href="#page246">246</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, chloroform, <a href="#page246">246</a></p>
+      <p class="i2">&mdash;&mdash;, aniline, phenol, <a href="#page250">250</a></p>
+      <p class="i2">&mdash;&mdash;, bivariant systems of, <a href="#page29">29</a></p>
+      <p class="i2">&mdash;&mdash;, critical pressure of, <a href="#page23">23</a></p>
+      <p class="i2">&mdash;&mdash;, critical temperature of, <a href="#page23">23</a></p>
+      <p class="i2">&mdash;&mdash;, equilibrium between ice and, <a href="#page25">25</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash; between vapour and, <a href="#page21">21</a></p>
+      <p class="i2">&mdash;&mdash;, ether, succinic nitrile, <a href="#page252">252</a></p>
+      <p class="i2">&mdash;&mdash;, supercooled, <a href="#page30">30</a></p>
+      <p class="i2">&mdash;&mdash;, &mdash;&mdash;, vapour pressure of, <a href="#page31">31</a></p>
+      <p class="i2">&mdash;&mdash;, vaporization curve of, <a href="#page21">21</a></p>
+      <p class="i2">&mdash;&mdash;, vapour pressure of, <a href="#page23">23</a></p>
+    </div>
+
+    <div class="stanza">
+      <p class="i8">Z</p>
+      <p>Zeolites, <a href="#page176">176</a></p>
+      <p>Zinc, lead, silver, <a href="#page246">246</a></p>
+      <p>&mdash;&mdash; chloride in water, solubility of, <a href="#page157">157</a></p>
+    </div>
+  </div>
+
+  <p>THE END</p>
+
+  <p>PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.</p>
+
+  <p><br style="clear:both" /></p>
+<hr class="full" />
+
+<h3>NOTES</h3>
+
+<div class="note">
+  <p><a name="Nt1" href="#NtA1">[1]</a> Except when the volume of the
+  liquid becomes exceedingly small, in which case the surface tension
+  exerts an influence on the vapour pressure.</p>
+
+  <p><a name="Nt2" href="#NtA2">[2]</a> For reasons which will appear later
+  (Chap. IV.), the volume of the vapour is supposed to be large in
+  comparison with that of the solid and liquid.</p>
+
+  <p><a name="Nt3" href="#NtA3">[3]</a> Ramsay and Young, <i>Phil.
+  Trans.</i>, 1886, 177. 87.</p>
+
+  <p><a name="Nt4" href="#NtA4">[4]</a> See, more especially, Vogt, <i>Die
+  Silikatschmelzlösungen</i>. (Christiania, 1903, 1904.)</p>
+
+  <p><a name="Nt5" href="#NtA5">[5]</a> <i>Trans. Connecticut Acad.</i>,
+  1874-1878.</p>
+
+  <p><a name="Nt6" href="#NtA6">[6]</a> Lehre von der chemischen
+  Verwandtschaft der Körper, 1777.</p>
+
+  <p><a name="Nt7" href="#NtA7">[7]</a> See Ostwald's <i>Klassiker</i>, No.
+  74.</p>
+
+  <p><a name="Nt8" href="#NtA8">[8]</a> Etudes sur les affinités chimiques,
+  1867; Ostwald's <i>Klassiker</i>, No. 104.</p>
+
+  <p><a name="Nt9" href="#NtA9">[9]</a> Died April, 1903.</p>
+
+  <p><a name="Nt10" href="#NtA10">[10]</a> For a mathematical treatment of
+  the Phase Rule the reader is referred to the volume in this series on
+  Thermodynamics, by F. G. Donnan.</p>
+
+  <p><a name="Nt11" href="#NtA11">[11]</a> Liebig's <i>Annalen</i>, 1873,
+  170, 192; Ostwald, <i>Lehrbuch</i>, II. 2. 111.</p>
+
+  <p><a name="Nt12" href="#NtA12">[12]</a> The action of gravity and other
+  forces being excluded (see p.&nbsp;<a href="#page5">5</a>).</p>
+
+  <p><a name="Nt13" href="#NtA13">[13]</a> It may seem as if this were a
+  contradiction to what was said on p.&nbsp;<a href="#page4">4</a> as to the
+  effect of the addition of ammonia or hydrogen chloride to the system
+  constituted by solid ammonium chloride in contact with its products of
+  dissociation. There is, however, no contradiction, because in the case of
+  ammonium chloride the gaseous phase consists of ammonia and hydrogen
+  chloride in equal proportions, and in adding ammonia or hydrogen chloride
+  alone we are not adding the gaseous phase, but only a constituent of it.
+  Addition of ammonia and hydrogen chloride together in the proportions in
+  which they are combined to form ammonium chloride would cause no change
+  in the equilibrium.</p>
+
+  <p><a name="Nt14" href="#NtA14">[14]</a> The vapour pressure of water in
+  small drops is greater than that of water in mass, and the solubility of
+  a solid is greater when in a state of fine subdivision than when in large
+  pieces (<i>cf.</i> Hulett, <i>Zeitschr. physikal. Chem.</i>, 1901, 37.
+  385).</p>
+
+  <p><a name="Nt15" href="#NtA15">[15]</a> See Ostwald, <i>Lehrbuch</i>,
+  II. 2. 476, 934; Roozeboom, <i>Zeitschr. physikal. Chem.</i>, 1894, 15.
+  150; <i>Heterogene Gleichgewichte</i>, I. p. 16; Wegscheider,
+  <i>Zeitschr. physikal. Chem.</i>, 1903, 43. 89.</p>
+
+  <p><a name="Nt16" href="#NtA16">[16]</a> Ostwald, <i>Lehrbuch</i>, II. 2.
+  478.</p>
+
+  <p><a name="Nt17" href="#NtA17">[17]</a> See also Hoitsema, <i>Zeitschr.
+  physikal. Chem.</i> 1895, <b>17</b>. 651.</p>
+
+  <p><a name="Nt18" href="#NtA18">[18]</a> The term "degree of freedom"
+  employed here must not be confused with the same term used to denote the
+  various movements of a gas molecule according to the kinetic theory.</p>
+
+  <p><a name="Nt19" href="#NtA19">[19]</a> Trevor, <i>Jour. Physical
+  Chem.</i>, 1902, <b>6</b>. 136.</p>
+
+  <p><a name="Nt20" href="#NtA20">[20]</a> Ostwald, <i>Principles of
+  Inorganic Chemistry</i>, translated by A. Findlay, 2nd edit., p. 7.
+  (Macmillan, 1904.)</p>
+
+  <p><a name="Nt21" href="#NtA21">[21]</a> See the volume in this series on
+  <i>Thermodynamics</i> by F. G. Donnan.</p>
+
+  <p><a name="Nt22" href="#NtA22">[22]</a> <i>Pogg. Annalen</i>, 1844,
+  <b>61.</b> 225.</p>
+
+  <p><a name="Nt23" href="#NtA23">[23]</a> <i>Mémoires de l'Acad.</i>,
+  <b>26.</b> 751.</p>
+
+  <p><a name="Nt24" href="#NtA24">[24]</a> <i>Phil. Trans.</i> 1884,
+  <b>175.</b> 461; 1892, A, <b>183.</b> 107.</p>
+
+  <p><a name="Nt25" href="#NtA25">[25]</a> <i>Bihang Svenska Akad.
+  Handl.</i> 1891, <b>17.</b> I. 1.</p>
+
+  <p><a name="Nt26" href="#NtA26">[26]</a> Abh<i>andl. physikal.-tech.
+  Reichsanstalt</i>, 1900, <b>3.</b> 71.</p>
+
+  <p><a name="Nt27" href="#NtA27">[27]</a> Ostwald-Luther,
+  <i>Physiko-chemische Messungen</i>, 2nd edit., p. 156.</p>
+
+  <p><a name="Nt28" href="#NtA28">[28]</a> <i>Annales chim. et phys.</i>,
+  1892 [6], <b>26.</b> 425.</p>
+
+  <p><a name="Nt29" href="#NtA29">[29]</a> The vapour pressure of water at
+  0° has recently been very accurately determined by Thiesen and Scheel
+  (<i>loc. cit.</i>), and found to be 4.579 ± 0.001 mm. of mercury (at 0°),
+  or equal to 0.006025 atm.</p>
+
+  <p><a name="Nt30" href="#NtA30">[30]</a> Juhlin, <i>Bihang Svenska Akad.
+  Handl.</i>, 1891, <b>17.</b> <b>I.</b> 58. See also Ramsay and Young,
+  <i>loc. cit.</i></p>
+
+  <p><a name="Nt31" href="#NtA31">[31]</a> <i>Trans. Roy. Soc. Edin.</i>,
+  1849, <b>16.</b> 575.</p>
+
+  <p><a name="Nt32" href="#NtA32">[32]</a> <i>Proc. Roy. Soc. Edin.</i>,
+  1850, <b>2</b>, 267.</p>
+
+  <p><a name="Nt33" href="#NtA33">[33]</a> <i>Annalen der Physik</i>, 1899
+  [3], 68. 564; 1900 [4], 2. 1, 424. See also Dewar, <i>Proc. Roy.
+  Soc.</i>, 1880, 30. 533.</p>
+
+  <p><a name="Nt34" href="#NtA34">[34]</a> The pressure of 1 atmosphere is
+  equal to 1.033 kilogm. per sq. cm.; or the pressure of 1 kilogm. per sq.
+  cm. is equal to 0.968 atm.</p>
+
+  <p><a name="Nt35" href="#NtA35">[35]</a> Tammann, <i>loc. cit.</i>, 1900,
+  2. 1, 424; cf. Goossens, <i>Arch. néerland</i>, 1886, 20. 449.</p>
+
+  <p><a name="Nt36" href="#NtA36">[36]</a> J. Thomson, <i>Proc. Roy.
+  Soc.</i>, 1874, 22. 28.</p>
+
+  <p><a name="Nt37" href="#NtA37">[37]</a> A field is "enclosed" by two
+  curves when these cut at an angle less than two right angles. It may be
+  useful to remember that an invariant system is represented by a
+  <i>point</i>, a univariant system by a <i>line</i>, and a bivariant
+  system by an <i>area</i>.</p>
+
+  <p><a name="Nt38" href="#NtA38">[38]</a> <i>Phil. Trans.</i>, 1724, 39.
+  78.</p>
+
+  <p><a name="Nt39" href="#NtA39">[39]</a> Juhlin, <i>loc. cit.</i>, p. 61;
+  cf. Ramsay and Young, <i>loc. cit.</i>: Thiesen and Scheel, <i>loc.
+  cit.</i></p>
+
+  <p><a name="Nt40" href="#NtA40">[40]</a> This small difference is due to
+  experimental errors in the determination of the vapour pressures; a
+  differential method betrayed no difference between the vapour pressure of
+  ice and of water at 0°.</p>
+
+  <p><a name="Nt41" href="#NtA41">[41]</a> <i>Phil. Mag.</i>, 1874 [4],
+  <b>47.</b> 447; <i>Proc. Roy. Soc.</i>, 1873, <b>22.</b> 27.</p>
+
+  <p><a name="Nt42" href="#NtA42">[42]</a> <i>Pogg. Annalen</i>, 1858,
+  <b>103</b>, 206.</p>
+
+  <p><a name="Nt43" href="#NtA43">[43]</a> See <i>Phil. Trans.</i>, 1884,
+  <b>175</b>, 461.</p>
+
+  <p><a name="Nt44" href="#NtA44">[44]</a> This phenomenon of distillation
+  from the supercooled liquid to the solid has been very clearly observed
+  in the case of furfuraldoxime (V. Goldschmidt, <i>Zeitschr. f.
+  Krystallographie</i>, 1897, <b>28.</b> 169).</p>
+
+  <p><a name="Nt45" href="#NtA45">[45]</a> <i>Annalen der Physik</i>, 1900
+  [4], <b>2.</b> 1, 424.</p>
+
+  <p><a name="Nt46" href="#NtA46">[46]</a> A similar triple point has been
+  determined by Tammann in the case of phenol (<i>Annalen der Physik</i>,
+  1902 [4], <b>9.</b> 249).</p>
+
+  <p><a name="Nt47" href="#NtA47">[47]</a> <i>Annales chim. et phys.</i>,
+  1821, <b>19.</b> 414.</p>
+
+  <p><a name="Nt48" href="#NtA48">[48]</a> Lehmann, <i>Molekularphysik</i>,
+  I. 153.; Arzruni, <i>Physikalische Chemie der Krystalle</i>.
+  (Graham-Otto, <i>Lehrbuch der Chemie</i>, I. 3.)</p>
+
+  <p><a name="Nt49" href="#NtA49">[49]</a> Brodie, <i>Proc. Roy. Soc.</i>,
+  1855, <b>7.</b> 24.</p>
+
+  <p><a name="Nt50" href="#NtA50">[50]</a> That solid sulphur does possess
+  a certain vapour pressure has been shown by Hallock, who observed the
+  formation at the ordinary temperature of copper sulphide in a tube
+  containing copper and sulphur (<i>Amer. Jour. Sci.</i>, 1889 [3],
+  <b>37.</b> 405). See also Zenghelis, <i>Zeitschr. physikal. Chem.</i>,
+  1904, <b>50.</b> 219.</p>
+
+  <p><a name="Nt51" href="#NtA51">[51]</a> <i>Zeitschr. für
+  Krystallographie</i>, 1884, <b>8.</b> 593.</p>
+
+  <p><a name="Nt52" href="#NtA52">[52]</a> Van't Hoff, <i>Studies on
+  Chemical Dynamics</i>, p. 163.</p>
+
+  <p><a name="Nt53" href="#NtA53">[53]</a> Reicher, <i>loc. cit.</i> See
+  also Tammann, <i>Annalen der Physik</i>, 1899 [3], <b>68.</b> 663.</p>
+
+  <p><a name="Nt54" href="#NtA54">[54]</a> Tammann, <i>Annalen der
+  Physik</i>, 1899 [3], <b>68.</b> 633.</p>
+
+  <p><a name="Nt55" href="#NtA55">[55]</a> Rec. Trav. <i>Chim.
+  Pays-Bas</i>, 1887, <b>6.</b> 314.</p>
+
+  <p><a name="Nt56" href="#NtA56">[56]</a> Cf. van't Hoff, <i>Lectures on
+  Physical Chemistry</i>, I., p. 27 (Arnold).</p>
+
+  <p><a name="Nt57" href="#NtA57">[57]</a> <i>Annalen der Physik</i>, 1899
+  [3], <b>68.</b> 663.</p>
+
+  <p><a name="Nt58" href="#NtA58">[58]</a> Brauns, <i>Jahrbuch für
+  Mineralogie</i>, 1899-1901, 13. Beilage, p. 39.</p>
+
+  <p><a name="Nt59" href="#NtA59">[59]</a> Fritsche, <i>Ber.</i>, 1869, 2.
+  112, 540.</p>
+
+  <p><a name="Nt60" href="#NtA60">[60]</a> <i>De mirabilibus
+  Auscultationibus</i>, Cap. 51 (<i>v.</i> Cohen, <i>Zeitschr. physikal.
+  Chem.</i>, 1901, <b>36.</b> 513).</p>
+
+  <p><a name="Nt61" href="#NtA61">[61]</a> E. Cohen and C. van Eyk,
+  <i>Zeitschr. physikal. Chem.</i>, 1899, <b>30.</b> 601; Cohen,
+  <i>ibid.</i>, 1900, <b>33.</b> 59; <b>35.</b> 588; 1901, <b>36.</b> 513;
+  Cohen and E. Goldschmidt, <i>ibid.</i>, 1904, <b>50.</b> 225.</p>
+
+  <p><a name="Nt62" href="#NtA62">[62]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1900, 33, 58.</p>
+
+  <p><a name="Nt63" href="#NtA63">[63]</a> Stortenbeker, <i>Zeitschr.
+  physikal. Chem.</i>, 1889, <b>3.</b> 11; <i>Rec. Trav. Chim.
+  Pays-Bas</i>, 1888, <b>7.</b> 152.</p>
+
+  <p><a name="Nt64" href="#NtA64">[64]</a> Zincke, <i>Ber.</i>, 1871,
+  <b>4.</b> 576.</p>
+
+  <p><a name="Nt65" href="#NtA65">[65]</a> Ostwald, <i>Zeitschr. physikal.
+  Chem.</i>, 1897, <b>22.</b> 313.</p>
+
+  <p><a name="Nt66" href="#NtA66">[66]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 177.</p>
+
+  <p><a name="Nt67" href="#NtA67">[67]</a> Roozeboom, <i>ibid.</i>, p.
+  179.</p>
+
+  <p><a name="Nt68" href="#NtA68">[68]</a> Schrötter, <i>Pogg. Annalen</i>,
+  1850, <b>81.</b> 276; Troost and Hautefeuille, <i>Annales de Chim. et
+  Phys.</i> 1874 [5], <b>2.</b> 153; <i>Ann. Scient. École Norm.</i> 1868
+  [2], II. 266.</p>
+
+  <p><a name="Nt69" href="#NtA69">[69]</a> Pedler, <i>Trans. Chem.
+  Soc.</i>, 1890, <b>57.</b> 599.</p>
+
+  <p><a name="Nt70" href="#NtA70">[70]</a> Brodie, <i>Trans. Chem.
+  Soc.</i>, 1853, <b>5,</b> 289.</p>
+
+  <p><a name="Nt71" href="#NtA71">[71]</a> This is a familiar fact in the
+  case of the solubility in carbon disulphide.</p>
+
+  <p><a name="Nt72" href="#NtA72">[72]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 170.</p>
+
+  <p><a name="Nt73" href="#NtA73">[73]</a> <i>Trans. Chem. Soc.</i>, 1899,
+  <b>57.</b> 734.</p>
+
+  <p><a name="Nt74" href="#NtA74">[74]</a> Carnelley, <i>Trans. Chem.
+  Soc.</i>, 1876, <b>29.</b> 489; 1878, <b>33.</b> 275. V. Meyer and
+  Riddle, <i>Ber.</i>, 1893, <b>26.</b> 2443.</p>
+
+  <p><a name="Nt75" href="#NtA75">[75]</a> Riecke, <i>Zeitschr. physikal.
+  Chem.</i>, 1890, <b>6.</b> 411.</p>
+
+  <p><a name="Nt76" href="#NtA76">[76]</a> <i>Annalen der Physik.</i>, 1898
+  [3], <b>66.</b> 492.</p>
+
+  <p><a name="Nt77" href="#NtA77">[77]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1899, <b>28.</b> 666.</p>
+
+  <p><a name="Nt78" href="#NtA78">[78]</a> See Naumann, <i>Ber.</i>, 1872,
+  <b>4.</b> 646; Troost and Hautefeuille, <i>Compt. rend.</i>, 1868,
+  <b>66.</b> 795; 1868, <b>67.</b> 1345; Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. pp. 62, 171.</p>
+
+  <p><a name="Nt79" href="#NtA79">[79]</a> Mitscherlich, <i>Lieb.
+  Annalen</i>, 1834, <b>12.</b> 137; Deville and Troost, <i>Compt.
+  rend.</i>, 1863, <b>56.</b> 891.</p>
+
+  <p><a name="Nt80" href="#NtA80">[80]</a> Beckmann, <i>Zeitschr. physikal.
+  Chem.</i>, 1890, <b>5.</b> 79; Hertz, <i>ibid.</i>, <b>6.</b> 358.</p>
+
+  <p><a name="Nt81" href="#NtA81">[81]</a> <i>Ber.</i>, 1902, <b>35.</b>
+  351. <i>Cf.</i> also, K. Schaum, <i>Annalen der Chem.</i>, 1898,
+  <b>300.</b> 221; R. Wegscheider and Kaufler, <i>Sitzungsber. kaiserl.
+  Akad. Wissensch. in Wien</i>, 1901, <b>110</b>, II. 606.</p>
+
+  <p><a name="Nt82" href="#NtA82">[82]</a> See also Roozeboom, <i>Das
+  Heterogene Gleichgewicht</i>, I. p. 177.</p>
+
+  <p><a name="Nt83" href="#NtA83">[83]</a> <i>Annales de Chim. et
+  Phys.</i>, 1874 [5], <b>2.</b> 154.</p>
+
+  <p><a name="Nt84" href="#NtA84">[84]</a> <i>Compt. rend.</i>, 1887,
+  <b>104.</b> 1505.</p>
+
+  <p><a name="Nt85" href="#NtA85">[85]</a> <i>Compt. rend.</i>, 1868,
+  <b>66.</b> 795.</p>
+
+  <p><a name="Nt86" href="#NtA86">[86]</a> <i>Phil. Mag.</i>, 1884 [5],
+  <b>18.</b> 210. See also Roozeboom, <i>Das Heterogene Gleichgewicht</i>,
+  I. p. 177.</p>
+
+  <p><a name="Nt87" href="#NtA87">[87]</a> Brauns, <i>Neues Jahrbuch für
+  Mineralogie</i>, 1900, <b>13.</b> Beilage-Band, p. 39; Roozeboom, <i>Das
+  Heterogene Gleichgewicht</i>, I. p. 181.</p>
+
+  <p><a name="Nt88" href="#NtA88">[88]</a> <i>Monatshefte</i>, 1888,
+  <b>9.</b> 435.</p>
+
+  <p><a name="Nt89" href="#NtA89">[89]</a> Gattermann, <i>Ber.</i>, 1890,
+  <b>53.</b> 1738.</p>
+
+  <p><a name="Nt90" href="#NtA90">[90]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1889, <b>4.</b> 468; <i>Annalen der Physik</i>, 1900 [4],
+  <b>2.</b> 649.</p>
+
+  <p><a name="Nt91" href="#NtA91">[91]</a> Quincke, <i>Annalen der
+  Physik</i>, 1894 [3], <b>53.</b> 613; Tammann, <i>Annalen der
+  Physik</i>, 1901 [4], <b>4.</b> 524; 1902, <b>8.</b> 103; Rotarski,
+  <i>ibid.</i>, <b>4.</b> 528.</p>
+
+  <p><a name="Nt92" href="#NtA92">[92]</a> <i>Annalen der Physik</i>, 1900
+  [4], <b>2.</b> 649.</p>
+
+  <p><a name="Nt93" href="#NtA93">[93]</a> <i>Annalen der Physik</i>, 1902
+  [4], <b>8.</b> 911.</p>
+
+  <p><a name="Nt94" href="#NtA94">[94]</a> See, more especially, O.
+  Lehmann, <i>Annalen der Physik</i>, 1900 [4], <b>2.</b> 649; Reinitzer,
+  <i>Sitzungsber. kaiserl. Akad. zu Wien.</i>, 1888, <b>94.</b> (2), 719;
+  <b>97.</b> (1), 167; Gattermann, <i>loc. cit.</i>; Schenck, <i>Zeitschr.
+  physikal. Chem.</i>, 1897, <b>23.</b> 703; 1898, <b>25.</b> 337;
+  <b>27.</b> 170; 1899, <b>28.</b> 280; Schenck and Schneider,
+  <i>ibid.</i>, 1899, <b>29.</b> 546; Abegg and Seitz, <i>ibid.</i>, 1899,
+  <b>29.</b> 491; Hulett, <i>ibid.</i>, 1899, <b>28.</b> 629; Coehn,
+  <i>Zeitschr. Elektrochem.</i>, 1904, <b>10.</b> 856: Bredig and
+  Schukowsky, <i>ibid.</i>, 3419. For a full account of the subject, the
+  reader is referred to the work by Lehmann, <i>Flüssige Kristalle</i>
+  (Engelmann, 1904), or the smaller monograph by Schenck,
+  <i>Kristallinische Flüssigkeiten und flüssige Kristalle</i> (Engelmann,
+  1905).</p>
+
+  <p><a name="Nt95" href="#NtA95">[95]</a> A. C. de Kock, <i>Zeitschr.
+  physikal. Chem.</i>, 1904, <b>48.</b> 129.</p>
+
+  <p><a name="Nt96" href="#NtA96">[96]</a> On account of the fact that all
+  grades of rigidity have been realized between the ordinary solid and the
+  liquid state, in the case both of crystalline and amorphous substances,
+  it has been proposed to abandon the terms "solid" and "liquid," and to
+  class bodies as "crystalline" or "amorphous," the passage from the one
+  condition to the other being discontinuous; crystalline bodies possess a
+  certain regular orientation of their molecules and a directive force,
+  while in amorphous bodies these are wanting (see Lehmann, <i>Annalen der
+  Physik</i>, 1900 [4], <b>2.</b> 696).</p>
+
+  <p><a name="Nt97" href="#NtA97">[97]</a> Hulett, <i>loc. cit.</i></p>
+
+  <p><a name="Nt98" href="#NtA98">[98]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 144. See also Schenck, <i>Kristallinische
+  Flüssigkeiten und flüssige Kristalle</i>, p. 8 (Engelmann, 1904).</p>
+
+  <p><a name="Nt99" href="#NtA99">[99]</a> The possible number of triple
+  points in a one-component system is given by the expression</p>
+
+<table class="math" summary="Formatted mathematical expression" title="Formatted mathematical expression"><tr><td><i>n</i>(<i>n</i> - 1)(<i>n</i> - 2)</td><td rowspan="2">,</td></tr><tr><td class="denom">1.2.3</td></tr></table>
+
+  <p>where <i>n</i> is the number of phases (Riecke, <i>Zeitschr. physikal.
+  Chem.</i>, 1890, <b>6</b>, 411). The number of triple points, therefore,
+  increases very rapidly as the number of possible phases increases.</p>
+
+  <p><a name="Nt100" href="#NtA100">[100]</a> Duhem, <i>Zeitschr. physikal.
+  Chem.</i>, 1891, <b>8</b>. 371. <i>Cf.</i> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, p. 94 ff.</p>
+
+  <p><a name="Nt101" href="#NtA101">[101]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 99.</p>
+
+  <p><a name="Nt102" href="#NtA102">[102]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 474.</p>
+
+  <p><a name="Nt103" href="#NtA103">[103]</a> These changes can be
+  predicted quantitatively by means of the thermodynamic equation,</p>
+
+<table class="math" summary="Formatted mathematical expression" title="Formatted mathematical expression"><tr><td><i>dp</i></td><td rowspan="2">=</td><td> Q</td><td rowspan="2">,</td></tr><tr><td class="denom"><i>dt</i></td><td class="denom">T(<i>v<sub>2</sub></i> - <i>v<sub>1</sub></i>)</td></tr></table>
+
+  <p>provided the specific volumes of the phases are known, and the heat
+  effect which accompanies the transformation of one phase into the
+  other.</p>
+
+  <p><a name="Nt104" href="#NtA104">[104]</a> <i>Studies on Chemical
+  Dynamics</i>, translated by Ewan, p. 218.</p>
+
+  <p><a name="Nt105" href="#NtA105">[105]</a> Le Chatelier, <i>Compt.
+  rend.</i>, 1884, <b>99</b>. 786.</p>
+
+  <p><a name="Nt106" href="#NtA106">[106]</a> See <i>Principles of
+  Inorganic Chemistry</i>, translated by Findlay, 2nd edit., p. 133.
+  (Macmillan, 1904.)</p>
+
+  <p><a name="Nt107" href="#NtA107">[107]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 474.</p>
+
+  <p><a name="Nt108" href="#NtA108">[108]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 189.</p>
+
+  <p><a name="Nt109" href="#NtA109">[109]</a> Roozeboom, <i>Das Heterogene
+  Gleichgewicht</i>, I. p. 125. See also Zawidski, <i>Zeitschr. physikal.
+  Chem.</i>, 1904, <b>47.</b> 727; van Eyk, <i>ibid.</i>, 1905, <b>51.</b>
+  720.</p>
+
+  <p><a name="Nt110" href="#NtA110">[110]</a> Roberts-Austen, <i>Proc. Roy.
+  Soc.</i>, <b>63.</b> 454; Spring, <i>Zeitschr. physikal. Chem.</i>, 1894,
+  <b>15.</b> 65. See also p.&nbsp;<a href="#page35">35</a>.</p>
+
+  <p><a name="Nt111" href="#NtA111">[111]</a> Ramsay and Young, <i>Phil.
+  Trans.</i>, 1884, <b>175.</b> 461; Allen, <i>Trans. Chem. Soc.</i>, 1900,
+  <b>77.</b> 413.</p>
+
+  <p><a name="Nt112" href="#NtA112">[112]</a> Ramsay and Young, <i>Phil.
+  Trans.</i> 1886, <b>177.</b> 87.</p>
+
+  <p><a name="Nt113" href="#NtA113">[113]</a> This is exemplified in the
+  well-known experiment with the cryophorus.</p>
+
+  <p><a name="Nt114" href="#NtA114">[114]</a> Tammann has, however, found
+  that the fusion curve (solid in contact with liquid) of phosphonium
+  chloride can be followed up to temperatures above the critical point
+  (<i>Arch. néer.</i>, 1901 [2], <b>6.</b> 244).</p>
+
+  <p><a name="Nt115" href="#NtA115">[115]</a> <i>Phil. Mag.</i>, 1886,
+  <b>21.</b> 33. See also S. A. Moss, <i>Physical Review</i>, 1903,
+  <b>16.</b> 356.</p>
+
+  <p><a name="Nt116" href="#NtA116">[116]</a> This is found also in the
+  case of bismuth. See Tammann, <i>Zeitschr. anorgan. Chem.</i>, 1904,
+  <b>40.</b> 54.</p>
+
+  <p><a name="Nt117" href="#NtA117">[117]</a> See p.&nbsp;57, <a
+  href="#Nt103">footnote</a>.</p>
+
+  <p><a name="Nt118" href="#NtA118">[118]</a> <i>Pogg. Annalen</i>, 1850,
+  <b>81.</b> 562.</p>
+
+  <p><a name="Nt119" href="#NtA119">[119]</a> Barus, <i>Amer. Jour.
+  Sci.</i>, 1892, <b>42.</b> 125; Mack, <i>Compt. rend.</i>, 1898,
+  <b>127.</b> 361; Hulett, <i>Zeitschr. physikal. Chem.</i>, 1899,
+  <b>38.</b> 629.</p>
+
+  <p><a name="Nt120" href="#NtA120">[120]</a> <i>Annalen der Physik</i>,
+  1899 [3], <b>68.</b> 553, 629; 1900 [4], <b>1.</b> 275; <b>2.</b> 1;
+  <b>3.</b> 161. See also Tammann, <i>Kristallisieren und Schmelzen</i>
+  (Leipzig, 1903).</p>
+
+  <p><a name="Nt121" href="#NtA121">[121]</a> Ostwald, <i>Lehrbuch</i>, II.
+  <b>2</b>. 373; Poynting, <i>Phil. Mag.</i>, 1881 [5], <b>12</b>. 2;
+  Planck, <i>Wied. Annalen</i>, 1882, <b>15</b>. 446.</p>
+
+  <p><a name="Nt122" href="#NtA122">[122]</a> Bakhuis Roozeboom, <i>Das
+  Heterogene Gleichgewicht</i>, I. p. 91.</p>
+
+  <p><a name="Nt123" href="#NtA123">[123]</a> Lussana, <i>Il nuovo
+  Cimento</i>, 1895 [4], <b>1</b>. 105.</p>
+
+  <p><a name="Nt124" href="#NtA124">[124]</a> Tammann, <i>Zeitschr.
+  physikal. Chem.</i>, 1903, <b>46</b>. 818.</p>
+
+  <p><a name="Nt125" href="#NtA125">[125]</a> Foote, <i>Zeitschr. physikal.
+  Chem.</i>, 1900, <b>33</b>. 740.</p>
+
+  <p><a name="Nt126" href="#NtA126">[126]</a> Ostwald, <i>Zeitschr.
+  physikal. Chem.</i>, 1897, <b>22</b>. 289.</p>
+
+  <p><a name="Nt127" href="#NtA127">[127]</a> Van't Hoff, <i>Arch,
+  néer.</i>, 1901, <b>6</b>. 471.</p>
+
+  <p><a name="Nt128" href="#NtA128">[128]</a> See, for example, the
+  determinations of the solubility of rhombic and monoclinic sulphur, by J.
+  Meyer, <i>Zeitschr. anorg. Chem.</i>, 1902, <b>33</b>. 140.</p>
+
+  <p><a name="Nt129" href="#NtA129">[129]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1899, <b>32</b>. 506.</p>
+
+  <p><a name="Nt130" href="#NtA130">[130]</a> Kastle and Reed, <i>Amer.
+  Chem. Jour.</i>, 1902, <b>27</b>. 209.</p>
+
+  <p><a name="Nt131" href="#NtA131">[131]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1900, <b>35</b>. 581.</p>
+
+  <p><a name="Nt132" href="#NtA132">[132]</a> <i>Compt. rend.</i>, 1882,
+  <b>95</b>. 1278; 1884, <b>97</b>. 1298, 1366, 1433.</p>
+
+  <p><a name="Nt133" href="#NtA133">[133]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1893, <b>12</b>. 545.</p>
+
+  <p><a name="Nt134" href="#NtA134">[134]</a> <i>Sitzungsber. Wiener
+  Akad.</i>, 1894, <b>103</b>. II<i>a</i>. 226.</p>
+
+  <p><a name="Nt135" href="#NtA135">[135]</a> <i>Zeitschr. physikal.
+  Chem.</i>, <b>23-29</b>. See also Küster, <i>ibid.</i>, <b>25-28</b>.</p>
+
+  <p><a name="Nt136" href="#NtA136">[136]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1897, <b>24</b>. 152.</p>
+
+  <p><a name="Nt137" href="#NtA137">[137]</a> <i>Ibid.</i>, 1898,
+  <b>27</b>. 585.</p>
+
+  <p><a name="Nt138" href="#NtA138">[138]</a> See W. Guertler, <i>Zeitschr.
+  anorgan. Chem.</i>, 1904, <b>40</b>. 268; Tammann, <i>Zeitschr.
+  Elektrochem.</i>, 1904, <b>10</b>. 532.</p>
+
+  <p><a name="Nt139" href="#NtA139">[139]</a> E. von Pickardt, <i>Zeitschr.
+  physikal. Chem.</i>, 1902, <b>42</b>. 17.</p>
+
+  <p><a name="Nt140" href="#NtA140">[140]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1904, <b>48</b>. 467.</p>
+
+  <p><a name="Nt141" href="#NtA141">[141]</a> M. Padoa, <i>Accad. Lincei,
+  Atti</i>, 1904, <b>13</b>. 329.</p>
+
+  <p><a name="Nt142" href="#NtA142">[142]</a> Deville, <i>Compt. rend.</i>,
+  1852, <b>34</b>. 561; Payen, <i>ibid.</i>, 1852, <b>34</b>. 508; Debray,
+  <i>ibid.</i>, 1858, <b>46</b>. 576. It has also been found by Jaffé
+  (<i>Zeitschr. physikal. Chem.</i>, 1903, <b>43</b>. 465) that when
+  spontaneous crystallization from solution occurs, the less stable form
+  always separates first when purification has been carried sufficiently
+  far.</p>
+
+  <p><a name="Nt143" href="#NtA143">[143]</a> Brauns, <i>Neues Jahrbuch für
+  Mineralogie</i>, 1899, <b>13</b>. (Beilage Band) 84.</p>
+
+  <p><a name="Nt144" href="#NtA144">[144]</a> <i>Lehrbuch</i>, II.
+  <b>2</b>. 445. See also <i>Principles of Inorganic Chemistry</i>, 2nd
+  edit., p. 210 ff.</p>
+
+  <p><a name="Nt145" href="#NtA145">[145]</a> Schaum and Schönbeck,
+  <i>Annalen der Physik</i>, 1902 [4], <b>8</b>. 652. See also Chr.
+  Füchtbauer, <i>Zeitschr. physikal. Chem.</i>, 1904, <b>48</b>. 549.</p>
+
+  <p><a name="Nt146" href="#NtA146">[146]</a> Ramsay and Young, <i>Phil.
+  Trans.</i>, 1886, <b>177</b>. 87.</p>
+
+  <p><a name="Nt147" href="#NtA147">[147]</a> See volume in this series on
+  <i>Chemical Dynamics</i>, by Dr. J. W. Mellor.</p>
+
+  <p><a name="Nt148" href="#NtA148">[148]</a> Isambert, <i>Compt.
+  rend.</i>, 1881, <b>92</b>. 919; 1882, <b>94</b>. 958; 1883, <b>96</b>.
+  643. Walker and Lumsden, <i>Jour. Chem. Soc.</i>, 1897, <b>71</b>.
+  428.</p>
+
+  <p><a name="Nt149" href="#NtA149">[149]</a> <i>Compt. rend.</i>, 1867,
+  <b>64</b>. 603.</p>
+
+  <p><a name="Nt150" href="#NtA150">[150]</a> <i>Compt. rend.</i>, 1883,
+  <b>102</b>. 1243.</p>
+
+  <p><a name="Nt151" href="#NtA151">[151]</a> <i>Compt. rend.</i>, 1868,
+  <b>66</b>, 1259.</p>
+
+  <p><a name="Nt152" href="#NtA152">[152]</a> Horstmann, <i>Ber.</i>, 1876,
+  <b>9</b>. 749.</p>
+
+  <p><a name="Nt153" href="#NtA153">[153]</a> <i>Loc. cit.</i></p>
+
+  <p><a name="Nt154" href="#NtA154">[154]</a> For the reasons for choosing
+  anhydrous salt and water instead of salt hydrate and water as components,
+  see p.&nbsp;<a href="#page14">14</a>.</p>
+
+  <p><a name="Nt155" href="#NtA155">[155]</a> See Ostwald, <i>Lehrbuch</i>,
+  II. <b>2.</b> 527.</p>
+
+  <p><a name="Nt156" href="#NtA156">[156]</a> Ostwald, <i>Lehrbuch</i>, II.
+  <b>2.</b> 538.</p>
+
+  <p><a name="Nt157" href="#NtA157">[157]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1889, <b>4</b>. 43.</p>
+
+  <p><a name="Nt158" href="#NtA158">[158]</a> <i>Ber.</i>, 1876, <b>9</b>.
+  749.</p>
+
+  <p><a name="Nt159" href="#NtA159">[159]</a> See, for example, van't Hoff,
+  <i>Lectures on Theoretical and Physical Chemistry</i>, I. p. 62
+  (Arnold).</p>
+
+  <p><a name="Nt160" href="#NtA160">[160]</a> <i>Jour. Chem. Soc.</i>,
+  1877, <b>32</b>. 395.</p>
+
+  <p><a name="Nt161" href="#NtA161">[161]</a> Hoitsema, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 1.</p>
+
+  <p><a name="Nt162" href="#NtA162">[162]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1887, <b>1</b>. 5; 1895, <b>17</b>. 52.</p>
+
+  <p><a name="Nt163" href="#NtA163">[163]</a> It is important to powder the
+  salt, since otherwise the dehydration of the hydrate and the production
+  of equilibrium occurs with comparatively great tardiness.</p>
+
+  <p><a name="Nt164" href="#NtA164">[164]</a> A chemical individual is a
+  substance which persists as a phase of constant composition when the
+  conditions of temperature, pressure, and composition of the other phases
+  present, undergo continuous alteration within certain limits&mdash;the
+  limits of existence of the substance (Wald, <i>Zeitschr. physikal.
+  Chem.</i>, 1897, <b>24</b>. 648).</p>
+
+  <p><a name="Nt165" href="#NtA165">[165]</a> Van't Hoff, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 323; Ostwald, <i>Lehrbuch</i>, I.
+  606.</p>
+
+  <p><a name="Nt166" href="#NtA166">[166]</a> That mercury does dissolve in
+  water can be argued from analogy, say, with mercury and bromonaphthalene.
+  At the ordinary temperature these two liquids appear to be quite
+  insoluble in one another, but at a temperature of 280° the mercury
+  dissolves in appreciable quantity; for on heating a tube containing
+  bromonaphthalene over mercury the latter sublimes <i>through</i> the
+  liquid bromonaphthalene and condenses on the upper surface of the
+  tube.</p>
+
+  <p><a name="Nt167" href="#NtA167">[167]</a> <i>Phil. Mag.</i>, 1884,
+  [5], <b>18</b>. 22; 495.</p>
+
+  <p><a name="Nt168" href="#NtA168">[168]</a> <i>Wied. Annalen</i>, 1886,
+  <b>28</b>. 305.</p>
+
+  <p><a name="Nt169" href="#NtA169">[169]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1898, <b>26</b>. 433.</p>
+
+  <p><a name="Nt170" href="#NtA170">[170]</a> Rothmund, <i>loc.
+  cit.</i></p>
+
+  <p><a name="Nt171" href="#NtA171">[171]</a> Rothmund, <i>loc.
+  cit.</i></p>
+
+  <p><a name="Nt172" href="#NtA172">[172]</a> A similar behaviour is found
+  in the case of diethylamine and water (R. T. Lattey, <i>Phil. Mag.</i>,
+  1905, [6], <b>10</b>, 397).</p>
+
+  <p><a name="Nt173" href="#NtA173">[173]</a> C. S. Hudson, <i>Zeitschr.
+  physikal. Chem.</i>, 1904, <b>47</b>. 113.</p>
+
+  <p><a name="Nt174" href="#NtA174">[174]</a> Konowaloff, <i>Wied.
+  Annalen</i>, 1881, <b>14</b>. 219. Ostwald, <i>Lehrbuch</i>, II.
+  <b>2</b>. 687. Bancroft, <i>Phase Rule</i>, p. 96.</p>
+
+  <p><a name="Nt175" href="#NtA175">[175]</a> Konowaloff, <i>loc.
+  cit.</i></p>
+
+  <p><a name="Nt176" href="#NtA176">[176]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1891, <b>8</b>. 526; <i>Rec. Trav. Chim.
+  Pays-Bas</i>, 1884, <b>3</b>. 38.</p>
+
+  <p><a name="Nt177" href="#NtA177">[177]</a> Konowaloff, <i>loc. cit.</i>
+  Cf. Bancroft, <i>Phase Rule</i>, p. 100.</p>
+
+  <p><a name="Nt178" href="#NtA178">[178]</a> <i>Phil. Mag.</i>, 1884 [5],
+  <b>18</b>. 503.</p>
+
+  <p><a name="Nt179" href="#NtA179">[179]</a> See, for example, Walker,
+  <i>Introduction to Physical Chemistry</i>, 3rd edit., p. 86 (Macmillan,
+  1903). Consult also Young, <i>Fractional Distillation</i> (Macmillan,
+  1903), or Kuenen, <i>Verdampfung und Verflüssigung von Gemischen</i>
+  (Barth, 1906), where the subject is fully treated.</p>
+
+  <p><a name="Nt180" href="#NtA180">[180]</a> Since this is the only phase
+  of variable composition present.</p>
+
+  <p><a name="Nt181" href="#NtA181">[181]</a> E. von Stackelberg,
+  <i>Zeitschr. physikal. Chem.</i>, 1896, <b>20</b>. 337. If the change of
+  volume which accompanies solution, and the heat effect are known, the
+  quantitative change of the solubility with the pressure can be calculated
+  (Braun, <i>Zeitschr. physikal. Chem.</i>, 1887, <b>1</b>. 259).</p>
+
+  <p><a name="Nt182" href="#NtA182">[182]</a> Van't Hoff, <i>Arch.
+  néerland.</i> 1901 [2], <b>6</b>. 471.</p>
+
+  <p><a name="Nt183" href="#NtA183">[183]</a> Tilden and Shenstone,
+  <i>Phil. Trans.</i> 1884, <b>175</b>. 23; Hulett and Allen, <i>Jour.
+  Amer. Chem. Soc.</i> 1902, <b>24</b>. 667; Andreä, <i>Jour. prak.
+  Chem.</i> <b>137</b>. 474; Lumsden, <i>Jour. Chem. Soc.</i>, 1902,
+  <b>81</b>. 350; Mylius and v. Wrochem, <i>Ber.</i> 1900, <b>33</b>.
+  3689.</p>
+
+  <p><a name="Nt184" href="#NtA184">[184]</a> E. von Stackelberg,
+  <i>Zeitschr. physikal. Chem.</i> 1896, <b>20</b>. 159; 1898, <b>26</b>.
+  533; Lumsden, <i>Jour. Chem. Soc.</i>, 1902, <b>81</b>. 350; Holsboer,
+  <i>Zeitschr. physikal. Chem.</i>, 1902, <b>39</b>. 691.</p>
+
+  <p><a name="Nt185" href="#NtA185">[185]</a> Reicher and van Deventer,
+  <i>Zeitschr. physikal. Chem.</i> 1890, <b>5</b>. 559; cf. Ostwald,
+  <i>Lehrbuch</i>, II. <b>2</b>. 803.</p>
+
+  <p><a name="Nt186" href="#NtA186">[186]</a> It has been shown that the
+  formula of Ramsay and Young (p.&nbsp;<a href="#page66">66</a>) can be applied
+  (with certain restrictions) to the interpolation and extrapolation of the
+  solubility curve of a substance provided two (or three) points on the
+  curve are known. In this case T, T<sub>1</sub>, etc., refer to the
+  temperatures at which the two substances&mdash;one the solubility curve
+  of which is known, the other the solubility curve of which is to be
+  calculated&mdash;have equal solubilities, instead of, as in the previous
+  case, equal vapour pressures. (Findlay, <i>Proc. Roy. Soc.</i>, 1902,
+  <b>69</b>. 471; <i>Zeitschr. physikal. Chem.</i>, 1903, <b>42</b>.
+  110.)</p>
+
+  <p><a name="Nt187" href="#NtA187">[187]</a> W. Müller and P. Kaufmann,
+  <i>Zeitschr. physikal. Chem.</i> 1903, <b>42</b>. 497.</p>
+
+  <p><a name="Nt188" href="#NtA188">[188]</a> W. O. Rabe, <i>Zeitschr.
+  physikal. Chem.</i>, 1901, <b>38</b>. 175.</p>
+
+  <p><a name="Nt189" href="#NtA189">[189]</a> With regard to the limits of
+  supersaturation and the spontaneous crystallization of the solute from
+  supersaturated solutions, see Jaffé, <i>Zeitschr. physikal. Chem.</i>,
+  1903, <b>43</b>. 565, and the very interesting paper by Miers and Isaac,
+  <i>Trans. Chem. Soc.</i>, 1906, <b>89</b>. 413.</p>
+
+  <p><a name="Nt190" href="#NtA190">[190]</a> <i>Annales chim. phys.</i>,
+  1894 [7], <b>2</b>. 524.</p>
+
+  <p><a name="Nt191" href="#NtA191">[191]</a> <i>Phil. Trans.</i>, 1884,
+  <b>175</b>. 23.</p>
+
+  <p><a name="Nt192" href="#NtA192">[192]</a> Hissink, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>32</b>. 543.</p>
+
+  <p><a name="Nt193" href="#NtA193">[193]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1903, <b>43</b>. 313.</p>
+
+  <p><a name="Nt194" href="#NtA194">[194]</a> Guthrie, <i>Phil. Mag.</i>,
+  1875, [4], <b>49</b>. 1; 1884, [5], <b>17</b>. 462.</p>
+
+  <p><a name="Nt195" href="#NtA195">[195]</a> See Roloff, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 325; Guthrie, <i>loc. cit.</i></p>
+
+  <p><a name="Nt196" href="#NtA196">[196]</a> Guthrie, <i>Phil. Mag.</i>,
+  <i>loc. cit.</i> Cf. Ostwald, <i>Lehrbuch</i>, II. <b>2</b>. 843.</p>
+
+  <p><a name="Nt197" href="#NtA197">[197]</a> Guthrie, <i>Phil. Mag.</i>,
+  1875 [4], <b>49</b>. 269.</p>
+
+  <p><a name="Nt198" href="#NtA198">[198]</a> <i>Ber.</i>, 1877, <b>20</b>.
+  2223.</p>
+
+  <p><a name="Nt199" href="#NtA199">[199]</a> <i>Silz-Ber. Wien. Akad.</i>,
+  1880, <b>81</b>. II. 1058.</p>
+
+  <p><a name="Nt200" href="#NtA200">[200]</a> Guthrie, <i>Phil. Mag.</i>,
+  1875 [4], <b>49</b>. 206.</p>
+
+  <p><a name="Nt201" href="#NtA201">[201]</a> If in the neighbourhood of
+  the cryohydric point solution should be accompanied by an evolution of
+  heat, then as the solubility would in that case increase with fall of
+  temperature, salt would pass into solution.</p>
+
+  <p><a name="Nt202" href="#NtA202">[202]</a> Walker, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 193.</p>
+
+  <p><a name="Nt203" href="#NtA203">[203]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1897, <b>23</b>. 418.</p>
+
+  <p><a name="Nt204" href="#NtA204">[204]</a> Provided the solid nitrile is
+  not present in too great excess.</p>
+
+  <p><a name="Nt205" href="#NtA205">[205]</a> <i>Wied. Annalen</i>, 1886,
+  <b>28</b>. 328. Cf. Ostwald, <i>Lehrbuch</i>, II. <b>2</b>. 872.</p>
+
+  <p><a name="Nt206" href="#NtA206">[206]</a> Walker, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 193. Schreinemakers, <i>ibid.</i>,
+  1897, <b>23</b>. 417. Roozeboom, <i>Rec. trav. chim. Pays-Bays</i>, 1889,
+  <b>8</b>. 257. Bruner, <i>Zeitschr. physikal. Chem.</i>, 1897, <b>23</b>.
+  542.</p>
+
+  <p><a name="Nt207" href="#NtA207">[207]</a> Van't Hoff, <i>Lectures on
+  Theoretical Chemistry</i>, I. p. 42. Ostwald, <i>Lehrbuch</i>, II.
+  <b>2</b>. 824.</p>
+
+  <p><a name="Nt208" href="#NtA208">[208]</a> Ostwald, <i>Principles of
+  Inorganic Chemistry</i>, translated by A. Findlay, 2nd edit., p. 453
+  (Macmillan, 1904); Skirrow and Calvert, <i>Zeitschr. physikal. Chem.</i>,
+  1901, <b>37</b>. 217.</p>
+
+  <p><a name="Nt209" href="#NtA209">[209]</a> <i>Vide</i> Loewel,
+  <i>Annales chim. phys.</i>, 1857 [3], <b>49</b>. 32. Cf. Löwenherz,
+  <i>Zeitschr. physikal. Chem.</i>, 1895, <b>18</b>. 82.</p>
+
+  <p><a name="Nt210" href="#NtA210">[210]</a> Loewel, <i>loc. cit.</i>
+  Gay-Lussac, <i>Annales chim. phys.</i>, 1819, <b>11</b>. 296. For the
+  solubility at higher temperatures, see Tilden and Shenstone, <i>Phil.
+  Trans.</i>, 1884, <b>175</b>. 23. Étard, <i>Annales chim. phys.</i>, 1894
+  [7], <b>2</b>. 548.</p>
+
+  <p><a name="Nt211" href="#NtA211">[211]</a> Richards, <i>Zeitschr.
+  physikal. Chem.</i>, 1898, <b>26</b>. 690; Richards and Wells,
+  <i>ibid.</i>, 1903, <b>43</b>. 465. This temperature is not quite the
+  same as that of the <i>quadruple point</i> anhydrous salt&mdash;hydrated
+  salt&mdash;solution&mdash;vapour, because the latter is the temperature
+  at which the system is under the pressure of its own vapour. Since,
+  however, the influence of pressure on the solubility is very slight
+  (p.&nbsp;<a href="#page107">107</a>), the position of the two points will not
+  be greatly different. The quadruple point was found by Cohen
+  (<i>Zeitschr. physikal. Chem.</i>, 1894, <b>14</b>. 90) to be 32.6° and
+  30.8 mm. of mercury.</p>
+
+  <p><a name="Nt212" href="#NtA212">[212]</a> Van't Hoff and van Deventer,
+  <i>Zeitschr. physikal. Chem.</i>, 1887, <b>1</b>. 185. Cf. Cohen,
+  <i>ibid.</i>, 1894, <b>14</b>. 88.</p>
+
+  <p><a name="Nt213" href="#NtA213">[213]</a> Debray, <i>Compt. rend.</i>,
+  1868, <b>66</b>. 194.</p>
+
+  <p><a name="Nt214" href="#NtA214">[214]</a> Richards, <i>Zeitschr.
+  physikal. Chem.</i>, 1898, <b>26</b>. 690. A number of other salt
+  hydrates, having transition-points ranging from 20° to 78°, which might
+  be used for the same purpose, have been given by Richards and Churchill,
+  <i>ibid.</i>, 1899, <b>28</b>. 313.</p>
+
+  <p><a name="Nt215" href="#NtA215">[215]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1903, <b>46</b>. 818.</p>
+
+  <p><a name="Nt216" href="#NtA216">[216]</a> Van't Hoff, <i>Lectures on
+  Physical Chemistry</i>, <b>I</b>. p. 67.</p>
+
+  <p><a name="Nt217" href="#NtA217">[217]</a> Cohen, <i>Zeitschr. physikal.
+  Chem.</i>, 1894, <b>14</b>. 90.</p>
+
+  <p><a name="Nt218" href="#NtA218">[218]</a> Ziz, <i>Schweigger's
+  Journal</i>, 1815, <b>15</b>. 166. See Ostwald, <i>Lehrbuch</i>, II.
+  <b>2</b>. 717.</p>
+
+  <p><a name="Nt219" href="#NtA219">[219]</a> See, for example, the
+  solubility determinations published in <i>Wissenschaftliche Abhandl. der
+  physikalisch-technischen Reichsanstalt</i>, Vol. III., or in the
+  <i>Berichte</i>, for the years 1897-1901.</p>
+
+  <p><a name="Nt220" href="#NtA220">[220]</a> Meusser, <i>Ber.</i>, 1901,
+  <b>34</b>. 2440.</p>
+
+  <p><a name="Nt221" href="#NtA221">[221]</a> Mylius and von Wrochem,
+  <i>Ber.</i>, 1900, <b>33</b>. 3693.</p>
+
+  <p><a name="Nt222" href="#NtA222">[222]</a> Walker and Fyffe, <i>Jour.
+  Chem. Soc.</i>, 1903, <b>83</b>. 180.</p>
+
+  <p><a name="Nt223" href="#NtA223">[223]</a> <i>Monatshefte</i>, 1887,
+  <b>8</b>. 601.</p>
+
+  <p><a name="Nt224" href="#NtA224">[224]</a> The equilibria between
+  calcium chloride and water have been most completely studied by Roozeboom
+  (<i>Zeitschr. physikal. Chem.</i>, 1889, <b>4</b>. 31).</p>
+
+  <p><a name="Nt225" href="#NtA225">[225]</a> Hammerl, <i>Sitzungsber.
+  Wien. Akad.</i>, 2<sup>te</sup> Abteil, 1878, <b>78</b>. 59. Roozeboom,
+  <i>Zeitschr. physikal. Chem.</i>, 1889, <b>4</b>. 31.</p>
+
+  <p><a name="Nt226" href="#NtA226">[226]</a> Lidbury, <i>Zeitschr.
+  physikal. Chem.</i>, 1902, <b>39</b>. 453. The curvature at the melting
+  point is all the greater the more the compound is dissociated into its
+  components in the liquid state. If the compound is <i>completely
+  undissociated</i>, even in the vapour phase, the two branches of the
+  curve will <i>intersect</i>, (<i>e.g.</i> pyridine and methyl iodide;
+  Aten, <i>Versl. Konink. Akad. Wetensch. Amsterdam</i>, 1905, <b>13</b>.
+  462). The smaller the degree of dissociation, therefore, the sharper will
+  be the bend. (See Stortenbeker, <i>Zeitschr. physikal. Chem.</i>, 1892,
+  <b>10</b>. 194.) From the extent of flattening of the curve, it is also
+  possible, with some degree of approximation, to calculate the degree of
+  dissociation of the substance in the fused state. (See Roozeboom and
+  Aten, <i>Zeitschr. physikal. Chem.</i>, 1905, <b>53</b>. 463; Kremann,
+  <i>Zeitschr. Elektrochem.</i>, 1906, <b>12</b>. 259.)</p>
+
+  <p><a name="Nt227" href="#NtA227">[227]</a> See Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1889, <b>4</b>. 31.</p>
+
+  <p><a name="Nt228" href="#NtA228">[228]</a> Tammann, <i>Wied.
+  Annalen</i>, 1899, <b>68</b>. 577.</p>
+
+  <p><a name="Nt229" href="#NtA229">[229]</a> Duhem, <i>Journ. Physical
+  Chem.</i>, 1898, <b>2</b>. 31.</p>
+
+  <p><a name="Nt230" href="#NtA230">[230]</a> Gibbs, <i>Trans. Conn.
+  Acad.</i>, <b>3</b>. 155; Saurel, <i>Journ. Phys. Chem.</i>, 1901,
+  <b>5</b>. 35.</p>
+
+  <p><a name="Nt231" href="#NtA231">[231]</a> In the case of the fusion of
+  a compound of two components with formation of a liquid phase of the same
+  composition, the temperature is a maximum; in the case of liquid mixtures
+  of constant boiling-point, the temperature may be a minimum (p.&nbsp;<a
+  href="#page105">105</a>).</p>
+
+  <p><a name="Nt232" href="#NtA232">[232]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1892, <b>10</b>. 477. The formula of ferric chloride
+  has been doubled, in order to avoid fractions in the expression of the
+  water of crystallization.</p>
+
+  <p><a name="Nt233" href="#NtA233">[233]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1892, <b>10</b>. 477.</p>
+
+  <p><a name="Nt234" href="#NtA234">[234]</a> A similar series of hydrates
+  is formed by zinc chloride and water (Dietz and Mylius, <i>Zeitschr.
+  anorg. Chem.</i>, 1905, <b>44</b>. 209).</p>
+
+  <p><a name="Nt235" href="#NtA235">[235]</a> Meyerhoffer, <i>Ber.</i>,
+  1897, <b>30</b>. 1810.</p>
+
+  <p><a name="Nt236" href="#NtA236">[236]</a> Walden, <i>Ber.</i>, 1899,
+  <b>32</b>. 2863.</p>
+
+  <p><a name="Nt237" href="#NtA237">[237]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1903, <b>42</b>. 432.</p>
+
+  <p><a name="Nt238" href="#NtA238">[238]</a> This composition was also
+  confirmed by measurements of the vapour pressure (cf. p.&nbsp;<a
+  href="#page90">90</a>).</p>
+
+  <p><a name="Nt239" href="#NtA239">[239]</a> Since all substances are no
+  doubt volatile to a certain extent at some temperature, it is to be
+  understood here that the substances are appreciably volatile at the
+  temperature of the experiment.</p>
+
+  <p><a name="Nt240" href="#NtA240">[240]</a> For a general discussion of
+  the partial pressures in a system of two components, see Bancroft,
+  <i>Journ. Physical Chem.</i>, 1899, <b>3</b>. 1.</p>
+
+  <p><a name="Nt241" href="#NtA241">[241]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1889, <b>3</b>. 11; <i>Rec. trav. chim. Pays-Bas</i>, 1888,
+  <b>7</b>. 152.</p>
+
+  <p><a name="Nt242" href="#NtA242">[242]</a> The composition of a solution
+  is represented symbolically by placing a double wavy line between the
+  symbols of the components, and indicating the number of atoms present in
+  the ordinary manner: thus, I <a href="images/wavy.png"><img
+  src="images/wavy.png" class="middle" style="height:1.5ex" alt="wavy"
+  /></a> Cl<sub><i>x</i></sub> represents a solution containing <i>x</i>
+  atoms of chlorine to one atom of iodine (Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 450).</p>
+
+  <p><a name="Nt243" href="#NtA243">[243]</a> Since iodine monochloride in
+  the liquid state is only very slightly dissociated, the bend at C is very
+  sharp (see p.&nbsp;<a href="#page147">147</a>, footnote). See also the
+  investigation of the system pyridine and methyl iodide (Aten, <i>Versl.
+  Konink. Akad. Wetensch. Amsterdam</i>, 1905, <b>13</b>. 462).</p>
+
+  <p><a name="Nt244" href="#NtA244">[244]</a> This upper branch of the
+  curve is not shown in the figure, as the ordinate corresponding to 30°
+  would be very great.</p>
+
+  <p><a name="Nt245" href="#NtA245">[245]</a> Stortenbeker, <i>Zeitschr.
+  physikal. Chem.</i>, 1889, <b>3</b>. 22.</p>
+
+  <p><a name="Nt246" href="#NtA246">[246]</a> Ramsay and Young, <i>Journ.
+  Chem. Soc.</i>, 1886, <b>49</b>. 458.</p>
+
+  <p><a name="Nt247" href="#NtA247">[247]</a> Van't Hoff, <i>Lectures on
+  Physical Chemistry</i>, I. p. 77 (Arnold).</p>
+
+  <p><a name="Nt248" href="#NtA248">[248]</a> This is different from what
+  we found in the case of non-volatile solutes (p.&nbsp;<a
+  href="#page126">126</a>). In the present case, the <i>partial
+  pressure</i> of the iodine in the vapour will be lowered by addition of
+  chlorine, but the <i>total pressure</i> is increased.</p>
+
+  <p><a name="Nt249" href="#NtA249">[249]</a> The diminution of volume is
+  supposed to be carried out at constant temperature. The pressure and the
+  composition of the phases must, therefore, remain unchanged, and only the
+  relative amounts of these can undergo alteration.</p>
+
+  <p><a name="Nt250" href="#NtA250">[250]</a> At point <i>b</i> the ratio
+  of chlorine to iodine in the solution is less than in the monochloride,
+  so that by the separation of this the excess of chlorine yielded by the
+  condensation of the vapour is removed.</p>
+
+  <p><a name="Nt251" href="#NtA251">[251]</a> Roozeboom, <i>Rec. trav.
+  chim. Pays-Bas</i>, 1884, <b>3</b>. 29; 1885, <b>4</b>. 65; <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 450.</p>
+
+  <p><a name="Nt252" href="#NtA252">[252]</a> Two curves "enclose" a field
+  when they form with one another an angle less than two right angles.</p>
+
+  <p><a name="Nt253" href="#NtA253">[253]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, <i>loc. cit.</i></p>
+
+  <p><a name="Nt254" href="#NtA254">[254]</a> Van't Hoff, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 323.</p>
+
+  <p><a name="Nt255" href="#NtA255">[255]</a> Bancroft has proposed to
+  restrict the term "occlusion" to the formation of solid solutions, and to
+  apply "adsorption" only to effects which are primarily due to surface
+  tension. Such a distinction, however, would probably be very difficult to
+  carry through, for although adsorption may, in large measure, be due to
+  surface tension, the behaviour of adsorbed substances is similar to that
+  of substances existing in solid solutions.</p>
+
+  <p><a name="Nt256" href="#NtA256">[256]</a> Tammann, <i>Wied.
+  Annalen</i>, 1897, <b>63</b>. 16; <i>Zeitschr. physikal. Chem.</i>, 1898,
+  <b>27</b>. 323.</p>
+
+  <p><a name="Nt257" href="#NtA257">[257]</a> See, for example, Chappuis,
+  <i>Wied. Annalen</i>, 1881, <b>12</b>. 161; Joulin, <i>Annal. chim.
+  phys.</i>, 1881, [5], <b>22</b>. 398; Kayser, <i>Wied. Annalen</i>,
+  1881, <b>12</b>. 526.</p>
+
+  <p><a name="Nt258" href="#NtA258">[258]</a> Hoitsema, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 1.</p>
+
+  <p><a name="Nt259" href="#NtA259">[259]</a> <i>Annales chim. phys.</i>,
+  1874, [5], <b>2</b>. 279.</p>
+
+  <p><a name="Nt260" href="#NtA260">[260]</a> Hoitsema, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 1; Dewar, <i>Phil. Mag.</i>, 1874,
+  [4], <b>47</b>, 324, 342; Mond, Ramsay and Shields, <i>Proc. Royal
+  Soc.</i>, 1897, <b>62</b>. 290.</p>
+
+  <p><a name="Nt261" href="#NtA261">[261]</a> <i>Loc. cit.</i></p>
+
+  <p><a name="Nt262" href="#NtA262">[262]</a> It is noteworthy that the
+  form of curve obtained for hydrogen and palladium bears a striking
+  resemblance to that for the dehydration of colloids containing absorbed
+  water, <i>e.g.</i> silicic acid (<i>vide</i> van Bemmelen, <i>Zeitschr.
+  anorg. Chem.</i>, 1897-1900. Cf. Zacharias, <i>Zeitschr. physikal.
+  Chem.</i>, 1902, <b>39</b>. 480).</p>
+
+  <p><a name="Nt263" href="#NtA263">[263]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1890, <b>5</b>. 322.</p>
+
+  <p><a name="Nt264" href="#NtA264">[264]</a> Küster, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 367. Bodländer, <i>Neues Jahrbuch
+  f. Mineralogie</i>, 1898-99, Beilage Band, <b>12</b>. 92.</p>
+
+  <p><a name="Nt265" href="#NtA265">[265]</a> Bruni and Padoa, <i>Atti
+  Accad. Lincei</i>, 1902 [5], <b>11</b>. 1; 565.</p>
+
+  <p><a name="Nt266" href="#NtA266">[266]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>30</b>. 385; Bruni, <i>Rend. Accad.
+  Lincei</i>, 1898, <b>2</b>. 138, 347. For a general account of "solid
+  solutions" the reader is referred to Bruni, "<i>Ueber feste Lösungen</i>"
+  (Ahrens'sche Sammlung), and to Bodländer, <i>loc. cit.</i> For the
+  formation and transformation of liquid mixed crystals, see A. C. de Kock,
+  <i>Zeitschr. physikal. Chem.</i>, 1904, <b>48</b>. 129.</p>
+
+  <p><a name="Nt267" href="#NtA267">[267]</a> In discussing the various
+  systems which may be obtained here, Roozeboom (<i>loc. cit.</i>) made use
+  of the variation of the thermodynamic potential (p.&nbsp;<a
+  href="#page29">29</a>) with the concentration. In spite of the advantages
+  which such a treatment affords, the temperature-concentration diagram has
+  been adopted as being more readily understood and as more suitable for an
+  elementary discussion of the subject.</p>
+
+  <p><a name="Nt268" href="#NtA268">[268]</a> These curves are also called
+  the "liquidus" and the "solidus" curve respectively.</p>
+
+  <p><a name="Nt269" href="#NtA269">[269]</a> Küster, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 360.</p>
+
+  <p><a name="Nt270" href="#NtA270">[270]</a> Küster, <i>ibid.</i>, 1891,
+  <b>8</b>. 589.</p>
+
+  <p><a name="Nt271" href="#NtA271">[271]</a> It should be remarked that
+  the behaviour described here will hold strictly only when the solid mixed
+  crystals undergo change sufficiently rapidly to be always in equilibrium
+  with the liquid. This, however, is not always the case (see Reinders,
+  <i>Zeitschr. physikal. Chem.</i>, 1900, <b>32</b>. 494; van Wyk,
+  <i>Zeitschr. anorg. Chem.</i>, 1905, <b>48</b>. 25), and complete
+  solidification will not in this case take place at the temperature
+  corresponding with the line <i>dc</i> in Fig. 50, but only at a lower
+  temperature.</p>
+
+  <p><a name="Nt272" href="#NtA272">[272]</a> Adriani, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>33</b>. 469.</p>
+
+  <p><a name="Nt273" href="#NtA273">[273]</a> Reinders, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>32</b>. 494.</p>
+
+  <p><a name="Nt274" href="#NtA274">[274]</a> Hissink, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>32</b>. 542.</p>
+
+  <p><a name="Nt275" href="#NtA275">[275]</a> Van Eyk, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>30</b>. 430.</p>
+
+  <p><a name="Nt276" href="#NtA276">[276]</a> Cady, <i>Journ. Physical.
+  Chem.</i>, 1899, <b>3</b>. 127.</p>
+
+  <p><a name="Nt277" href="#NtA277">[277]</a> See Roberts-Austen and
+  Stansfield, <i>Rapports du congrès international de physique</i>, 1900,
+  I. 363.</p>
+
+  <p><a name="Nt278" href="#NtA278">[278]</a> Heycock and Neville, <i>Proc.
+  Roy. Soc.</i>, 1903, <b>71</b>. 409. For the partial liquefaction of
+  mixed crystals on cooling, see also A. C. de Kock (<i>Zeitschr. physikal.
+  Chem.</i>, 1904, <b>48</b>. 129).</p>
+
+  <p><a name="Nt279" href="#NtA279">[279]</a> Armstrong, <i>Watt's
+  Dictionary of Chemistry</i> (Morley and Muir), III., p. 88. See also
+  Lowry, <i>Jour. Chem. Soc.</i>, 1899, <b>75</b>. 211.</p>
+
+  <p><a name="Nt280" href="#NtA280">[280]</a> See Bancroft, <i>Journ.
+  Physical Chem.</i>, 1898, <b>2</b>. 143; Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>28</b>. 288.</p>
+
+  <p><a name="Nt281" href="#NtA281">[281]</a> Hylotropic substances are
+  such as can undergo transformation into other substances of the same
+  composition (Ostwald, <i>Lehrbuch</i>, II. <b>2</b>. 298).</p>
+
+  <p><a name="Nt282" href="#NtA282">[282]</a> Also called Equilibrium Point
+  (Lowry).</p>
+
+  <p><a name="Nt283" href="#NtA283">[283]</a> For a discussion of these
+  systems, see Roozeboom, <i>Zeitschr. physikal. Chem.</i>, <i>loc.
+  cit</i>.</p>
+
+  <p><a name="Nt284" href="#NtA284">[284]</a> See Bancroft, <i>loc.
+  cit.</i>, p. 147; Wegscheider, <i>Sitzungsber. Wiener Akad.</i>, 1902,
+  <b>110</b>. 908.</p>
+
+  <p><a name="Nt285" href="#NtA285">[285]</a> Reference may be made here to
+  the term "stability limit," introduced by Knorr (<i>Annalen</i>, 1896,
+  <b>293</b>. 88) to indicate that temperature above which liquefaction and
+  isomeric change takes place. As employed by Knorr and others, the term
+  does not appear to have a very precise meaning, since it is used to
+  denote, not the temperature at which these changes can occur, but the
+  temperature at which the change is rapid (vide <i>Annalen</i>, 1896,
+  <b>293</b>. 91; 1899, <b>306</b>. 334); and the introduction of an
+  indefinite velocity of change renders the temperature of the stability
+  limit also somewhat indefinite. The definiteness of the term is also not
+  a little diminished by the fact that the "limit" can be altered by means
+  of catalytic agents. Since, as we have seen, the stable modification can
+  always undergo isomeric change and liquefy at temperatures above the
+  natural freezing point, but not below that point; and, further, the less
+  stable modification can undergo isomeric transformation and liquefy at
+  temperatures above the eutectic point, but will not liquefy at
+  temperatures below that; it seems to the author that it would be more
+  precise to identify these two points&mdash;the natural freezing point and
+  the eutectic point&mdash;which are not altered by catalytic agents, with
+  the "stability limits" of the stable and unstable modification
+  respectively. A perfectly definite meaning would thereby be given to the
+  term. In the case of those substances which do not undergo appreciable
+  isomeric change at the temperature of the melting point, the stability
+  limits would be the points G and H, Fig. 60.</p>
+
+  <p><a name="Nt286" href="#NtA286">[286]</a> Cameron, <i>Journ. Physical
+  Chem.</i>, 1898, <b>2</b>. 409.</p>
+
+  <p><a name="Nt287" href="#NtA287">[287]</a> Carveth, <i>Journ. Phys.
+  Chem.</i>, 1898, <b>2</b>. 159. See also Dutoit and Fath, <i>Journ. chim.
+  phys</i>., 1903, <b>1</b>. 358; Findlay, <i>Trans. Chem. Soc.</i>, 1904,
+  <b>85</b>. 403.</p>
+
+  <p><a name="Nt288" href="#NtA288">[288]</a> Hollmann, <i>Zeitschr.
+  physikal. Chem.</i>, 1903, <b>43</b>. 129.</p>
+
+  <p><a name="Nt289" href="#NtA289">[289]</a> For other examples of the
+  application of the Phase Rule to isomeric substances, see <i>Journ.
+  Physical Chem.</i>, vols. <b>2</b>. <i>et seq.</i>; Findlay, <i>Trans.
+  Chem. Soc.</i>, 1904, <b>85</b>. 403.</p>
+
+  <p><a name="Nt290" href="#NtA290">[290]</a> See Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>30</b>. 410.</p>
+
+  <p><a name="Nt291" href="#NtA291">[291]</a> See also Saposchnikoff,
+  <i>Zeitschr. physikal. Chem.</i>, <b>49</b>. 688; Kremann,
+  <i>Monatshefte</i>, 1904, <b>25</b>. 1215, 1271, 1311.</p>
+
+  <p><a name="Nt292" href="#NtA292">[292]</a> J. C. Philip, <i>Journ. Chem.
+  Soc.</i>, 1903, <b>83</b>. 821.</p>
+
+  <p><a name="Nt293" href="#NtA293">[293]</a> <i>Cf.</i> also Paterno and
+  Ampolla, <i>Gazzetta chim. ital.</i>, 1897, <b>27</b>. 481.</p>
+
+  <p><a name="Nt294" href="#NtA294">[294]</a> Philip, <i>loc. cit.</i>, p.
+  826.</p>
+
+  <p><a name="Nt295" href="#NtA295">[295]</a> Philip, <i>loc. cit.</i>, p.
+  829. Compare curves for iodine monochloride, Fig. 42, p. 162.</p>
+
+  <p><a name="Nt296" href="#NtA296">[296]</a> Kuriloff, <i>Zeitschr.
+  physikal. Chem.</i>, 1897, <b>23</b>. 676.</p>
+
+  <p><a name="Nt297" href="#NtA297">[297]</a> Ladenburg, <i>Ber.</i>, 1895,
+  <b>28</b>. 163; 1991.</p>
+
+  <p><a name="Nt298" href="#NtA298">[298]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>28</b>. 494; Adriani, <i>ibid.</i>, 1900,
+  <b>33</b>. 453.</p>
+
+  <p><a name="Nt299" href="#NtA299">[299]</a> Adriani, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>33</b>. 453.</p>
+
+  <p><a name="Nt300" href="#NtA300">[300]</a> A. Findlay and Miss E.
+  Hickmans.</p>
+
+  <p><a name="Nt301" href="#NtA301">[301]</a> Kipping and Pope, <i>Journ.
+  Chem. Soc.</i>, 1897, <b>71</b>. 993.</p>
+
+  <p><a name="Nt302" href="#NtA302">[302]</a> See Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>28</b>. 512; Adriani, <i>ibid.</i>, 1900,
+  <b>33</b>. 473; 1901, <b>36</b>. 168.</p>
+
+  <p><a name="Nt303" href="#NtA303">[303]</a> In this connection reference
+  should be made more especially to the paper by Roberts-Austen and
+  Stansfield, "Sur la constitution des alliages métalliques," in the
+  <i>Rapports du congrès international de physique</i>, 1900, I. 363; J. A.
+  Mathews, <i>Journ. of the Franklin Inst.</i>, 1902; Gautier, <i>Compt.
+  rend.</i>, 1896, <b>123</b>. 109; Roberts-Austen, "Reports of the Alloys
+  Research Committee," in <i>Journ. Inst. Mechan. Engineers</i>, from 1891
+  to 1904; and the papers by Heycock and Neville, published in the
+  <i>Journ. Chem. Soc.</i>, and the <i>Trans. Roy. Soc.</i> since 1897;
+  also Neville, <i>Reports of the British Association</i>, 1900, p. 131.
+  Reference must also be made to the important metallographic
+  investigations by Tammann and his pupils, and of Kurnakoff (<i>Zeitschr.
+  anorgan. Chem.</i>, vol. <b>40</b> and onwards), and also to those of
+  Shepherd, <i>Journ. Physical Chem.</i>, <b>8</b>. A bibliography of the
+  alloys is given in <i>Zeitschr. anorgan. Chem.</i>, 1903, <b>35</b>.
+  249.</p>
+
+  <p><a name="Nt304" href="#NtA304">[304]</a> Kurnakoff and Puschin,
+  <i>Zeitschr. anorgan. Chem.</i>, 1902, <b>30</b>. 104.</p>
+
+  <p><a name="Nt305" href="#NtA305">[305]</a> Gautier, <i>Bull. Soc.
+  d'Encouragement</i>, 1896 [5], <b>1</b>. 1312.</p>
+
+  <p><a name="Nt306" href="#NtA306">[306]</a> Heycock and Neville, <i>Phil.
+  Trans.</i>, 1900, <b>194</b>. 201.</p>
+
+  <p><a name="Nt307" href="#NtA307">[307]</a> Gautier, <i>loc. cit.</i> See
+  also Roberts-Austen and Rose, <i>Proc. Roy. Soc.</i>, 1903, <b>71</b>.
+  161.</p>
+
+  <p><a name="Nt308" href="#NtA308">[308]</a> Heycock and Neville,
+  <i>Journ. Chem. Soc.</i>, 1897, <b>71</b>. 414.</p>
+
+  <p><a name="Nt309" href="#NtA309">[309]</a> See Roberts-Austen,
+  <i>Introduction to Metallurgy</i>, 5th edit., p. 102; Bakhuis Roozeboom,
+  <i>Journ. Iron and Steel Inst.</i>, 1900, II. 311; <i>Zeitschr. physikal.
+  Chem.</i>, 1900, <b>34</b>. 437; von Jüptner, <i>Siderology</i>, p. 223
+  (translation by C. Salter); van't Hoff, <i>Zinn, Gips, und Stahl</i>, p.
+  24, or <i>Acht Vorträge über physikalische Chemie</i>, p. 37. Further,
+  Roozeboom, <i>Zeitschr. Elektrochem.</i>, 1904, <b>10</b>. 489; E. Heyn,
+  <i>ibid.</i>, p. 491; Carpenter and Keeling, <i>Journ. Iron and Steel
+  Inst.</i>, 1904, <b>65</b>. 224.</p>
+
+  <p><a name="Nt310" href="#NtA310">[310]</a> The melting point of pure
+  iron is given by Carpenter and Keeling (<i>Journ. Iron and Steel
+  Inst.</i>, 1904, <b>65</b>. 224) as 1505°.</p>
+
+  <p><a name="Nt311" href="#NtA311">[311]</a> <i>Zeitschr. für
+  Elektrochem.</i>, 1904, <b>10</b>. 491.</p>
+
+  <p><a name="Nt312" href="#NtA312">[312]</a> See also Hiorns, <i>Journ.
+  Soc. Chem. Ind.</i>, 1906, <b>25</b>. 50.</p>
+
+  <p><a name="Nt313" href="#NtA313">[313]</a> Bancroft, <i>Jour. Physical
+  Chem.</i>, 1902, <b>6</b>. 178; Bell and Taber, <i>ibid.</i>, 1906,
+  <b>10</b>. 120.</p>
+
+  <p><a name="Nt314" href="#NtA314">[314]</a> The method to be followed
+  when the third component enters into the solid phase will be explained
+  later.</p>
+
+  <p><a name="Nt315" href="#NtA315">[315]</a> Tammann, <i>Zeitschr. anorg.
+  Chem.</i>, 1903, <b>37</b>. 303; 1905, <b>45</b>. 24. Reference may be
+  made here to the registering pyrometer of Kurnakoff, <i>Zeitschr. anorg.
+  Chem.</i>, 1904, <b>42</b>. 184.</p>
+
+  <p><a name="Nt316" href="#NtA316">[316]</a> In this connection, see
+  Doelter, <i>Physikalisch-chemisch Mineralogie</i> (Barth, 1901);
+  Meyerhoffer, <i>Zeitschr. f. Kristallographie</i>, 1902, <b>36</b>. 593;
+  Guthrie, <i>Phil. Mag.</i>, 1884 [5], <b>17</b>. 479; Le Chatelier,
+  <i>Compt. rend.</i>, 1900, <b>130</b>. 85; and especially E. Baur,
+  <i>Zeitschr. physikal. Chem.</i>, 1903, <b>42</b>. 567; J. H. L. Vogt,
+  <i>Zeitschr. Elektrochem.</i>, 1903, <b>9</b>. 852, and <i>Die
+  Silikatschmelzlösungen</i>, Parts I. and II. (Christiania, 1903, 1904).
+  See also N. V. Kultascheff, <i>Zeitschr. anorg. Chem.</i>, 1903,
+  <b>35</b>. 187.</p>
+
+  <p><a name="Nt317" href="#NtA317">[317]</a> G. G. Stokes, <i>Proc. Roy.
+  Soc.</i>, 1891, <b>49</b>. 174; Gibbs, <i>Trans. Conn. Acad.</i>, 1876,
+  <b>3</b>. 176; Roozeboom, <i>Zeitschr. physikal. Chem.</i>, 1894,
+  <b>15</b>. 147.</p>
+
+  <p><a name="Nt318" href="#NtA318">[318]</a> This figure has been taken
+  from Ostwald's <i>Lehrbuch</i>, II. 2. 984.</p>
+
+  <p><a name="Nt319" href="#NtA319">[319]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1893, <b>12</b>. 369.</p>
+
+  <p><a name="Nt320" href="#NtA320">[320]</a> C. R. A. Wright, <i>Proc.
+  Roy. Soc.</i>, 1891, <b>49</b>. 174; 1892, <b>50</b>. 375.</p>
+
+  <p><a name="Nt321" href="#NtA321">[321]</a> The distribution coefficient
+  will not remain constant because, apart from other reasons, the mutual
+  solubility of chloroform and water is altered by the addition of the
+  acid.</p>
+
+  <p><a name="Nt322" href="#NtA322">[322]</a> Bancroft, <i>Physical
+  Review</i>, 1895, <b>3</b>. 21; Schreinemakers, <i>Zeitschr. physikal.
+  Chem.</i>, 1897, <b>23</b>. 652, and subsequent volumes.</p>
+
+  <p><a name="Nt323" href="#NtA323">[323]</a> C. R. A. Wright, <i>Proc.
+  Roy. Soc.</i>, 1889-1893.</p>
+
+  <p><a name="Nt324" href="#NtA324">[324]</a> C. R. A. Wright, <i>Proc.
+  Roy. Soc.</i>, 1892, <b>50</b>. 390.</p>
+
+  <p><a name="Nt325" href="#NtA325">[325]</a> Bodländer, <i>Berg- und
+  Hüttenmänn. Ztg.</i>, 1897, <b>56</b>. 331.</p>
+
+  <p><a name="Nt326" href="#NtA326">[326]</a> C. R. A. Wright, <i>Proc.
+  Roy. Soc.</i>, <i>loc. cit.</i></p>
+
+  <p><a name="Nt327" href="#NtA327">[327]</a> Schreinemakers, <i>Zeitschr.
+  physikal. Chem.</i>, 1900, <b>33</b>. 78.</p>
+
+  <p><a name="Nt328" href="#NtA328">[328]</a> Schreinemakers, <i>Zeitschr.
+  physikal. Chem.</i>, 1898, <b>27</b>. 95.</p>
+
+  <p><a name="Nt329" href="#NtA329">[329]</a> Schreinemakers, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>29</b>. 577.</p>
+
+  <p><a name="Nt330" href="#NtA330">[330]</a> Schreinemakers, <i>Zeitschr.
+  physikal. Chem.</i>, 1898, <b>25</b>. 543.</p>
+
+  <p><a name="Nt331" href="#NtA331">[331]</a> Charpy, <i>Compt. rend.</i>,
+  1898, <b>126</b>. 1569. Compare the curves for the system
+  KNO<sub>3</sub>&mdash;NaNO<sub>3</sub>&mdash;LiNO<sub>3</sub> (H. R.
+  Carveth, <i>Journ. Physical Chem.</i>, 1898, <b>2</b>. 209). Also alloys
+  of Pb&mdash;Sn&mdash;Bi (E. S. Shepherd, <i>Journ. Physical Chem.</i>,
+  1902, <b>6</b>. 527).</p>
+
+  <p><a name="Nt332" href="#NtA332">[332]</a> It should be remembered that
+  in the triangular diagram a <i>line</i> parallel to one of the sides
+  indicates, at a given temperature, a constant amount of the component
+  represented by the opposite corner of the triangle; and, hence, points in
+  a <i>plane</i>, parallel to one face of a right prism, will indicate for
+  different temperatures, variation in the amounts of two components, but
+  constancy in the amount of the third.</p>
+
+  <p><a name="Nt333" href="#NtA333">[333]</a> <i>Gazzetta chim. ital.</i>,
+  1898, <b>28</b>. II. 520.</p>
+
+  <p><a name="Nt334" href="#NtA334">[334]</a> Bruni, <i>Gazzetta chim.
+  ital.</i>, 1898, <b>28</b>. II. 508; 1900, <b>30</b>. I. 35.</p>
+
+  <p><a name="Nt335" href="#NtA335">[335]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1900, <b>36</b>. 168.</p>
+
+  <p><a name="Nt336" href="#NtA336">[336]</a> For a discussion of these
+  systems, see van't Hoff, <i>Bildung und Spaltung von Doppelsalzen</i>
+  (Leipzig, 1897).</p>
+
+  <p><a name="Nt337" href="#NtA337">[337]</a> Van Leeuwen, <i>Zeitschr.
+  physikal. Chem.</i>, 1897, <b>23</b>. 35.</p>
+
+  <p><a name="Nt338" href="#NtA338">[338]</a> Meyerhoffer, <i>Zeitschr.
+  physikal. Chem.</i>, 1889, <b>3</b>. 336; 1890, <b>5</b>. 97.</p>
+
+  <p><a name="Nt339" href="#NtA339">[339]</a> Reicher, <i>Zeitschr.
+  physikal. Chem.</i>, 1887, <b>1</b>. 220.</p>
+
+  <p><a name="Nt340" href="#NtA340">[340]</a> For other examples of the
+  formation and decomposition of double salts at a transition point, the
+  reader is referred to the work by van't Hoff, already cited, on the
+  <i>Bildung und Spaltung von Doppelsalzen</i>; or to Bancroft, <i>Phase
+  Rule</i>, p. 180.</p>
+
+  <p><a name="Nt341" href="#NtA341">[341]</a> Bancroft, <i>Phase Rule</i>,
+  p. 183.</p>
+
+  <p><a name="Nt342" href="#NtA342">[342]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 514.</p>
+
+  <p><a name="Nt343" href="#NtA343">[343]</a> The influence of pressure on
+  the transition point in the case of tachydrite has been determined by
+  van't Hoff, Kenrick, and Dawson (<i>Zeitschr. physikal. Chem.</i>, 1901,
+  <b>39</b>. 27, 34; van't Hoff, <i>Zur Bildung der ozeanischen
+  Salzablagerungen</i>, I. p. 66&mdash;Brunswick, 1905). This salt is
+  formed from magnesium chloride and calcium chloride at 22°, in accordance
+  with the equation&mdash;</p>
+
+<p class="cenhead">2MgCl<sub>2</sub>.6H<sub>2</sub>O + CaCl<sub>2</sub>.6H<sub>2</sub>O = Mg<sub>2</sub>CaCl<sub>6</sub>.12H<sub>2</sub>O + 6H<sub>2</sub>O</p>
+
+  <p>Increase of pressure raises the transition point, because the
+  formation of tachydrite is accompanied by increase of volume; the
+  elevation being 0.016° for an increase of pressure of 1 atm. The number
+  calculated from the theoretical formula (p.&nbsp;<a href="#page57">57</a>) is
+  0.013° for 1 atm.</p>
+
+  <p>If one calculates the influence of the pressure of sea-water on the
+  temperature of formation of tachydrite (which is of interest on account
+  of the natural occurrence of this salt), it is found that a depth of
+  water of 1500 metres, exerting a pressure of 180 atm., would alter the
+  temperature of formation of tachydrite by only 3°. The effect is,
+  therefore, comparatively unimportant.</p>
+
+  <p><a name="Nt344" href="#NtA344">[344]</a> Roozeboom, <i>Zeitschr.
+  physical. Chem.</i>, 1887, <b>1</b>. 227.</p>
+
+  <p><a name="Nt345" href="#NtA345">[345]</a> <i>Zeitschr. physical.
+  Chem.</i>, 1887, <b>1</b>. 227.</p>
+
+  <p><a name="Nt346" href="#NtA346">[346]</a> Van't Hoff and Müller,
+  <i>Ber.</i>, 1898, <b>31</b>. 2206.</p>
+
+  <p><a name="Nt347" href="#NtA347">[347]</a> Van't Hoff and van Deventer,
+  <i>Zeitschr. physikal. Chem.</i>, 1887, <b>1</b>. 165.</p>
+
+  <p><a name="Nt348" href="#NtA348">[348]</a> For a full discussion of the
+  solubility relations of sodium ammonium racemate, see van't Hoff,
+  <i>Bildung und Spaltung von Doppelsalzen</i>, p. 81.</p>
+
+  <p><a name="Nt349" href="#NtA349">[349]</a> <i>Annales chim. phys.</i>,
+  1848 [3], <b>24</b>. 442.</p>
+
+  <p><a name="Nt350" href="#NtA350">[350]</a> See Van't Hoff and van
+  Deventer, <i>Zeitschr. phys. Chem.</i>, 1887, <b>1</b>. 165.</p>
+
+  <p><a name="Nt351" href="#NtA351">[351]</a> Meyerhoffer, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 121.</p>
+
+  <p><a name="Nt352" href="#NtA352">[352]</a> Roozeboom, <i>Zeitschr.
+  physikal. Chem.</i>, 1888, <b>2</b>. 518.</p>
+
+  <p><a name="Nt353" href="#NtA353">[353]</a> Meyerhoffer, <i>Zeitschr.
+  physikal. Chem.</i>, 1890, <b>5</b>. 109. On the importance of the
+  transition interval in the case of optically active substances, see
+  Meyerhoffer, <i>Ber.</i>, 1904, <b>37</b>. 2604.</p>
+
+  <p><a name="Nt354" href="#NtA354">[354]</a> In connection with this
+  chapter, see, more especially, van't Hoff, <i>Bildung und Spaltung von
+  Doppelsalzen</i>, p. 3, <i>ff.</i>; Roozeboom, <i>Zeitschr. physikal
+  Chem.</i>, 1892, <b>10</b>. 158; Bancroft, <i>Phase Rule</i>, p. 201;
+  209.</p>
+
+  <p><a name="Nt355" href="#NtA355">[355]</a> The same restriction must be
+  made here as was imposed in the preceding chapter, namely, that the two
+  salts in solution give a common ion.</p>
+
+  <p><a name="Nt356" href="#NtA356">[356]</a> For example, addition of
+  ammonium chloride to solutions of ferric chloride (Roozeboom,
+  <i>Zeitschr. physikal. Chem.</i>, 1892, <b>10</b>. 149).</p>
+
+  <p><a name="Nt357" href="#NtA357">[357]</a> It must, of course, be
+  understood that the temperature is on that side of the transition point
+  on which the double salt is stable.</p>
+
+  <p><a name="Nt358" href="#NtA358">[358]</a> Excess of the double salt
+  must be taken, because otherwise an unsaturated solution might be formed,
+  and this would, of course, not deposit any salt.</p>
+
+  <p><a name="Nt359" href="#NtA359">[359]</a> Meyerhoffer, <i>Ber.</i>,
+  1904, <b>37</b>. 2605.</p>
+
+  <p><a name="Nt360" href="#NtA360">[360]</a> Meyerhoffer, <i>Ber.</i>,
+  1897, <b>30</b>. 1809.</p>
+
+  <p><a name="Nt361" href="#NtA361">[361]</a> Meyerhoffer, <i>Ber.</i>,
+  1904, <b>37</b>. 2604.</p>
+
+  <p><a name="Nt362" href="#NtA362">[362]</a> Bancroft, <i>Phase Rule</i>,
+  p. 203; Roozeboom, <i>Zeitschr. physikal. Chem.</i>, 1891, <b>8</b>. 504,
+  531; Stortenbeker, <i>ibid.</i>, 1895, <b>17</b>. 643; 1897, <b>22</b>.
+  60; 1900, <b>34</b>. 108.</p>
+
+  <p><a name="Nt363" href="#NtA363">[363]</a> Roozeboom, <i>Zeitschr. phys.
+  Chem.</i>, 1899, <b>28</b>. 494; <i>Ber.</i>, 1899, <b>32</b>. 537.</p>
+
+  <p><a name="Nt364" href="#NtA364">[364]</a> As, for instance, strychnine
+  racemate, a compound of racemic acid with the <i>optically active</i>
+  strychnine. This would be resolved into strychnine <i>d</i>-tartrate and
+  strychnine <i>l</i>-tartrate, which are not enantiomorphous forms.</p>
+
+  <p><a name="Nt365" href="#NtA365">[365]</a> Van't Hoff and Meyerhoffer,
+  <i>Zeitschr. physikal Chem.</i>, 1898, <b>27</b>. 75; 1899, <b>30</b>.
+  86. Fig. 113 is taken from the latter paper.</p>
+
+  <p><a name="Nt366" href="#NtA366">[366]</a> Solid models constructed of
+  plaster of Paris can be obtained from Max Kaehler and Martini,
+  Berlin.</p>
+
+  <p><a name="Nt367" href="#NtA367">[367]</a> Instead of the present method
+  of obtaining potassium chloride by decomposing carnallite with water,
+  advantage might be taken of the fact that carnallite when heated to 168°
+  undergoes decomposition with separation of three-fourths of the potassium
+  chloride (van't Hoff, <i>Acht Vorträge über physikalische Chemie</i>,
+  1902, p. 32).</p>
+
+  <p><a name="Nt368" href="#NtA368">[368]</a> Roozeboom and Schreinemakers,
+  <i>Zeitschr. physikal. Chem.</i>, 1894, <b>15</b>. 588.</p>
+
+  <p><a name="Nt369" href="#NtA369">[369]</a> These curves represent only
+  portions of the isotherms, since the systems in which a ternary solution
+  is in equilibrium with solid hydrogen chloride or a hydrate, have not
+  been investigated.</p>
+
+  <p><a name="Nt370" href="#NtA370">[370]</a> The numbers printed beside
+  the points on the curves refer to the number of the experiment in the
+  original paper.</p>
+
+  <p><a name="Nt371" href="#NtA371">[371]</a> Lash, Miller and Kenrick,
+  <i>Journ. Physical. Chem.</i>, 1903, <b>7</b>. 259; Allan, <i>Amer. Chem.
+  Journ.</i>, 1901, <b>25</b>. 307.</p>
+
+  <p><a name="Nt372" href="#NtA372">[372]</a> Allan, <i>Amer. Chem.
+  Journ.</i>, 1901, <b>25</b>. 307.</p>
+
+  <p><a name="Nt373" href="#NtA373">[373]</a> Hoitsema, <i>Zeitschr.
+  physikal. Chem.</i>, 1895, <b>17</b>. 651; Allan, <i>loc. cit.</i></p>
+
+  <p><a name="Nt374" href="#NtA374">[374]</a> Rutten, <i>Zeitschr. anorgan.
+  Chem.</i>, 1902, <b>30</b>. 342. Compare the system
+  BeO&mdash;SO<sub>3</sub>&mdash;H<sub>2</sub>O; Parsons, <i>Zeitschr.
+  anorgan. Chem.</i>, 1904, <b>42</b>. 250.</p>
+
+  <p><a name="Nt375" href="#NtA375">[375]</a> <i>Zeitschr. anorgan.
+  Chem.</i>, 1904, <b>40</b>. 146.</p>
+
+  <p><a name="Nt376" href="#NtA376">[376]</a> Schreinemakers, <i>Zeitschr.
+  physikal. Chem.</i>, 1893, <b>11</b>. 76; Bancroft, <i>Journ. Physical
+  Chem.</i>, 1902, <b>6</b>. 179.</p>
+
+  <p><a name="Nt377" href="#NtA377">[377]</a> <i>Zeitschr. anorgan.
+  Chem.</i>, 1904, <b>40</b>. 148.</p>
+
+  <p><a name="Nt378" href="#NtA378">[378]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1903, <b>43</b>. 354.</p>
+
+  <p><a name="Nt379" href="#NtA379">[379]</a> These equilibria were
+  obtained by Boudouard, <i>Annales chim. phys.</i>, 1901 [7], <b>24</b>.
+  5. See also Hahn, <i>Zeitschr. physikal. Chem.</i>, 1903, <b>42</b>. 705;
+  <b>44</b>. 513.</p>
+
+  <p><a name="Nt380" href="#NtA380">[380]</a> G. Preuner, <i>Zeitschr.
+  physikal. Chem.</i>, 1903, <b>47</b>. 385.</p>
+
+  <p><a name="Nt381" href="#NtA381">[381]</a> See Hahn, <i>Zeitschr.
+  physikal. Chem.</i>, 1903, <b>42</b>. 705; <b>44</b>. 513; Boudouard,
+  <i>Bull. Soc. chim.</i>, [3], <b>25</b>. 484; Bodländer, <i>Zeitschr. f.
+  Elektrochem.</i>, 1902, <b>8</b>. 833; R. Schenck and Zimmermann,
+  <i>Ber.</i>, 1903, <b>36</b>. 1231, 3663; Schenck and Heller,
+  <i>ibid.</i>, 1905, <b>38</b>. 2132; <i>Zeitschr. f. Elektrochem.</i>,
+  1903, <b>9</b>. 691; Haber, <i>Thermodynamik technischer
+  Gasreaktionen</i>, p. 293 (Munich, 1903).</p>
+
+  <p><a name="Nt382" href="#NtA382">[382]</a> A very useful summary of the
+  investigations carried out by van't Hoff and his pupils on the formation
+  of the Stassfurt salt-beds is given by E. F. Armstrong, in the <i>Reports
+  of the British Association for 1901</i>, p. 262. See also van't Hoff,
+  <i>Zur Bildung der ozeanischen Salzablagerungen</i> (Brunswick,
+  1905).</p>
+
+  <p><a name="Nt383" href="#NtA383">[383]</a> See especially Meyerhoffer,
+  <i>Silzungsber. Wien. Akad.</i>, 1895, <b>104</b>. II. <i>b</i>, 840;
+  Meyerhoffer and Saunders, <i>Zeitschr. physikal. Chem.</i>, 1899,
+  <b>28</b>. 453; <b>31</b>. 370. The investigation of the equilibria
+  between reciprocal salt-pairs alone (three-component systems) is of great
+  importance for the artificial preparations of minerals, as also in
+  analytical chemistry for the proper understanding of the methods of
+  conversion of insoluble systems into soluble by fusion (see Meyerhoffer,
+  <i>Zeitschr. physikal. Chem.</i>, 1901, <b>38</b>. 307).</p>
+
+  <p><a name="Nt384" href="#NtA384">[384]</a> See Meyerhoffer, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>28</b>. 459.</p>
+
+  <p><a name="Nt385" href="#NtA385">[385]</a> Compare the reciprocal
+  salt-pair NaCl&mdash;NH<sub>4</sub>HCO<sub>3</sub> (p.&nbsp;<a
+  href="#page321">321</a>). In this case the upper limit of the transition
+  interval was found by extrapolation of the solubility curve for
+  NaHCO<sub>3</sub> + NH<sub>4</sub>Cl + NH<sub>4</sub>HCO<sub>3</sub> and
+  NaHCO<sub>3</sub> + NH<sub>4</sub>Cl + NaCl to be 32° (Fedotieff,
+  <i>Zeitschr. phys. Chem.</i>, 1904, <b>49</b>. 179).</p>
+
+  <p><a name="Nt386" href="#NtA386">[386]</a> Löwenherz, <i>Zeitschr.
+  physikal. Chem.</i>, 1894, <b>13</b>. 464.</p>
+
+  <p><a name="Nt387" href="#NtA387">[387]</a> Meyerhoffer and Saunders,
+  <i>Zeitschr. physikal. Chem.</i>, 1899, <b>28</b>. 479.</p>
+
+  <p><a name="Nt388" href="#NtA388">[388]</a> As the quantities of the
+  salts are expressed in <i>equivalent</i> gram-molecules, the molecule of
+  sodium and potassium chloride must be doubled in order to be equivalent
+  to sodium sulphate and potassium sulphate.</p>
+
+  <p><a name="Nt389" href="#NtA389">[389]</a> <i>Sitz-Ber. der kgl. preuss.
+  Akad. der Wiss.</i>, 1903, p. 359. Van't Hoff, <i>Zur Bildung der
+  ozeanischen Salzablagerungen</i>, I. p. 34 (Brunswick, 1905).</p>
+
+  <p><a name="Nt390" href="#NtA390">[390]</a> <i>Zeitschr. für
+  Kristallographie</i>, 1904, <b>39</b>. 155.</p>
+
+  <p><a name="Nt391" href="#NtA391">[391]</a> Meyerhoffer and Saunders,
+  <i>Zeitschr. physikal. Chem.</i>, 1899, <b>28</b>. 479.</p>
+
+  <p><a name="Nt392" href="#NtA392">[392]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1904, <b>49</b>. 162.</p>
+
+  <p><a name="Nt393" href="#NtA393">[393]</a> Another commercial process,
+  in the study of which good service is done by the Phase Rule, is the
+  caustification of the alkali salts (G. Bodländer, <i>Zeitschr. für
+  Elektrochem.</i>, 1905, <b>11</b>. 186; J. Herold, <i>ibid.</i>,
+  418).</p>
+
+  <p><a name="Nt394" href="#NtA394">[394]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1900, <b>35</b>. 32.</p>
+
+  <p><a name="Nt395" href="#NtA395">[395]</a> Mention may also be made here
+  of the equilibria between magnesium carbonate and potassium carbonate,
+  although these do not form a reciprocal salt-pair (Auerbach, <i>Zeitschr.
+  für Elektrochem.</i>, 1904, <b>10</b>. 161).</p>
+
+  <p><a name="Nt396" href="#NtA396">[396]</a> O. N. Witt and K. Ludwig,
+  <i>Ber.</i>, 1903, <b>36</b>. 4384; Meyerhoffer, <i>ibid.</i>, 1904,
+  <b>37</b>. 261, 1116.</p>
+
+  <p><a name="Nt397" href="#NtA397">[397]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1905, <b>53</b>. 513. Compare also, <i>ibid.</i>, 1903,
+  <b>38</b>. 307.</p>
+
+  <p><a name="Nt398" href="#NtA398">[398]</a> See Schwarz, <i>Beiträge zur
+  Kenntnis der umkehrbaren Umwandlungen polymorpher Korper</i> (Göttingen,
+  1892); or, Roozeboom, <i>Heterogen. Gleichgewicht</i>, I. p. 125. Also
+  Barnes and Cooke, <i>Journ. Physical Chem.</i>, 1902, <b>6</b>. 172.</p>
+
+  <p><a name="Nt399" href="#NtA399">[399]</a> Van't Hoff and van Deventer,
+  <i>Zeitschr. physikal. Chem.</i>, 1887, <b>1</b>. 173.</p>
+
+  <p><a name="Nt400" href="#NtA400">[400]</a> Reicher, <i>Zeitschr. für
+  Krystallographie</i>, 1884, <b>8</b>. 593.</p>
+
+  <p><a name="Nt401" href="#NtA401">[401]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1895, <b>17</b>. 153.</p>
+
+  <p><a name="Nt402" href="#NtA402">[402]</a> <i>Zeitschr. physikal.
+  Chem.</i>, 1899, <b>28</b>. 464.</p>
+
+  <p><a name="Nt403" href="#NtA403">[403]</a> Meyerhoffer and Saunders,
+  <i>ibid.</i>, p. 466.</p>
+
+  <p><a name="Nt404" href="#NtA404">[404]</a> See Van Eyk, <i>Zeitschr.
+  physikal. Chem.</i>, 1899, <b>30</b>. 446.</p>
+
+  <p><a name="Nt405" href="#NtA405">[405]</a> See in this connection the
+  volume in this series on <i>Electro-chemistry</i>, by Dr. R. A.
+  Lehfeldt.</p>
+
+  <p><a name="Nt406" href="#NtA406">[406]</a> Barnes and Cooke, <i>Journ.
+  Physical Chem.</i>, 1902, <b>6</b>. 172.</p>
+
+  <p><a name="Nt407" href="#NtA407">[407]</a> For a description and
+  explanation of these, the reader should consult the volume in this series
+  by Dr. Lehfeldt on <i>Electro-chemistry</i>; and van't Hoff, <i>Bildung
+  und Spaltung von Doppelsalzen</i>, p. 48 <i>ff.</i></p>
+
+</div>
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+The Project Gutenberg eBook, The Phase Rule and Its Applications, by
+Alexander Findlay
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
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+
+
+
+
+
+Title: The Phase Rule and Its Applications
+
+
+Author: Alexander Findlay
+
+
+
+Release Date: November 27, 2010  [eBook #34457]
+
+Language: English
+
+Character set encoding: ISO-646-US (US-ASCII)
+
+
+***START OF THE PROJECT GUTENBERG EBOOK THE PHASE RULE AND ITS
+APPLICATIONS***
+
+
+E-text prepared by Juliet Sutherland, Keith Edkins, and the Online
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+Note: Project Gutenberg also has an HTML version of this
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+
+Transcriber's note:
+
+      Page numbers enclosed by curly braces (example: {25}) have been
+      incorporated to facilitate the use of the Table of Contents.
+
+      In chemical formulas an underscore is used to indicate that
+      the following number enclosed by curly braces is a subscript.
+      For example, the formula of water is indicated by H_{2}O.
+
+      A carat character followed by a number enclosed by curly
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+      A few typographical errors have been corrected and are listed
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+
+
+
+
+THE PHASE RULE AND ITS APPLICATIONS
+
+by
+
+ALEX. FINDLAY, M.A., PH.D., D.SC.
+
+       *       *       *       *       *
+
+TEXT-BOOKS OF PHYSICAL CHEMISTRY.
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+    THE PHASE RULE AND ITS APPLICATIONS. By ALEX. FINDLAY, M.A., Ph.D.,
+    D.Sc., Lecturer and Demonstrator in Chemistry, University of
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+       *       *       *       *       *
+
+
+THE PHASE RULE AND ITS APPLICATIONS
+
+BY
+
+ALEX. FINDLAY, M.A., PH.D., D.SC.
+
+Lecturer on Physical Chemistry, University of Birmingham
+
+With One Hundred and Thirty-Four Figures in the Text
+
+Third Impression
+
+
+
+
+
+
+
+Longmans, Green, and Co. 39 Paternoster Row, London
+New York, Bombay, and Calcutta
+1908
+
+All rights reserved
+
+
+
+
+DEDICATED
+
+TO
+
+FRANCIS ROBERT JAPP, LL.D., F.R.S.
+
+PROFESSOR OF CHEMISTRY, UNIVERSITY OF ABERDEEN,
+
+IN GRATITUDE FOR EARLY TRAINING
+
+AND ADVICE
+
+
+
+{vii}
+
+PREFACE TO THE SECOND EDITION.
+
+During the two years which have elapsed since the first edition of this
+book appeared, the study of chemical equilibria has been prosecuted with
+considerable activity, and valuable additions have been made to our
+knowledge in several departments of this subject. In view of the scope of
+the present work, it has been, of course, impossible to incorporate all
+that has been done; but several new sections have been inserted, notably
+those on the study of basic salts; the interpretation of cooling curves,
+and the determination of the composition of solid phases without analysis;
+the equilibria between iron, carbon monoxide, and carbon dioxide, which are
+of importance in connection with the processes occurring in the blast
+furnace; and the Phase Rule study of the ammonia-soda process. I have also
+incorporated a short section on the reciprocal salt-pair barium
+carbonate--potassium sulphate, which had been written for the German
+edition of this book by the late Professor W. Meyerhoffer. The section on
+the iron-carbon alloys, which in the first edition was somewhat
+unsatisfactory, has been rewritten.
+
+A. F.
+
+_September, 1906._
+
+
+
+{viii}
+
+PREFACE
+
+Although we are indebted to the late Professor Willard Gibbs for the first
+enunciation of the Phase Rule, it was not till 1887 that its practical
+applicability to the study of Chemical Equilibria was made apparent. In
+that year Roozeboom disclosed the great generalization, which for upwards
+of ten years had remained hidden and unknown save to a very few, by
+stripping from it the garb of abstract Mathematics in which it had been
+clothed by its first discoverer. The Phase Rule was thus made generally
+accessible; and its adoption by Roozeboom as the basis of classification of
+the different cases of chemical equilibrium then known established its
+value, not only as a means of co-ordinating the large number of isolated
+cases of equilibrium and of giving a deeper insight into the relationships
+existing between the different systems, but also as a guide in the
+investigation of unknown systems.
+
+While the revelation of the principle embedded in the Phase Rule is
+primarily due to Roozeboom, it should not be forgotten that, some years
+previously, van't Hoff, in ignorance of the work of Willard Gibbs, had
+enunciated his "law of the incompatibility of condensed systems," which in
+some respects coincides with the Phase Rule; and it is only owing to the
+more general applicability of the latter that the very {ix} important
+generalization of van't Hoff has been somewhat lost sight of.
+
+The exposition of the Phase Rule and its applications given in the
+following pages has been made entirely non-mathematical, the desire having
+been to explain as clearly as possible the principles underlying the Phase
+Rule, and to illustrate their application to the classification and
+investigation of equilibria, by means of a number of cases actually
+studied. While it has been sought to make the treatment sufficiently
+elementary to be understood by the student just commencing the study of
+chemical equilibria, an attempt has been made to advance his knowledge to
+such a stage as to enable him to study with profit the larger works on the
+subject, and to follow with intelligence the course of investigation in
+this department of Physical Chemistry. It is also hoped that the volume may
+be of use, not only to the student of Physical Chemistry, or of the other
+branches of that science, but also to the student of Metallurgy and of
+Geology, for whom an acquaintance with at least the principles of the Phase
+Rule is becoming increasingly important.
+
+In writing the following account of the Phase Rule, it is scarcely
+necessary to say that I have been greatly indebted to the larger works on
+Chemical Equilibria by Ostwald ("Lehrbuch"), Roozeboom ("Die Heterogenen
+Gleichgewichte"), and Bancroft ("The Phase Rule"); and in the case of the
+first-named, to the inspiration also of personal teaching. My indebtedness
+to these and other authors I have indicated in the following pages.
+
+In conclusion, I would express my thanks to Sir William Ramsay, whose
+guidance and counsel have been constantly {x} at my disposal; and to my
+colleagues, Dr. T. Slater Price and Dr. A. McKenzie, for their friendly
+criticism and advice. To Messrs. J. N. Friend, M.Sc., and W. E. S. Turner,
+B.Sc., I am also indebted for their assistance in reading the proof-sheets.
+
+A. F.
+
+_November, 1903._
+
+
+
+
+{xi}
+
+CONTENTS
+
+                                                                       PAGE
+
+  CHAPTER I
+
+  INTRODUCTION                                                            1
+
+  General, I. Homogeneous and heterogeneous equilibrium,
+  5. Real and apparent equilibrium, 5.
+
+  CHAPTER II
+
+  THE PHASE RULE                                                          7
+
+  Phases, 8. Components, 10. Degree of freedom. Variability
+  of a system, 14. The Phase Rule, 16. Classification of systems
+  according to the Phase Rule, 17. Deduction of the Phase
+  Rule, 18.
+
+  CHAPTER III
+
+  TYPICAL SYSTEMS OF ONE COMPONENT                                       21
+
+  A. _Water._ Equilibrium between liquid and vapour. Vaporization
+  curve, 21. Upper limit of vaporization curve, 23.
+  Sublimation curve of ice, 24. Equilibrium between ice and
+  water. Curve of fusion, 25. Equilibrium between ice, water,
+  and vapour. The triple point, 27. Bivariant systems of water,
+  29. Supercooled water. Metastable state, 30. Other systems
+  of the substance water, 32. B. _Sulphur_, 33. Polymorphism, 33.
+  Sulphur, 34. Triple point--Rhombic and monoclinic sulphur
+  and vapour. Transition point, 34. Condensed systems, 36.
+  Suspended transformation, 37. Transition curve--Rhombic
+  and monoclinic sulphur, 37. Triple point--Monoclinic sulphur,
+  liquid, and vapour. Melting point of monoclinic sulphur, 38.
+  Triple point--Rhombic and monoclinic sulphur and liquid, 38.
+  Triple point--Rhombic sulphur, liquid, and vapour. Metastable
+  triple point, 38. Fusion curve of rhombic sulphur, 39.
+  Bivariant systems, 39. C. _Tin_, 41. Transition point, 41.
+  {xii}
+  Enantiotropy and monotropy, 44. D. _Phosphorus_, 46. Enantiotropy
+  combined with monotropy, 51. E. _Liquid Crystals_, 51.
+  Phenomena observed, 51. Nature of liquid crystals, 52. Equilibrium
+  relations in the case of liquid crystals, 53.
+
+  CHAPTER IV
+
+  GENERAL SUMMARY                                                        55
+
+  Triple point, 55. Theorems of van't Hoff and of Le Chatelier,
+  57. Changes at the triple point, 58. Triple point solid--solid--vapour,
+  62. Sublimation and vaporization curves,
+  63. Fusion curve--Transition curve, 66. Suspended transformation.
+  Metastable equilibria, 69. Velocity of transformation,
+  70. Law of successive reactions, 73.
+
+  CHAPTER V
+
+  SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION                   76
+
+  Different systems of two components, 77. PHENOMENA OF
+  DISSOCIATION. Bivariant systems, 79. Univariant systems,
+  80. Ammonia compounds of metal chlorides, 82. Salts with
+  water of crystallization, 85. Efflorescence, 86. Indefiniteness
+  of the vapour pressure of a hydrate, 87. Suspended transformation,
+  89. Range of existence of hydrates, 90. Constancy
+  of vapour pressure and the formation of compounds, 90.
+  Measurement of the vapour pressure of hydrates, 91.
+
+  CHAPTER VI
+
+  SOLUTIONS                                                              92
+
+  Definition, 92. SOLUTIONS OF GASES IN LIQUIDS, 93.
+  SOLUTIONS OF LIQUIDS IN LIQUIDS, 95. Partial or limited
+  miscibility, 96. Phenol and water, 97. Methylethylketone
+  and water, 100. Triethylamine and water, 101. General form
+  of concentration-temperature curve, 101. Pressure-concentration
+  diagram, 102. Complete miscibility, 104. Pressure-concentration
+  diagram, 104.
+
+  CHAPTER VII
+
+  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
+  VOLATILE                                                              106
+
+  General, 106. The saturated solution, 108. Form of the
+  solubility curve, 108. A. ANHYDROUS SALT AND WATER.
+  {xiii}
+  The solubility curve, 111. Suspended transformation and
+  supersaturation, 113. Solubility curve at higher temperatures,
+  114. (1) _Complete miscibility of the fused components._ Ice as
+  solid phase, 116. Cryohydrates, 117. Changes at the quadruple
+  point, 119. Freezing mixtures, 120. (2) _Partial miscibility of
+  the fused components._ Supersaturation, 124. Pressure-temperature
+  diagram, 126. Vapour pressure of solid--solution--vapour,
+  126. Other univariant systems, 127. Bivariant systems, 129.
+  Deliquescence, 130. Separation of salt on evaporation, 130.
+  General summary, 131.
+
+  CHAPTER VIII
+
+  SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING
+  VOLATILE                                                              133
+
+  B. HYDRATED SALT AND WATER, (1) _The compounds
+  formed do not have a definite melting point._ Concentration-temperature
+  diagram, 133. Sodium sulphate and water, 134.
+  Suspended transformation, 137. Dehydration by means of
+  anhydrous sodium sulphate, 138. Pressure-temperature diagram,
+  138. (2) _The compounds formed have a definite melting point._
+  Solubility curve of calcium chloride hexahydrate, 145.
+  Pressure-temperature diagram, 149. The indifferent point, 150.
+  The hydrates of ferric chloride, 151. Suspended transformation,
+  155. Evaporation of solutions at constant temperature, 155.
+  Inevaporable solutions, 157. Illustration, 158.
+
+  CHAPTER IX
+
+  EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS                            161
+
+  General, 161. Iodine and chlorine, 161. Concentration-temperature
+  diagram, 162. Pressure-temperature diagram, 165.
+  Bivariant systems, 167. Sulphur dioxide and water, 169.
+  Pressure-temperature diagram, 170. Bivariant systems, 173.
+
+  CHAPTER X
+
+  SOLID SOLUTIONS. MIXED CRYSTALS                                       175
+
+  General, 175. Solution of gases in solids, 176. Palladium
+  and hydrogen, 178. Solutions of solids in solids. Mixed
+  crystals, 180. Formation of mixed crystals of isomorphous
+  substances, 182. I. The two components can form an unbroken
+  series of mixed crystals. (_a_) _The freezing points of all mixtures
+  lie between the freezing points of the pure components._ Examples,
+  183. Melting-point curve, 183. (_b_) _The freezing-point curve passes
+  through a maximum._ Example, 186. (_c_) _The freezing-point
+  curve passes through a minimum._ Example, 188. Fractional
+  {xiv}
+  crystallization of mixed crystals, 188. II. The two components
+  do not form a continuous series of mixed crystals. (_a_) _The
+  freezing-point curve exhibits a transition point_, 190. Example,
+  190. (_b_) _The freezing-point curve exhibits a eutectic point_, 191.
+  Examples, 192. Changes in mixed crystals with the temperature,
+  192.
+
+  CHAPTER XI
+
+  EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES                                195
+
+  Temperature-concentration diagram, 196. Transformation
+  of the unstable into the stable form, 201. Examples, 203.
+  _Benzaldoximes_, 203. _Acetaldehyde and paraldehyde_, 204.
+
+  CHAPTER XII
+
+  SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF
+  TWO COMPONENTS                                                        207
+
+  Summary of the different systems of two components, 208.
+  (1) _Organic compounds_, 212. (2) _Optically active substances_,
+  213. Examples, 216. Transformations, 217. (3) _Alloys_, 220.
+  Iron--carbon alloys, 223. Determination of the composition of
+  compounds without analysis, 228. Formation of minerals, 232.
+
+  CHAPTER XIII
+
+  SYSTEMS OF THREE COMPONENTS                                           234
+
+  General, 234. Graphic representation, 235.
+
+  CHAPTER XIV
+
+  SOLUTIONS OF LIQUIDS IN LIQUIDS                                       240
+
+  1. _The three components form only one pair of partially
+  miscible liquids_, 240. Retrograde solubility, 245. The influence
+  of temperature, 247. 2. _The three components can form two
+  pairs of partially miscible liquids_, 249. 3. _The three components
+  form three pairs of partially miscible liquids_, 251.
+
+  CHAPTER XV
+
+  PRESENCE OF SOLID PHASES                                              253
+
+  A. The ternary eutectic point, 253. Formation of compounds,
+  255. B. Equilibria at higher temperatures. Formation
+  of double salts, 258. Transition point, 258. Vapour pressure.
+  {xv}
+  Quintuple point, 261. Solubility curves at the transition point,
+  264. Decomposition of the double salt by water, 267. Transition
+  interval, 270. Summary, 271.
+
+  CHAPTER XVI
+
+  ISOTHERMAL CURVES AND THE SPACE MODEL                                 272
+
+  Non-formation of double salts, 272. Formation of double
+  salt, 273. Transition interval, 277. Isothermal evaporation,
+  278. Crystallization of double salt from solutions containing
+  excess of one component, 280. Formation of mixed crystals,
+  281. Application to the characterization of racemates, 282.
+  _Representation in space._ Space model for carnallite, 284.
+  Summary and numerical data, 287. Ferric chloride--hydrogen
+  chloride--water, 290. Ternary systems, 291. The isothermal
+  curves, 294. Basic Salts, 296. Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O, 298.
+  Basic mercury salts, 301. Indirect determination of the composition
+  of the solid phase, 302.
+
+  CHAPTER XVII
+
+  ABSENCE OF LIQUID PHASE                                               305
+
+  Iron, carbon monoxide, carbon dioxide, 305.
+
+  CHAPTER XVIII
+
+  SYSTEMS OF FOUR COMPONENTS                                            312
+
+  Reciprocal salt-pairs. Choice of components, 313. Transition
+  point, 314. Formation of double salts, 315. Transition
+  interval, 315. Graphic representation, 316. Example, 317.
+  Ammonia-soda process, 320. Preparation of barium nitrite, 327.
+  Barium carbonate and potassium sulphate, 328.
+
+  APPENDIX
+
+  EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT                    331
+
+  I. The dilatometric method, 331. II. Measurement of
+  the vapour pressure, 334. III. Solubility measurements, 335.
+  IV. Thermometric method, 337. V. Optical method, 338.
+  VI. Electrical methods, 338.
+
+  NAME INDEX                                                            341
+
+  SUBJECT INDEX                                                         345
+
+       *       *       *       *       *
+
+
+{1}
+
+THE PHASE RULE
+
+CHAPTER I
+
+INTRODUCTION
+
+General.--Before proceeding to the more systematic treatment of the Phase
+Rule, it may, perhaps, be not amiss to give first a brief forecast of the
+nature of the subject we are about to study, in order that we may gain some
+idea of what the Phase Rule is, of the kind of problem which it enables us
+to solve, and of the scope of its application.
+
+It has long been known that if water is placed in a closed, exhausted
+space, vapour is given off and a certain pressure is created in the
+enclosing vessel. Thus, when water is placed in the Torricellian vacuum of
+the barometer, the mercury is depressed, and the amount of depression
+increases as the temperature is raised. But, although the pressure of the
+vapour increases as the temperature rises, its value at any given
+temperature is constant, no matter whether the amount of water present or
+the volume of the vapour is great or small; if the pressure on the vapour
+is altered while the temperature is maintained constant, either the water
+or the vapour will ultimately disappear; the former by evaporation, the
+latter by condensation. At any given temperature within certain limits,
+therefore, water and vapour can exist permanently in contact with one
+another--or, as it is said, be in equilibrium with one another--only when
+the pressure has a certain definite value. The same law of constancy of
+vapour pressure at a given {2} temperature, quite irrespective of the
+volumes of liquid and vapour,[1] holds good also in the case of alcohol,
+ether, benzene, and other pure liquids. It is, therefore, not unnatural to
+ask the question, Does it hold good for all liquids? Is it valid, for
+example, in the case of solutions?
+
+We can find the answer to these questions by studying the behaviour of a
+solution--say, a solution of common salt in water--when placed in the
+Torricellian vacuum. In this case, also, it is observed that the pressure
+of the vapour increases as the temperature is raised, but the pressure is
+no longer independent of the volume; as the volume increases, the pressure
+slowly diminishes. If, however, solid salt is present in contact with the
+solution, then the pressure again becomes constant at constant temperature,
+even when the volume of the vapour is altered. As we see, therefore,
+solutions do not behave in the same way as pure liquids.
+
+Moreover, on lowering the temperature of water, a point is reached at which
+ice begins to separate out; and if heat be now added to the system or
+withdrawn from it, no change will take place in the temperature or vapour
+pressure of the latter until either the ice or the water has
+disappeared.[2] Ice, water, and vapour, therefore, can be in equilibrium
+with one another only at one definite temperature and one definite
+pressure.
+
+In the case of a solution of common salt, however, we may have ice in
+contact with the solution at different temperatures and pressures. Further,
+it is possible to have a solution in equilibrium not only with anhydrous
+salt (NaCl), but also with the hydrated salt (NaCl, 2H_{2}O), as well as
+with ice, and the question, therefore, arises: Is it possible to state in a
+general manner the conditions under which such different systems can exist
+in equilibrium; or to obtain some insight {3} into the relations which
+exist between pure liquids and solutions? As we shall learn, the Phase Rule
+enables us to give an answer to this question.
+
+The preceding examples belong to the class of so-called "physical"
+equilibria, or equilibria depending on changes in the physical state. More
+than a hundred years ago, however, it was shown by Wenzel and Berthollet
+that "chemical" equilibria can also exist; that chemical reactions do not
+always take place completely in one direction as indicated by the usual
+chemical equation, but that before the reacting substances are all used up
+the reaction ceases, and there is a condition of equilibrium between the
+reacting substances and the products of reaction. As an example of this,
+there may be taken the process of lime-burning, which depends on the fact
+that when calcium carbonate is heated, carbon dioxide is given off and
+quicklime is produced. If the carbonate is heated in a closed vessel it
+will be found, however, not to undergo entire decomposition. When the
+pressure of the carbon dioxide reaches a certain value (which is found to
+depend on the temperature), decomposition ceases, and calcium carbonate
+exists side by side with calcium oxide and carbon dioxide. Moreover, at any
+given temperature the pressure is constant and independent of the amount of
+carbonate or oxide present, or of the volume of the gas; _nor does the
+addition of either of the products of dissociation, carbon dioxide or
+calcium oxide, cause any change in the equilibrium_. Here, then, we see
+that, although there are three different substances present, and although
+the equilibrium is no longer due to physical, but to chemical change, it
+nevertheless obeys the same law as the vapour pressure of a pure volatile
+liquid, such as water.
+
+It might be supposed, now, that this behaviour would be shown by other
+dissociating substances, _e.g._ ammonium chloride. When this substance is
+heated it dissociates into ammonia and hydrogen chloride, and at any given
+temperature the pressure of these gases is constant,[3] and is independent
+of the amounts of solid and gas present. So far, therefore, ammonium
+chloride behaves like calcium carbonate. If, however, one of the {4}
+products of dissociation be added to the system, it is found that the
+pressure is no longer constant at a given temperature, but varies with the
+amount of gas, ammonia or hydrogen chloride, which is added. In the case of
+certain dissociating substances, therefore, addition of one of the products
+of dissociation alters the equilibrium, while in other cases it does not.
+With the help of the Phase Rule, however, a general interpretation of this
+difference of behaviour can be given--an interpretation which can be
+applied not only to the two cases cited, but to all cases of dissociation.
+
+Again, it is well known that sulphur exists in two different crystalline
+forms, octahedral and prismatic, each of which melts at a different
+temperature. The problem here is, therefore, more complicated than in the
+case of ice, for there is now a possibility not only of one solid form, but
+of two different forms of the same substance existing in contact with
+liquid. What are the conditions under which these two forms can exist in
+contact with liquid, either singly or together, and under what conditions
+can the two solid forms exist together without the presence of liquid
+sulphur? To these questions an answer can also be given with the help of
+the Phase Rule.
+
+These cases are, however, comparatively simple; but when we come, for
+instance, to study the conditions under which solutions are formed, and
+especially when we inquire into the solubility relations of salts capable
+of forming, perhaps, a series of crystalline hydrates; and when we seek to
+determine the conditions under which these different forms can exist in
+contact with the solution, the problem becomes more complicated, and the
+necessity of some general guide to the elucidation of the behaviour of
+these different systems becomes more urgent.
+
+It is, now, to the study of such physical and chemical equilibria as those
+above-mentioned that the Phase Rule finds application; to the study, also,
+of the conditions regulating, for example, the formation of alloys from
+mixtures of the fused metals, or of the various salts of the Stassfurt
+deposits; the behaviour of iron and carbon in the formation of steel and
+the {5} separation of different minerals from a fused rock-mass.[4] With
+the help of the Phase Rule we can group together into classes the large
+number of different isolated cases of systems in equilibrium; with its aid
+we are able to state, in a general manner at least, the conditions under
+which a system can be in equilibrium, and by its means we can gain some
+insight into the relations existing between different kinds of systems.
+
+Homogeneous and Heterogeneous Equilibrium.--Before passing to the
+consideration of this generalization, it will be well to first make mention
+of certain restrictions which must be placed on its treatment, and also of
+the limitations to which it is subject. If a system is uniform throughout
+its whole extent, and possesses in every part identical physical properties
+and chemical composition, it is called _homogeneous_. Such is, for example,
+a solution of sodium chloride in water. An equilibrium occurring in such a
+homogeneous system (such as the equilibrium occurring in the formation of
+an ester in alcoholic solution) is called _homogeneous equilibrium_. If,
+however, the system consists of parts which have different physical
+properties, perhaps also different chemical properties, and which are
+marked off and separated from one another by bounding surfaces, the system
+is said to be _heterogeneous_. Such a system is formed by ice, water, and
+vapour, in which the three portions, each in itself homogeneous, can be
+mechanically separated from one another. When equilibrium exists between
+different, physically distinct parts, it is known as _heterogeneous
+equilibrium_. It is, now, with heterogeneous equilibria, with the
+conditions under which a heterogeneous system can exist, that we shall deal
+here.
+
+Further, we shall not take into account changes of equilibrium due to the
+action of electrical, magnetic, or capillary forces, or of gravity; but
+shall discuss only those which are due to changes of pressure, temperature,
+and volume (or concentration).
+
+Real and Apparent Equilibrium.--In discussing equilibria, also, a
+distinction must be drawn between real and {6} apparent equilibria. In the
+former case there is a state of rest which undergoes continuous change with
+change of the conditions (_e.g._ change of temperature or of pressure), and
+for which the chief criterion is that _the same condition of equilibrium is
+reached from whichever side it is approached_. Thus in the case of a
+solution, if the temperature is maintained constant, the same concentration
+will be obtained, no matter whether we start with an unsaturated solution
+to which we add more solid, or with a supersaturated solution from which we
+allow solid to crystallize out; or, in the case of water in contact with
+vapour, the same vapour pressure will be obtained, no matter whether we
+heat the water up to the given temperature or cool it down from a higher
+temperature. In this case, water and vapour are in _real_ equilibrium. On
+the other hand, water in contact with hydrogen and oxygen at the ordinary
+temperature is a case only of _apparent_ equilibrium; on changing the
+pressure and temperature continuously within certain limits there is no
+continuous change observed in the relative amounts of the two gases. On
+heating beyond these limits there is a sudden and not a continuous change,
+and the system no longer regains its former condition on being cooled to
+the ordinary temperature. In all such cases the system may be regarded as
+undergoing change and as tending towards a state of true or real
+equilibrium, but with such slowness that no change is observed.
+
+Although the case of water in contact with hydrogen and oxygen is an
+extreme one, it must be borne in mind that the condition of true
+equilibrium may not be reached instantaneously or even with measurable
+velocity, and in all cases it is necessary to be on one's guard against
+mistaking apparent (or false) for real (or true) equilibrium. The
+importance of this will be fully illustrated in the sequel.
+
+       *       *       *       *       *
+
+
+{7}
+
+CHAPTER II
+
+THE PHASE RULE
+
+Although the fact that chemical reactions do not take place completely in
+one direction, but proceed only to a certain point and there make a halt,
+was known in the last quarter of the eighteenth century (Wenzel, 1777;
+Berthollet, 1799); and although the opening and subsequent decades of the
+following century brought many further examples of such equilibria to our
+knowledge, it was not until the last quarter of the nineteenth century that
+a theorem, general in its application and with foundations weakened by no
+hypothetical assumptions as to the nature or constitution of matter, was
+put forward by Willard Gibbs;[5] a generalization which serves at once as a
+golden rule by which the condition of equilibrium of a system can be
+tested, and as a guide to the similarities and dissimilarities existing in
+different systems.
+
+Before that time, certainly, attempts had been made to bring the different
+known cases of equilibria--chemical and physical--under general laws. From
+the very first, both Wenzel[6] and Berthollet[7] recognized the influence
+exercised by the _mass_ of the substances on the equilibrium of the system.
+It was reserved, however, for Guldberg and Waage, by their more general
+statement and mathematical treatment of the Law of Mass Action,[8] to
+inaugurate the period of quantitative study of equilibria. The law which
+these investigators enunciated {8} served satisfactorily to summarize the
+conditions of equilibrium in many cases both of homogeneous and, with the
+help of certain assumptions and additions, of heterogeneous equilibrium. By
+reason, however, of the fact that it was developed on the basis of the
+kinetic and molecular theories, and involved, therefore, certain
+hypothetical assumptions as to the nature and condition of the substances
+taking part in the equilibrium, the law of mass action failed, as it
+necessarily must, when applied to those systems in which neither the number
+of different molecular aggregates nor the degree of their molecular
+complexity was known.
+
+Ten years after the law of mass action was propounded by Guldberg and
+Waage, Willard Gibbs,[9] Professor of Physics in Yale University, showed
+how, in a perfectly general manner, free from all hypothetical assumptions
+as to the molecular condition of the participating substances, all cases of
+equilibrium could be surveyed and grouped into classes, and how
+similarities in the behaviour of apparently different kinds of systems, and
+differences in apparently similar systems, could be explained.
+
+As the basis of his theory of equilibria, Gibbs adopted the laws of
+thermodynamics,[10] a method of treatment which had first been employed by
+Horstmann.[11] In deducing the law of equilibrium, Gibbs regarded a system
+as possessing only three independently variable factors[12]--temperature,
+pressure, and the concentration of the components of the system--and he
+enunciated the general theorem now usually known as the _Phase Rule_, by
+which he defined the conditions of equilibrium as a relationship between
+the number of what are called the phases and the components of the system.
+
+Phases.--Before proceeding farther we shall first consider what exactly is
+meant by the terms _phase_ and _component_. We have already seen (p. 5)
+that a heterogeneous system is made {9} up of different portions, each in
+itself homogeneous, but marked off in space and separated from the other
+portions by bounding surfaces. These homogeneous, physically distinct and
+mechanically separable portions are called _phases_. Thus ice, water, and
+vapour, are three phases of the same chemical substance--water. A phase,
+however, whilst it must be physically and chemically homogeneous, need not
+necessarily be chemically simple. Thus, a gaseous mixture or a solution may
+form a phase; but a heterogeneous mixture of solid substances constitutes
+as many phases as there are substances present. Thus when calcium carbonate
+dissociates under the influence of heat, calcium oxide and carbon dioxide
+are formed. There are then _two_ solid phases present, viz. calcium
+carbonate and oxide, and one gas phase, carbon dioxide.
+
+The _number of phases_ which can exist side by side may vary greatly in
+different systems. In all cases, however, there can be but one gas or
+vapour phase on the account of the fact that all gases are miscible with
+one another in all proportions. In the case of liquid and solid phases the
+number is indefinite, since the above property does not apply to them. The
+number of phases which can be formed by any given substance or group of
+substances also differs greatly, and in general increases with the number
+of participating substances. Even in the case of a single substance,
+however, the number may be considerable; in the case of sulphur, for
+example, at least eight different solid phases are known (_v._ Chap. III.).
+
+It is of importance to bear in mind that equilibrium is _independent of the
+amounts_ of the phases present.[13] Thus it is a familiar fact that the
+pressure of a vapour in contact with a {10} liquid (_i.e._ the pressure of
+the saturated vapour) is unaffected by the amounts, whether relative or
+absolute, of the liquid and vapour; also the amount of a substance
+dissolved by a liquid is independent of the amount of solid in contact with
+the solution. It is true that deviations from this general law occur when
+the amount of liquid or the size of the solid particles is reduced beyond a
+certain point,[14] owing to the influence of surface energy; but we have
+already (p. 5) excluded such cases from consideration.
+
+Components.--Although the conception of phases is one which is readily
+understood, somewhat greater difficulty is experienced when we come to
+consider what is meant by the term _component_; for the components of a
+system are not synonymous with the chemical elements or compounds present,
+_i.e._ with the _constituents_ of the system, although both elements and
+compounds may be components. By the latter term there are meant only those
+constituents the concentration of which can undergo _independent_ variation
+in the different phases, and it is only with these that we are concerned
+here.[15]
+
+To understand the meaning of this term we shall consider briefly some cases
+with which the reader will be familiar, and at the outset it must be
+emphasized that the Phase Rule is concerned merely with those constituents
+which take part in the state of real equilibrium (p. 5); for it is only to
+the final state, not to the processes by which that state is reached, that
+the Phase Rule applies.
+
+Consider now the case of the system water--vapour or ice--water--vapour.
+The number of constituents taking part in the equilibrium here is only one,
+viz. the chemical substance, water. Hydrogen and oxygen, the constituents
+of water, are not to be regarded as components, because, in the first
+place, they are {11} not present in the system in a state of real
+equilibrium (p. 6); in the second place, they are combined in definite
+proportions to form water, and their amounts, therefore, cannot be varied
+independently. A variation in the amount of hydrogen necessitates a
+definite variation in the amount of oxygen.
+
+In the case, already referred to, in which hydrogen and oxygen are present
+along with water at the ordinary temperature, we are not dealing with a
+condition of true equilibrium. If, however, the temperature is raised to a
+certain point, a state of true equilibrium between hydrogen, oxygen, and
+water-vapour will be possible. In this case hydrogen and oxygen will be
+components, because now they do take part in the equilibrium; also, they
+need no longer be present in definite proportions, but excess of one or the
+other may be added. Of course, if the restriction be arbitrarily made that
+the free hydrogen and oxygen shall be present always and only in the
+proportions in which they are combined to form water, there will be, as
+before, only one component, water. From this, then, we see that a change in
+the conditions of the experiment (in the present case a rise of
+temperature) may necessitate a change in the number of the components.
+
+It is, however, only in the case of systems of more than one component that
+any difficulty will be found; for only in this case will a choice of
+components be possible. Take, for instance, the dissociation of calcium
+carbonate into calcium oxide and carbon dioxide. At each temperature, as we
+have seen, there is a definite state of equilibrium. When equilibrium has
+been established, there are three different substances present--calcium
+carbonate, calcium oxide, and carbon dioxide; and these are the
+constituents of the system between which equilibrium exists. Now, although
+these constituents take part in the equilibrium, they are not all to be
+regarded as components, for they are not mutually independent. On the
+contrary, the different phases are related to one another, and if two of
+these are taken, the composition of the third is defined by the equation
+
+  CaCO_{3} = CaO + CO_{2}
+
+{12} Now, in deciding the number of components in any given system, not
+only must the constituents chosen be capable of independent variation, but
+a further restriction is imposed, and we obtain the following rule: _As the
+components of a system there are to be chosen the_ smallest number _of
+independently variable constituents by means of which the composition of
+each phase participating in the state of equilibrium can be expressed in
+the form of a chemical equation._
+
+Applying this rule to the case under consideration, we see that of the
+three constituents present when the system is in a state of equilibrium,
+only two, as already stated, are independently variable. It will further be
+seen that in order to express the composition of each phase present, two of
+these constituents are necessary. The system is, therefore, one of _two
+components_, or a system of the second order.
+
+When, now, we proceed to the actual choice of components, it is evident
+that any two of the constituents can be selected. Thus, if we choose as
+components CaCO_{3} and CaO, the composition of each phase can be expressed
+by the following equations:--
+
+  CaCO_{3} = CaCO_{3} + 0CaO
+       CaO = CaO + 0CaCO_{3}
+    CO_{2} = CaCO_{3} - CaO
+
+As we see, then, both zero and negative quantities of the components have
+been introduced; and similar expressions would be obtained if CaCO_{3} and
+CO_{2} were chosen as components. The matter can, however, be simplified
+and the use of negative quantities avoided if CaO and CO_{2} are chosen;
+and it is, therefore, customary to select these as the components.
+
+While it is possible in the case of systems of the second order to choose
+the two components in such a way that the composition of each phase can be
+expressed by positive quantities of these, such a choice is not always
+possible when dealing with systems of a higher order (containing three or
+four components).
+
+From the example which has just been discussed, it might {13} appear as if
+the choice of the components was rather arbitrary. On examining the point,
+however, it will be seen that the arbitrariness affects only the _nature_,
+not the _number_, of the components; a choice could be made with respect to
+which, not to how many, constituents were to be regarded as components. As
+we shall see presently, however, it is only the number, not the nature of
+the components that is of importance.
+
+After the discussion of the conditions which the substances chosen as
+components must satisfy, another method may be given by which the number of
+components present in a system can be determined. Suppose a system
+consisting of several phases in equilibrium, and the composition of each
+phase determined by analysis. If each phase present, regarded as a whole,
+has the same composition, the system contains only one component, or is of
+the first order. If two phases must be mixed in suitable quantities in
+order that the composition of a third phase may be obtained, the system is
+one of two components or of the second order; and if three phases are
+necessary to give the composition of a fourth coexisting phase, the system
+is one of three components, or of the third order.[16]
+
+Although the examples to be considered in the sequel will afford sufficient
+illustration of the application of the rules given above, one case may
+perhaps be discussed to show the application of the method just given for
+determining the number of components.
+
+Consider the system consisting of Glauber's salt in equilibrium with
+solution and vapour. If these three phases are analyzed, the composition of
+the solid will be expressed by Na_{2}SO_{4}, 10H_{2}O; that of the solution
+by Na_{2}SO_{4} + _x_H_{2}O, while the vapour phase will be H_{2}O. The
+system evidently cannot be a one-component system, for the phases have not
+all the same composition. By varying the amounts of two phases, however
+(_e.g._ Na_{2}SO_{4}, 10H_{2}O and H_{2}O), the composition of the third
+phase--the solution--can be obtained. The system is, therefore, one of _two
+components_.
+
+But sodium sulphate can also exist in the anhydrous form and as the hydrate
+Na_{2}SO_{4}, 7H_{2}O. In these cases there may {14} be chosen as
+components Na_{2}SO_{4} and H_{2}O, and Na_{2}SO_{4}, 7H_{2}O and H_{2}O
+respectively. In both cases, therefore, there are two components. But the
+two systems (Na_{2}SO_{4}, 10H_{2}O--H_{2}O, and Na_{2}SO_{4},
+7H_{2}O--H_{2}O) can be regarded as special cases of the system
+Na_{2}SO_{4}--H_{2}O, and these two components will apply to all systems
+made up of sodium sulphate and water, no matter whether the solid phase is
+anhydrous salt or one of the hydrates. In all three cases, of course, the
+_number_ of components is the same; but by choosing Na_{2}SO_{4} and H_{2}O
+as components, the possible occurrence of negative quantities of components
+in expressing the composition of the phases is avoided; and, further, these
+components apply over a much larger range of experimental conditions.
+Again, therefore, we see that, although the number of the components of a
+system is definite, a certain amount of liberty is allowed in the choice of
+the substances; and we also see that the choice will be influenced by the
+conditions of experiment.
+
+Summing up, now, we may say--
+
+(1) The components are to be chosen from among the constituents which are
+present when the system is in a state of true equilibrium, and which take
+part in that equilibrium.
+
+(2) As components are to be chosen the _smallest number_ of such
+constituents necessary to express the composition of each phase
+participating in the equilibrium, zero and negative quantities of the
+components being permissible.
+
+(3) In any given system the _number_ of the components is definite, but may
+alter with alteration of the conditions of experiment. A certain freedom of
+choice, however, is allowed in the (qualitative, not quantitative)
+selection of the components, the choice being influenced by considerations
+of simplicity, suitability, or generality of application.[17]
+
+Degree of Freedom. Variability of a System.--It is well known that in
+dealing with a certain mass of gas or vapour, _e.g._ water vapour, if only
+one of the independently variable factors--temperature, pressure, and
+concentration (or volume)--is fixed, the state of the gas or vapour is
+undefined; while occupying the same volume (the concentration, therefore,
+remaining {15} unchanged), the temperature and the pressure may be altered;
+at a given temperature, a gas can exist under different pressures and
+occupy different volumes, and under any given pressure the temperature and
+volume may vary. If, however, two of the factors are arbitrarily fixed,
+then the third factor can only have a certain definite value; at any given
+values of temperature and pressure a given mass of gas can occupy only a
+definite volume.
+
+Suppose, however, that the system consists of water in contact with vapour.
+The condition of the system then becomes perfectly defined on arbitrarily
+giving one of the variables a certain value. If the temperature is fixed,
+the pressure under which water and water vapour can coexist is also
+determined; and conversely, if a definite pressure is chosen, the
+temperature is also defined. Water and vapour can coexist under a given
+pressure only at a definite temperature.
+
+Finally, let the water and vapour be cooled down until ice begins to
+separate out. So soon as the third phase, ice, appears, the state of the
+system as regards temperature and pressure of the vapour is perfectly
+defined, and none of the variables can be arbitrarily changed without
+causing the disappearance of one of the phases, ice, water, or vapour.
+
+We see, therefore, that in the case of some systems two, in other cases,
+only one of the independent variables (temperature, pressure,
+concentration) can be altered without destroying the nature of the system;
+while in other systems, again, these variables have all fixed and definite
+values. We shall therefore define the number of degrees of freedom[18] of a
+system as the _number of the variable factors, temperature, pressure, and
+concentration of the components, which must be arbitrarily fixed in order
+that the condition of the system may be perfectly defined_. From what has
+been said, therefore, we shall describe a gas or vapour as having two
+degrees of freedom; the system water--vapour as having only one; and the
+system ice--water--vapour as having no degrees of freedom. We may also
+speak of the {16} _variability_ or _variance_ of a system, and describe a
+system as being invariant, univariant, bivariant, multivariant,[19]
+according as the number of degrees of freedom is nought, one, two, or more
+than two.
+
+A knowledge of its variability is, therefore, of essential importance in
+studying the condition and behaviour of a system, and it is the great merit
+of the Phase Rule that _the state of a system is defined entirely by the
+relation existing between the number of the components and the phases
+present_, no account being taken of the molecular complexity of the
+participating substances, nor any assumption made with regard to the
+constitution of matter. It is, further, as we see, quite immaterial whether
+we are dealing with "physical" or "chemical" equilibrium; in principle,
+indeed, no distinction need be drawn between the two classes, although it
+is nevertheless often convenient to make use of the terms, in spite of a
+certain amount of indefiniteness which attaches to them--an indefiniteness,
+indeed, which attaches equally to the terms "physical" and "chemical"
+process.[20]
+
+The Phase Rule.--The Phase Rule of Gibbs, which defines the condition of
+equilibrium by the relation between the number of coexisting phases and the
+components, may be stated as follows: A system consisting of n components
+can exist in _n_ + 2 phases only when the temperature, pressure, and
+concentration have fixed and definite values; if there are _n_ components
+in _n_ + 1 phases, equilibrium can exist while one of the factors varies,
+and if there are only _n_ phases, two of the varying factors may be
+arbitrarily fixed. This rule, the application of which, it is hoped, will
+become clear in the sequel, may be very concisely and conveniently
+summarized in the form of the equation--
+
+  P + F = C + 2, or F = C + 2 - P
+
+where P denotes the number of the phases, F the degrees of freedom, and C
+the number of components. From the second form of the equation it can be
+readily seen that the greater the number of the phases, the fewer are the
+degrees of freedom. With increase in the number of the phases, therefore,
+the {17} condition of the system becomes more and more defined, or less and
+less variable.
+
+Classification of Systems according to the Phase Rule.--We have already
+learned in the introductory chapter that systems which are apparently quite
+different in character may behave in a very similar manner. Thus it was
+stated that the laws which govern the equilibrium between water and its
+vapour are quite analogous to those which are obeyed by the dissociation of
+calcium carbonate into carbon dioxide and calcium oxide; in each case a
+certain temperature is associated with a definite pressure, no matter what
+the relative or absolute amounts of the respective substances are. And
+other examples were given of systems which were apparently similar in
+character, but which nevertheless behaved in a different manner. The
+relations between the various systems, however, become perfectly clear and
+intelligible in the light of the Phase Rule. In the case first mentioned,
+that of water in equilibrium with its vapour, we have one
+component--water--present in two phases, _i.e._ in two physically distinct
+forms, viz. liquid and vapour. According to the Phase Rule, therefore,
+since C = 1, and P = 2, the degree of freedom F is equal to 1 + 2 - 2 = 1;
+the system possesses one degree of freedom, as has already been stated. But
+in the case of the second system mentioned above there are two components,
+viz. calcium oxide and carbon dioxide (p. 12), and three phases, viz. two
+solid phases, CaO and CaCO_{3}, and the gaseous phase, CO_{2}. The number
+of degrees of freedom of the system, therefore, is 2 + 2 - 3 = 1; this
+system, therefore, also possesses one degree of freedom. We can now
+understand why these two systems behave in a similar manner; both are
+univariant, or possess only one degree of freedom. We shall therefore
+expect a similar behaviour in the case of all univariant systems, no matter
+how dissimilar the systems may outwardly appear. Similarly, all bivariant
+systems will exhibit analogous behaviour; and generally, systems possessing
+the same degree of freedom will show a like behaviour. In accordance with
+the Phase Rule, therefore, we may classify the different systems which may
+be found into invariant, univariant, bivariant, multivariant, {18}
+according to the relation which obtains between the number of the
+components and the number of coexisting phases; and we shall expect that in
+each case the members of any particular group will exhibit a uniform
+behaviour. By this means we are enabled to obtain an insight into the
+general behaviour of any system, so soon as we have determined the number
+of the components and the number of the coexisting phases.
+
+The adoption of the Phase Rule for the purposes of classification has been
+of great importance in studying changes in the equilibrium existing between
+different substances; for not only does it render possible the grouping
+together of a large number of isolated phenomena, but the guidance it
+affords has led to the discovery of new substances, has given the clue to
+the conditions under which these substances can exist, and has led to the
+recognition of otherwise unobserved resemblances existing between different
+systems.
+
+Deduction of the Phase Rule.--In the preceding pages we have restricted
+ourselves to the statement of the Phase Rule, without giving any indication
+of how it has been deduced. At the close of this chapter, therefore, the
+mathematical deduction of the generalization will be given, but in brief
+outline only, the reader being referred to works on Thermodynamics for a
+fuller treatment of the subject.[21]
+
+All forms of energy can be resolved into two factors, the _capacity_ factor
+and the _intensity_ factor; but for the production of equilibrium, only the
+intensity factor is of importance. Thus, if two bodies having the same
+temperature are brought in contact with each other, they will be in
+equilibrium as regards heat energy, no matter what may be the amounts of
+heat (capacity factor) contained in either, because the intensity
+factor--the temperature--is the same. But if the temperature of the two
+bodies is different, _i.e._ if the intensity factor of heat energy is
+different, the two bodies will no longer be in equilibrium; but heat will
+pass from the hotter to the colder until both have the same temperature.
+
+As with heat energy, so with chemical energy. If we have a substance
+existing in two different states, or in two different {19} phases of a
+system, equilibrium can occur only when the intensity factor of chemical
+energy is the same. This intensity factor may be called the _chemical
+potential_; and we can therefore say that a system will be in equilibrium
+when the chemical potential of each component is the same in all the phases
+in which the component occurs. Thus, for example, ice, water, and vapour
+have, at the triple point, the same chemical potential.
+
+The potential of a component in any phase depends not only on the
+composition of the phase, but also on the temperature and the pressure (or
+volume). If, therefore, we have a system of C components existing in P
+phases, then, in order to fix the composition of unit mass of each phase,
+it is necessary to know the masses of (C - 1) components in each of the
+phases. As regards the composition, therefore, each phase possesses (C - 1)
+variables. Since there are P phases, it follows that, as regards
+composition, the whole system possesses P(C - 1) variables. Besides these
+there are, however, two other variables, viz. temperature and pressure, so
+that altogether a system of C components in P phases possesses P(C - 1) + 2
+variables.
+
+In order to define the state of the system completely, it will be necessary
+to have as many equations as there are variables. If, therefore, there are
+fewer equations than there are variables, then, according to the deficiency
+in the number of the equations, one or more of the variables will have an
+undefined value; and values must be assigned to these variables before the
+system is entirely defined. The number of these undefined values gives us
+the variability or the degree of freedom of the system.
+
+The equations by which the system is to be defined are obtained from the
+relationship between the potential of a component and the composition of
+the phase, the temperature and the pressure. Further, as has already been
+stated, equilibrium occurs when the potential of each component is the same
+in the different phases in which it is present. If, therefore, we choose as
+standard one of the phases in which all the components occur, then in any
+other phase in equilibrium with {20} it, the potential of each component
+must be the same as in the standard phase. For each phase in equilibrium
+with the standard phase, therefore, there will be a definite equation of
+state for each component in the phase; so that, if there are P phases, we
+obtain for each component (P - 1) equations; and for C components,
+therefore, we obtain C(P - 1) equations.
+
+But we have seen above that there are P(C - 1) + 2 variables, and as we
+have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2
+- P variables undefined. That is to say, the degree of freedom (F) of a
+system consisting of C components in P phases is--
+
+  F = C + 2 - P
+
+       *       *       *       *       *
+
+
+{21}
+
+CHAPTER III
+
+TYPICAL SYSTEMS OF ONE COMPONENT
+
+A. _Water._
+
+For the sake of rendering the Phase Rule more readily intelligible, and at
+the same time also for the purpose of obtaining examples by which we may
+illustrate the general behaviour of systems, we shall in this chapter
+examine in detail the behaviour of several well-known systems consisting of
+only one component.
+
+The most familiar examples of equilibria in a one-component system are
+those furnished by the three phases of water, viz. ice, water, water
+vapour. The system consists of one component, because all three phases have
+the same chemical composition, represented by the formula H_{2}O. As the
+criterion of equilibrium we shall choose a definite pressure, and shall
+study the variation of the pressure with the temperature; and for the
+purpose of representing the relationships which we obtain we shall employ a
+temperature-pressure diagram, in which the temperatures are measured as
+abscissae and the pressures as ordinates. In such a diagram invariant
+systems will be represented by points; univariant systems by lines, and
+bivariant systems by areas.
+
+Equilibrium between Liquid and Vapour. Vaporization Curve.--Consider in the
+first place the conditions for the coexistence of liquid and vapour.
+According to the Phase Rule (p. 16), a system consisting of one component
+in two phases has one degree of freedom, or is univariant. We should
+therefore expect that it will be possible for liquid water to coexist with
+water vapour at different values of temperature and {22} pressure, but that
+if we arbitrarily fix one of the variable factors, pressure, temperature,
+or volume (in the case of a given mass of substance), the state of the
+system will then be defined. If we fix, say, the temperature, then the
+pressure will have a definite value; or if we adopt a certain pressure, the
+liquid and vapour can coexist only at a certain definite temperature. Each
+temperature, therefore, will correspond to a definite pressure; and if in
+our diagram we join by a continuous line all the points indicating the
+values of the pressure corresponding to the different temperatures, we
+shall obtain a curve (Fig. 1) representing the variation of the pressure
+with the temperature. This is the curve of vapour pressure, or the
+_vaporization curve_ of water.
+
+[Illustration: FIG. 1.]
+
+Now, the results of experiment are quite in agreement with the requirements
+of the Phase Rule, and at any given temperature the system water--vapour
+can exist in equilibrium only under a definite pressure.
+
+The vapour pressure of water at different temperatures has been subjected
+to careful measurement by Magnus,[22] Regnault,[23] Ramsay and Young,[24]
+Juhlin,[25] Thiesen and Scheel,[26] and others. In the following table the
+values of the vapour pressure from -10deg to +100deg are those calculated
+from the measurements of Regnault, corrected by the measurements of Wiebe
+and Thiesen and Scheel;[27] those from 120deg to 270deg were determined
+{23} by Ramsay and Young, while the values of the critical pressure and
+temperature are those determined by Battelli.[28]
+
+  VAPOUR PRESSURE OF WATER.
+
+  -------------+-----------------+--------------+--------------------
+               |                 |              |
+  Temperature. | Pressure in cm. | Temperature. |   Pressure in cm.
+               |    mercury.     |              |      mercury.
+  -------------+-----------------+--------------+--------------------
+               |                 |              |
+     -10deg    |     0.213       |    120deg    |   148.4
+       0deg    |     0.458[29]   |    130deg    |   201.9
+     +20deg    |     1.752       |    150deg    |   356.8
+      40deg    |     5.516       |    200deg    |  1162.5
+      60deg    |    14.932       |    250deg    |  2973.4
+      80deg    |    35.54        |    270deg    |  4110.1
+     100deg    |    76.00        |    364.3deg  | 14790.4 (194.6 atm.)
+               |                 | (critical    | (critical pressure).
+               |                 | temperature) |
+  -------------+-----------------+--------------+--------------------
+
+The pressure is, of course, independent of the relative or absolute volumes
+of the liquid and vapour; on increasing the volume at constant temperature,
+a certain amount of the liquid will pass into vapour, and the pressure will
+regain its former value. If, however, the pressure be permanently
+maintained at a value different from that corresponding to the temperature
+employed, then either all the liquid will pass into vapour, or all the
+vapour will pass into liquid, and we shall have either vapour alone or
+liquid alone.
+
+Upper Limit of Vaporization Curve.--On continuing to add heat to water
+contained in a closed vessel, the pressure of the vapour will gradually
+increase. Since with increase of pressure the density of the vapour must
+increase, and since with rise of temperature the density of the liquid must
+decrease, a point will be reached at which the density of liquid and vapour
+become identical; the system ceases to be heterogeneous, and passes into
+one homogeneous phase. The temperature at which this occurs is called the
+_critical temperature_. To this temperature there will, of course,
+correspond a certain definite pressure, called the _critical pressure_. The
+curve representing the {24} equilibrium between liquid and vapour must,
+therefore, end abruptly at the critical point. At temperatures above this
+point no pressure, however great, can cause the formation of the liquid
+phase; at temperatures above the critical point the vapour becomes a gas.
+In the case of water, the critical temperature is 364.3deg, and the
+critical pressure 194.6 atm.; at the point representing these conditions
+the vapour-pressure curve of water must cease.
+
+Sublimation Curve of Ice.--Vapour is given off not only by liquid water,
+but also by solid water, or ice. That this is so is familiar to every one
+through the fact that ice or snow, even at temperatures below the melting
+point, gradually disappears in the form of vapour. Even at temperatures
+considerably lower than 0deg, the vapour pressure of ice, although small,
+is quite appreciable; and it is possible, therefore, to have ice and vapour
+coexisting in equilibrium. When we inquire into the conditions under which
+such a system can exist, we see again that we are dealing with a univariant
+system--one component existing in two phases--and that, therefore, just as
+in the case of the system water and vapour, there will be for each
+temperature a certain definite pressure of the vapour, and this pressure
+will be independent of the relative or absolute amounts of the solid or
+vapour present, and will depend solely on the temperature. Further, just as
+in the case of the vapour pressure of water, the condition of equilibrium
+between ice and water vapour will be represented by a line or curve showing
+the change of pressure with the temperature. Such a curve, representing the
+conditions of equilibrium between a solid and its vapour, is called a
+_sublimation curve_. At temperatures represented by any point on this
+curve, the solid (ice) will sublime or pass into vapour without previously
+fusing. Since ice melts at 0deg (_vide infra_), the sublimation curve must
+end at that temperature.
+
+The following are the values of the vapour pressure of ice between 0deg and
+-50deg.[30]
+
+{25}
+
+  VAPOUR PRESSURE OF ICE.
+
+  ---------------------------------------------------------------
+  Temperature. | Pressure in mm. | Temperature. | Pressure in mm.
+               |     mercury.    |              |     mercury.
+  -------------+-----------------+--------------+----------------
+    -50deg     |     0.050       |   -8deg      |     2.379
+    -40deg     |     0.121       |   -6deg      |     2.821
+    -30deg     |     0.312       |   -4deg      |     3.334
+    -20deg     |     0.806       |   -2deg      |     3.925
+    -15deg     |     1.279       |    0deg      |     4.602
+    -10deg     |     1.999       |              |
+  ----------------------------------------------------------------
+
+Equilibrium between Ice and Water. Curve of Fusion.--There is still another
+univariant system of the one component water, the existence of which, at
+definite values of temperature and pressure, the Phase Rule allows us to
+predict. This is the system solid--liquid. Ice on being heated to a certain
+temperature melts and passes into the liquid state; and since this system
+solid--liquid is univariant, there will be for each temperature a certain
+definite pressure at which ice and water can coexist or be in equilibrium,
+independently of the amounts of the two phases present. Since now the
+temperature at which the solid phase is in equilibrium with the liquid
+phase is known as the melting point or point of fusion of the solid, the
+curve representing the temperatures and pressures at which the solid and
+liquid are in equilibrium will represent the change of the melting point
+with the pressure. Such a curve is called the _curve of fusion_, or the
+melting-point curve.
+
+It was not until the middle of the nineteenth century that this connection
+between the pressure and the melting point, or the change of the melting
+point with the pressure, was observed. The first to recognize the existence
+of such a relationship was James Thomson,[31] who in 1849 showed that from
+theoretical considerations such a relationship must exist, and predicted
+that in the case of ice the melting point would be lowered by pressure.
+This prediction was fully confirmed by his brother, W. Thomson[32] (Lord
+Kelvin), who found that under a pressure {26} of 8.1 atm. the melting point
+of ice was -0.059deg; under a pressure of 16.8 atm. the melting point was
+-0.129deg.
+
+The experiments which were first made in this connection were more of a
+qualitative nature, but in recent years careful measurements of the
+influence of pressure on the melting point of ice have been made more
+especially by Tammann,[33] and the results obtained by him are given in the
+following table and represented graphically in Fig. 2.
+
+  FUSION PRESSURE OF ICE.
+
+  ---------------------------------------------------------------------
+                  | Pressure in kilogms. per | Change of melting point for
+  Temperature.    |  sq. cm.[34]             | an increase of pressure of
+                  |                          | 1 kilogm. per sq. cm.
+  ---------------------------------------------------------------------
+     -0deg        |        1                 |
+     -2.5deg      |      336                 |       0.0074deg
+     -5deg        |      615                 |       0.0090deg
+     -7.5deg      |      890                 |       0.0091deg
+     -10.0deg     |     1155                 |       0.0094deg
+     -12.5deg     |     1410                 |       0.0100deg
+     -15.0deg     |     1625                 |       0.0116deg
+     -17.5deg     |     1835                 |       0.0119deg
+     -20.0deg     |     2042                 |       0.0121deg
+     -22.1deg     |     2200                 |       0.0133deg
+  ---------------------------------------------------------------------
+
+From the numbers in the table and from the figure we see that as the
+pressure is increased the melting point of ice is lowered; but we also
+observe that a very large change of pressure is required in order to
+produce a very small change in the melting point. The curve, therefore, is
+very steep. Increase of pressure by one atmosphere lowers the melting point
+by only 0.0076deg,[35] or an increase of pressure of 135 atm. is required
+to produce a lowering of the melting point of 1deg. We see further that the
+fusion curve bends slightly as the pressure is increased, which signifies
+that the variation of {27} the melting point with the pressure changes; at
+-15deg, when the pressure is 1625 kilogm. per sq. cm., increase of pressure
+by 1 kilogm. per sq. cm. lowers the melting point by 0.012deg. This
+curvature of the fusion curve we shall later (Chap. IV.) see to be an
+almost universal phenomenon.
+
+[Illustration: FIG. 2.]
+
+[Illustration: FIG. 3.]
+
+Equilibrium between Ice, Water, and Vapour. The Triple Point.--On examining
+the vapour-pressure curves of ice and water (Fig. 3), we see that at a
+temperature of about 0deg and under a pressure of about 4.6 mm. mercury,
+the two curves cut. At this point liquid water and solid ice are each in
+equilibrium with vapour at the same pressure. Since this is so, they must,
+of course, be in equilibrium {28} with one another, as experiment also
+shows. At this point, therefore, ice, water, and vapour can be in
+equilibrium, and as there are three phases present, the point is called a
+_triple point_.[36]
+
+The triple point, however, does not lie exactly at 0deg C., for this
+temperature is defined as the melting point of ice under atmospheric
+pressure. At the triple point, however, the pressure is equal to the vapour
+pressure of ice and water, and this pressure, as we see from the tables on
+pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less than in the
+previous case. Now, we have just seen that a change of pressure of 1 atm.
+corresponds to a change of the melting point of 0.0076deg; the melting
+point of ice, therefore, when under the pressure of its own vapour, will be
+very nearly +0.0076deg, and the pressure of the vapour will be very
+slightly greater than 4.579 mm., which is the pressure at 0deg (p. 21). The
+difference is, however, slight, and may be neglected here. At the
+temperature, then, of +0.0076deg, and under a pressure of 4.6 mm. of
+mercury, ice, water, and vapour will be in equilibrium; the point in our
+diagram representing this particular temperature and pressure is,
+therefore, the triple point of the system ice--water--vapour.
+
+Since at the triple point we have three phases of one component, the system
+at this point is invariant--it possesses no degrees of freedom. If the
+temperature is changed, the system will undergo alteration in such a way
+that one of the phases will disappear, and a univariant system will result;
+if heat be added, ice will melt, and we shall have left water and vapour;
+if heat be abstracted, water will freeze, and we shall have left ice and
+vapour; if, when the temperature is altered, the pressure is kept constant,
+then we shall ultimately obtain only one phase (see Chap. IV.).
+
+The triple point is not only the point of intersection of the vaporization
+and sublimation curves, but it is also the end-point of the fusion curve.
+The fusion curve, as we have seen, is the curve of equilibrium between ice
+and water; and since at the triple point ice and water are each in
+equilibrium with {29} vapour of the same pressure, they must, of course,
+also be in equilibrium with one another.
+
+[Illustration: FIG. 4.]
+
+Bivariant Systems of Water.--If we examine Fig. 4, we see that the curves
+OA, OB, OC, which represent diagrammatically the conditions under which
+water and vapour, ice and vapour, and water and ice are in equilibrium,
+form the boundaries of three "fields," or areas, I., II., III. These areas,
+now, represent the conditions for the existence of the single phases,
+solid, liquid, and vapour respectively. At temperatures and pressures
+represented by any point in the field I., solid only can exist as a stable
+phase. Since we have here one component in only one phase, the system is
+bivariant, and at any given temperature, therefore, ice can exist under a
+series of pressures; and under any given pressure, at a series of
+temperatures, these pressures and temperatures being limited only by the
+curves OB, OC. Similarly also with the areas II. and III.
+
+We see, further, that the different areas are the regions of stability of
+the phase common to the two curves by which the area is enclosed.[37] Thus,
+the phase common to the two systems {30} represented by BO (ice and
+vapour), and OA (water and vapour) is the vapour phase; and the area BOA is
+therefore the area of the vapour phase. Similarly, BOC is the area of the
+ice phase, and COA the area of the water phase.
+
+Supercooled Water. Metastable State.--When heated under the ordinary
+atmospheric pressure, ice melts when the temperature reaches 0deg, and it
+has so far not been found possible to raise the temperature of ice above
+this point without liquefaction taking place. On the other hand, it has
+long been known that water can be cooled below zero without solidification
+occurring. This was first discovered in 1724 by Fahrenheit,[38] who found
+that water could be exposed to a temperature of -9.4deg without
+solidifying; so soon, however, as a small particle of ice was brought in
+contact with the water, crystallization commenced. Superfused or
+supercooled water--_i.e._ water cooled below 0deg--is unstable only in
+respect of the solid phase; so long as the presence of the solid phase is
+carefully avoided, the water can be kept for any length of time without
+solidifying, and the system supercooled water and vapour behaves in every
+way like a stable system. A system, now, which in itself is stable, and
+which becomes instable only in contact with a particular phase, is said to
+be _metastable_, and the region throughout which this condition exists is
+called the metastable region. Supercooled water, therefore, is in a
+metastable condition. If the supercooling be carried below a certain
+temperature, solidification takes place spontaneously without the addition
+of the solid phase; the system then ceases to be metastable, and becomes
+_instable_.
+
+Not only has water been cooled to temperatures considerably below the
+melting point of ice, but the vapour pressure of the supercooled water has
+been measured. It is of interest and importance, now, to see what
+relationship exists between the vapour pressure of ice and that of
+supercooled water at the same temperature. This relationship is clearly
+shown by the numbers in the following table,[39] and is represented in Fig.
+3, {31} p. 27., and diagrammatically in Fig. 4, the vapour pressures of
+supercooled water being represented by the curve OA', which is the unbroken
+continuation of AO.
+
+  VAPOUR PRESSURE OF ICE AND OF SUPERCOOLED WATER.
+
+  ---------------------------------------------------------------------
+                 |       Pressure in mm. mercury.
+                 ------------------------------------------------------
+  Temperature.   |               |               |
+                 |     Water.    |     Ice.      |    Difference.
+  ---------------------------------------------------------------------
+        0deg     |     4.618     |     4.602     |    0.016[40]
+       -2deg     |     3.995     |     3.925     |    0.070
+       -4deg     |     3.450     |     3.334     |    0.116
+       -8deg     |     2.558     |     2.379     |    0.179
+      -10deg     |     2.197     |     1.999     |    0.198
+      -15deg     |     1.492     |     1.279     |    0.213
+      -20deg     |     1.005     |     0.806     |    0.199
+  ---------------------------------------------------------------------
+
+At all temperatures below 0deg (more correctly +0.0076deg), at which
+temperature water and ice have the same vapour pressure, the vapour
+pressure of supercooled water is _greater_ than that of ice at the same
+temperature.
+
+From the relative positions of the curves OB and OA (Fig. 4) we see that at
+all temperatures above 0deg, the (metastable) sublimation curve of ice, if
+it could be obtained, would be higher than the vaporization curve of water.
+This shows, therefore, that at 0deg a "break" must occur in the curve of
+states, and that in the neighbourhood of this break the curve above that
+point must ascend less rapidly than the curve below the break. Since,
+however, the differences in the vapour pressures of supercooled water and
+of ice are very small, the change in the direction of the vapour-pressure
+curve on passing from ice to water was at first not observed, and Regnault
+regarded the sublimation curve as passing continuously into {32} the
+vaporization curve. The existence of a break was, however, shown by James
+Thomson[41] and by Kirchhoff[42] to be demanded by thermo-dynamical
+considerations, and the prediction of theory was afterwards realized
+experimentally by Ramsay and Young in their determinations of the vapour
+pressure of water and ice, as well as in the case of other substances.[43]
+
+From what has just been said, we can readily understand why ice and water
+cannot exist in equilibrium below 0deg. For, suppose we have ice and water
+in the same closed space, but not in contact with one another, then since
+the vapour pressure of the supercooled water is higher than that of ice,
+the vapour of the former must be supersaturated in contact with the latter;
+vapour must, therefore, condense on the ice; and in this way there will be
+a slow distillation from the water to the ice, until at last all the water
+will have disappeared, and only ice and vapour remain.[44]
+
+Other Systems of the Substance Water.--We have thus far discussed only
+those systems which are constituted by the three phases--ice, water, and
+water vapour. It has, however, been recently found that at a low
+temperature and under a high pressure ordinary ice can pass into two other
+crystalline varieties, called by Tammann[45] ice II. and ice III., ordinary
+ice being ice I. According to the Phase Rule, now, since each of these
+solid forms constitutes a separate phase (p. 9), it will be possible to
+have the following (and more) systems of water, in addition to those
+already studied, viz. water, ice I., ice II.; water, ice I., ice III.;
+water, ice II., ice III., forming invariant systems and existing in
+equilibrium only at a definite triple point; further, water, ice II.;
+water, ice III.; ice I., ice II.; ice I., ice III.; ice II., ice III.,
+forming univariant systems, existing, therefore, at definite corresponding
+values of {33} temperature and pressure; and lastly, the bivariant systems,
+ice II. and ice III. Several of these systems have been investigated by
+Tammann. The triple point for water, ice I., ice III., lies at -22deg, and
+a pressure of 2200 kilogms. per sq. cm. (2130 atm.), as indicated in Fig.
+2, p. 27.[46] In contrast with the behaviour of ordinary ice, the
+temperature of equilibrium in the case of water--ice II., and water--ice
+III., is _raised_ by increase of pressure.
+
+B. _Sulphur._
+
+Polymorphism.--Reference has just been made to the fact that ice can exist
+not only in the ordinary form, but in at least two other crystalline
+varieties. This phenomenon, the existence of a substance in two or more
+different crystalline forms, is called _polymorphism_. Polymorphism was
+first observed by Mitscherlich[47] in the case of sodium phosphate, and
+later in the case of sulphur. To these two cases others were soon added, at
+first of inorganic, and later of organic substances, so that polymorphism
+is now recognized as of very frequent occurrence indeed.[48] These various
+forms of a substance differ not only in crystalline shape, but also in
+melting point, specific gravity, and other physical properties. In the
+liquid state, however, the differences do not exist.
+
+According to our definition of phases (p. 9), each of these polymorphic
+forms constitutes a separate phase of the particular substance. As is
+readily apparent, the number of possible systems formed of one component
+may be considerably increased when that component is capable of existing in
+different crystalline forms. We have, therefore, to inquire what are the
+conditions under which different polymorphic forms can coexist, either
+alone or in presence of the liquid and vapour phase. For the purpose of
+illustrating the general behaviour of such systems, we shall study the
+systems formed by the different crystalline forms of sulphur, tin, and
+benzophenone.
+
+{34}
+
+Sulphur exists in two well-known crystalline forms--rhombic, or octahedral,
+and monoclinic, or prismatic sulphur. Of these, the former melts at
+114.5deg; the latter at 120deg.[49] Further, at the ordinary temperature,
+rhombic sulphur can exist unchanged, whereas, on being heated to
+temperatures somewhat below the melting point, it passes into the prismatic
+variety. On the other hand, at temperatures above 96deg, prismatic sulphur
+can remain unchanged, whereas at the ordinary temperature it passes slowly
+into the rhombic form.
+
+If, now, we examine the case of sulphur with the help of the Phase Rule, we
+see that the following systems are theoretically possible:--
+
+  I. _Bivariant Systems: One component in one phase._
+    (_a_) Rhombic sulphur.
+    (_b_) Monoclinic sulphur.
+    (_c_) Sulphur vapour.
+    (_d_) Liquid sulphur.
+
+  II. _Univariant Systems: One component in two phases._
+    (_a_) Rhombic sulphur and vapour.
+    (_b_) Monoclinic sulphur and vapour.
+    (_c_) Rhombic sulphur and liquid.
+    (_d_) Monoclinic sulphur and liquid.
+    (_e_) Rhombic and monoclinic sulphur.
+    (_f_) Liquid and vapour.
+
+  III. _Invariant Systems: One component in three phases._
+    (_a_) Rhombic and monoclinic sulphur and vapour.
+    (_b_) Rhombic sulphur, liquid and vapour.
+    (_c_) Monoclinic sulphur, liquid and vapour.
+    (_d_) Rhombic and monoclinic sulphur and liquid.
+
+[Illustration: FIG. 5.]
+
+Triple Point--Rhombic and Monoclinic Sulphur and Vapour. Transition
+Point.--In the case of ice, water and vapour, we saw that at the triple
+point the vapour pressures of ice and water are equal; below this point,
+ice is stable; above this point, water is stable. We saw, further, that
+below 0deg the vapour pressure of the stable system is lower than that of
+the metastable, and therefore that at the triple point there is a break in
+the vapour pressure curve of such a kind that above {35} the triple point
+the vapour-pressure curve ascends more slowly than below it. Now, although
+the vapour pressure of solid sulphur has not been determined, we can
+nevertheless consider that it does possess a certain, even if very small,
+vapour pressure,[50] and that at the temperature at which the vapour
+pressures of rhombic and monoclinic sulphur become equal, we can have these
+two solid forms existing in equilibrium with the vapour. Below that point
+only one form, that with the lower vapour pressure, will be stable; above
+that point only the other form will be stable. On passing through the
+triple point, therefore, there will be a change of the one form into the
+other. This point is represented in our diagram (Fig. 5) by the point O,
+the two curves AO and OB representing diagrammatically the vapour pressures
+of rhombic and monoclinic sulphur respectively. If the vapour phase is
+absent and the system maintained under a constant pressure, _e.g._ {36}
+atmospheric pressure, there will also be a definite temperature at which
+the two solid forms are in equilibrium, and on passing through which
+complete and reversible transformation of one form into the other occurs.
+This temperature, which refers to equilibrium in absence of the vapour
+phase, is known as the _transition temperature_ or _inversion temperature_.
+
+Were we dependent on measurements of pressure and temperature, the
+determination of the transition point might be a matter of great
+difficulty. When we consider, however, that the other physical properties
+of the solid phases, _e.g._ the density, undergo an abrupt change on
+passing through the transition point, owing to the transformation of one
+form into the other, then any method by which this abrupt change in the
+physical properties can be detected may be employed for determining the
+transition point. A considerable number of such methods have been devised,
+and a description of the most important of these is given in the Appendix.
+
+In the case of sulphur, the transition point of rhombic into monoclinic
+sulphur was found by Reicher[51] to lie at 95.5deg. Below this temperature
+the octahedral, above it the monoclinic, is the stable form.
+
+Condensed Systems.--We have already seen that in the change of the melting
+point of water with the pressure, a very great increase of the latter was
+necessary in order to produce a comparatively small change in the
+temperature of equilibrium. This is a characteristic of all systems from
+which the vapour phase is absent, and which are composed only of solid and
+liquid phases. Such systems are called _condensed systems_,[52] and in
+determining the temperature of equilibrium of such systems, practically the
+same point will be obtained whether the measurements are carried out under
+atmospheric pressure or under the pressure of the vapour of the solid or
+liquid phases. The transition point, therefore, as determined in open
+vessels at atmospheric pressure, will differ only by a very slight amount
+from the triple point, or point at which the two solid or liquid phases are
+in equilibrium under the pressure of their vapour. {37} The determination
+of the transition point is thereby greatly simplified.
+
+Suspended Transformation.--In many respects the transition point of two
+solid phases is analogous to the melting point of a solid, or point at
+which the solid passes into a liquid. In both cases the change of phase is
+associated with a definite temperature and pressure in such a way that
+below the point the one phase, above the point the other phase, is stable.
+The transition point, however, differs in so far from a point of fusion,
+that while it is possible to supercool a liquid, no definite case is known
+where the solid has been heated above the triple point without passing into
+the liquid state. Transformation, therefore, is suspended only on one side
+of the melting point. In the case of two solid phases, however, the
+transition point can be overstepped in both directions, so that each phase
+can be obtained in the metastable condition. In the case of supercooled
+water, further, we saw that the introduction of the stable, solid phase
+caused the speedy transformation of the metastable to the stable condition
+of equilibrium; but in the case of two solid phases the change from the
+metastable to the stable modification may occur with great slowness, even
+in presence of the stable form. This tardiness with which the stable
+condition of equilibrium is reached greatly increases in many cases the
+difficulty of accurately determining the transition point. The phenomena of
+suspended transformation will, however, receive a fuller discussion later
+(p. 68).
+
+Transition Curve--Rhombic and Monoclinic Sulphur.--Just as we found the
+melting point of ice to vary with the pressure, so also do we find that
+change of pressure causes an alteration in the transition point. In the
+case of the transition point of rhombic into monoclinic sulphur, increase
+of pressure by 1 atm. raises the transition point by 0.04deg-0.05deg.[53]
+The transition curve, or curve representing the change of the transition
+point with pressure, will therefore slope to the right away from the
+pressure axis. This is curve OC (Fig. 5).
+
+{38}
+
+Triple Point--Monoclinic Sulphur, Liquid, and Vapour. Melting Point of
+Monoclinic Sulphur.--Above 95.5deg, monoclinic sulphur is, as we have seen,
+the stable form. On being heated to 120deg, under atmospheric pressure, it
+melts. This temperature is, therefore, the point of equilibrium between
+monoclinic sulphur and liquid sulphur under atmospheric pressure. Since we
+are dealing with a condensed system, this temperature may be regarded as
+very nearly that at which the solid and liquid are in equilibrium with
+their vapour, _i.e._ the triple point, solid (monoclinic)--liquid--vapour.
+This point is represented in the diagram by B.
+
+Triple Point--Rhombic and Monoclinic Sulphur and Liquid.--In contrast with
+that of ice, the fusion point of monoclinic sulphur is _raised_ by increase
+of pressure, and the fusion curve, therefore, slopes to the right. The
+transition curve of rhombic and monoclinic sulphur, as we have seen, also
+slopes to the right, and more so than the fusion curve of monoclinic
+sulphur. There will, therefore, be a certain pressure and temperature at
+which the two curves will cut. This point lies at 151deg, and a pressure of
+1320 kilogm. per sq. cm., or about 1288 atm.[54] It, therefore, forms
+another triple point, the existence of which had been predicted by
+Roozeboom,[55] at which rhombic and monoclinic sulphur are in equilibrium
+with liquid sulphur. It is represented in our diagram by the point C.
+_Beyond this point monoclinic sulphur ceases to exist in a stable
+condition._ At temperatures and pressures above this triple point, rhombic
+sulphur will be the stable modification, and this fact is of mineralogical
+interest, because it explains the occurrence in nature of well-formed
+rhombic crystals. Under ordinary conditions, prismatic sulphur separates
+out on cooling fused sulphur, but at temperatures above 151deg and under
+pressures greater than 1288 atm., the rhombic form would be produced.[56]
+
+Triple Point--Rhombic Sulphur, Liquid, and Vapour. Metastable Triple
+Point.--On account of the slowness with {39} which transformation of one
+form into the other takes place on passing the transition point, it has
+been found possible to heat rhombic sulphur up to its melting point
+(114.5deg). At this temperature, not only is rhombic sulphur in a
+metastable condition, but the liquid is also metastable, its vapour
+pressure being greater than that of solid monoclinic sulphur. This point is
+represented in our diagram by the point b.
+
+From the relative positions of the metastable melting point of rhombic
+sulphur and the stable melting point of monoclinic sulphur at 120deg, we
+see that, of the two forms, the metastable form has the lower melting
+point. This, of course, is valid only for the relative stability in the
+neighbourhood of the melting point; for we have already learned that at
+lower temperatures rhombic sulphur is the stable, monoclinic sulphur the
+metastable (or unstable) form.
+
+Fusion Curve of Rhombic Sulphur.--Like any other melting point, that of
+rhombic sulphur will be displaced by increase of pressure; increase of
+pressure raises the melting point, and we can therefore obtain a metastable
+fusion curve representing the conditions under which rhombic sulphur is in
+equilibrium with liquid sulphur. This metastable fusion curve must pass
+through the triple point for rhombic sulphur--monoclinic sulphur--liquid
+sulphur, and on passing this point it becomes a stable fusion curve. The
+continuation of this curve, therefore, above 151deg forms the stable fusion
+curve of rhombic sulphur (curve CD).
+
+These curves have been investigated at high pressures by Tammann, and the
+results are represented according to scale in Fig. 6,[57] _a_ being the
+curve for monoclinic sulphur and liquid; _b_, that for rhombic sulphur and
+liquid; and _c_, that for rhombic and monoclinic sulphur.
+
+Bivariant Systems.--Just as in the case of the diagram of states of water,
+the areas in Fig. 5 represent the conditions for the stable existence of
+the single phases: rhombic sulphur in the area to the left of AOCD;
+monoclinic sulphur in the area OBC; liquid sulphur in the area EBCD;
+sulphur vapour below the curves AOBE. As can be seen from the diagram, {40}
+the existence of monoclinic sulphur is limited on all sides, its area being
+bounded by the curves OB, OC, BC. At any point outside this area,
+monoclinic sulphur can exist only in a metastable condition.
+
+[Illustration: FIG. 6.]
+
+Other crystalline forms of sulphur have been obtained,[58] so that the
+existence of other systems of the one-component sulphur besides those
+already described is possible. Reference will be made to these later
+(p. 51).
+
+{41}
+
+C. _Tin._
+
+Another substance capable of existing in more than one crystalline form, is
+the metal tin, and although the general behaviour, so far as studied, is
+analogous to that of sulphur, a short account of the two varieties of tin
+may be given here, not only on account of their metallurgical interest, but
+also on account of the importance which the phenomena possess for the
+employment of this metal in everyday life.
+
+After a winter of extreme severity in Russia (1867-1868), the somewhat
+unpleasant discovery was made that a number of blocks of tin, which had
+been stored in the Customs House at St. Petersburg, had undergone
+disintegration and crumbled to a grey powder.[59] That tin undergoes change
+on exposure to extreme cold was known, however, before that time, even as
+far back as the time of Aristotle, who spoke of the tin as "melting."[60]
+Ludicrous as that term may now appear, Aristotle nevertheless unconsciously
+employed a strikingly accurate analogy, for the conditions under which
+ordinary white tin passes into the grey modification are, in many ways,
+quite analogous to those under which a substance passes from the solid to
+the liquid state. The knowledge of this was, however, beyond the wisdom of
+the Greek philosopher.
+
+For many years there existed considerable confusion both as to the
+conditions under which the transformation of white tin into its allotropic
+modification occurs, and to the reason of the change. Under the guidance of
+the Phase Rule, however, the confusion which obtained has been cleared
+away, and the "mysterious" behaviour of tin brought into accord with other
+phenomena of transformation.[61]
+
+Transition Point.--Just as in the case of sulphur, so also in the case of
+tin, there is a transition point above which the {42} one form, ordinary
+white tin, and below which the other form, grey tin, is the stable variety.
+In the case of this metal, the transition point was found by Cohen and van
+Eyk, who employed both the dilatometric and the electrical methods
+(Appendix) to be 20deg. Below this temperature, grey tin is the stable
+form. But, as we have seen in the case of sulphur, the change of the
+metastable into the stable solid phase occurs with considerable slowness,
+and this behaviour is found also in the case of tin. Were it not so, we
+should not be able to use this metal for the many purposes to which it is
+applied in everyday life; for, with the exception of a comparatively small
+number of days in the year, the temperature of our climate is below 20deg,
+and _white tin is, therefore, at the ordinary temperature, in a metastable
+condition_. The change, however, into the stable form at the ordinary
+temperature, although slow, nevertheless takes place, as is shown by the
+partial or entire conversion of articles of tin which have lain buried for
+several hundreds of years.
+
+On lowering the temperature, the velocity with which the transformation of
+the tin occurs is increased, and Cohen and van Eyk found that the
+temperature of maximum velocity is about -50deg. Contact with the stable
+form will, of course, facilitate the transformation.
+
+The change of white tin into grey takes place also with increased velocity
+in presence of a solution of tin ammonium chloride (pink salt), which is
+able to dissolve small quantities of tin. In presence of such a solution
+also, it was found that the temperature at which the velocity of
+transformation was greatest was raised to 0deg. At this temperature, white
+tin in contact with a solution of tin ammonium chloride, and the grey
+modification, undergoes transformation to an appreciable extent in the
+course of a few days.
+
+Fig. 7 is a photograph of a piece of white tin undergoing transformation
+into the grey variety.[62] The bright surface of the tin becomes covered
+with a number of warty masses, formed of the less dense grey form, and the
+number and size of these continue to grow until the whole of the white tin
+has passed {43} into a grey powder. On account of the appearance which is
+here seen, this transformation of tin has been called by Cohen the "tin
+plague."
+
+[Illustration: FIG. 7.]
+
+{44}
+
+Enantiotropy and Monotropy.--In the case of sulphur and tin, we have met
+with two substances existing in polymorphic forms, and we have also learned
+that these forms exhibit a definite transition point at which their
+relative stability is reversed. Each form, therefore, possesses a definite
+range of stable existence, and is capable of undergoing transformation into
+the other, at temperatures above or below that of the transition point.
+
+Another class of dimorphous substances is, however, met with as, for
+instance, in the case of the well-known compounds iodine monochloride and
+benzophenone. Each crystalline form has its own melting point, the
+dimorphous forms of iodine monochloride melting at 13.9deg and 27.2deg,[63]
+and those of benzophenone at 26deg and 48deg.[64] This class of substance
+differs from that which we have already studied (_e.g._ sulphur and tin),
+in that at all temperatures up to the melting point, only one of the forms
+is stable, the other being metastable. There is, therefore, no transition
+point, and transformation of the crystalline forms can be observed _only in
+one direction_. These two classes of phenomena are distinguished by the
+names _enantiotropy_ and _monotropy_; enantiotropic substances being such
+that the change of one form into the other is a reversible process (_e.g._
+rhombic sulphur into monoclinic, and monoclinic sulphur into rhombic), and
+monotropic substances, those in which the transformation of the crystalline
+forms is irreversible.
+
+[Illustration: FIG. 8.]
+
+[Illustration: FIG. 9.]
+
+These differences in the behaviour can be explained very well in many cases
+by supposing that in the case of enantiotropic substances the transition
+point lies below the melting point, while in the case of monotropic
+substances, it lies above the melting point.[65] These conditions would be
+represented by the Figs. 8 and 9.
+
+In these two figures, O_{3} is the transition point, O_{1} and O_{2} the
+melting points of the metastable and stable forms {45} respectively. From
+Fig. 9 we see that the crystalline form I. at all temperatures up to its
+melting point is metastable with respect to the form II. In such cases the
+transition point could be reached only at higher pressures.
+
+Although, as already stated, this explanation suffices for many cases, it
+does not prove that in all cases of monotropy the transition point is above
+the melting point of the two forms. It is also quite possible that the
+transition point may lie below the melting points;[66] in this case we have
+what is known as _pseudomonotropy_. It is possible that graphite and
+diamond,[67] perhaps also the two forms of phosphorus, stand in the
+relation of pseudomonotropy (_v._ p. 49).
+
+The disposition of the curves in Figs. 8 and 9 also explains the phenomenon
+sometimes met with, especially in organic chemistry, that the substance
+first melts, then solidifies, and remelts at a higher temperature. On again
+determining the melting point after re-solidification, only the higher
+melting point is obtained.
+
+The explanation of such a behaviour is, that if the determination of the
+melting point is carried out rapidly, the point O_{1}, the melting point of
+the metastable solid form, may be realized. At this temperature, however,
+the liquid is metastable with respect to the stable solid form, and if the
+temperature is {46} not allowed to rise above the melting point of the
+latter, the liquid may solidify. The stable solid modification thus
+obtained will melt only at a higher temperature.
+
+D. _Phosphorus._
+
+An interesting case of a monotropic dimorphous substance is found in
+phosphorus, which occurs in two crystalline forms; white phosphorus
+belonging to the regular system, and red phosphorus belonging to the
+hexagonal system. From determinations of the vapour pressures of liquid
+white phosphorus, and of solid red phosphorus,[68] it was found that the
+vapour pressure of red phosphorus was considerably lower than that of
+liquid white phosphorus at the same temperature, the values obtained being
+given in the following table.
+
+  VAPOUR PRESSURES OF WHITE AND RED PHOSPHORUS.
+
+  -------------------------------------------------------------------------
+    Vapour pressure of liquid white phosphorus.    | Vapour pressure of red
+                                                   |     phosphorus.
+  -------------------------------------------------+-----------------------
+  Temperature.| Pressure | Temperature.|  Pressure | Temperature.| Pressure
+              |  in cm.  |             |   in atm. |             |  in atm.
+  ------------+----------+-------------+-----------+-------------+---------
+   165deg     |  12      |   360deg    |    3.2    |   360deg    |   0.1
+   180deg     |  20.4    |   440deg    |    7.5    |   440deg    |   1.75
+   200deg     |  26.6    |   494deg    |   18.0    |   487deg    |   6.8
+   219deg     |  35.9    |   503deg    |   21.9    |   510deg    |  10.8
+   230deg     |  51.4    |   511deg    |   26.2    |   531deg    |  16.0
+   290deg     |  76.0    |     --      |     --    |   550deg    |  31.0
+    --        |   --     |     --      |     --    |   577deg    |  56.0
+  -------------------------------------------------------------------------
+
+These values are also represented graphically in Fig. 10.
+
+[Illustration: FIG. 10.]
+
+At all temperatures above about 260deg, transformation of the white into
+the red modification takes place with appreciable velocity, and this
+velocity increases as the temperature is raised. Even at lower
+temperatures, _e.g._ at the ordinary temperature, the velocity of
+transformation is increased under the influence {47} of light,[69] or by
+the presence of certain substances, _e.g._ iodine,[70] just as the velocity
+of transformation of white tin into the grey modification was increased by
+the presence of a solution of tin ammonium chloride (p. 40). At the
+ordinary temperature, therefore, white phosphorus must be considered as the
+less stable (metastable) form, for although it can exist in contact with
+red phosphorus for a long period, its vapour pressure, as we have seen, is
+greater than that of the red modification, and also, its solubility in
+different solvents is greater[71] than that of the red modification; as we
+shall find later, the solubility of the metastable form is always greater
+than that of the stable.
+
+The relationships which are met with in the case of phosphorus can be best
+represented by the diagram, Fig. 11.[72]
+
+In this figure, BO_{1} represents the conditions of equilibrium of the
+univariant system red phosphorus and vapour, which ends at O_{1}, the
+melting point of red phosphorus. By heating in capillary tubes of hard
+glass, Chapman[73] found that red phosphorus melts at the melting point of
+potassium iodide, _i.e._ about 630deg,[74] but the pressure at this
+temperature is unknown.
+
+At O_{1}, then, we have the triple point, red phosphorus, liquid, and
+vapour, and starting from it, we should have the {48} vaporization curve of
+liquid phosphorus, O_{1}A, and the fusion curve of red phosphorus, O_{1}F.
+Although these have not been determined, the latter curve must, from
+theoretical considerations (_v._ p. 58), slope slightly to the right;
+_i.e._ increase of pressure raises the melting point of red phosphorus.
+
+[Illustration: FIG. 11.]
+
+When white phosphorus is heated to 44deg, it melts. At this
+point, therefore, we shall have another triple point, white
+phosphorus--liquid--vapour; the pressure at this point has been calculated
+to be 3 mm.[75] This point is the intersection of three curves, viz.
+sublimation curve, vaporization curve, and the fusion curve of white
+phosphorus. The fusion curve, O_{2}E, has been determined by Tammann[76]
+and by G. A. Hulett,[77] and it was found that increase of pressure by 1
+atm. raises the melting point by 0.029deg. The sublimation curve of white
+phosphorus has not yet been determined.
+
+As can be seen from the table of vapour pressures (p. 46), the vapour
+pressure of white phosphorus has been determined up to 500deg; at
+temperatures above this, however, the velocity with which transformation
+into red phosphorus takes place is so great as to render the determination
+of the vapour pressure {49} at higher temperatures impossible. Since,
+however, the difference between white phosphorus and red phosphorus
+disappears in the liquid state, the vapour pressure curve of white
+phosphorus must pass through the point O_{1}, the melting point of red
+phosphorus, and must be continuous with the curve O_{1}A, the vapour
+pressure curve of liquid phosphorus (_vide infra_). Since, as Fig. 10
+shows, the vapour pressure curve of white phosphorus ascends very rapidly
+at higher temperatures, the "break" between BO_{1} and O_{1}A must be very
+slight.
+
+As compared with monotropic substances like benzophenone, phosphorus
+exhibits the peculiarity that transformation of the metastable into the
+stable modification takes place with great slowness; and further, the time
+required for the production of equilibrium between red phosphorus and
+phosphorus vapour is great compared with that required for establishing the
+same equilibrium in the case of white phosphorus. This behaviour can be
+best explained by the assumption that change in the molecular complexity
+(polymerization) occurs in the conversion of white into red phosphorus, and
+when red phosphorus passes into vapour (depolymerization).[78]
+
+This is borne out by the fact that measurements of the vapour density of
+phosphorus vapour at temperatures of 500deg and more, show it to have the
+molecular weight represented by P_{4},[79] and the same molecular weight
+has been found for phosphorus in solution.[80] On the other hand, it has
+recently been shown by R. Schenck,[81] that the molecular weight of red
+phosphorus is at least P_{8}, and very possibly higher.
+
+In the case of phosphorus, therefore, it is more than possible that we are
+dealing, not simply with two polymorphic {50} forms of the same substance,
+but with polymeric forms, and that there is no transition point at
+temperatures above the absolute zero, unless we assume the molecular
+complexity of the two forms to become the same. The curve for red
+phosphorus would therefore lie below that of white phosphorus, for the
+vapour pressure of the polymeric form, if produced from the simpler form
+with evolution of heat, must be lower than that of the latter. A transition
+point would, of course, become possible if the sign of the heat effect in
+the transformation of the one modification into the other should change.
+If, further, the liquid which is produced by the fusion of red phosphorus
+at 630deg under high pressure also exists in a polymeric form, greater than
+P_{4}, then the metastable vaporization curve of white phosphorus would not
+pass through the melting point of red phosphorus, as was assumed above.[82]
+
+We have already seen in the case of water (p. 31) that the vapour pressure
+of supercooled water is greater than that of ice, and that therefore it is
+possible, theoretically at least, by a process of distillation, to transfer
+the water from one end of a closed tube to the other, and to there condense
+it as ice. On account of the very small difference between the vapour
+pressure of supercooled water and ice, this distillation process has not
+been experimentally realized. In the case of phosphorus, however, where the
+difference in the vapour pressures is comparatively great, it has been
+found possible to distil white phosphorus from one part of a closed tube to
+another, and to there condense it as red phosphorus; and since the vapour
+pressure of red phosphorus at 350deg is less than the vapour pressure of
+white phosphorus at 200deg, it is possible to carry out the distillation
+from a _colder_ part of the tube to a _hotter_, by having white phosphorus
+at the former and red phosphorus at the latter. Such a process of
+distillation has been carried out by Troost and Hautefeuille between 324deg
+and 350deg.[83]
+
+Relationships similar to those found in the case of phosphorus are also met
+with in the case of cyanogen and {51} paracyanogen, which have been studied
+by Chappuis,[84] Troost and Hautefeuille,[85] and Dewar,[86] and also in
+the case of other organic substances.
+
+Enantiotropy combined with Monotropy.--Not only can polymorphic substances
+exhibit enantiotropy or monotropy, but, if the substance is capable of
+existing in more than two crystalline forms, both relationships may be
+found, so that some of the forms may be enantiotropic to one another, while
+the other forms exhibit only monotropy. This behaviour is seen in the case
+of sulphur, which can exist in as many as eight different crystalline
+varieties. Of these only monoclinic and rhombic sulphur exhibit the
+relationship of enantiotropy, _i.e._ they possess a definite transition
+point, while the other forms are all metastable with respect to rhombic and
+monoclinic sulphur, and remain so up to the melting point; that is to say,
+they are monotropic modifications.[87]
+
+E. _Liquid Crystals._
+
+Phenomena observed.--In 1888 it was discovered by Reinitzer[88] that the
+two substances, cholesteryl acetate and cholesteryl benzoate, possess the
+peculiar property of melting sharply at a definite temperature to milky
+liquids; and that the latter, on being further heated, suddenly become
+clear, also at a definite temperature. Other substances, more especially
+_p_-azoxyanisole and _p_-azoxyphenetole, were, later, found to possess the
+same property of having apparently a double melting point.[89] On cooling
+the clear liquids, the reverse series of changes occurred.
+
+The turbid liquids which were thus obtained were found to possess not only
+the usual properties of liquids (such as the {52} property of flowing and
+of assuming a perfectly spherical shape when suspended in a liquid of the
+same density), but also those properties which had hitherto been observed
+only in the case of solid crystalline substances, viz. the property of
+double refraction and of giving interference colours when examined by
+polarized light; the turbid liquids are _anisotropic_. To such liquids, the
+optical properties of which were discovered by O. Lehmann,[90] the name
+_liquid crystals_, or crystalline liquids, was given.
+
+Nature of Liquid Crystals.--During the past ten years the question as to
+the nature of liquid crystals has been discussed by a number of
+investigators, several of whom have contended strongly against the idea of
+the term "liquid" being applied to the crystalline condition; and various
+attempts have been made to prove that the turbid liquids are in reality
+heterogeneous and are to be classed along with emulsions.[91] This view was
+no doubt largely suggested by the fact that the anisotropic liquids were
+turbid, whereas the "solid" crystals were clear. Lehmann found, however,
+that, when examined under the microscope, the "simple" liquid crystals were
+also clear,[92] the apparent turbidity being due to the aggregation of a
+number of differently oriented crystals, in the same way as a piece of
+marble does not appear transparent although composed of transparent
+crystals.[93]
+
+Further, no proof of the heterogeneity of liquid crystals has yet been
+obtained, but rather all chemical and physical investigations indicate that
+they are homogeneous.[94] No separation {53} of a solid substance from the
+milky, anisotropic liquids has been effected; the anisotropic liquid is in
+some cases less viscous than the isotropic liquid formed at a higher
+temperature; and the temperature of liquefaction is constant, and is
+affected by pressure and admixture with foreign substances exactly as in
+the case of a pure substance.[95]
+
+[Illustration: FIG. 12.]
+
+Equilibrium Relations in the Case of Liquid Crystals.--Since, now, we have
+seen that we are dealing here with substances in two crystalline forms
+(which we may call the solid and liquid[96] crystalline form), which
+possess a definite transition point, at which, transformation of the one
+form into the other occurs in both directions, we can represent the
+conditions of equilibrium by a diagram in all respects similar to that
+employed in the case of other enantiotropic substances, _e.g._ sulphur
+(p. 35).
+
+{54}
+
+In Fig. 12 there is given a diagrammatic representation of the
+relationships found in the case of _p_-azoxyanisole.[97]
+
+Although the vapour pressure of the substance in the solid, or liquid
+state, has not been determined, it will be understood from what we have
+already learned, that the curves AO, OB, BC, representing the vapour
+pressure of solid crystals, liquid crystals, isotropic liquid, must have
+the relative positions shown in the diagram. Point O, the transition point
+of the solid into the liquid crystals, lies at 118.27deg, and the change of
+the transition point with the pressure is +0.032deg pro 1 atm. The
+transition curve OE slopes, therefore, slightly to the right. The point B,
+the melting point of the liquid crystals, lies at 135.85deg, and the
+melting point is raised 0.0485deg pro 1 atm. The curve BD, therefore, also
+slopes to the right, and more so than the transition curve. In this respect
+azoxyanisole is different from sulphur.
+
+The areas bounded by the curves represent the conditions for the stable
+existence of the four single phases, solid crystals, liquid crystals,
+isotropic liquid and vapour.
+
+The most important substances hitherto found to form liquid crystals
+are[98]:--
+
+  ----------------------------------+------------+--------
+                                    |            |
+          Substance.                | Transition | Melting
+                                    |   point.   | point.
+  ----------------------------------+------------+--------
+                                    |            |
+  Cholesteryl benzoate              |  145.5deg  | 178.5deg
+  Azoxyanisole                      |  118.3deg  | 135.9deg
+  Azoxyphenetole                    |  134.5deg  | 168.1deg
+  Condensation product from         |            |
+   benzaldehyde and benzidine       |  234deg    | 260deg
+  Azine of _p_-oxyethylbenzaldehyde |  172deg    | 196deg
+  Condensation product from         |            |
+   _p_-tolylaldehyde and benzidine  |  231deg    |   --
+  _p_-Methoxycinnamic acid          |  169deg    | 185deg
+  ----------------------------------+------------+--------
+
+       *       *       *       *       *
+
+
+{55}
+
+CHAPTER IV
+
+GENERAL SUMMARY
+
+In the preceding pages we have learned how the principles of the Phase Rule
+can be applied to the elucidation of various systems consisting of one
+component. In the present chapter it is proposed to give a short summary of
+the relationships we have met with, and also to discuss more generally how
+the Phase Rule applies to other one-component systems. On account of the
+fact that beginners are sometimes inclined to expect too much of the Phase
+Rule; to expect, for example, that it will inform them as to the exact
+behaviour of a substance, it may here be emphasized that the Phase Rule is
+a general rule; it informs us only as to the general conditions of
+equilibrium, and leaves the determination of the definite, numerical data
+to experiment.
+
+Triple Point.--We have already (p. 28) defined a triple point in a
+one-component system, as being that pressure and temperature at which three
+phases coexist in equilibrium; it represents, therefore, an invariant
+system (p. 16). At the triple point also, three curves cut, viz. the curves
+representing the conditions of equilibrium of the three univariant systems
+formed by the combination of the three phases in pairs. The most common
+triple point of a one-component system is, of course, the triple point,
+solid, liquid, vapour (S-L-V), but other triple points[99] are also
+possible when, as in the case of {56} sulphur or benzophenone, polymorphic
+forms occur. Whether or not all the triple points can be experimentally
+realized will, of course, depend on circumstances. We shall, in the first
+place, consider only the triple point S-L-V.
+
+As to the general arrangement of the three univariant curves around the
+triple point, the following rules may be given. (1) The prolongation of
+each of the curves beyond the triple point must lie between the other two
+curves. (2) The middle position at one and the same temperature in the
+neighbourhood of the triple point is taken by that curve (or its metastable
+prolongation) which represents the two phases of most widely differing
+specific volume.[100] That is to say, if a line of constant temperature is
+drawn immediately above or below the triple point so as to cut the three
+curves--two stable curves and the metastable prolongation of the third--the
+position of the curves at that temperature will be such that the middle
+position is occupied by that curve (or its metastable prolongation) which
+represents the two phases of most widely differing specific volume.
+
+Now, although these rules admit of a considerable variety of possible
+arrangements of curves around the triple point,[101] only two of these have
+been experimentally obtained in the case of the triple point
+solid--liquid--vapour. At present, therefore, we shall consider only these
+two cases (Figs. 13 and 14).
+
+[Illustration: FIG. 13.]
+
+[Illustration: FIG. 14.]
+
+An examination of these two figures shows that they satisfy the rules laid
+down. Each of the curves on being prolonged passes between the other two
+curves. In the case of substances of the first type (Fig. 13), the specific
+volume of the solid is greater than that of the liquid (the substance
+contracts on fusion); the difference of specific volume will, therefore, be
+greatest between liquid and vapour. The curve, therefore, for liquid and
+vapour (or its prolongation) must lie between the other two curves; this is
+seen from the figure to be the case. Similarly, the rule is satisfied by
+the arrangement of curves in Fig. 14, where the difference of specific
+volumes is {57} greatest between the solid and vapour. In this case the
+curve S-V occupies the intermediate position.
+
+As we see, the two figures differ from one another only in that the fusion
+curve OC in one case slopes to the right away from the pressure axis, thus
+indicating that the melting point is raised by increase of pressure; in the
+other case, to the left, indicating a lowering of the melting point with
+the pressure. These conditions are found exemplified in the case of sulphur
+and ice (pp. 29 and 35). We see further from the two figures, that O in
+Fig. 13 gives the highest temperature at which the solid can exist, for the
+curve for solid--liquid slopes back to regions of lower temperature; in
+Fig. 14, O gives the lowest temperature at which the liquid phase can exist
+as stable phase.[102]
+
+Theorems of van't Hoff and of Le Chatelier.--So far we have studied only
+the conditions under which various systems exist in equilibrium; and we now
+pass to a consideration of the changes which take place in a system when
+the external conditions of temperature and pressure are altered. For all
+such changes there exist two theorems, based on the laws of thermodynamics,
+by means of which the alterations in a system can be qualitatively
+predicted.[103] The first of these, usually {58} known as van't Hoff's _law
+of movable equilibrium_,[104] states: When the temperature of a system in
+equilibrium is raised, that reaction takes place which is accompanied by
+absorption of heat; and, conversely, when the temperature is lowered, that
+reaction occurs which is accompanied by an evolution of heat.
+
+The second of the two theorems refers to the effect of change of pressure,
+and states:[105] When the pressure on a system in equilibrium is increased,
+that reaction takes place which is accompanied by a diminution of volume;
+and when the pressure is diminished, a reaction ensues which is accompanied
+by an increase of volume.
+
+The demonstration of the universal applicability of these two theorems is
+due chiefly to Le Chatelier, who showed that they may be regarded as
+consequences of the general law of action and reaction. For this reason
+they are generally regarded as special cases of the more general law, known
+as the _theorem of Le Chatelier_, which may be stated in the words of
+Ostwald, as follows:[106] _If a system in equilibrium is subjected to a
+constraint by which the equilibrium is shifted, a reaction takes place
+which opposes the constraint, _i.e._ one by which its effect is partially
+destroyed._
+
+This theorem of Le Chatelier is of very great importance, for it applies to
+all systems and changes of the condition of equilibrium, whether physical
+or chemical; to vaporization and fusion; to solution and chemical action.
+In all cases, whenever changes in the external condition of a system in
+equilibrium are produced, processes also occur within the system which tend
+to counteract the effect of the external changes.
+
+_Changes at the Triple Point._--If now we apply this theorem to equilibria
+at the triple point S-L-V, and ask what changes will occur in such a system
+when the external conditions of pressure and temperature are altered, the
+general answer to the question will be: So long as the three phases are
+present, no {59} change in the temperature or pressure of the system can
+occur, but _only changes in the relative amounts of the phases_; that is to
+say, the effect on the system of change in the external conditions is
+opposed by the reactions or changes which take place within the system
+(according to the theorems of van't Hoff and Le Chatelier). We now proceed
+to discuss what these changes are, and shall consider first the effect of
+alteration of the temperature at constant volume and constant pressure, and
+then the effect of alteration of the pressure both when the temperature
+remains constant and when it varies.
+
+When the volume is kept constant, the effect of the addition of heat to a
+system at the triple point S-L-V differs somewhat according as there is an
+increase or diminution of volume when the solid passes into the liquid
+state. In the former and most general case (Fig. 14), addition of heat will
+cause a certain amount of the solid phase to melt, whereby the heat which
+is added becomes latent; the temperature of the system therefore does not
+rise. Since, however, the melting of the solid is accompanied by an
+increase of volume, whereby an increase of pressure would result, a certain
+portion of the vapour must condense to liquid, in order that the pressure
+may remain constant. The total effect of addition of heat, therefore, is to
+cause both solid and vapour to pass into liquid, _i.e._ there occurs the
+change S + V --> L. It will, therefore, depend on the relative quantities
+of solid and vapour, which will disappear first. If the solid disappears
+first, then we shall pass to the system L-V; if vapour disappears first, we
+shall obtain the system S-L. Withdrawal of heat causes the reverse change,
+L --> S + V; at all temperatures below the triple point the liquid is
+unstable or metastable (p. 30).
+
+When fusion is accompanied by a diminution of volume (_e.g._ ice, Fig. 13),
+then, since the melting of the solid phase would decrease the total volume,
+_i.e._ would lower the pressure, a certain quantity of the solid must also
+pass into vapour in order that the pressure may be maintained constant. On
+addition of heat, therefore, there occurs the reaction S --> L + V;
+withdrawal of heat causes the reverse change L + V --> S. Above the
+temperature of the triple point the {60} solid cannot exist; below the
+triple point both systems, S-L and S-V, can exist, and it will therefore
+depend on the relative amounts of liquid and vapour which of these two
+systems is obtained on withdrawing heat from the system at constant volume.
+
+The same changes in the phases occur when heat is added or withdrawn at
+constant pressure, so long as the three phases are present. Continued
+addition of heat, however, at constant pressure will ultimately cause the
+formation of the bivariant system vapour alone; continued withdrawal of
+heat will ultimately cause the formation of solid alone. This will be
+readily understood from Fig. 15. The dotted line D'OD is a line of constant
+pressure; on adding heat, the system passes along the line OD into the
+region of vapour; on heat being withdrawn, the system passes along OD' into
+the area of solid.
+
+[Illustration: FIG. 15.]
+
+Similar changes are produced when the volume of the system is altered.
+Alteration of volume may take place either while transference of heat to or
+from the system is cut off (adiabatic change), or while such transference
+may occur (isothermal change). In the latter case, the temperature of the
+system will remain constant; in the former case, since at the triple point
+the pressure must be constant so long as the three phases are present,
+increase of volume must be compensated by the evaporation of liquid. This,
+however, would cause the temperature to fall (since communication of heat
+from the outside is supposed to be cut off), and a portion of the liquid
+must therefore freeze. In this way the latent heat of evaporation is
+counterbalanced by the latent heat of fusion. As the result of increase of
+volume, therefore, the process occurs L --> S + V. Diminution of volume,
+without transference of heat, will bring about the opposite change, S + V
+--> L. In the former case there is ultimately obtained the univariant
+system S-V; in the latter case there will be {61} obtained either S-L or
+L-V according as the vapour or solid phase disappears first.
+
+This argument holds good for both types of triple point shown in Figs. 13
+and 14 (p. 57). A glance at these figures will show that increase of volume
+(diminution of pressure) will lead ultimately to the system S-V, for at
+pressures lower than that of the triple point, the liquid phase cannot
+exist. Decrease of volume (increase of pressure), on the other hand, will
+lead either to the system S-L or L-V, because these systems can exist at
+pressures higher than that of the triple point. If the vapour phase
+disappears and we pass to the curve S-L, continued diminution of volume
+will be accompanied by a fall in temperature in the case of systems of the
+first type (Fig. 13), and by a rise in temperature in the case of systems
+of the second type (Fig. 14).
+
+[Illustration: FIG. 16.]
+
+[Illustration: FIG. 17.]
+
+Lastly, if the temperature is maintained constant, _i.e._ if heat can pass
+into or out of the system, then on changing the volume the same changes in
+the phases will take place as described above until one of the phases has
+disappeared. Continued increase of volume (decrease of pressure) will then
+cause the disappearance of a second phase, the system passing along the
+dotted line OE' (Figs. 16, 17), so that ultimately there remains only the
+vapour phase. Conversely, diminution of volume (increase of pressure) will
+ultimately lead either to solid (Fig. 16) or to liquid alone (Fig. 17), the
+system passing along the dotted line OE. {62}
+
+In discussing the alterations which may take place at the triple point with
+change of temperature and pressure, we have considered only the triple
+point S-L-V. The same reasoning, however, applies, _mutatis mutandis_, to
+all other triple points, so that if the specific volumes of the phases are
+known, and the sign of the heat effects which accompany the transformation
+of one phase into the other, it is possible to predict (by means of the
+theorem of Le Chatelier) the changes which will be produced in the system
+by alteration of the pressure and temperature.
+
+In all cases of transformation at the triple point, it should be noted that
+all _three phases are involved in the change_,[107] and not two only; the
+fact that in the case, say, of the transformation from solid to liquid, or
+liquid to solid, at the melting point with change of temperature, only
+these two phases appear to be affected, is due to there generally being a
+large excess of the vapour phase present and to the prior disappearance
+therefore of the solid or liquid phase.
+
+In the case of triple points at which two solid phases are in equilibrium
+with liquid, other arrangements of the curves around the triple point are
+found. It is, however, unnecessary to give a general treatment of these
+here, since the principles which have been applied to the triple point
+S-L-V can also be applied to the other triple points.[108]
+
+Triple Point Solid--Solid--Vapour.--The triple point solid--solid--vapour
+is one which is of considerable importance. Examples of such a triple point
+have already been given in sulphur and tin, and a list of other substances
+capable of yielding two solid phases is given below. The triple point S-S-V
+is not precisely the same as the transition point, but is very nearly so.
+The transition point is the temperature at which the relative stability of
+the two solid phases undergoes change, when the vapour phase is absent and
+the pressure is 1 atm.; whereas at the triple point the pressure is that of
+the system itself. The transition point, therefore, bears the same relation
+to the triple point S-S-V as the melting point to the triple point S-L-V.
+
+{63}
+
+In the following table is given a list of the most important polymorphous
+substances, and the temperatures of the transition point.[109]
+
+  ------------------------------------+-------------
+                                      |
+              Substance.              |  Transition
+                                      | temperature.
+  ------------------------------------+-------------
+                                      |
+  Ammonium nitrate--                  |
+   [beta]-rhombic --> [alpha]-rhombic |     35deg
+   [alpha]-rhombic --> rhombohedral   |     83deg
+   Rhombohedral --> regular           |    125deg
+  Mercuric iodide                     |    126deg
+  Potassium nitrate                   |    129deg
+  Silver iodide                       |    145deg
+  Silver nitrate                      |    160deg
+  Sulphur                             |     95.5deg
+  Tetrabrommethane                    |     46.8deg
+  Thallium nitrate--                  |
+   Rhombic --> rhombohedral           |     80deg
+   Rhombohedral --> regular           |    142.5deg
+  Thallium picrate                    |     46deg
+  Tin                                 |     20deg
+  ------------------------------------+-------------
+
+Sublimation and Vaporization Curves.--We have already seen, in the case of
+ice and liquid water, that the vapour pressure increases as the temperature
+rises, the increase of pressure per degree being greater the higher the
+temperature. The sublimation and vaporization curves, therefore, are not
+straight lines, but are bent, the convex side of the curve being towards
+the temperature axis in the ordinary _pt_-diagram.
+
+In the case of sulphur and of tin, we assumed vapour to be given off by the
+solid substance, although the pressure of the vapour has not hitherto been
+measured. The assumption, however, is entirely justified, not only on
+theoretical grounds, but also because the existence of a vapour pressure
+has been observed in the case of many solid substances at temperatures much
+below the melting point,[110] and in some cases, _e.g._ camphor,[111] the
+vapour pressure is considerable.
+
+{64}
+
+As the result of a large number of determinations, it has been found that
+all vapour pressure curves have the same general form alluded to above.
+Attempts have also been made to obtain a general expression for the
+quantitative changes in the vapour pressure with change of temperature, but
+without success. Nevertheless, the _qualitative_ changes, or the general
+direction of the curves, can be predicted by means of the theorem of Le
+Chatelier.
+
+As we have already learned (p. 16), the Phase Rule takes no account of the
+molecular complexity of the substances participating in an equilibrium. A
+dissociating substance, therefore, in contact with its vaporous products of
+dissociation (_e.g._ ammonium chloride in contact with ammonia and hydrogen
+chloride), will likewise constitute a univariant system of one component,
+provided the composition of the vapour phase as a whole is the same as that
+of the solid or liquid phase (p. 13). For all such substances, therefore,
+the conditions of equilibrium will be represented by a curve of the same
+general form as the vapour pressure curve of a non-dissociating
+substance.[112] The same behaviour is also found in the case of substances
+which polymerize on passing into the solid or liquid state (_e.g._ red
+phosphorus). Where such changes in the molecular state occur, however, the
+time required for equilibrium to be established is, as a rule, greater than
+when the molecular state is the same in both phases.
+
+From an examination of Figs. 13 and 14, it will be easy to predict the
+effect of change of pressure and temperature on the univariant systems S-V
+or L-V. If the volume is kept constant, addition of heat will cause an
+increase of pressure, the system S-V moving along the curve AO until at the
+triple point the liquid phase is formed, and the system L-V moving along
+the curve OB; so long as two phases are present, the condition of the
+system must be represented by these two curves. Conversely, withdrawal of
+heat will cause condensation of vapour, and therefore diminution of
+pressure; the system will therefore move along the vaporization or
+sublimation curve to lower temperatures and pressures, so long as the
+system remains univariant.
+
+{65}
+
+If transference of heat to or from the system is prevented, increase of
+volume (diminution of pressure) will cause the system L-V to pass along the
+curve BO; liquid will pass into vapour and the temperature will fall.[113]
+At O solid may appear, and the temperature of the system will then remain
+constant until the liquid phase has disappeared (p. 57); the system will
+then follow the curve OA until the solid phase disappears, and we are
+ultimately left with vapour. On the other hand, diminution of volume
+(increase of pressure) will cause condensation of vapour, and the system
+S-V will pass along the curve AO to higher temperatures and pressures; at O
+the solid will melt, and the system will ultimately pass to the curve OB or
+to OC (p. 57).
+
+Addition or withdrawal of heat at constant pressure, and increase or
+diminution of the pressure at constant temperature, will cause the system
+to pass along lines parallel to the temperature and the pressure axis
+respectively; the working out of these changes may be left to the reader,
+guided by what has been said on pp. 60 and 61.
+
+The sublimation curve of all substances, so far as yet found, has its upper
+limit at the melting point (triple point), although the possibility of the
+existence of a superheated solid is not excluded. The lower limit is,
+theoretically at least, at the absolute zero, provided no new phase, _e.g._
+a different crystalline modification, is formed. If the sublimation
+pressure of a substance is greater than the atmospheric pressure at any
+temperature below the point of fusion, then the substance will _sublime
+without melting_ when heated in an open vessel; and fusion will be possible
+only at a pressure higher than the atmospheric. This is found, for example,
+in the case of red phosphorus (p. 47). If, however, the sublimation
+pressure of a substance at its triple point S-L-V is less than one
+atmosphere, then the substance will melt when heated in an open vessel.
+
+In the case of the vaporization curve, the upper limit lies at the critical
+point where the liquid ceases to exist;[114] the {66} lower limit is
+determined by the range of the metastable state of the supercooled liquid.
+
+The interpolation and extrapolation of vapour-pressure curves is rendered
+very easy by means of a relationship which Ramsay and Young[115] found to
+exist between the vapour-pressure curves of different substances. It was
+observed that in the case of closely related substances, the ratio of the
+absolute temperatures corresponding to equal vapour pressures is constant,
+_i.e._ T_{1}/T'_{1} = T_{2}/T'_{2}. When the two substances are not closely
+related, it was found that the relationship could be expressed by the
+equation T_{1}/T'_{1} = T_{2}/T'_{2} + _c_(_t_' - _t_) where _c_ is a
+constant having a small positive or negative value, and _t_' and _t_ are
+the temperatures at which one of the substances has the two values of the
+vapour pressure in question. By means of this equation, if the
+vapour-pressure curve of one substance is known, the vapour-pressure curve
+of any other substance can be calculated from the values at any two
+temperatures of the vapour pressure of that substance.
+
+Fusion Curve--Transition Curve.--The fusion curve represents the conditions
+of equilibrium between the solid and liquid phase; it shows the change of
+the melting point of a substance with change of pressure.
+
+As shown in Figs. 13 and 14, the fusion curve is inclined either towards
+the pressure axis or away from it; that is, increase of pressure can either
+lower or raise the melting point. It is easy to predict in a qualitative
+manner the different effect of pressure on the melting point in the two
+cases mentioned, if we consider the matter in the light of the theorem of
+Le Chatelier (p. 58). Water, on passing into ice, expands; therefore, if
+the pressure on the system ice--water be increased, a reaction will take
+place which is accompanied by a diminution in volume, _i.e._ the ice will
+melt. Consequently, a lower temperature will be required in order to
+counteract the effect of increase of pressure; or, in other words, the
+melting point will {67} be lowered by pressure.[116] In the second case,
+the passage of the liquid to the solid state is accompanied by a diminution
+of volume; the effect of increase of pressure will therefore be the reverse
+of that in the previous case.
+
+If the value of the heat of fusion and the alteration of volume
+accompanying the change of state are known, it is possible to calculate
+_quantitatively_ the effect of pressure.[117]
+
+We have already seen (p. 25) that the effect of pressure on the melting
+point of a substance was predicted as the result of theoretical
+considerations, and was first proved experimentally in the case of ice.
+Soon after, Bunsen[118] showed that the melting point of other substances
+is also affected by pressure; and in more recent years, ample experimental
+proof of the change of the melting point with the pressure has been
+obtained. The change of the melting point is, however, small; as a rule,
+increase of pressure by 1 atm. changes the melting point by about 0.03deg,
+but in the case of water the change is much less (0.0076deg), and in the
+case of camphor much more (0.13deg). In other words, if we take the mean
+case, an increase of pressure of more than 30 atm. is required to produce a
+change in the melting point of 1deg.
+
+Investigations which were made of the influence of pressure on the
+melting-point, showed that up to pressures of several hundred atmospheres
+the fusion curve is a straight line.[119] Tammann[120] has, however, found
+that on increasing the pressure the fusion curve no longer remains
+straight, but bends towards the pressure axis, so that, on sufficiently
+increasing the pressure, a maximum temperature might at length be reached.
+This maximum has, so far, however, not been attained, although the melting
+point curves of various substances have been studied up to pressures of
+4500 atm. This is to be accounted for partly {68} by the fact that the
+probable maximum temperature in the case of most substances lies at very
+great pressures, and also by the fact that other solid phases make their
+appearance, as, for example, in the case of ice (p. 32).
+
+As to the upper limit of the fusion curve, the view has been expressed[121]
+that just as in the case of liquid and vapour, so also in the case of solid
+and liquid, there exists a critical point at which the solid and the liquid
+phase become identical. Experimental evidence, however, does not appear to
+favour this view.[122]
+
+The _transition point_, like the melting point, is also influenced by the
+pressure, and in this case also it is found that pressure may either raise
+or lower the transition point, so that the transition curve may be inclined
+either away from or towards the pressure axis. The direction of the
+transition curve can also be predicted if the change of volume accompanying
+the passage of one form into the other is known. In the case of sulphur, we
+saw that the transition point is raised by increase of pressure; in the
+case of the transition of rhombohedral into [alpha]-rhombic form of
+ammonium nitrate, however, the transition point is lowered by pressure, as
+shown by the following table.[123]
+
+  -------------+----------
+               |
+  Temperature. | Pressure.
+  -------------+----------
+               |
+    85.85deg   |   1 atm.
+    84.38deg   | 100  "
+    83.03deg   | 200  "
+    82.29deg   | 250  "
+  -------------+----------
+
+So far as investigations have been carried out, it appears that in most
+cases the transition curve is practically a straight line.
+
+It has, however, been found in the case of Glauber's salt, that with
+increase of pressure the transition curve passes through a point of maximum
+temperature, and exhibits, therefore, a form similar to that assumed by
+Tammann for the fusion curve.[124]
+
+{69}
+
+Suspended Transformation. Metastable Equilibria.--Hitherto we have
+considered only systems in stable equilibrium. We have, however, already
+seen, in the case of water, that on cooling the liquid down to the triple
+point, solidification did not necessarily take place, although the
+conditions were such as to allow of its formation. Similarly, we saw that
+rhombic sulphur can be heated above the transition point, and monoclinic
+sulphur can be obtained at temperatures below the transition point,
+although in both cases transformation into a more stable form is possible;
+the system becomes metastable.
+
+The same reluctance to form a new phase is observed also in the phenomena
+of superheating of liquids, and in the "hanging" of mercury in barometers,
+in which case the vapour phase is not formed. In general, then, we may say
+that _a new phase will not necessarily be formed immediately the system
+passes into such a condition that the existence of that phase is possible_;
+but rather, instead of the system undergoing transformation so as to pass
+into the most stable condition under the existing pressure and temperature,
+this transformation will be "suspended" or delayed, and the system will
+become metastable. Only in the case of the formation of the liquid from the
+solid phase, in a one-component system, has this reluctance to form a new
+phase not been observed.
+
+_To ensure the formation of the new phase, it is necessary to have that
+phase present._ The presence of the solid phase will prevent the
+supercooling of the liquid; and the presence of the vapour phase will
+prevent the superheating of the liquid. However, even in the presence of
+the more stable phase, transformation of the metastable phase occurs with
+very varying velocity; in some cases so quickly as to appear almost
+instantaneous; while in other cases, the change takes place so slowly as to
+require hundreds of years for its achievement. It is this slow rate of
+transformation that renders the existence of metastable forms possible,
+when in contact with the more stable phase. Thus, for example, although
+calcite is the most stable form of calcium carbonate at the ordinary
+temperature,[125] the less stable {70} modification, aragonite,
+nevertheless exists under the ordinary conditions in an apparently very
+stable state.
+
+As to the amount of the new phase required to bring about the
+transformation of the metastable phase, quantitative measurements have been
+carried out only in the case of the initiation of crystallization in a
+supercooled liquid.[126] As the result of these investigations, it was
+found that, in the case of superfused salol, the very small amount of 1 x
+10^{-7} gm. of the solid phase was sufficient to induce crystallization.
+Crystallization of a supercooled liquid, however, can be initiated only by
+a "nucleus" of the same substance in the solid state, or, as has also been
+found, by a nucleus of an isomorphous solid phase; it is not brought about
+by the presence of any chance solid.
+
+Velocity of Transformation.--Attention has already been drawn to the
+sluggishness with which reciprocal transformation of the polymorphic forms
+of a substance may occur. In the case of tin, for example, it was found
+that the white modification, although apparently possessing permanence, is
+in reality in a metastable state, under the ordinary conditions of
+temperature and pressure. This great degree of stability is due to the
+tardiness with which transformation into the grey form occurs.
+
+What was found in the case of tin, is met with also in the case of all
+transformations in the solid state, but the velocity of the change is less
+in some cases than in others, and appears to decrease with increase of the
+valency of the element.[127] To this fact van't Hoff attributes the great
+permanence of many really unstable (or metastable) carbon compounds.
+
+Reference has been made to the fact that the velocity of transformation can
+be accelerated by various means. One of the most important of these is the
+employment of a liquid which has a solvent action on the solid phases. Just
+as we have seen that at any given temperature the less stable form has the
+higher vapour pressure, but that at the transition point the vapour
+pressure of both forms becomes identical, so also it can be proved
+theoretically, and be shown experimentally, that {71} at a given
+temperature the solubility of the less stable form is greater than that of
+the more stable, but that at the transition point the solubility of the two
+forms becomes identical.[128]
+
+If, then, the two solid phases are brought into contact with a solvent, the
+less stable phase will dissolve more abundantly than the more stable; the
+solution will therefore become supersaturated with respect to the latter,
+which will be deposited. A gradual change of the less stable form,
+therefore, takes place through the medium of the solvent. In this way the
+more rapid conversion of white tin into grey in presence of a solution of
+tin ammonium chloride (p. 42) is to be explained. Although, as a rule,
+solvents accelerate the transformation of one solid phase into the other,
+they may also have a retarding influence on the velocity of transformation,
+as was found by Reinders in the case of mercuric iodide.[129]
+
+The velocity of inversion, also, is variously affected by different
+solvents, and in some cases, at least, it appears to be slower the more
+viscous the solvent;[130] indeed, Kastle and Reed state that yellow
+crystals of mercuric iodide, which, ordinarily, change with considerable
+velocity into the red modification, have been preserved for more than a
+year under vaseline.
+
+Temperature, also, has a very considerable influence on the velocity of
+transformation. The higher the temperature, and the farther it is removed
+from the equilibrium point (transition point), the greater is the velocity
+of change. Above the transition point, these two factors act in the same
+direction, and the velocity of transformation will therefore go on
+increasing indefinitely the higher the temperature is raised. Below the
+transition point, however, the two factors act in opposite directions, and
+the more the temperature is lowered, the more is the effect of removal from
+the equilibrium point counteracted. A point will therefore be reached at
+which the velocity is a maximum. Reduction of the temperature {72} below
+this point causes a rapid falling off in the velocity of change. The point
+of maximum velocity, however, is not definite, but may be altered by
+various causes. Thus, Cohen found that in the case of tin, the point of
+maximum velocity was altered if the metal had already undergone
+transformation; and also by the presence of different liquids.[131]
+
+Lastly, the presence of small quantities of different substances--catalytic
+agents or catalyzers--has a great influence on the velocity of
+transformation. Thus, _e.g._, the conversion of white to red phosphorus is
+accelerated by the presence of iodine (p. 47).
+
+Greater attention, however, has been paid to the study of the velocity of
+crystallization of a supercooled liquid, the first experiments in this
+direction having been made by Gernez[132] on the velocity of
+crystallization of phosphorus and sulphur. Since that time, the velocity of
+crystallization of other supercooled liquids has been investigated; such as
+acetic acid and phenol by Moore;[133] supercooled water by Tumlirz;[134]
+and a number of organic substances by Tammann,[135] Friedlaender and
+Tammann,[136] and by Bogojawlenski.[137]
+
+In measuring the velocity of crystallization, the supercooled liquids were
+contained in narrow glass tubes, and the time required for the
+crystallization to advance along a certain length of the tube was
+determined, the velocity being expressed in millimetres per minute. The
+results which have so far been obtained may be summarized as follows. For
+any given degree of supercooling of a substance, the velocity of
+crystallization is constant. As the degree of supercooling increases, the
+velocity of crystallization also increases, until a certain point is
+reached at which the velocity is a maximum, which has a definite
+characteristic value for each substance. This maximum velocity remains
+constant over a certain range of {73} temperature; thereafter, the velocity
+diminishes fairly rapidly, and, with sufficient supercooling, may become
+zero. The liquid then passes into a glassy mass, which will remain
+(practically) permanent even in contact with the crystalline solid.
+
+In ordinary glass we have a familiar example of a liquid which has been
+cooled to a temperature at which crystallization takes place with very
+great slowness. If, however, glass is heated, a temperature is reached,
+much below the melting point of the glass, at which crystallization occurs
+with appreciable velocity, and we observe the phenomenon of
+devitrification.[138]
+
+When the velocity of crystallization is studied at temperatures above the
+maximum point, it is found that the velocity is diminished by the addition
+of foreign substances; and in many cases, indeed, it has been found that
+the diminution is the same for equimolecular quantities of different
+substances. It would hence appear possible to utilize this behaviour as a
+method for determining molecular weights.[139] The rule is, however, by no
+means a universal one. Thus it has been found by F. Dreyer,[140] in
+studying the velocity of crystallization of formanilide, that the
+diminution in the velocity produced by equivalent amounts of different
+substances is not the same, but that the foreign substances exercise a
+specific influence. Further, von Pickardt's rule does not hold when the
+foreign substance forms mixed crystals (Chap. X.) with the crystallizing
+substance.[141]
+
+Law of Successive Reactions.--When sulphur vapour is cooled at the ordinary
+temperature, it first of all condenses to drops of liquid, which solidify
+in an amorphous form, and only after some time undergo crystallization; or,
+when phosphorus vapour is condensed, white phosphorus is first formed, and
+not the more stable form--red phosphorus. It has also been observed that
+even at the ordinary temperature (therefore much below the transition
+point) sulphur may crystallize out from solution in benzene, alcohol,
+carbon disulphide, and other {74} solvents, in the prismatic form, the less
+stable prismatic crystals then undergoing transformation into the rhombic
+form;[142] a similar behaviour has also been observed in the transformation
+of the monotropic crystalline forms of sulphur.[143]
+
+Many other examples might be given. In organic chemistry, for instance, it
+is often found that when a substance is thrown out of solution, it is first
+deposited as a liquid, which passes later into the more stable crystalline
+form. In analysis, also, rapid precipitation from concentrated solution
+often causes the separation of a less stable and more soluble amorphous
+form.
+
+On account of the great frequency with which the prior formation of the
+less stable form occurs, Ostwald[144] has put forward the _law of
+successive reactions_, which states that when a system passes from a less
+stable condition it does not pass directly into the most stable of the
+possible states; but into the next more stable, and so step by step into
+the most stable. This law explains the formation of the metastable forms of
+monotropic substances, which would otherwise not be obtainable. Although it
+is not always possible to observe the formation of the least stable form,
+it should be remembered that that may quite conceivably be due to the great
+velocity of transformation of the less stable into the more stable form.
+From what we have learned about the velocity of transformation of
+metastable phases, we can understand that rapid cooling to a low
+temperature will tend to preserve the less stable form; and, on account of
+the influence of temperature in increasing the velocity of change, it can
+be seen that the formation of the less stable form will be more difficult
+to observe in superheated than in supercooled systems. The factors,
+however, which affect the readiness with which {75} the less stable
+modification is produced, appear to be rather various.[145]
+
+Although a number of at least apparent exceptions to Ostwald's law have
+been found, it may nevertheless be accepted as a very useful generalization
+which sums up very frequently observed phenomena.
+
+       *       *       *       *       *
+
+
+{76}
+
+CHAPTER V
+
+SYSTEMS OF TWO COMPONENTS--PHENOMENA OF DISSOCIATION
+
+In the preceding pages we have studied the behaviour of systems consisting
+of only one component, or systems in which all the phases, whether solid,
+liquid, or vapour, had the same chemical composition (p. 13). In some
+cases, as, for example, in the case of phosphorus and sulphur, the
+component was an elementary substance; in other cases, however, _e.g._
+water, the component was a compound. The systems which we now proceed to
+study are characterized by the fact that the different phases have no
+longer all the same chemical composition, and cannot, therefore, according
+to definition, be considered as one-component systems.
+
+In most cases, little or no difficulty will be experienced in deciding as
+to the _number_ of the components, if the rules given on pp. 12 and 13 are
+borne in mind. If the composition of all the phases, each regarded as a
+whole, is the same, the system is to be regarded as of the first order, or
+a one-component system; if the composition of the different phases varies,
+the system must contain more than one component. If, in order to _express_
+the composition of all the phases present when the system is in
+equilibrium, two of the constituents participating in the equilibrium are
+necessary and sufficient, the system is one of two components. Which two of
+the possible substances are to be regarded as components will, however, be
+to a certain extent a matter of arbitrary choice.
+
+The principles affecting the choice of components will best be learned by a
+study of the examples to be discussed in the sequel. {77}
+
+Different Systems of Two Components.--Applying the Phase Rule
+
+  P + F = C + 2
+
+to systems of two components, we see that in order that the system may be
+invariant, there must be four phases in equilibrium together; two
+components in three phases constitute a univariant, two components in two
+phases a bivariant system. In the case of systems of one component, the
+highest degree of variability found was two (one component in one phase);
+but, as is evident from the formula, there is a higher degree of freedom
+possible in the case of two-component systems. Two components existing in
+only one phase constitute a tervariant system, or a system with three
+degrees of freedom. In addition to the pressure and temperature, therefore,
+a third variable factor must be chosen, and as such there is taken the
+_concentration of the components_. In systems of two components, therefore,
+not only may there be change of pressure and temperature, as in the case of
+one-component systems, but the concentration of the components in the
+different phases may also alter; a variation which did not require to be
+considered in the case of one-component systems.
+
+[Illustration: FIG. 18.]
+
+Since a two-component system may undergo three possible {78} independent
+variations, we should require for the graphic representation of all the
+possible conditions of equilibrium a system of three co-ordinates in space,
+three axes being chosen, say, at right angles to one another, and
+representing the three variables--pressure, temperature, and concentration
+of components (Fig. 18). A curve (_e.g._ AB) in the plane containing the
+pressure and temperature axes would then represent the change of pressure
+with the temperature, the concentration remaining unaltered (_pt_-diagram);
+one in the plane containing the pressure and concentration axes (_e.g._ AF
+or DF), the change of pressure with the concentration, the temperature
+remaining constant (_pc_-diagram), while in the plane containing the
+concentration and the temperature axes, the simultaneous change of these
+two factors at constant pressure would be represented (_tc_-diagram). If
+the points on these three curves are joined together, a surface, ABDE, will
+be formed, and any line on that surface (_e.g._ FG, or GH, or GI) would
+represent the simultaneous variation of the three factors--pressure,
+temperature, concentration. Although we shall at a later point make some
+use of these solid figures, we shall for the present employ the more
+readily intelligible plane diagram.
+
+The number of different systems which can be formed from two components, as
+well as the number of the different phenomena which can there be observed,
+is much greater than in the case of one component. In the case of no two
+substances, however, have all the possible relationships been studied; so
+that for the purpose of gaining an insight into the very varied behaviour
+of two-component systems, a number of different examples will be discussed,
+each of which will serve to give a picture of some of the relationships.
+
+Although the strict classification of the different systems according to
+the Phase Rule would be based on the variability of the systems, the study
+of the many different phenomena, and the correlation of the comparatively
+large number of different systems, will probably be rendered easiest by
+grouping these different phenomena into classes, each of these classes
+being studied with the help of one or more typical examples. The order of
+treatment adopted here is, of course, quite arbitrary; {79} but has been
+selected from considerations of simplicity and clearness.
+
+PHENOMENA OF DISSOCIATION.
+
+Bivariant Systems.--As the first examples of the equilibria between a
+substance and its products of dissociation, we shall consider very briefly
+those cases in which there is one solid phase in equilibrium with vapour.
+Reference has already been made to such systems in the case of ammonium
+chloride. On being heated, ammonium chloride dissociates into ammonia and
+hydrogen chloride. Since, however, in that case the vapour phase has the
+same total composition as the solid phase, viz. NH_{3} + HCl = NH_{4}Cl,
+the system consists of only one component existing in two phases; it is
+therefore univariant, and to each temperature there will correspond a
+definite vapour pressure (dissociation pressure).[146]
+
+If, however, excess of one of the products of dissociation be added, the
+system becomes one of two components.
+
+In the first place, analysis of each of the two phases yields as the
+composition of each, solid: NH_{4}Cl (= NH_{3} + HCl); vapour: _m_NH_{3} +
+_n_HCl. Obviously the smallest number of substances by which the
+composition of the two phases can be expressed is two; that is, the number
+of components is two. What, then, are the components? The choice lies
+between NH_{3} + HCl, NH_{4}Cl + NH_{3}, and NH_{4}Cl + HCl; for the three
+substances, ammonium chloride, ammonia, hydrogen chloride, are the only
+ones taking part in the equilibrium of the system.
+
+Of these three pairs of components, we should obviously choose as the most
+simple NH_{3} and HCl, for we can then represent the composition of the two
+phases as the _sum_ of the two components. If one of the other two possible
+pairs of components be chosen, we should have to introduce negative
+quantities of one of the components, in order to represent the composition
+of the vapour phase. Although it must be allowed that the introduction of
+negative quantities of a component in such cases is quite permissible,
+still it will be {80} better to adopt the simpler and more direct choice,
+whereby the composition of each of the phases is represented as a sum of
+two components in varying proportions (p. 12).
+
+If, therefore, we have a solid substance, such as ammonium chloride, which
+dissociates on volatilization, and if the products of dissociation are
+added in varying amounts to the system, we shall have, in the sense of the
+Phase Rule, a _two-component system existing in two phases_. Such a system
+will possess two degrees of freedom. At any given temperature, not only the
+pressure, but also the composition, of the vapour-phase, _i.e._ the
+concentration of the components, can vary. Only after one of these
+independent variables, pressure or composition, has been arbitrarily fixed
+does the system become univariant, and exhibit a definite, constant
+pressure at a given temperature.
+
+Now, although the Phase Rule informs us that at a given temperature change
+of composition of the vapour phase will be accompanied by change of
+pressure, it does not cast any light on the relation between these two
+variables. This relationship, however, can be calculated theoretically by
+means of the Law of Mass Action.[147] From this we learn that in the case
+of a substance which dissociates into equivalent quantities of two gases,
+the product of the partial pressures of the gases is constant at a given
+temperature.
+
+This has been proved experimentally in the case of ammonium hydrosulphide,
+ammonium cyanide, phosphonium bromide, and other substances.[148]
+
+Univariant Systems.--In order that a system of two components shall possess
+only one degree of freedom, three phases must be present. Of such systems,
+there are seven possible, viz. S-S-S, S-S-L, S-S-V, L-L-L, S-L-L, L-L-V,
+S-L-V; S denoting solid, L liquid, and V vapour. In the present chapter we
+shall consider only the systems S-S-V, _i.e._ those systems in which there
+are two solid phases and a vapour phase present.
+
+{81}
+
+As an example of this, we may first consider the well-known case of the
+dissociation of calcium carbonate. This substance on being heated
+dissociates into calcium oxide, or quick-lime, and carbon dioxide, as shown
+by the equation CaCO_{3} <--> CaO + CO_{2}. In accordance with our
+definition (p. 9), we have here two solid phases, the carbonate and the
+quick-lime, and one vapour phase; the system is therefore univariant. To
+each temperature, therefore, there will correspond a certain, definite
+maximum pressure of carbon dioxide (dissociation pressure), and this will
+follow the same law as the vapour pressure of a pure liquid (p. 21). More
+particularly, it will be independent of the relative or absolute amounts of
+the two solid phases, and of the volume of the vapour phase. If the
+temperature is maintained constant, increase of volume will cause the
+dissociation of a further amount of the carbonate until the pressure again
+reaches its maximum value corresponding to the given temperature.
+Diminution of volume, on the other hand, will bring about the combination
+of a certain quantity of the carbon dioxide with the calcium oxide until
+the pressure again reaches its original value.
+
+The dissociation pressure of calcium carbonate was first studied by
+Debray,[149] but more exact measurements have been made by Le
+Chatelier,[150] who found the following corresponding values of temperature
+and pressure:--
+
+  -------------+-------------------------
+               |
+  Temperature. | Pressure in cm. mercury.
+  -------------+-------------------------
+               |
+     547deg    |             2.7
+     610deg    |             4.6
+     625deg    |             5.6
+     740deg    |            25.5
+     745deg    |            28.9
+     810deg    |            67.8
+     812deg    |            76.3
+     865deg    |           133.3
+  -------------+-------------------------
+
+From this table we see that it is only at a temperature of about 812deg
+that the pressure of the carbon dioxide becomes equal to atmospheric
+pressure. In a vessel open to {82} the air, therefore, the complete
+decomposition of the calcium carbonate would not take place below this
+temperature by the mere heating of the carbonate. If, however, the carbon
+dioxide is removed as quickly as it is formed, say by a current of air,
+then the entire decomposition can be made to take place at a much lower
+temperature. For the dissociation equilibrium of the carbonate depends only
+on the partial pressure of the carbon dioxide, and if this is kept small,
+then the decomposition can proceed, even at a temperature below that at
+which the pressure of the carbon dioxide is less than atmospheric pressure.
+
+Ammonia Compounds of Metal Chlorides.--Ammonia possesses the property of
+combining with various substances, chiefly the halides of metals, to form
+compounds which again yield up the ammonia on being heated. Thus, for
+example, on passing ammonia over silver chloride, absorption of the gas
+takes place with formation of the substances AgCl,3NH_{3} and
+2AgCl,3NH_{3}, according to the conditions of the experiment. These were
+the first known substances belonging to this class, and were employed by
+Faraday in his experiments on the liquefaction of ammonia. Similar
+compounds have also been obtained by the action of ammonia on silver
+bromide, iodide, cyanide, and nitrate; and with the halogen compounds of
+calcium, zinc, and magnesium, as well as with other salts. The behaviour of
+the ammonia compounds of silver chloride is typical for the compounds of
+this class, and may be briefly considered here.
+
+It was found by Isambert[151] that at temperatures below 15deg, silver
+chloride combined with ammonia to form the compound AgCl,3NH_{3}, while at
+temperatures above 20deg the compound 2AgCl,3NH_{3} was produced. On
+heating these substances, ammonia was evolved, and the pressure of this gas
+was found in the case of both compounds to be constant at a given
+temperature, but was greater in the case of the former than in the case of
+the latter substance; the pressure, further, was independent of the amount
+decomposed. The behaviour of these two substances is, therefore, exactly
+analogous to that shown by calcium carbonate, and the explanation is also
+similar.
+
+{83}
+
+Regarded from the point of view of the Phase Rule, we see that we are here
+dealing with two components, AgCl and NH_{3}. On being heated, the
+compounds decompose according to the equations:--
+
+  2(AgCl,3NH_{3}) <--> 2AgCl,3NH_{3} + 3NH_{3}.
+    2AgCl,3NH_{3} <--> 2AgCl + 3NH_{3}.
+
+There are, therefore, three phases, viz. AgCl,3NH_{3}; 2AgCl,3NH_{3}, and
+NH_{3}, in the one case; and 2AgCl,3NH_{3}; AgCl, and NH_{3} in the other.
+These two systems are therefore univariant, and to each temperature there
+must correspond a definite pressure of dissociation, quite irrespective of
+the amounts of the phases present. Similarly, if, at constant temperature,
+the volume is increased (or if the ammonia which is evolved is pumped off),
+the pressure will remain constant so long as two solid phases, AgCl,3NH_{3}
+and 2AgCl,3NH_{3}, are present, _i.e._ until the compound richer in ammonia
+is completely decomposed, when there will be a sudden fall in the pressure
+to the value corresponding to the system 2AgCl,3NH_{3}--AgCl--NH_{3}. The
+pressure will again remain constant at constant temperature, until all the
+ammonia has been pumped off, when there will again be a sudden fall in the
+pressure to that of the system formed by solid silver chloride in contact
+with its vapour.
+
+The reverse changes take place when the pressure of the ammonia is
+gradually increased. If the volume is continuously diminished, the pressure
+will first increase until it has reached a certain value; the compound
+2AgCl,3NH_{3} can then be formed, and the pressure will now remain constant
+until all the silver chloride has disappeared. The pressure will again
+rise, until it has reached the value at which the compound AgCl,3NH_{3} can
+be formed, when it will again remain constant until the complete
+disappearance of the lower compound. _There is no gradual change of
+pressure_ on passing from one system to another; but the changes are
+abrupt, as is demanded by the Phase Rule, and as experiment has
+conclusively proved.[152]
+
+The dissociation pressures of the two compounds of silver {84} chloride and
+ammonia, as determined by Isambert,[153] are given in the following
+table:--
+
+  -------------------------+-------------------------
+                           |
+          AgCl,3NH_{3}.    |        2AgCl,3NH_{3}.
+  -------------+-----------+--------------+----------
+               |           |              |
+  Temperature. | Pressure. | Temperature. | Pressure.
+  -------------+-----------+--------------+----------
+               |           |              |
+        0deg   |  29.3 cm. |    20.0deg   |   9.3 cm.
+     10.6deg   |  50.5 "   |    31.0deg   |  12.5 "
+     17.5deg   |  65.5 "   |    47.0deg   |  26.8 "
+     24.0deg   |  93.7 "   |    58.5deg   |  52.8 "
+     28.0deg   | 135.5 "   |    69.0deg   |  78.6 "
+     34.2deg   | 171.3 "   |    71.5deg   |  94.6 "
+     48.5deg   | 241.4 "   |    77.5deg   | 119.8 "
+     51.5deg   | 413.2 "   |    83.5deg   | 159.3 "
+     54.0deg   | 464.1 "   |    86.1deg   | 181.3 "
+               |           |    88.5deg   | 201.3 "
+  -------------+-----------+--------------+----------
+
+The conditions for the formation of these two compounds, by passing ammonia
+over silver chloride, to which reference has already been made, will be
+readily understood from the above tables. In the case of the triammonia
+mono-chloride, the dissociation pressure becomes equal to atmospheric
+pressure at a temperature of about 20deg; above this temperature,
+therefore, it cannot be formed by the action of ammonia at atmospheric
+pressure on silver chloride. The triammonia dichloride can, however, be
+formed, for its dissociation pressure at this temperature amounts to only 9
+cm., and becomes equal to the atmospheric pressure only at a temperature of
+about 68deg; and this temperature, therefore, constitutes the limit above
+which no combination can take place between silver chloride and ammonia
+under atmospheric pressure.
+
+Attention may be here drawn to the fact, to which reference will also be
+made later, that _two_ solid phases are necessary in order that the
+dissociation pressure at a given temperature shall be definite; _and for
+the exact definition of this pressure it is necessary to know, not merely
+what is the substance undergoing dissociation, but also what is the solid
+product of dissociation formed_. For the definition of the equilibrium, the
+latter is as important as the former. We shall presently find proof of this
+in the case {85} of an analogous class of phenomena, viz. the dissociation
+of salt hydrates.
+
+Salts with Water of Crystallization.--In the case of the dehydration of
+crystalline salts containing water of crystallization, we meet with
+phenomena which are in all respects similar to those just studied. A salt
+hydrate on being heated dissociates into a lower hydrate (or anhydrous
+salt) and water vapour. Since we are dealing with two components--salt and
+water[154]--in three phases, viz. hydrate _a_, hydrate _b_ (or anhydrous
+salt), and vapour, the system is univariant, and to each temperature there
+will correspond a certain, definite vapour pressure (the dissociation
+pressure), which will be independent of the relative or absolute amounts of
+the phases, _i.e._ of the amount of hydrate which has already undergone
+dissociation or dehydration.
+
+[Illustration: FIG. 19.]
+
+The constancy of the dissociation pressure had been proved experimentally
+by several investigators[155] a number of years before the theoretical
+basis for its necessity had been given. In the case of salts capable of
+forming more than one hydrate, we should obtain a series of dissociation
+curves (_pt_-curves), as in the case of the different hydrates of copper
+sulphate. In Fig. 19 there are represented diagrammatically the
+vapour-pressure curves of the following univariant systems of copper
+sulphate and water:--
+
+  Curve OA: CuSO_{4},5H_{2}O <--> CuSO_{4},3H_{2}O + 2H_{2}O.
+  Curve OB: CuSO_{4},3H_{2}O <--> CuSO_{4},H_{2}O + 2H_{2}O.
+  Curve OC: CuSO_{4},H_{2}O   <--> CuSO_{4} + H_{2}O.
+
+Let us now follow the changes which take place on {86} increasing the
+pressure of the aqueous vapour in contact with anhydrous copper sulphate,
+the temperature being meanwhile maintained constant. If, starting from the
+point D, we slowly add water vapour to the system, the pressure will
+gradually rise, without formation of hydrate taking place; for at pressures
+below the curve OC only the anhydrous salt can exist. At E, however, the
+hydrate CuSO_{4},H_{2}O will be formed, and as there are now three phases
+present, viz. CuSO_{4}, CuSO_{4},H_{2}O, and vapour, the system becomes
+_univariant_; and since the temperature is constant, the pressure must also
+be constant. Continued addition of vapour will result merely in an increase
+in the amount of the hydrate, and a decrease in the amount of the anhydrous
+salt. When the latter has entirely disappeared, _i.e._ has passed into
+hydrated salt, the system again becomes _bivariant_, and passes along the
+line EF; the pressure gradually increases, therefore, until at F the
+hydrate 3H_{2}O is formed, and the system again becomes univariant; the
+three phases present are CuSO_{4},H_{2}O, CuSO_{4},3H_{2}O, vapour. The
+pressure will remain constant, therefore, until the hydrate 1H_{2}O has
+disappeared, when it will again increase till G is reached; here the
+hydrate 5H_{2}O is formed, and the pressure once more remains constant
+until the complete disappearance of the hydrate 3H_{2}O has taken place.
+
+Conversely, on dehydrating CuSO_{4},5H_{2}O at constant temperature, we
+should find that the pressure would maintain the value corresponding to the
+dissociation pressure of the system
+CuSO_{4},5H_{2}O--CuSO_{4},3H_{2}O--vapour, until all the hydrate 5H_{2}O
+had disappeared; further removal of water would then cause the pressure to
+fall _abruptly_ to the pressure of the system
+CuSO_{4},3H_{2}O--CuSO_{4},H_{2}O--vapour, at which value it would again
+remain constant until the tri-hydrate had passed into the monohydrate, when
+a further sudden diminution of the pressure would occur. This behaviour is
+represented diagrammatically in Fig. 20, the values of the pressure being
+those at 50deg.
+
+Efflorescence.--From Fig. 19 we are enabled to predict the conditions under
+which a given hydrated salt will effloresce when exposed to the air. We
+have just learned that copper {87} sulphate pentahydrate, for example, will
+not be formed unless the pressure of the aqueous vapour reaches a certain
+value; and that conversely, if the vapour pressure falls below the
+dissociation pressure of the pentahydrate, this salt will undergo
+dehydration. From this, then, it is evident that a crystalline salt hydrate
+will effloresce when exposed to the air, if the partial pressure of the
+water vapour in the air is lower than the dissociation pressure of the
+hydrate. At the ordinary temperature the dissociation pressure of copper
+sulphate is less than the pressure of water vapour in the air, and
+therefore copper sulphate does not effloresce. In the case of sodium
+sulphate decahydrate, however, the dissociation pressure is greater than
+the normal vapour pressure in a room, and this salt therefore effloresces.
+
+[Illustration: FIG. 20.]
+
+Indefiniteness of the Vapour Pressure of a Hydrate.--Reference has already
+been made (p. 84), in the case of the ammonia compounds of the metal
+chlorides, to the importance of the solid product of dissociation for the
+definition of the dissociation pressure. Similarly also in the case of a
+hydrated salt. A salt hydrate in contact with vapour constitutes only a
+bivariant system, and can exist therefore at different values of
+temperature and pressure of vapour, as is seen from the diagram, Fig. 19.
+Anhydrous copper sulphate can exist in contact with water vapour at all
+values of temperature and pressure lying in the field below the curve OC;
+and the hydrate CuSO_{4},H_{2}O can exist in contact with vapour at all
+values of temperature and pressure in the field BOC. Similarly, each of the
+other hydrates can exist in contact with vapour at different values of
+temperature and pressure.
+
+From the Phase Rule, however, we learn that, in order that at a given
+temperature the pressure of a two-component system {88} may be constant,
+there must be three phases present. Strictly, therefore, we can speak only
+of the vapour pressure of a _system_; and since, in the cases under
+discussion, the hydrates dissociate into a solid and a vapour, any
+statement as to the vapour pressure of a hydrate has a definite meaning
+_only when the second solid phase produced by the dissociation is given_.
+The everyday custom of speaking of the vapour pressure of a hydrated salt
+acquires a meaning only through the assumption, tacitly made, that the
+second solid phase, or the solid produced by the dehydration of the
+hydrate, is the _next lower_ hydrate, where more hydrates than one exist.
+That a hydrate always dissociates in such a way that the next lower hydrate
+is formed is, however, by no means certain; indeed, cases have been met
+with where apparently the anhydrous salt, and not the lower hydrate (the
+existence of which was possible), was produced by the dissociation of the
+higher hydrate.[156]
+
+That a salt hydrate can exhibit different vapour pressures according to the
+solid product of dissociation, can not only be proved theoretically, but it
+has also been shown experimentally to be a fact. Thus CaCl_{2},6H_{2}O can
+dissociate into water vapour and either of two lower hydrates, each
+containing four molecules of water of crystallization, and designated
+respectively as CaCl_{2},4H_{2}O[alpha], and CaCl_{2},4H_{2}O[beta].
+Roozeboom[157] has shown that the vapour pressure which is obtained differs
+according to which of these two hydrates is formed, as can be seen from the
+following figures:--
+
+  -------------+----------------------------------------------------------
+               |                  Pressure of System.
+  Temperature. +-----------------------------+----------------------------
+               | CaCl_{2},6H_{2}O; CaCl_{2}, | CaCl_{2},6H_{2}O; CaCl_{2},
+               | 4H_{2}O[alpha]; vapour.     | 4H_{2}O[beta]; vapour.
+  -------------+-----------------------------+----------------------------
+       -15deg  |         0.027 cm.           |         0.022 cm.
+         0     |         0.092  "            |         0.076  "
+       +10     |         0.192  "            |         0.162  "
+        20     |         0.378  "            |         0.315  "
+        25     |         0.508  "            |         0.432  "
+        29.2   |            --               |         0.567  "
+        29.8   |         0.680  "            |            --
+  -------------+-----------------------------+---------------------------
+
+{89}
+
+By reason of the non-recognition of the importance of the solid
+dissociation product for the definition of the dissociation pressure of a
+salt hydrate, many of the older determinations lose much of their value.
+
+Suspended Transformation.--Just as in systems of one component we found
+that a new phase was not necessarily formed when the conditions for its
+existence were established, so also we find that even when the vapour
+pressure is lowered below the dissociation pressure of a system,
+dissociation does not necessarily occur. This is well known in the case of
+Glauber's salt, first observed by Faraday. Undamaged crystals of
+Na_{2}SO_{4},10H_{2}O could be kept unchanged in the open air, although the
+vapour pressure of the system Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
+is greater than the ordinary pressure of aqueous vapour in the air. That is
+to say, the possibility of the formation of the new phase Na_{2}SO_{4} was
+given; nevertheless this new phase did not appear, and the system therefore
+became metastable, or unstable with respect to the anhydrous salt. When,
+however, a trace of the new phase--the anhydrous salt--was brought in
+contact with the hydrate, transformation occurred; the hydrate effloresced.
+
+The possibility of suspended transformation or the non-formation of the new
+phases must also be granted in the case where the vapour pressure is raised
+above that corresponding to the system hydrate--anhydrous salt (or lower
+hydrate)--vapour; in this case the formation of the higher hydrate becomes
+a possibility, but not a certainty. Although there is no example of this
+known in the case of hydrated salts, the suspension of the transformation
+has been observed in the case of the compounds of ammonia with the metal
+chlorides (p. 82). Horstmann,[158] for example, found that the pressure of
+ammonia in contact with 2AgCl,3NH_{3} could be raised to a value higher
+than the dissociation pressure of AgCl,3NH_{3} without this compound being
+formed. We see, therefore, that even when the existence of the higher
+compound in contact with the lower became possible, the higher compound was
+not immediately formed.
+
+Range of Existence of Hydrates.--In Fig. 19 the vapour {90} pressure curves
+of the different hydrates of copper sulphate are represented as maintaining
+their relative positions throughout the whole range of temperatures. But
+this is not necessarily the case. It is possible that at some temperature
+the vapour pressure curve of a lower hydrate may cut that of a higher
+hydrate. At temperatures above the point of intersection, the lower hydrate
+would have a higher vapour pressure than the higher hydrate, and would
+therefore be metastable with respect to the latter. The range of stable
+existence of the lower hydrate would therefore end at the point of
+intersection. This appears to be the case with the two hydrates of sodium
+sulphate, to which reference will be made later.[159]
+
+Constancy of Vapour Pressure and the Formation of Compounds.--We have seen
+in the case of the salt hydrates that the continued addition of the vapour
+phase to the system caused an increase in the pressure until at a definite
+value of the pressure a hydrate is formed; the pressure then becomes
+constant, and remains so, until one of the solid phases has disappeared.
+Conversely, on withdrawing the vapour phase, the pressure remained constant
+so long as any of the dissociating compound was present, independently of
+the degree of the decomposition (p. 86). This behaviour, now, has been
+employed for the purpose of determining whether or not definite chemical
+compounds are formed. Should compounds be formed between the vapour phase
+and the solid, then, on continued addition or withdrawal of the vapour
+phase, it will be found that the vapour pressure remains constant for a
+certain time, and will then suddenly assume a new value, at which it will
+again remain constant. By this method, Ramsay[160] found that no definite
+hydrates were formed in the case of ferric and aluminium oxides, but that
+two are formed in the case of lead oxide, viz. 2PbO,H_{2}O and 3PbO,H_{2}O.
+
+The method has also been applied to the investigation of the so-called
+palladium hydride,[161] and the results obtained appear to show that no
+compound is formed. Reference will, however, be made to this case later
+(Chap. X.).
+
+{91}
+
+Measurement of the Vapour Pressure of Hydrates.--For the purpose of
+measuring the small pressures exerted by the vapour of salt hydrates, use
+is very generally made of a differential manometer called the
+_Bremer-Frowein tensimeter_.[162]
+
+This apparatus has the form shown in Fig. 21. It consists of a U-tube, the
+limbs of which are bent close together, and placed in front of a millimetre
+scale. The bend of the tube is filled with oil or other suitable liquid,
+_e.g._ bromonaphthalene. If it is desired to measure the dissociation
+pressure of, say, a salt hydrate, concentrated sulphuric acid is placed in
+the flask _e_, and a quantity of the hydrate, well dried and powdered,[163]
+in the bulb d. The necks of the bulbs _d_ and _e_ are then sealed off.
+Since, as we have learned, suspended transformation may occur, it is
+advisable to first partially dehydrate the salt, in order to ensure the
+presence of the second solid product of dissociation; the value of the
+dissociation pressure being independent of the degree of dissociation of
+the hydrate (p. 86). The small bulbs _d_ and _e_ having been filled, the
+apparatus is placed on its side, so as to allow the liquid to run from the
+bend of the tube into the bulbs _a_ and _b_; it is then exhausted through
+_f_ by means of a mercury pump, and sealed off. The apparatus is now placed
+in a perpendicular position in a thermostat, and kept at constant
+temperature until equilibrium is established. Since the vapour pressure on
+the side containing the sulphuric acid may be regarded as zero, the
+difference in level of the two surfaces of liquid in the U-tube gives
+directly the dissociation pressure of the hydrate in terms of the
+particular liquid employed; if the density of the latter is known, the
+pressure can then be calculated to cm. of mercury.
+
+[Illustration: FIG. 21.]
+
+       *       *       *       *       *
+
+
+{92}
+
+CHAPTER VI
+
+SOLUTIONS
+
+Definition.--In all the cases which have been considered in the preceding
+pages, the different phases--with the exception of the vapour
+phase--consisted of a single substance of definite composition, or were
+definite chemical individuals.[164] But this invariability of the
+composition is by no means imposed by the Phase Rule; on the contrary, we
+shall find in the examples which we now proceed to study, that the
+participation of phases of variable composition in the equilibrium of a
+system is in no way excluded. To such phases of variable composition there
+is applied the term _solution_. A solution, therefore, is to be defined as
+_a homogeneous mixture, the composition of which can undergo continuous
+variation within certain limits_; the limits, namely, of its
+existence.[165]
+
+From this definition we see that the term solution is not restricted to any
+particular physical state of substances, but includes within its range not
+only the liquid, but also the gaseous and solid states. We may therefore
+have solutions of gases in liquids, and of gases in solids; of liquids in
+liquids or in solids; of solids in liquids, or of solids in solids.
+Solutions of gases in gases are, of course, also possible; since, however,
+gas solutions never give rise to more than one phase, their {93} treatment
+does not come within the scope of the Phase Rule, which deals with
+heterogeneous equilibria.
+
+It should also be emphasized that the definition of solution given above,
+neither creates nor recognizes any distinction between solvent and
+dissolved substance (solute); and, indeed, a too persistent use of these
+terms and the attempt to permanently label the one or other of two
+components as the solvent or the solute, can only obscure the true
+relationships and aggravate the difficulty of their interpretation. In all
+cases it should be remembered that we are dealing with equilibria between
+two components (we confine our attention in the first instance to such),
+the solution being constituted of these components in variable and varying
+amounts. The change from the case where the one component is in great
+excess (ordinarily called the solvent) to that in which the other component
+predominates, may be quite gradual, so that it is difficult or impossible
+to say at what point the one component ceases to be the solvent and becomes
+the solute. The adoption of this standpoint need not, however, preclude one
+from employing the conventional terms solvent and solute in ordinary
+language, especially when reference is made only to some particular
+condition of equilibrium of the system, when the concentration of the two
+components in the solution is widely different.
+
+SOLUTIONS OF GASES IN LIQUIDS.
+
+As the first class of solutions to which we shall turn our attention, there
+may be chosen the solutions of gases in liquids, or the equilibria between
+a liquid and a gas. These equilibria really constitute a part of the
+equilibria to be studied more fully in Chapter VIII.; but since the
+two-phase systems formed by the solutions of gases in liquids are among the
+best-known of the two-component systems, a short section may be here
+allotted to their treatment.
+
+When a gas is passed into a liquid, absorption takes place to a greater or
+less extent, and a point is at length reached when the liquid absorbs no
+more of the gas; a condition of equilibrium is attained, and the liquid is
+said to be saturated {94} with the gas. In the light of the Phase Rule,
+now, such a system is bivariant (two components in two phases); and two of
+the variable factors, pressure, temperature, and concentration of the
+components, must therefore be chosen in order that the condition of the
+system may be defined. If the concentration and the temperature are fixed,
+then the pressure is also defined; or under given conditions of temperature
+and pressure, the concentration of the gas in the solution must have a
+definite value. If, however, the temperature alone is fixed, the
+concentration and the pressure can alter; a fact so well known that it does
+not require to be further insisted on.
+
+As to the way in which the solubility of a gas in a liquid varies with the
+pressure, the Phase Rule of course does not state; but guidance on this
+point is again yielded by the theorem of van't Hoff and Le Chatelier. Since
+the absorption of a gas is in all cases accompanied by a diminution of the
+total volume, this process must take place with increase of pressure. This,
+indeed, is stated in a quantitative manner in the law of Henry, according
+to which the amount of a gas absorbed is proportional to the pressure. But
+this law must be modified in the case of gases which are very readily
+absorbed; the _direction of change_ of concentration with the pressure
+will, however, still be in accordance with the theorem of Le Chatelier.
+
+If, on the other hand, the pressure is fixed, then the concentration will
+vary with the temperature; and since the absorption of gases is in all
+cases accompanied by the evolution of heat, the solubility is found, in
+accordance with the theorem of Le Chatelier, to diminish with rise of
+temperature.
+
+In considering the changes of pressure accompanying changes of
+concentration and temperature, a distinction must be drawn between the
+total pressure and the partial pressure of the dissolved gas, in cases
+where the solvent is volatile. In these cases, the law of Henry applies not
+to the total pressure of the vapour, but only to the partial pressure of
+the dissolved gas. {95}
+
+SOLUTIONS OF LIQUIDS IN LIQUIDS.
+
+When mercury and water are brought together, the two liquids remain side by
+side without mixing. Strictly speaking, mercury undoubtedly dissolves to a
+certain extent in the water, and water no doubt dissolves, although to a
+less extent, in the mercury; the amount of substance passing into solution
+is, however, so minute, that it may, for all practical purposes, be left
+out of account, so long as the temperature does not rise much above the
+ordinary.[166] On the other hand, if alcohol and water be brought together,
+complete miscibility takes place, and one homogeneous solution is obtained.
+Whether water be added in increasing quantities to pure alcohol, or pure
+alcohol be added in increasing amount to water, at no point, at no degree
+of concentration, is a system obtained containing more than one liquid
+phase. At the ordinary temperature, water and alcohol can form only two
+phases, liquid and vapour. If, however, water be added to ether, or if
+ether be added to water, solution will not occur to an indefinite extent;
+but a point will be reached when the water or the ether will no longer
+dissolve more of the other component, and a further addition of water on
+the one hand, or ether on the other, will cause the formation of two liquid
+layers, one containing excess of water, the other excess of ether. We
+shall, therefore, expect to find all grades of miscibility, from almost
+perfect immiscibility to perfect miscibility, or miscibility in all
+proportions. In cases of perfect immiscibility, the components do not
+affect one another, and the system therefore remains unchanged. Such cases
+do not call for treatment here. We have to concern ourselves here only with
+the second and third cases, viz. with cases of complete and of partial
+miscibility. There is no essential difference between the two classes, for,
+as we shall see, {96} the one passes into the other with change of
+temperature. The formal separation into two groups is based on the
+miscibility relations at ordinary temperatures.
+
+Partial or Limited Miscibility.--In accordance with the Phase Rule, a pure
+liquid in contact with its vapour constitutes a univariant system. If,
+however, a small quantity of a second substance is added, which is capable
+of dissolving in the first, a bivariant system will be obtained; for there
+are now two components and, as before, only two phases--the homogeneous
+liquid solution and the vapour. At constant temperature, therefore, both
+the composition of the solution and the pressure of the vapour can undergo
+change; or, if the composition of the solution remains unchanged, the
+pressure and the temperature can alter. If the second (liquid) component is
+added in increasing amount, the liquid will at first remain homogeneous,
+and its composition and pressure will undergo a continuous change; when,
+however, the concentration has reached a definite value, solution no longer
+takes place; two liquid phases are produced. Since there are now three
+phases present, two liquids and vapour, the system is univariant; at a
+given temperature, therefore, the concentration of the components in the
+two liquid phases, as well as the vapour pressure, must have definite
+values. Addition of one of the components, therefore, cannot alter the
+concentrations or the pressure, but can only cause a change in the relative
+amounts of the phases.
+
+The two liquid phases can be regarded, the one as a solution of the
+component I. in component II., the other as a solution of component II. in
+component I. If the pressure is maintained constant, then to each
+temperature there will correspond a definite concentration of the
+components in the two liquid phases; and addition of excess of one will
+merely alter the relative amounts of the two solutions. As the temperature
+changes, the composition of the two solutions will change, and there will
+therefore be obtained two solubility curves, one showing the solubility of
+component I. in component II., the other showing the solubility of
+component II. in component I. Since heat may be either evolved or absorbed
+when one liquid dissolves in another, the solubility may diminish or
+increase {97} with rise of temperature. The two solutions which at a given
+temperature correspond to one another are known as _conjugate solutions_.
+
+The solubility relations of partially miscible liquids have been studied by
+Guthrie,[167] and more especially by Alexejeff[168] and by Rothmund.[169] A
+considerable variety of curves have been obtained, and we shall therefore
+discuss only a few of the different cases which may be taken as typical of
+the rest.
+
+Phenol and Water.--When phenol is added to water at the ordinary
+temperature, solution takes place, and a homogeneous liquid is produced.
+When, however, the concentration of the phenol in the solution has risen to
+about 8 per cent., phenol ceases to be dissolved; and a further addition of
+it causes the formation of a second liquid phase, which consists of excess
+of phenol and a small quantity of water. In ordinary language it may be
+called a solution of water in phenol. If now the temperature is raised,
+this second liquid phase will disappear, and a further amount of phenol
+must be added in order to produce a separation of the liquid into two
+layers. In this way, by increasing the amount of phenol and noting the
+temperature at which the two layers disappear, the so-called solubility
+curve of phenol in water can be obtained. By noting the change of the
+solubility with the temperature in this manner, it is found that at all
+temperatures below 68.4deg, the addition of more than a certain amount of
+phenol causes the formation of two layers; at temperatures above this,
+however, two layers cannot be formed, no matter how much phenol is added.
+At temperatures above 68.4deg, therefore, water and phenol are miscible in
+all proportions.
+
+On the other hand, if water is added to phenol at the ordinary temperature,
+a liquid is produced which consists chiefly of phenol, and on increasing
+the amount of water beyond a certain point, two layers are formed. On
+raising the temperature these two layers disappear, and a homogeneous
+solution is again obtained. The phenomena are exactly analogous to those
+already described. Since, now, in the second {98} case the concentration of
+the phenol in the solution gradually decreases, while in the former case it
+gradually increases, a point must at length be reached at which the
+composition of the two solutions becomes the same. On mixing the two
+solutions, therefore, one homogeneous liquid will be obtained. But the
+point at which two phases become identical is called a critical point, so
+that, in accordance with this definition, the temperature at which the two
+solutions of phenol and water become identical may be called the _critical
+solution temperature_, and the concentration at this point may be called
+the _critical concentration_.
+
+[Illustration: FIG. 22.]
+
+From what has been said above, it will be seen that at any temperature
+below the critical solution temperature, two conjugate solutions containing
+water and phenol in different concentration can exist together, one
+containing excess of water, the other excess of phenol. The following table
+gives the composition of the two layers, and the values are represented
+graphically in Fig. 22.[170]
+
+  PHENOL AND WATER.
+
+  C_{1} is the percentage amount of phenol in the first layer.
+  C_{2}         "          "          "           second layer.
+  -------------+--------+--------
+  Temperature. |  C_{1}.|  C_{2}.
+  -------------+--------+--------
+     20deg     |   8.5  |  72.2
+     30deg     |   8.7  |  69.9
+     40deg     |   9.7  |  66.8
+     50deg     |  12.0  |  62.7
+     55deg     |  14.2  |  60.0
+     60deg     |  17.5  |  56.2
+     65deg     |  22.7  |  49.7
+     68.4deg   |  36.1  |  36.1
+  -------------+--------+--------
+
+{99}
+
+The critical solution temperature for phenol and water is 68.4deg, the
+critical concentration 36.1 per cent. of phenol. At all temperatures above
+68.4deg, only homogeneous solutions of phenol and water can be obtained;
+water and phenol are then miscible in all proportions.
+
+At the critical solution point the system exists in only two phases--liquid
+and vapour. It ought, therefore, to possess two degrees of freedom. The
+restriction is, however, imposed that the composition of the two liquid
+phases, coexisting at a point infinitely near to the critical point,
+becomes the same, and this disposes of one of the degrees of freedom. The
+system is therefore univariant; and at a given temperature the pressure
+will have a definite value. Conversely, if the pressure is fixed (as is the
+case when the system is under the pressure of its own vapour), then the
+temperature will also be fixed; that is, the critical solution temperature
+has a definite value depending only on the substances. If the vapour phase
+is omitted, the temperature will alter with the pressure; in this case,
+however, as in the case of other condensed systems, the effect of pressure
+is slight.
+
+From Fig. 22 it is easy to predict the effect of bringing together water
+and phenol in any given quantities at any temperature. Start with a
+solution of phenol and water having the composition represented by the
+point _x_. If to this solution phenol is added at constant temperature, it
+will dissolve, and the composition of the solution will gradually change,
+as shown by the dotted line _xy_. When, however, the concentration has
+reached the value represented by the point _y_, two liquid layers will be
+formed, the one solution having the composition represented by _y_, the
+other that represented by _y'_. The system is now univariant, and on
+further addition of phenol, the composition of the two liquid phases will
+remain unchanged, but their relative amounts will alter. The phase richer
+in phenol will increase in amount; that richer in water will decrease, and
+ultimately disappear, and there will remain the solution _y'_. Continued
+addition of phenol will then lead to the point _x'_, there being now only
+one liquid phase present.
+
+Since the critical solution point represents the highest temperature at
+which two liquid phases consisting of phenol and {100} water can exist
+together, these two substances can be brought together in any amount
+whatever at temperatures higher than 68.4deg, without the formation of two
+layers. It will therefore be possible to pass from a system represented by
+_x_ to one represented by _x'_, without at any time two liquid phases
+appearing. Starting with _x_, the temperature is first raised above the
+critical solution temperature; phenol is then added until the concentration
+reaches the point _x__{2}. On allowing the temperature to fall, the system
+will then pass into the condition represented by _x'_.
+
+[Illustration: FIG. 23.]
+
+Methylethylketone and Water.--In the case just described, the solubility of
+each component in the other increased continuously with the temperature.
+There are, however, cases where a maximum or minimum of solubility is
+found, _e.g._ methylethylketone and water. The curve which represents the
+equilibria between these two substances is given in Fig. 23, the
+concentration values being contained in the following table:[171]--
+
+  METHYLETHYLKETONE AND WATER.
+
+  --------------+-----------------+-----------------
+  Temperature.  | C_{1} per cent. |  C_{2} per cent.
+  --------------+-----------------+-----------------
+      -10deg    |      34.5       |     89.7
+      +10deg    |      26.1       |     90.0
+       30deg    |      21.9       |     89.9
+       50deg    |      17.5       |     89.0
+       70deg    |      16.2       |     85.7
+       90deg    |      16.1       |     84.8
+      110deg    |      17.7       |     80.0
+      130deg    |      21.8       |     71.9
+      140deg    |      26.0       |     64.0
+      151.8deg  |      44.2       |     44.2
+  --------------+-----------------+-----------------
+
+{101}
+
+These numbers and Fig. 23 show clearly the occurrence of a minimum in the
+solubility of the ketone in water, and also a minimum (at about 10deg) in
+the solubility of water in methylethylketone. Minima of solubility have
+also been found in other cases.
+
+[Illustration: FIG. 24.]
+
+Triethylamine and Water.--Although in most of the cases studied the
+solubility of one liquid in another increases with rise of temperature,
+this is not so in all cases. Thus, at temperatures below 18deg,
+triethylamine and water mix together in all proportions; but, on raising
+the temperature, the homogeneous solution becomes turbid and separates into
+two layers. In this case, therefore, the critical solution temperature is
+found in the direction of lower temperature, not in the direction of
+higher.[172] This behaviour is clearly shown by the graphic representation
+in Fig. 24, and also by the numbers in the following table:--
+
+  TRIETHYLAMINE AND WATER.
+
+  -------------+-----------------+----------------
+  Temperature. | C_{1} per cent. |  C_{2} per cent.
+  -------------+-----------------+----------------
+      70deg    |      1.6        |     --
+      50deg    |      2.9        |     --
+      30deg    |      5.6        |     96
+      25deg    |      7.3        |     95.5
+      20deg    |     15.5        |     73
+   +/-18.5deg  |  +/-30          |  +/-30
+  -------------+-----------------+----------------
+
+General Form of Concentration-Temperature Curve.--From the preceding
+figures it will be seen that the general {102} form of the solubility curve
+is somewhat parabolic in shape; in the case of triethylamine and water, the
+closed end of the curve is very flat. Since for all liquids there is a
+point (critical point) at which the liquid and gaseous states become
+identical, and since all gases are miscible in all proportions, it follows
+that there must be some temperature at which the liquids become perfectly
+miscible. In the case of triethylamine and water, which has just been
+considered, there must therefore be an upper critical solution temperature,
+so that the complete solubility relations would be represented by a closed
+curve of an ellipsoidal aspect. An example of such a curve is furnished by
+nicotine and water. At temperatures below 60deg and above 210deg, nicotine
+and water mix in all proportions.[173] Although it is possible that this is
+the general form of the curve for all pairs of liquids, there are as yet
+insufficient data to prove it.
+
+With regard to the closed end of the curve it may be said that it is
+continuous; the critical solution point is not the intersection of two
+curves, for such a break in the continuity of the curve could occur only if
+there were some discontinuity in one of the phases. No such discontinuity
+exists. The curve is, therefore, not to be considered as two solubility
+curves cutting at a point; it is a curve of equilibrium between two
+components, and so long as the phases undergo continuous change, the curve
+representing the equilibrium must also be continuous. As has already been
+emphasized, a distinction between solvent and solute is merely conventional
+(p. 93).
+
+Pressure-Concentration Diagram.--In considering the pressure-concentration
+diagram of a system of two liquid components, a distinction must be drawn
+between the total pressure of the system and the partial pressures of the
+components. On studying the total pressure of a system, it is found that
+two cases can be obtained.[174]
+
+So long as there is only one liquid phase, the system is bivariant. The
+pressure therefore can change with the concentration and the temperature.
+If the temperature is maintained {103} constant, the pressure will vary
+only with the concentration, and this variation can therefore be
+represented by a curve. If, however, two liquid phases are formed, the
+system becomes univariant: and if one of the variables, say the
+temperature, is arbitrarily fixed, the system no longer possesses any
+degree of freedom. _When two liquid phases are formed, therefore, the
+concentrations and the vapour pressure have definite values, which are
+maintained so long as the two liquid phases are present_; the temperature
+being supposed constant.
+
+In Fig. 25 is given a diagrammatic representation of the two kinds of
+pressure-concentration curves which have so far been obtained. In the one
+case, the vapour pressure of the invariant system (at constant temperature)
+lies higher than the vapour pressure of either of the pure components; a
+phenomenon which is very generally found in the case of partially miscible
+liquids, _e.g._ ether and water.[175] Accordingly, by the addition of water
+to ether, or of ether to water, there is an increase in the _total_ vapour
+pressure of the system.
+
+[Illustration: FIG. 25.]
+
+With regard to the second type, the vapour pressure of the systems with two
+liquid phases lies between that of the two single components. An example of
+this is found in sulphur dioxide and water.[176] On adding sulphur dioxide
+to water there is an increase of the total vapour pressure; but on adding
+water to liquid sulphur dioxide, the total vapour pressure is diminished.
+
+The case that the vapour pressure of the system with two {104} liquid
+phases is _less_ than that of each of the components is not possible.
+
+With regard to the _partial pressure_ of the components, the behaviour is
+more uniform. The partial pressure of one component is in all cases lowered
+by the addition of the other component, the diminution being approximately
+proportional to the amount added. If two liquid phases are present, the
+partial pressure of the components, as well as the total pressure, is
+constant, and is the same for both phases. That is to say, in the case of
+the two liquids, saturated solution of water in ether, and of ether in
+water, the partial pressure of the ether in the vapour in contact with the
+one solution is the same as that in the vapour over the other
+solution.[177]
+
+Complete Miscibility.--Although the phenomena of complete miscibility are
+here treated under a separate heading, it must not be thought that there is
+any essential difference between those cases where the liquids exhibit
+limited miscibility and those in which only one homogeneous solution is
+formed. As has been already pointed out, the solubility relations alter
+with the temperature; and liquids which at one temperature can dissolve in
+one another only to a limited extent, are found at some other temperature
+to possess the property of complete miscibility. Conversely, we may expect
+that liquids which at one temperature, say at the ordinary temperature, are
+miscible in all proportions, will be found at some other temperature to be
+only partially miscible. Thus, for example, it was found by Guthrie that
+ethyl alcohol and carbon disulphide, which are miscible in all proportions
+at the ordinary temperature, possess only limited miscibility at
+temperatures below -14.4deg.[178] Nevertheless, it is doubtful if the
+critical solution temperature is in all cases experimentally realizable.
+
+Pressure-Concentration Diagram.--Since, in the cases of complete
+miscibility of two liquid components, there are never more than two phases
+present, the system must always be bivariant; and two of the variables
+pressure, temperature or concentration of the components, must be
+arbitrarily chosen {105} before the system becomes defined. For this reason
+the Phase Rule affords only a slight guidance in the study of such
+equilibria; and we shall therefore not enter in detail into the behaviour
+of these homogeneous mixtures. All that the Phase Rule can tell us in
+connection with these solutions, is that at constant temperature the vapour
+pressure of the solution varies with the composition of the liquid phase;
+and if the composition of the liquid phase remains unchanged, the pressure
+also must remain unchanged. This constancy of composition is exhibited not
+only by pure liquids, but also by liquid solutions in all cases where the
+vapour pressure of the solution reaches a maximum or minimum value. This is
+the case, for example, with mixtures of constant boiling point.[179]
+
+       *       *       *       *       *
+
+
+{106}
+
+CHAPTER VII
+
+SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE
+
+General.--When a solid is brought into contact with a liquid in which it
+can dissolve, a certain amount of it passes into solution; and the process
+continues until the concentration reaches a definite value independent of
+the amount of solid present. A condition of equilibrium is established
+between the solid and the solution; the solution becomes _saturated_. Since
+the number of components is two, and the number of phases three, viz.
+solid, liquid solution, vapour, the system is univariant. If, therefore,
+one of the factors, pressure, temperature, or concentration of the
+components (in the solution[180]), is arbitrarily fixed, the state of the
+system becomes perfectly defined. Thus, at any given temperature, the
+vapour pressure of the system and the concentration of the components have
+a definite value. If the temperature is altered, the vapour pressure and
+also, in general, the concentration will undergo change. Likewise, if the
+pressure varies, while the system is isolated so that no heat can pass
+between it and its surroundings, the concentration and the temperature must
+also undergo variation until they attain values corresponding to the
+particular pressure.
+
+That the temperature has an influence, sometimes a very considerable
+influence, on the amount of substance passing into solution, is
+sufficiently well known; the effect of pressure, although less apparent, is
+no less certain. If at any given temperature the volume of the vapour phase
+is diminished, {107} vapour will condense to liquid, in order that the
+pressure may remain constant, and so much of the solid will pass into
+solution that the concentration may remain unchanged; for, so long as the
+three phases are present, the state of the system cannot alter. If,
+however, one of the phases, _e.g._ the vapour phase, disappears, the system
+becomes bivariant; at any given temperature, therefore, there may be
+different values of concentration and pressure.
+
+The direction in which change of concentration will occur with change of
+pressure can be predicted by means of the theorem of Le Chatelier, if it is
+known whether solution is accompanied by increase or diminution of the
+total volume. If diminution of the total volume of the system occurs on
+solution, increase of pressure will increase the solubility; in the reverse
+case, increase of pressure will diminish the solubility.
+
+This conclusion has also been verified by experiment, as is shown by the
+following figures.[181]
+
+  ---------------------------------------------------------------
+                    |Change of       |  Solubility (at 18deg) (grams salt
+                    |volume by       |  in 1 gram of solution).
+                    |dissolving 1 gm.|---------------------------
+       Salt.        |of salt in the  |          |
+                    |saturated       | Pressure | Pressure
+                    |solution.       | = 1 atm. | = 500 atm.
+  ------------------+----------------+----------+----------------
+  Sodium chloride   |     -0.07      |  0.264   |   0.270
+  Ammonium chloride |     +0.10      |  0.272   |   0.258
+  Alum              |     -0.067     |  0.115   |   0.142
+                    |                |          |(_p_ = 400 atm.)
+  -------------------------------------------------------------
+
+As can be seen, a large increase of the pressure brings about a no more
+than appreciable alteration of the solubility; a result which is due, as in
+the case of the alteration of the fusion point with the pressure, to the
+small change in volume accompanying solution or increase of pressure. For
+all practical purposes, therefore, the solubility as determined under
+atmospheric pressure may be taken as equal to the true {108} solubility,
+that is, the solubility when the system is under the pressure of its own
+vapour.
+
+The Saturated Solution.--From what has been said above, it will be seen
+that the condition of saturation of a solution can be defined only with
+respect to a certain solid phase; if no solid is present, the system is
+undefined, for it then consists of only two phases, and is therefore
+bivariant. Under such circumstances not only can there be at one given
+temperature solutions of different concentration, all containing less of
+one of the components than when that component is present in the solid
+form, but there can also exist solutions containing more of that component
+than corresponds to the equilibrium when the solid is present. In the
+former case the solutions are _unsaturated_, in the latter case they are
+_supersaturated with respect to a certain solid phase_; in themselves, the
+solutions are stable, and are neither unsaturated nor supersaturated.
+Further, if the solid substance can exist in different allotropic
+modifications, the particular form of the substance which is in equilibrium
+with the solution must be known, in order that the statement of the
+solubility may be definite; for each form has its own solubility, and, as
+we shall see presently, the less stable form has the greater solubility
+(cf. p. 47). In all determinations of the solubility, therefore, not only
+must the concentration of the components in the solution be determined, but
+equal importance should be attached to the characterisation of the solid
+phase present.
+
+In this connection, also, one other point may be emphasised. For the
+production of the equilibrium between a solid and a liquid, time is
+necessary, and this time not only varies with the state of division of the
+solid and the efficiency of the stirring, but is also dependent on the
+nature of the substance.[182] Considerable care must therefore be taken
+that sufficient time is allowed for equilibrium to be established. Such
+care is more especially needful when changes may occur in the solid phase,
+and neglect of it has greatly diminished the value of many of the older
+determinations of solubility.
+
+Form of the Solubility Curve.--The solubility curve--that {109} is, the
+curve representing the change of concentration of the components in the
+solution with the temperature--differs markedly from the curve of vapour
+pressure (p. 63), in that it possesses no general form, but may vary in the
+most diverse manner. Not only may the curve have an almost straight and
+horizontal course, or slope or curve upwards at varying angles; but it may
+even slope downwards, corresponding to a decrease in the solubility with
+rise of temperature; may exhibit maxima or minima of solubility, or may, as
+in the case of some hydrated salts, pass through a point of maximum
+temperature. In the latter case the salt may possess two values of
+solubility at the same temperature. We shall consider these cases in the
+following chapter.
+
+[Illustration: FIG. 26.]
+
+The great variety of form shown by solubility curves is at once apparent
+from Fig. 26, in which the solubility curves of various substances (not,
+however, drawn to scale) are reproduced.[183]
+
+Varied as is the form of the solubility curve, its _direction_,
+nevertheless, can be predicted by means of the theorem of van't Hoff and Le
+Chatelier; for in accordance with that theorem (p. 57) increase of
+solubility with the temperature must occur in those cases where the process
+of solution is accompanied by an _absorption_ of heat; and a decrease in
+the solubility with rise of temperature will be found in cases where
+solution occurs with _evolution_ of heat. Where there is no heat effect
+accompanying solution, {110} change of temperature will be without
+influence on the solubility; and if the sign of the heat of solution
+changes, the direction of the solubility curve must also change, _i.e._
+must show a maximum or minimum point. This has in all cases been verified
+by experiment.[184]
+
+In applying the theorem of Le Chatelier to the course of the solubility
+curve, it should be noted that by heat of solution there is meant, not the
+heat effect produced on dissolving the salt in a large amount of solvent
+(which is the usual signification of the expression), but the heat which is
+absorbed or evolved when the salt is dissolved in the almost saturated
+solution (the so-called last heat of solution). Not only does the heat
+effect in the two cases have a different value, but it may even have a
+different sign. A striking example of this is afforded by cupric chloride,
+as the following figures show:[185]--
+
+  -----------------------------------------------------------
+  Number of gram-molecules of        |
+  CuCl_{2}, 2H_{2}O dissolved in 198 |       Heat effect.
+  gram-molecules of water.           |
+  -----------------------------------+-----------------------
+              1                      |          +37 K
+              2.02                   |          +66 "
+              4.15                   |         +105 "
+              7.07                   |         +117 "
+              9.95                   |         +117 "
+             11                      |          +91 "
+             18.8                    |          -10 "
+             19.6                    |          -31 "
+             24.75                   |         -198 "
+  ------------------------------------------------------------
+
+In the above table the positive sign indicates evolution of heat, the
+negative sign, absorption of heat; and the values of the heat effect are
+expressed in centuple calories. Judging from the heat effect produced on
+dissolving cupric chloride in a large bulk of water, we should predict that
+the solubility of that salt would diminish with rise of temperature; as a
+matter of fact, it increases. This is in accordance with the fact that
+{111} the last heat of solution is _negative_ (as expressed above), _i.e._
+solution of the salt in the almost saturated solution is accompanied by
+absorption of heat. We are led to expect this from the fact that the heat
+of solution changes sign from positive to negative as the concentration
+increases; experiment also showed it to be the case.
+
+Despite its many forms, it should be particularly noted that the solubility
+curve of any substance is _continuous_, so long as the solid phase, or
+solid substance in contact with the solution, remains unchanged. If any
+"break" or discontinuous change in the direction of the curve occurs, it is
+a sign that the _solid phase has undergone alteration_. Conversely, if it
+is known that a change takes place in the solid phase, a break in the
+solubility curve can be predicted. We shall presently meet with examples of
+this.[186]
+
+A.--ANHYDROUS SALT AND WATER.
+
+The Solubility Curve.--In studying the equilibria in those systems of two
+components in which the liquid phase is a solution or phase of varying
+composition, we shall in the present chapter limit the discussion to those
+cases where no compounds are formed, but where the components crystallise
+out in the pure state. Since some of the best-known examples of such
+systems are yielded by the solutions of anhydrous salts in water, we shall
+first of all briefly consider some of the results which have been obtained
+with them.
+
+For the most part the solubility curves have been studied only at
+temperatures lying between 0deg and 100deg, the solid phase in contact with
+the solution being the anhydrous salt. For the representation of these
+equilibria, the concentration-temperature {112} diagram is employed, the
+concentration being expressed as the number of grams of the salt dissolved
+in 100 grams of water, or as the number of gram-molecules of salt in 100
+gram-molecules of water. The curves thus obtained exhibit the different
+forms to which reference has already been made. So long as the salt remains
+unchanged the curve will be continuous, but if the salt alters its form,
+then the solubility curve will show a break.
+
+[Illustration: FIG. 27.]
+
+Now, we have already seen in Chapter III. that certain substances are
+capable of existing in various crystalline forms, and these forms are so
+related to one another that at a given temperature the relative stability
+of each pair of polymorphic forms undergoes change. Since each crystalline
+variety of a substance must have its own solubility, there must be a break
+in the solubility curve at the temperature of transition of the two
+enantiotropic forms. At this point the two solubility curves must cut, for
+since the two forms are in equilibrium with respect to their vapour, they
+must also be in equilibrium with respect to their solutions. From the table
+on p. 63 it is seen that potassium nitrate, ammonium nitrate, silver
+nitrate, thallium nitrate, thallium picrate, are capable of existing in two
+or more different enantiotropic crystalline forms, the range of stability
+of these forms being limited by definite temperatures (transition
+temperature). Since the transition point is not altered by a solvent
+(provided the latter is not absorbed by the solid phase), we should find on
+studying the solubility of these substances in water that the solubility
+curve would exhibit a change in direction at the temperature of transition.
+As a matter of fact this has been verified, more especially in the case of
+ammonium nitrate[187] {113} and thallium picrate.[188] The following table
+contains the values of the solubility of ammonium nitrate obtained by
+Mueller and Kaufmann, the solubility being expressed in gram-molecules
+NH_{4}NO_{3} in 100 gram-molecules of water. In Fig. 27 these results are
+represented graphically. The equilibrium point was approached both from the
+side of unsaturation and of supersaturation, and the condition of
+equilibrium was controlled by determinations of the density of the
+solution.
+
+  SOLUBILITY OF AMMONIUM NITRATE.
+
+  ------------------------------------------------------------
+   Temperature. | Solubility. | Temperature. | Solubility.
+  --------------+-------------+--------------+----------------
+     12.2deg    |    34.50    |   32.7deg    |    57.90
+     20.2deg    |    43.30    |   34.0deg    |    58.89
+     25.05deg   |    48.19    |   35.0deg    |    59.80
+     28.0deg    |    51.86    |   36.0deg    |    61.00
+     30.0deg    |    54.40    |   37.5deg    |    62.90
+     30.2deg    |    54.61    |   38.0deg    |    63.60
+     31.9deg    |    57.20    |   39.0deg    |    65.09
+     32.1deg    |    57.60    |   40.0deg    |    66.80
+  ------------------------------------------------------------
+
+From the graphic representation of the solubility given in Fig. 27, there
+is seen to be a distinct change in the direction of the curve at a
+temperature of 32deg; and this break in the curve corresponds to the
+transition of the [beta]-rhombic into the [alpha]-rhombic form of ammonium
+nitrate (p. 63).
+
+Suspended Transformation and Supersaturation.--As has already been learned,
+the transformation of the one crystalline form into the other does not
+necessarily take place immediately the transition point has been passed;
+and it has therefore been found possible in a number of cases to follow the
+solubility curve of a given crystalline form beyond the point at which it
+ceases to be the most stable modification. Now, it will be readily seen
+from Fig. 27 that if the two solubility curves be prolonged beyond the
+point of intersection, the solubility of the less stable form is greater
+than that of the more stable. A solution, therefore, which is saturated
+with respect to the less stable form, _i.e._ which is in equilibrium with
+that form, is _supersaturated with respect to the more stable
+modification_. If, {114} therefore, a small quantity of the more stable
+form is introduced into the solution, the latter must deposit such an
+amount of the more stable form that the concentration of the solution
+corresponds to the solubility of the stable form at the particular
+temperature. Since, however, the solution is now _unsaturated_ with respect
+to the less stable variety, the latter, if present, must pass into
+solution; and the two processes, deposition of the stable and solution of
+the metastable form, must go on until the latter form has entirely
+disappeared and a saturated solution of the stable form is obtained. There
+will thus be a conversion, through the medium of the solvent, of the less
+stable into the more stable modification. This behaviour is of practical
+importance in the determination of transition points (_v._ Appendix).
+
+From the above discussion it will be seen how important is the statement of
+the solid phase for the definition of saturation and supersaturation.[189]
+
+Solubility Curve at Higher Temperatures.--On passing to the consideration
+of the solubility curves at higher temperatures, two chief cases must be
+distinguished.
+
+    (1) The two components in the fused state can mix in all proportions.
+
+    (2) The two components in the fused state cannot mix in all
+    proportions.
+
+1. _Complete Miscibility of the Fused Components._
+
+[Illustration: FIG. 28.]
+
+The best example of this which has been studied, so far as anhydrous salts
+and water are concerned, is that of silver nitrate and water. The
+solubility of this salt at temperatures {115} above 100deg has been studied
+chiefly by Etard[190] and by Tilden and Shenstone.[191] The values obtained
+by Etard are given in the following table, and represented graphically in
+Fig. 28.
+
+  SOLUBILITY OF SILVER NITRATE.
+
+  ---------------------------------------------------
+      Temperature.    |Parts of dry salt in 100 parts
+                      |   of solution.
+  --------------------+------------------------------
+         -7deg        |         46.2
+         -1deg        |         52.1
+         +5deg        |         56.3
+         10deg        |         61.2
+         20deg        |         67.8
+         40.5deg      |         76.8
+         73deg        |         84.0
+        135deg        |         92.8
+        182deg        |         96.9
+  ---------------------------------------------------
+
+In this figure the composition of the solution is expressed in parts of
+silver nitrate in 100 parts by weight of the solution, so that 100 per
+cent. represents pure silver nitrate. As can be seen, the solubility
+increases with the temperature. At a temperature of about 160deg there
+should be a break in the curve due to change of crystalline form (p. 63).
+Such a change in the direction of the solubility curve, however, does not
+in any way alter the essential nature of the relationships discussed here,
+and may for the present be left out of account. On following the solubility
+curve of silver nitrate to higher temperatures, therefore, the
+concentration of silver nitrate in the solution gradually increases, until
+at last, at a temperature of 208deg,[192] the melting point of pure silver
+nitrate is reached, and the concentration of the water has become zero. The
+curve throughout its whole extent represents the equilibrium between silver
+nitrate, solution, and vapour. Conversely, starting with pure silver
+nitrate in contact with the fused salt, addition of water will lower the
+melting point, _i.e._ will lower the temperature at which the solid salt
+can exist in contact with the liquid; {116} and the depression will be all
+the greater the larger the amount of water added. As the concentration of
+the water in the liquid phase is increased, therefore, the system will pass
+back along the curve from higher to lower temperatures, and from greater to
+smaller concentrations of silver nitrate in the liquid phase. The curve in
+Fig. 28 may, therefore, be regarded either as the solubility curve of
+silver nitrate in water, or as the freezing point curve for silver nitrate
+in contact with a solution consisting of that salt and water.
+
+As the temperature of the saturated solution falls, silver nitrate is
+deposited, and on lowering the temperature sufficiently a point will at
+last be reached at which ice also begins to separate out. Since there are
+now four phases co-existing, viz. silver nitrate, ice, solution, vapour,
+the system is invariant, and the point is a _quadruple point_. This
+quadruple point, therefore, forms the lower limit of the solubility curve
+of silver nitrate. Below this point the solution becomes metastable.
+
+Ice as Solid Phase.--Ice melts or is in equilibrium with water at a
+temperature of 0deg. The melting point, will, however, be lowered by the
+solution of silver nitrate in the water; and the greater the concentration
+of the salt in the solution the greater will be the depression of the
+temperature of equilibrium. On continuing the addition of silver nitrate, a
+point will at length be reached at which the salt is no longer dissolved,
+but remains in the solid form along with the ice. We again obtain,
+therefore, the invariant system ice--salt--solution--vapour. The
+temperature at which this invariant system can exist has been found by
+Middelberg[193] to be -7.3deg, the solution at this point containing 47.1
+per cent. of silver nitrate.
+
+The same general behaviour will be found in the case of all other systems
+of two components belonging to this class; that is, in the case of systems
+from which the components crystallise out in the pure state, and in which
+the fused components are miscible in all proportions. In all such cases,
+therefore, the solubility curves (curves of equilibrium) can be represented
+diagrammatically as in Fig. 29. In this figure OA represents the solubility
+curve of the salt, and OB the freezing {117} point curve of ice. O is the
+quadruple point at which the invariant system exists, and may be regarded
+as the point of intersection of the solubility curve with the
+freezing-point curve. Since this point is fixed, the condition of the
+system as regards temperature, vapour pressure, and concentration of the
+components (or composition of the solution), is perfectly definite. From
+the way, also, in which the condition is attained, it is evident that the
+quadruple point is the lowest temperature that can be obtained with
+mixtures of the two components in presence of vapour. It is known as the
+_cryohydric point_, or, generally, the _eutectic point_.[194]
+
+[Illustration: FIG. 29.]
+
+Cryohydrates.[195]--On cooling a solution of common salt in water to a
+temperature of -3deg, Guthrie observed that the hydrate NaCl,2H_{2}O
+separated out. This salt continued to be deposited until at a temperature
+of -22deg opaque crystals made their appearance, and the liquid passed into
+the solid state without change of temperature. A similar behaviour was
+found by Guthrie in the case of a large number of other salts, a
+temperature below that of the melting point of ice being reached at which
+on continued withdrawal of heat, the solution solidified at a constant
+temperature. When the system had attained this minimum temperature, it was
+found that the composition of the solid and the liquid phases was the same,
+and remained unchanged throughout the period of solidification. This is
+shown by the following figures, which give the composition of different
+samples of the solid phase deposited from the solution at constant
+temperature.[196]
+
+{118}
+
+  ---------------------------------
+  No. | Temperature of  |   NaCl.
+      | solidification. | Per cent.
+  ----|-----------------|----------
+   1  |-21deg to -22deg |   23.72
+   2  |     -22deg      |   23.66
+   3  |     -22deg      |   23.73
+   4  |     -23deg      |   23.82
+   5  |     -23deg      |   23.34
+   6  |     -23deg      |   23.35
+  ---------------------------------
+             Mean           23.6
+  ---------------------------------
+
+Conversely, a mixture of ice and salt containing 23.6 per cent. of sodium
+chloride will melt at a definite and constant temperature, and exhibit,
+therefore, a behaviour supposed to be characteristic of a pure chemical
+compound. This, then, combined with the fact that the solid which was
+deposited was crystalline, and that the same constant temperature was
+attained, no matter with what proportions of water and salt one started,
+led Guthrie to the belief that the solids which thus separated at constant
+temperature were definite chemical compounds, to which he gave the general
+name _cryohydrate_. A large number of such cryohydrates were prepared and
+analysed by Guthrie, and a few of these are given in the following table,
+together with the temperature of the cryohydric point:[197]--
+
+  CRYOHYDRATES.
+
+  ------------------------------------------------------------------
+         Salt.       | Cryohydric point. | Percentage of anhydrous
+                     |                   | salt in the cryohydrate.
+  ------------------------------------------------------------------
+  Sodium bromide     |       -24deg      |          41.33
+  Sodium chloride    |       -22deg      |          23.60
+  Potassium iodide   |       -22deg      |          52.07
+  Sodium nitrate     |       -17.5deg    |          40.80
+  Ammonium sulphate  |       -17deg      |          41.70
+  Ammonium chloride  |       -15deg      |          19.27
+  Sodium iodide      |       -15deg      |          59.45
+  Potassium bromide  |       -13deg      |          32.15
+  Potassium chloride |       -11.4deg    |          20.03
+  Magnesium sulphate |        -5deg      |          21.86
+  Potassium nitrate  |        -2.6deg    |          11.20
+  Sodium sulphate    |        -0.7deg    |           4.55
+  ------------------------------------------------------------------
+
+{119}
+
+The chemical individuality of these cryohydrates was, however, called in
+question by Pfaundler,[198] and disproved by Offer,[199] who showed that in
+spite of the constancy of the melting point, the cryohydrates had the
+properties, not of definite chemical compounds, but of mixtures; the
+arguments given being that the heat of solution and the specific volume are
+the same for the cryohydrate as for a mixture of ice and salt of the same
+composition; and it was further shown that the cryohydrate had not a
+definite crystalline form, but separated out as an opaque mass containing
+the two components in close juxtaposition. The heterogeneous nature of
+cryohydrates can also be shown by a microscopical examination.
+
+At the cryohydric point, therefore, we are not dealing with a single solid
+phase, but with two solid phases, ice and salt; the cryohydric point,
+therefore, as already stated, is a quadruple point and represents an
+invariant system.
+
+Although on cooling a solution to the cryohydric point, separation of ice
+may occur, it will not necessarily take place; the system may become
+metastable. Similarly, separation of salt may not take place immediately
+the cryohydric point is reached. It will, therefore, be possible to follow
+the curves BO and AO beyond the quadruple point,[200] which is thereby
+clearly seen to be the point of intersection of the solubility curve of the
+salt and the freezing-point curve of ice. At this point, also, the curves
+of the univariant systems ice--salt--vapour and ice--salt--solution
+intersect.
+
+Changes at the Quadruple Point.--Since the invariant system
+ice--salt--solution--vapour can exist only at a definite temperature,
+addition or withdrawal of heat must cause the disappearance of one of the
+phases, whereby the system will become univariant. So long as all four
+phases are present the temperature, pressure, and concentration of the
+components in the solution must remain constant. When, therefore, heat is
+added to or withdrawn from the system, mutually compensatory changes will
+take place within the system whereby the {120} condition of the latter is
+preserved. These changes can in all cases be foreseen with the help of the
+theorem of van't Hoff and Le Chatelier; and, after what was said in Chap.
+IV., need only be briefly referred to here. In the first place, addition of
+heat will cause ice to melt, and the concentration of the solution will be
+thereby altered; salt must therefore dissolve until the original
+concentration is reached, and the heat of fusion of ice will be
+counteracted by the heat of solution of the salt. Changes of volume of the
+solid and liquid phases must also be taken into account; an alteration in
+the volume of these phases being compensated by condensation or
+evaporation. All four phases will therefore be involved in the change, and
+the final state of the system will be dependent on the amounts of the
+different phases present; the ultimate result of addition or withdrawal of
+heat or of change of pressure at the quadruple point will be one of the
+four univariant systems: ice--solution--vapour; salt--solution--vapour;
+ice--salt--vapour; ice--salt--solution. If the vapour phase disappear,
+there will be left the univariant system ice--salt--solution, and the
+temperature at which this system can exist will alter with the pressure.
+Since in this case the influence of pressure is comparatively slight, the
+temperature of the quadruple point will differ only slightly from that of
+the cryohydric point as determined under atmospheric pressure.
+
+Freezing Mixtures.--Not only will the composition of a univariant system
+undergo change when the temperature is varied, but, conversely, if the
+_composition_ of the system is caused to change, corresponding changes of
+temperature must ensue. Thus, if ice is added to the univariant system
+salt--solution--vapour, the ice must melt and the temperature fall; and if
+sufficient ice is added, the temperature of the cryohydric point must be at
+length reached, for it is only at this temperature that the four phases
+ice--salt--solution--vapour can coexist. Or, on the other hand, if salt is
+added to the system ice--solution--vapour, the concentration of the
+solution will increase, ice must melt, and the temperature must thereby
+fall; and this process also will go on until the cryohydric point is
+reached. In both cases ice melts and there is a change in the {121}
+composition of the solution; in the former case, salt will be
+deposited[201] because the solubility diminishes as the temperature falls;
+in the latter, salt will pass into solution. This process may be
+accompanied either by an evolution or, more generally, by absorption of
+heat; in the former case the effect of the addition of ice will be
+partially counteracted; in the latter case it will be augmented.
+
+These principles are made use of in the preparation of _freezing mixtures_.
+The lowest temperature which can be reached by means of these (under
+atmospheric pressure) is the cryohydric point. This temperature-minimum is,
+however, not always attained in the preparation of a freezing mixture, and
+that for various reasons. The chief of these are radiation and the heat
+absorbed in cooling the solution produced. The lower the temperature falls,
+the more rapid does the radiation become; and the rate at which the
+temperature sinks decreases as the amount of solution increases. Both these
+factors counteract the effect of the latent heat of fusion and the heat of
+solution, so that a point is reached (which may lie considerably above the
+cryohydric point) at which the two opposing influences balance. The
+absorption of heat by the solution can be diminished by allowing the
+solution to drain off as fast as it is produced; and the effect of
+radiation can be partially annulled by increasing the rate of cooling. This
+can be done by the more intimate mixing of the components. Since, under
+atmospheric pressure, the temperature of the cryohydric point is constant,
+the cryohydrates are very valuable for the production of baths of constant
+low temperature.
+
+2. _Partial Miscibility of the Fused Components._
+
+On passing to the study of the second class of systems of two components
+belonging to this group, namely, those in which the fused components are
+not miscible in all proportions, we find that the relationships are not
+quite so simple as {122} in the case of silver nitrate and water. In the
+latter case, only one liquid phase was possible; in the cases now to be
+studied, two liquid phases can be formed, and there is a marked
+discontinuity in the solubility curve on passing from the cryohydric point
+to the melting point of the second (non-volatile) component.
+
+Paratoluidine dissolves in water, and the solubility increases as the
+temperature rises.[202] At 44.2deg, however, paratoluidine in contact with
+water melts, and two liquid phases are formed, viz. a solution of water in
+fused paratoluidine and a solution of fused paratoluidine in water. We
+have, therefore, the phenomenon of _melting under the solvent_. This
+melting point will, of course, be lower than the melting point of the pure
+substance, because the solid is now in contact with a solution, and, as we
+have already seen, addition of a foreign substance lowers the melting
+point. Such cases of melting under the solvent are by no means rare, and a
+review of the relationships met with may, therefore, be undertaken here. As
+an example, there may be chosen the equilibrium between succinic nitrile,
+C_{2}H_{4}(CN)_{2} and water, which has been fully studied by
+Schreinemakers.[203]
+
+[Illustration: FIG. 30.]
+
+If to the system ice--water at 0deg succinic nitrile is added, the
+temperature will fall; and continued addition of the nitrile will lead at
+last to the cryohydric point _b_ (Fig. 30), at which solid nitrile, ice,
+solution, and vapour can coexist. The temperature of the cryohydric point
+is -1.2deg, and the composition of the solution is 1.29 mol. of nitrile in
+100 mol. of solution. From _a_ to _b_ the solid phase in contact with the
+solution is ice. {123} If the temperature be now raised so as to cause the
+disappearance of the ice, and the addition of nitrile be continued, the
+concentration of the nitrile in the solution will increase as represented
+by the curve _bc_. At the point _c_ (18.5deg), when the concentration of
+the nitrile in the solution has increased to 2.5 molecules per cent., the
+nitrile melts and two liquid phases are formed; the concentration of the
+nitrile in these two phases is given by the points _c_ and _c'_. As there
+are now four phases present, viz. solid nitrile, solution of fused nitrile
+in water, solution of water in fused nitrile, and vapour, the system is
+_invariant_. Since at this point the concentration, temperature, and
+pressure are completely defined, addition or withdrawal of heat can only
+cause a change in the relative amounts of the phases, _but no variation of
+the concentrations_ of the respective phases. As a matter of fact,
+continued addition of nitrile and addition of heat will cause an increase
+in the amount of the liquid phase containing excess of nitrile (_i.e._ the
+solution of water in fused nitrile), whereas the other liquid phase, the
+solution of fused nitrile in water, will gradually disappear. When it has
+completely disappeared, the system will be represented by the point _c'_,
+where the molecular concentration of nitrile is now 75 per cent., and again
+becomes univariant, the three phases being solid nitrile, liquid phase
+containing excess of nitrile, and vapour; and as the amount of the water is
+diminished the temperature of equilibrium rises, until at 54deg the melting
+point of the pure nitrile is reached.
+
+Return now to the point c. At this point there exists the invariant system
+solid nitrile, two liquid phases, vapour. If heat be added, the solid
+nitrile will disappear, and there will be left the univariant system,
+consisting of two liquid phases and vapour.[204] Such a system will exhibit
+relationships similar to those already studied in the previous chapter. As
+the temperature rises, the mutual solubility of the two fused components
+becomes greater, until at _d_ (55.5deg) the critical solution temperature
+is reached, and the fused components become miscible in all proportions.
+
+At all temperatures and concentrations lying to the right {124} of the
+curve _abcdc'e_ there can be only one liquid phase; in the field _cdc'_
+there are two liquid phases.
+
+From the figure it will be easy to see what will be the result of bringing
+together succinic nitrile and water at different temperatures and in
+different amounts. Since _b_ is the lowest temperature at which liquid can
+exist in stable equilibrium with solid, ice and succinic nitrile can be
+mixed in any proportions at temperatures below _b_ without undergoing
+change. Between _b_ and _c_ succinic nitrile will be dissolved until the
+concentration reaches the value on the curve _bc_, corresponding to the
+given temperature. On adding the nitrile to water at temperatures between
+_c_ and _d_, it will dissolve until a concentration lying on the curve _cd_
+is attained; at this point two liquid phases will be formed, and further
+addition of nitrile will cause the one liquid phase (that containing excess
+of nitrile) to increase, while the other liquid phase will decrease, until
+it finally disappears and there is only one liquid phase left, that
+containing excess of nitrile. This can dissolve further quantities of the
+nitrile, and the concentration will increase until the curve _c'e_ is
+reached, when the concentration will remain unchanged, and addition of
+solid will merely increase the amount of the solid phase.
+
+If a solution represented by any point in the field lying below the curve
+_bcd_ is heated to a temperature above _d_, the critical solution
+temperature, then the concentration of the nitrile can be increased to any
+desired amount without at any time two liquid phases making their
+appearance; the system can then be cooled down to a temperature represented
+by any point between the curves _dc'e_. In this way it is possible to pass
+continuously from a solution containing excess of one component to
+solutions containing excess of the other, as represented by the dotted line
+_xxxx_ (_v._ p. 100). At no point is there formation of two liquid phases.
+
+Supersaturation.--Just as suspended transformation is rarely met with in
+the passage from the solid to the liquid state, so also it is found in the
+case of the melting of substances under the solvent that suspended fusion
+does not occur; but that when the temperature of the invariant point is
+reached at which, therefore, the formation of two liquid layers is
+possible, {125} these two liquid layers, as a matter of fact, make their
+appearance. Suspended transformation can, however, take place from the side
+of the liquid phase, just as water or other liquid can be cooled below the
+normal freezing point without solidification occurring. The question,
+therefore, arises as to the relative solubilities of the solid and the
+supercooled liquid at the same temperature.
+
+[Illustration: FIG. 31.]
+
+The answer to this question can at once be given from what we have already
+learned (p. 113), if we recollect that at temperatures below the point of
+fusion under the solvent, the solid form, at temperatures above that point,
+the liquid form, is the more stable; at this temperature, therefore, the
+relative stability of the solid and liquid forms changes. Since, as we have
+already seen, the less stable form has the greater solubility, it follows
+that the supercooled liquid, being the less stable form, must have the
+greater solubility. This was first proved experimentally by Alexejeff[205]
+in the case of benzoic acid and water, the solubility curves for which are
+given in Fig. 31. As can be seen from the figure, the prolongation of the
+curve for liquid--liquid, which represents the solubility of the
+supercooled liquid benzoic acid, lies above that for the solubility of the
+{126} solid benzoic acid in water; the solution saturated with respect to
+the supercooled liquid is therefore supersaturated with respect to the
+solid form. A similar behaviour has been found in the case of other
+substances.[206]
+
+Pressure-Temperature Diagram.--Having considered the changes which occur in
+the concentration of the components in a solution with the temperature, we
+may conclude the discussion of the equilibrium between a salt and water by
+studying the variation of the vapour pressure.
+
+Since in systems of two components the two phases, solution and vapour,
+constitute a bivariant system, the vapour pressure is undefined, and may
+have different values at the same temperature, depending on the
+concentration. In order that there may be for each temperature a definite
+corresponding pressure of the vapour, a third phase must be present. This
+condition is satisfied by the system solid--liquid (solution)--vapour; that
+is, by the saturated solution (p. 108). In the case of a saturated
+solution, therefore, the pressure of the vapour at any given temperature is
+constant.
+
+Vapour Pressure of Solid--Solution--Vapour.--It has long been known that
+the addition of a non-volatile solid to a liquid in which it is soluble
+lowers the vapour pressure of the solvent; and the diminution of the
+pressure is approximately proportional to the amount of substance dissolved
+(Law of Babo). The vapour-pressure curve, therefore, of a solution of a
+salt in water must lie below that for pure water. Further, in the case of a
+pure liquid, the vaporization curve is a function only of the temperature
+(p. 63), whereas, in the case of a solution, the pressure varies both with
+the temperature and the _concentration_. These two factors, however, act in
+opposite directions; for although the vapour pressure in all cases
+increases as the temperature rises, increase of concentration, as we have
+seen, lowers the vapour pressure. Again, since the concentration itself
+varies with the temperature, two cases have to be considered, viz. where
+the concentration increases with rise of {127} temperature, and where the
+concentration diminishes with rise of temperature.
+
+The relations which are found here will be best understood with the help of
+Fig. 32.[207] In this figure, OB represents the sublimation curve of ice,
+and BC the vaporization curve of water; the curve for the solution must lie
+below this, and must cut the sublimation curve of ice at some temperature
+below the melting point. The point of intersection A is the cryohydric
+point. If the solubility increases with rise of temperature, the increase
+of the vapour pressure due to the latter will be partially annulled. Since
+at first the effect of increase of temperature more than counteracts the
+depressing action of increase of concentration, the vapour pressure will
+increase on raising the temperature above the cryohydric point. If the
+elevation of temperature is continued, however, to the melting point of the
+salt, the effect of increasing concentration makes itself more and more
+felt, so that the vapour-pressure curve of the solution falls more and more
+below that of the pure liquid, and the pressure will ultimately become
+equal to that of the pure salt; that is to say, practically equal to zero.
+The curve will therefore be of the general form AMF shown in Fig. 32. If
+the solubility should diminish with rise of temperature, the two factors,
+temperature and concentration, will act in the same direction, and the
+vapour-pressure curve will rise relatively more rapid than that of the pure
+liquid; since, however, the pure salt is ultimately obtained, the
+vapour-pressure curve must in this case also finally approach the value
+zero.
+
+[Illustration: FIG. 32.]
+
+Other Univariant Systems.--Besides the univariant system {128}
+salt--solution--vapour already considered, three others are possible, viz.
+ice--solution--vapour, ice--salt--solution, and ice--salt--vapour.
+
+The fusion point of a substance is lowered, as we have seen, by the
+addition of a foreign substance, and the depression is all the greater the
+larger the quantity of substance added. The vapour pressure of the water,
+also, is lowered by the solution in it of other substances, so that the
+vapour pressure of the system ice--solution--vapour must decrease as the
+temperature falls from the fusion point of ice to the cryohydric point.
+This curve is represented by BA (Fig. 32), and is coincident with the
+sublimation curve of ice.
+
+This, at first sight, strange fact will be readily understood when we
+consider that since ice and solution are together in equilibrium with the
+same vapour, they must have the same vapour pressure. For suppose at any
+given temperature equilibrium to have been established in the system
+ice--solution--vapour, removal of the ice will not alter this equilibrium.
+Suppose, now, the ice and the solution placed under a bell-jar so that they
+have a common vapour, but are not themselves in contact; then, if they do
+not have the same vapour pressure, distillation must take place and the
+solution will become more dilute or more concentrated. Since, at the
+completion of this process, the ice and solution are now in equilibrium
+when they are not in contact, they must also be in equilibrium when they
+are in contact (p. 32). But if distillation has taken place the
+concentration of the solution must have altered, so that the ice will now
+be in equilibrium with a solution of a different concentration from before.
+But according to the Phase Rule ice cannot at one and the same temperature
+be in equilibrium with two solutions of different concentration, for the
+system ice--solution--vapour is univariant, and at any given temperature,
+therefore, not only the pressure but also the _concentration of the
+components in the solution must be constant_. Distillation could not,
+therefore, take place from the ice to the solution or _vice versa_; that is
+to say, the solution and the ice must have the same vapour pressure--the
+sublimation pressure of ice. The reason of the coincidence is the
+non-volatility of the salt: had {129} the salt a measurable vapour pressure
+itself, the sublimation curve of ice and the curve for
+ice--solution--vapour would no longer fall together.
+
+The curve AO represents the pressures of the system ice--salt--vapour. This
+curve will also be coincident with the sublimation curve of ice, on account
+of the non-volatility of the salt.
+
+The equilibria of the fourth univariant system ice--salt--solution are
+represented by AE. Since this is a condensed system, the effect of a small
+change of temperature will be to cause a large change of pressure, as in
+the case of the fusion point of a pure substance. The direction of this
+curve will depend on whether there is an increase or diminution of volume
+on solidification; but the effect in any given case can be predicted with
+the help of the theorem of Le Chatelier.
+
+Since the cryohydric point is a quadruple point in a two-component system,
+it represents an invariant system. The condition of the system is,
+therefore, completely defined; the four phases, ice, salt, solution,
+vapour, can co-exist only when the temperature, pressure, and concentration
+of the solution have constant and definite values. Addition or withdrawal
+of heat, therefore, can cause no alteration of the condition of the system
+except a variation of the relative amounts of the phases. Addition
+of heat at constant volume will ultimately lead to the system
+salt--solution--vapour or the system ice--solution--vapour, according as
+ice or salt disappears first. This is readily apparent from the diagram
+(Fig. 32), for the systems ice--salt--solution and ice--salt--vapour can
+exist only at temperatures below the cryohydric point (provided the curve
+for ice--salt--solution slopes towards the pressure axis).
+
+Bivariant Systems.--Besides the univariant systems already discussed,
+various bivariant systems are possible, the conditions for the existence of
+which are represented by the different areas of Fig. 32. They are as
+follows:--
+
+  _Area._             _System._
+
+  OAMF     Salt--vapour.
+  CBAMF    Solution--vapour; salt--solution.
+  EABD     Salt--solution; ice--solution.
+  EAO      Ice--salt.
+
+{130}
+
+Deliquescence.--As is evident from Fig. 32, salt can exist in contact with
+water vapour at pressures under those represented by OAMF. If, however, the
+pressure of the vapour is increased until it reaches a value lying on this
+curve at temperatures above the cryohydric point, solution will be formed;
+for the curve AMF represents the equilibria between salt--solution--vapour.
+From this, therefore, it is clear that if the pressure of the aqueous
+vapour in the atmosphere is greater than that of the saturated solution of
+a salt, that salt will, on being placed in the air, form a solution; it
+will _deliquesce_.
+
+Separation of Salt on Evaporation.--With the help of Fig. 32 it is possible
+to state in a general manner whether or not salt will be deposited when a
+solution is evaporated under a constant pressure.[208]
+
+The curve AMF (Fig. 32) is the vapour-pressure curve of the saturated
+solutions of the salt, _i.e._ it represents, as we have seen, the maximum
+vapour pressure at which salt can exist in contact with solution and
+vapour. The dotted line _aa_ represents atmospheric pressure. If, now, an
+unsaturated solution, the composition of which is represented by the point
+_x_, is heated in an open vessel, the temperature will rise, and the vapour
+pressure of the solution will increase. The system will, therefore, pass
+along a line represented diagrammatically by _xx'_. At the point _x'_ the
+vapour pressure of the system becomes equal to 1 atm.; and as the vessel is
+open to the air, the pressure cannot further rise; the solution boils. If
+the heating is continued, water passes off, the concentration increases,
+and the boiling point rises. The system will therefore pass along the line
+_x'm_, until at the point _m_ solid salt separates out (provided
+supersaturation is excluded). The system is now univariant, and continued
+heating will no longer cause an alteration of the concentration; as water
+passes off, solid salt will be deposited, and the solution will evaporate
+to dryness.
+
+If, however, the atmospheric pressure is represented not by _aa_ but by
+_bb_, then, as Fig. 32 shows, the maximum vapour {131} pressure of the
+system salt--solution--vapour never reaches the pressure of 1 atm. Further,
+since the curve _bb_ lies in the area of the bivariant system
+solution--vapour there can at no point be a separation of the solid form;
+for the system solid--solution--vapour can exist only along the curve AMF.
+
+On evaporating the solution of a salt in an open vessel, therefore, salt
+can be deposited only if at some temperature the pressure of the saturated
+solution is equal to the atmospheric pressure. This is found to be the case
+with most salts. In the case of aqueous solutions of sodium and potassium
+hydroxide, however, the vapour pressure of the saturated solution never
+reaches the value of 1 atm., and on evaporating these solutions, therefore,
+in an open vessel, there is no separation of the solid. Only a homogeneous
+fused mass is obtained. If, however, the evaporation be carried out under a
+pressure which is lower than the maximum pressure of the saturated
+solution, separation of the solid substance will be possible.
+
+General Summary.--The systems which have been discussed in the present
+chapter contained water as one of their components, and an anhydrous salt
+as the other. It will, however, be clear that the relationships which were
+found in the case of these will be found also in other cases where it is a
+question of the equilibria between two components, which crystallize out in
+the pure state, and only one of which possesses a measurable vapour
+pressure. A similar behaviour will, for example, be found in the case of
+many pairs of organic substances; and in all cases the equilibria will be
+represented by a diagram of the general appearance of Fig. 29 or Fig. 30.
+That is to say: Starting from the fusion point of component I., the system
+will pass, by progressive addition of component II., to regions of lower
+temperature, until at last the cryohydric or eutectic point is reached. On
+further addition of component II., the system will pass to regions of
+higher temperature, the solid phase now being component II. If the fused
+components are miscible with one another in all proportions a continuous
+curve will be obtained leading up to the point of fusion of component II.
+Slight changes of direction, it is true, due to changes in the crystalline
+form, may be found along this curve, {132} but throughout its whole course
+there will be but one liquid phase. If, on the other hand, the fused
+components are not miscible in all proportions, then the second curve will
+exhibit a marked discontinuity, and two liquid phases will make their
+appearance.
+
+       *       *       *       *       *
+
+
+{133}
+
+CHAPTER VIII
+
+SOLUTIONS OF SOLIDS IN LIQUIDS, ONLY ONE OF THE COMPONENTS BEING VOLATILE
+
+B.--HYDRATED SALT AND WATER.
+
+In the preceding chapter we discussed the behaviour of systems formed of
+two components, only one of which was volatile, in those cases where the
+two components separated from solution in the pure state. In the present
+chapter we shall consider those systems in which combination between the
+components can occur with the formation of definite compounds; such as are
+found in the case of crystalline salt hydrates. Since a not inconsiderable
+amount of study has been devoted to the systems formed by hydrated salts
+and water, systems which are of great chemical interest and importance, the
+behaviour of these will first call for discussion in some detail, and it
+will be found later that the relationships which exist in such systems
+appear also in a large number of other two-component systems.
+
+The systems belonging to this group may be divided into two classes
+according as the compounds formed possess a definite melting point, _i.e._
+form a liquid phase of the same composition, or do not do so. We shall
+consider the latter first.
+
+1. _The Compounds formed do not have a Definite Melting Point._
+
+Concentration-Temperature Diagram.--In the case of salts which can form
+crystalline hydrates, the temperature-concentration diagram, representing
+the equilibria of the {134} different possible systems, must necessarily be
+somewhat more complicated than where no such combination of the components
+occurs. For, as has already been pointed out, each substance has its own
+solubility curve; and there will therefore be as many solubility curves as
+there are solid phases possible, _the curve for each particular solid phase
+being continuous so long as it remains unchanged in contact with the
+solution_. As an example of the relationships met with in such cases, we
+shall first of all consider the systems formed of sodium sulphate and
+water.
+
+[Illustration: FIG. 33.]
+
+Sodium Sulphate and Water.--At the ordinary temperatures, sodium sulphate
+crystallises from water with ten molecules of water of crystallisation,
+forming Glauber's salt. On determining the solubility of this salt in
+water, it is found that the solubility increases as the temperature rises,
+the values of the solubility, represented graphically by the curve AC (Fig.
+33), being given in the following table.[209] The numbers denote grams of
+sodium sulphate, calculated as anhydrous salt, dissolved by 100 grams of
+water.
+
+  SOLUBILITY OF Na_{2}SO_{4},10H_{2}O.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+      0deg     |     5.02
+     10deg     |     9.00
+     15deg     |    13.20
+     18deg     |    16.80
+     20deg     |    19.40
+     25deg     |    28.00
+     30deg     |    40.00
+     33deg     |    50.76
+     34deg     |    55.00
+  --------------------------
+
+{135}
+
+On continuing the investigation at higher temperatures, it was found that
+the solubility no longer increased, but _decreased with rise of
+temperature_. At the same time, it was observed that the solid phase was
+now different from that in contact with the solution at temperatures below
+33deg; for whereas in the latter case the solid phase was sodium sulphate
+decahydrate, at temperatures above 33deg the solid phase was the anhydrous
+salt. The course of the solubility curve of anhydrous sodium sulphate is
+shown by BD, and the values of the solubility are given in the following
+table:--[210]
+
+  SOLUBILITY OF ANHYDROUS SODIUM SULPHATE.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+   18deg       |    53.25
+   20deg       |    52.76
+   25deg       |    51.53
+   30deg       |    50.37
+   33deg       |    49.71
+   34deg       |    49.53
+   36deg       |    49.27
+   40.15deg    |    48.78
+   50.40deg    |    46.82
+  --------------------------
+
+As is evident from the figure, the solubility curve which is obtained when
+anhydrous sodium sulphate is present as the solid phase, cuts the curve
+representing the solubility of the decahydrate, at a temperature of about
+33deg.
+
+If a solution of sodium sulphate which has been saturated at a temperature
+of about 34deg be cooled down to a temperature below 17deg, while care is
+taken that the solution is protected against access of particles of
+Glauber's salt, crystals of a second hydrate of sodium sulphate, having the
+composition Na_{2}SO_{4},7H_{2}O, separate out. On determining the
+composition of the solutions in equilibrium with this hydrate at different
+temperatures, the following values were obtained, these values being
+represented by the curve FE (Fig. 33):--
+
+{136}
+
+  SOLUBILITY OF Na_{2}SO_{4},7H_{2}O.
+
+  --------------------------
+  Temperature. | Solubility.
+  --------------------------
+      0deg     |    19.62
+     10deg     |    30.49
+     15deg     |    37.43
+     18deg     |    41.63
+     20deg     |    44.73
+     25deg     |    52.94
+     26deg     |    54.97
+  --------------------------
+
+Since, as has already been stated, each solid substance has its own
+solubility curve, there are three separate curves to be considered in the
+case of sodium sulphate and water. Where two curves cut, the solution must
+be saturated with respect to two solid phases; at the point B, therefore,
+the point of intersection of the solubility curve of anhydrous sodium
+sulphate with that of the decahydrate, the solution must be saturated with
+respect to these two solid substances. But a system of two components
+existing in four phases, anhydrous salt--hydrated salt--solution--vapour,
+is invariant; and this invariability will remain even if only three phases
+are present, provided that one of the factors, pressure, temperature, or
+concentration of components retains a constant value. This is the case when
+solubilities are determined in open vessels; the pressure is then equal to
+atmospheric pressure. Under these circumstances, then, the system,
+anhydrous sodium sulphate--decahydrate--solution, will possess no degree of
+freedom, and can exist, therefore, only at one definite temperature and
+when the solution has a certain definite composition. The temperature of
+this point is 32.482deg on a mercury thermometer, or 32.379deg on the
+hydrogen thermometer.[211]
+
+{137}
+
+Suspended Transformation.--Although it is possible for the anhydrous salt
+to make its appearance at the temperature of the quadruple point, it will
+not necessarily do so; and it is therefore possible to follow the
+solubility curve of sodium sulphate decahydrate to a higher temperature.
+Since, however, the solubility of the decahydrate at temperatures above the
+quadruple point is greater than that of the anhydrous salt, the solution
+which is _saturated_ with respect to the former will be _supersaturated_
+with respect to the latter. On bringing a small quantity of the anhydrous
+salt in contact with the solution, therefore, anhydrous salt will be
+deposited; and all the hydrated salt present will ultimately undergo
+conversion into the anhydrous salt, through the medium of the solution. In
+this case, as in all cases, the solid phase, which is the most stable at
+the temperature of the experiment, has at that temperature the least
+solubility.
+
+Similarly, the solubility curve of anhydrous sodium sulphate has been
+followed to temperatures below 32.5deg. Below this temperature, however,
+the solubility of this salt is greater than that of the decahydrate, and
+the saturated solution of the anhydrous salt will therefore be
+supersaturated for the decahydrate, and will deposit this salt if a
+"nucleus" is added to the solution. From this we see that at temperatures
+above 32.5deg the anhydrous salt is the stable form, while the decahydrate
+is unstable (or metastable); at temperatures below 32.5deg the decahydrate
+is stable. This temperature, therefore, is the _transition temperature_ for
+decahydrate and anhydrous salt.
+
+From Fig. 33 we see further that the solubility curve of the anhydrous salt
+(which at all temperatures below 32.5deg is metastable) is cut by the
+solubility curve of the heptahydrate; and this point of intersection (at a
+temperature of 24.2deg) must be the _transition point_ for heptahydrate and
+anhydrous salt. Since at all temperatures the solubility of the
+heptahydrate is greater than that of the decahydrate, the former hydrate
+must be metastable with respect to the latter; so that throughout its whole
+course the solubility curve of the heptahydrate {138} represents only
+metastable equilibria. Sodium sulphate, therefore, forms only one stable
+hydrate, the decahydrate.
+
+The solubility relations of sodium sulphate illustrate very clearly the
+importance of the solid phase for the definition of saturation and
+supersaturation. Since the solubility curve of the anhydrous salt has been
+followed backwards to a temperature of about 18deg, it is readily seen,
+from Fig. 33, that at a temperature of, say, 20deg three different
+_saturated_ solutions of sodium sulphate are possible, according as the
+anhydrous salt, the heptahydrate or the decahydrate, is present as the
+solid phase. Two of these solutions, however, would be metastable and
+_supersaturated with respect to the decahydrate_.
+
+Further, the behaviour of sodium sulphate and water furnishes a very good
+example of the fact that a "break" in the solubility curve occurs when, and
+only when, the solid phase undergoes change. So long as the decahydrate,
+for example, remained unaltered in contact with the solution, the
+solubility curve was continuous; but when the anhydrous salt appeared in
+the solid phase, a distinct change in the direction of the solubility curve
+was observed.
+
+Dehydration by Means of Anhydrous Sodium Sulphate.--The change in the
+relative stability of sodium sulphate decahydrate and anhydrous salt in
+presence of water at a temperature of 32.5deg explains why the latter salt
+cannot be employed for dehydration purposes at temperatures above the
+transition point. The dehydrating action of the anhydrous salt depends on
+the formation of the decahydrate; but since at temperatures above 33deg the
+latter is unstable, and cannot be formed in presence of the anhydrous salt,
+this salt cannot, of course, effect a dehydration above that temperature.
+
+Pressure-Temperature Diagram.--The consideration of the
+pressure-temperature relations of the two components, sodium sulphate and
+water, must include not only the vapour pressure of the saturated
+solutions, but also that of the crystalline hydrates. The vapour pressures
+of salt hydrates have already been treated in a general manner (Chap. V.),
+so that it is only necessary here to point out the connection between the
+two classes of systems. {139}
+
+In most cases the vapour pressure of a salt hydrate, _i.e._ the vapour
+pressure of the system hydrate--anhydrous salt (or lower hydrate)--vapour,
+is at all temperatures lower than that of the system anhydrous salt (or
+lower hydrate)--solution--vapour. This, however, is not a necessity; and
+cases are known where the vapour pressure of the former system is, under
+certain circumstances, equal to or higher than that of the latter. An
+example of this is found in sodium sulphate decahydrate.
+
+On heating Na_{2}SO_{4},10H_{2}O, a point is reached at which the
+dissociation pressure into anhydrous salt and water vapour becomes equal to
+the vapour pressure of the saturated solution of the anhydrous salt, as is
+apparent from the following measurements;[212] the differences in pressure
+being expressed in millimetres of a particular oil.
+
+  Temperature:      29.0deg 30.83deg 31.79deg 32.09deg 32.35deg 32.6deg
+  Difference of
+  pressure:         23.8    10.8      5.6      3.6      1.6      0
+
+At 32.6deg, therefore, the vapour pressures of the two systems
+
+  Na_{2}SO_{4},10H_{2}O--Na_{2}SO_{4}--vapour
+  Na_{2}SO_{4}--solution--vapour
+
+are equal; at this temperature the four phases, Na_{2}SO_{4},10H_{2}O;
+Na_{2}SO_{4}; solution; vapour, can coexist. From this it is evident that
+when sodium sulphate decahydrate is heated to 32.6deg, the two new phases
+anhydrous salt and solution will be formed (suspended transformation being
+supposed excluded), and the hydrate will appear to undergo _partial
+fusion_; and during the process of "melting" the vapour pressure and
+temperature will remain constant.[213] This is, however, not a true but a
+so-called _incongruent_ melting point; for the composition of the liquid
+phase is not the same as that of the solid. As has already been pointed out
+(p. 137), we are dealing here with the _transition point_ of the
+decahydrate and anhydrous salt, _i.e._ with the reaction
+Na_{2}SO_{4},10H_{2}O <--> Na_{2}SO_{4} + 10H_{2}O.
+
+Since at the point of partial fusion of the decahydrate four {140} phases
+can coexist, the point is a quadruple point in a two-component system, and
+the system at this point is therefore invariant. The temperature of this
+point is therefore perfectly definite, and on this account the proposal has
+been made to adopt this as a fixed point in thermometry.[214] The
+temperature is, of course, practically the same as that at which the two
+solubility curves intersect (p. 112). If, however, the vapour phase
+disappears, the system becomes univariant, and the equilibrium temperature
+undergoes change with change of pressure. The transition curve has been
+determined by Tammann,[215] and shown to pass through a point of maximum
+temperature.
+
+[Illustration: FIG. 34.]
+
+The vapour pressure of the different systems of sodium sulphate and water
+can best be studied with the help of the diagram in Fig. 34.[216] The curve
+ABCD represents the vapour-pressure curve of the saturated solution of
+anhydrous sodium sulphate. GC is the pressure curve of decahydrate +
+anhydrous salt, which, as we have seen, cuts the curve ABCD at the
+transition temperature, 32.6deg. Since at this point the solution is
+saturated with respect to both the anhydrous salt and the decahydrate, the
+vapour-pressure curve of the saturated solution of the latter must also
+pass through the point C.[217] As at temperatures below this point the
+solubility of the decahydrate is less than that of the anhydrous salt, the
+vapour pressure of the solution will, in accordance with Babo's law
+(p. 126), be higher than that of the solution of the anhydrous salt; which
+was also found experimentally to be the case (curve HC).
+
+{141}
+
+In connection with the vapour pressure of the saturated solutions of the
+anhydrous salt and the decahydrate, attention must be drawn to a
+conspicuous deviation from what was found to hold in the case of
+one-component systems in which a vapour phase was present (p. 31). There,
+it was seen that the vapour pressure of the more stable system was always
+_lower_ than that of the less stable; in the present case, however, we find
+that this is no longer so. We have already learned that at temperatures
+below 32.5deg the system decahydrate--solution--vapour is more stable than
+the system anhydrous salt--solution--vapour; but the vapour pressure of the
+latter system is, as has just been stated, lower than that of the former.
+At temperatures above the transition point the vapour pressure of the
+saturated solution of the decahydrate will be lower than that of the
+saturated solution of the anhydrous salt.
+
+This behaviour depends on the fact that the less stable form is the more
+soluble, and that the diminution of the vapour pressure increases with the
+amount of salt dissolved.
+
+With regard to sodium sulphate heptahydrate the same considerations will
+hold as in the case of the decahydrate. Since at 24deg the four phases
+heptahydrate, anhydrous salt, solution, vapour can coexist, the
+vapour-pressure curves of the systems hydrate--anhydrous salt--vapour
+(curve EB) and hydrate--solution--vapour (curve FB) must cut the pressure
+curve of the saturated solution of the anhydrous salt at the above
+temperature, as represented in Fig. 34 by the point B. This constitutes,
+therefore, a second quadruple point, which is, however, metastable.
+
+From the diagram it is also evident that the dissociation pressure of the
+heptahydrate is higher than that of the decahydrate, although it contains
+less water of crystallization. The system heptahydrate--anhydrous
+salt--vapour must be metastable with respect to the system
+decahydrate--anhydrous salt--vapour, and will pass into the latter.[218]
+Whether or not there is a temperature at which the vapour-pressure curves
+of the two systems intersect, and below which the heptahydrate becomes the
+more stable form, is not known.
+
+{142}
+
+In the case of sodium sulphate there is only one stable hydrate. Other
+salts are known which exhibit a similar behaviour; and we shall therefore
+expect that the solubility relationships will be represented by a diagram
+similar to that for sodium sulphate. A considerable number of such cases
+have, indeed, been found,[219] and in some cases there is more than one
+metastable hydrate. This is found, for example, in the case of nickel
+iodate,[220] the solubility curves for which are given in Fig. 35. As can
+be seen from the figure, suspended transformation occurs, the solubility
+curves having in some cases been followed to a considerable distance beyond
+the transition point. One of the most brilliant examples, however, of
+suspended transformation in the case of salt hydrates, and the sluggish
+transition from the less stable to the more stable form, is found in the
+case of the hydrates of calcium chromate.[221]
+
+[Illustration: FIG. 35.]
+
+In the preceding cases, the dissociation-pressure curve of the hydrated
+salt cuts the vapour-pressure curve of the saturated {143} solution of the
+anhydrous salt. It can, however, happen that the dissociation-pressure
+curve of one hydrate cuts the solubility curve, not of the anhydrous salt,
+but of a lower hydrate; in this case there will be more than one stable
+hydrate, each having a stable solubility curve; and these curves will
+intersect at the temperature of the transition point. Various examples of
+this behaviour are known, and we choose for illustration the solubility
+relationships of barium acetate and its hydrates[222] (Fig. 36).
+
+[Illustration: FIG. 36.]
+
+At temperatures above 0deg, barium acetate can form two stable hydrates, a
+trihydrate and a monohydrate. The solubility of the trihydrate increases
+very rapidly with rise of temperature, and has been determined up to
+26.1deg. At temperatures above 24.7deg, however, the trihydrate is
+metastable with respect to the monohydrate; for at this temperature the
+solubility curve of the latter hydrate cuts that of the former. This is,
+therefore, the transition temperature for the trihydrate and monohydrate.
+The solubility curve of the monohydrate succeeds that of the trihydrate,
+and exhibits a conspicuous point of minimum solubility at about 30deg.
+Below 24.7deg the {144} monohydrate is the less stable hydrate, but its
+solubility has been determined to a temperature of 22deg. At 41deg the
+solubility curve of the monohydrate intersects that of the anhydrous salt,
+and this is therefore the transition temperature for the monohydrate and
+anhydrous salt. Above this temperature the anhydrous salt is the stable
+solid phase. Its solubility curve also passes through a minimum.
+
+The diagram of solubilities of barium acetate not only illustrates the way
+in which the solubility curves of the different stable hydrates of a salt
+succeed one another, but it has also an interest and importance from
+another point of view. In Fig. 36 there is also shown a faintly drawn curve
+which is continuous throughout its whole course. This curve represents the
+solubility of barium acetate as determined by Krasnicki.[223] Since,
+however, three different solid phases can exist under the conditions of
+experiment, it is evident, from what has already been stated (p. 111), that
+the different equilibria between barium acetate and water could not be
+represented by one _continuous_ curve.
+
+Another point which these experiments illustrate and which it is of the
+highest importance to bear in mind is, that in making determinations of the
+solubility of salts which are capable of forming hydrates, it is not only
+necessary to determine the composition of the solution, but _it is of equal
+importance to determine the composition of the solid phase in contact with
+it_. In view of the fact, also, that the solution equilibrium is in many
+cases established with comparative slowness, it is necessary to confirm the
+point of equilibrium, either by approaching it from higher as well as from
+lower temperatures, or by actually determining the rate with which the
+condition of equilibrium is attained. This can be accomplished by actual
+weighing of the dissolved salt or by determinations of the density of the
+solution, as well as by other methods.
+
+{145}
+
+2. _The Compounds formed have a Definite Melting Point._
+
+In the cases which have just been considered we saw that the salt hydrates
+on being heated did not undergo complete fusion, but that a solid was
+deposited consisting of a lower hydrate or of the anhydrous salt. It has,
+however, been long known that certain crystalline salt hydrates (_e.g._
+sodium thiosulphate, Na_{2}S_{2}O_{3},5H_{2}O, sodium acetate,
+NaC_{2}H_{3}O_{2},3H_{2}O) melt completely in their water of
+crystallization, and yield a liquid of the _same composition_ as the
+crystalline salt. In the case of sodium thiosulphate pentahydrate the
+temperature of liquefaction is 56deg; in the case of sodium acetate
+trihydrate, 58deg. These two salts, therefore, have a definite melting
+point. For the purpose of studying the behaviour of such salt hydrates, we
+shall choose not the cases which have just been mentioned, but two others
+which have been more fully studied, viz. the hydrates of calcium chloride
+and of ferric chloride.
+
+Solubility Curve of Calcium Chloride Hexahydrate.[224]--Although calcium
+chloride forms several hydrates, each of which possesses its own
+solubility, it is nevertheless the solubility curve of the hexahydrate
+which will chiefly interest us at present, and we shall therefore first
+discuss that curve by itself.
+
+[Illustration: FIG. 37.]
+
+The solubility of this salt has been determined from the cryohydric point,
+which lies at about -55deg, up to the melting point of the salt.[225] The
+solubility increases with rise of temperature, as is shown by the figures
+in the following table, and by the (diagrammatic) curve AB in Fig. 37. In
+the table, the numbers under the heading "solubility" denote the number of
+grams of CaCl_{2} dissolved in 100 grams {146} of water; those under the
+heading "composition," the number of gram-molecules of water in the
+solution to one gram-molecule of CaCl_{2}.
+
+  SOLUBILITY OF CALCIUM CHLORIDE HEXAHYDRATE.
+
+  -----------------------------------------
+  Temperature. | Solubility. | Composition.
+  -----------------------------------------
+     -55deg    |     42.5    |    14.5
+     -25deg    |     50.0    |    12.3
+     -10deg    |     55.0    |    11.2
+       0deg    |     59.5    |    10.37
+      10deg    |     65.0    |     9.49
+      20deg    |     74.5    |     8.28
+      25deg    |     82.0    |     7.52
+      28.5deg  |     90.5    |     6.81
+      29.5deg  |     95.5    |     6.46
+      30.2deg  |    102.7    |     6.00
+      29.6deg  |    109.0    |     5.70
+      29.2deg  |    112.8    |     5.41
+  -----------------------------------------
+
+So far as the first portion of the curve is concerned, it resembles the
+most general type of solubility curve. In the present case the solubility
+is so great and increases so rapidly with rise of temperature, that a point
+is reached at which the water of crystallization of the salt is sufficient
+for its complete solution. This temperature is 30.2deg; and since the
+composition of the solution is the same as that of the solid salt, viz. 1
+mol. of CaCl_{2} to 6 mols. of water, this temperature must be the melting
+point of the hexahydrate. At this point the hydrate will fuse or the
+solution will solidify without change of temperature and without change of
+composition. Such a melting point is called a _congruent_ melting point.
+
+But the solubility curve of calcium chloride hexahydrate differs markedly
+from the other solubility curves hitherto considered in that it possesses a
+_retroflex portion_, represented in the figure by BC. As is evident from
+the figure, therefore, calcium chloride hexahydrate exhibits the peculiar
+and, as it was at first thought, impossible behaviour that it can be in
+equilibrium at one and the same temperature with two different solutions,
+one of which contains more, the other less, water than the solid hydrate;
+for it must be remembered that {147} throughout the whole course of the
+curve ABC the solid phase present in equilibrium with the solution is the
+hexahydrate.
+
+Such a behaviour, however, on the part of calcium chloride hexahydrate will
+appear less strange if one reflects that the melting point of the hydrate
+will, like the melting point of other substances, be lowered by the
+addition of a second substance. If, therefore, water is added to the
+hydrate at its melting point, the temperature at which the solid hydrate
+will be in equilibrium with the liquid phase (solution) will be lowered; or
+if, on the other hand, anhydrous calcium chloride is added to the hydrate
+at its melting point (or what is the same thing, if water is removed from
+the solution), the temperature at which the hydrate will be in equilibrium
+with the liquid will also be lowered; _i.e._ the hydrate will melt at a
+lower temperature. In the former case we have the hydrate in equilibrium
+with a solution containing more water, in the latter case with a solution
+containing less water than is contained in the hydrate itself.
+
+It has already been stated (p. 109) that the solubility curve (in general,
+the equilibrium curve) is continuous so long as the solid phase remains
+unchanged; and we shall therefore expect that the curve ABC will be
+continuous. Formerly, however, it was considered by some that the curve was
+not continuous, but that the melting point is the point of intersection of
+two curves, a solubility curve and a fusion curve. Although the earlier
+solubility determinations were insufficient to decide this point
+conclusively, more recent investigation has proved beyond doubt that the
+curve is continuous and exhibits no break.[226]
+
+{148}
+
+Although in taking up the discussion of the equilibria between calcium
+chloride and water, it was desired especially to call attention to the form
+of the solubility curve in the case of salt hydrates possessing a definite
+melting point, nevertheless, for the sake of completeness, brief mention
+may be made of the other systems which these two components can form.
+
+[Illustration: FIG. 38.]
+
+Besides the hexahydrate, the solubility curve of which has already been
+described, calcium chloride can also crystallize in two different forms,
+each of which contains four molecules {149} of water of crystallization;
+these are distinguished as [alpha]-tetrahydrate, and [beta]-tetrahydrate.
+Two other hydrates are also known, viz. a dihydrate and a monohydrate. The
+solubility curves of these different hydrates are given in Fig. 38.
+
+On following the solubility curve of the hexahydrate from the ordinary
+temperature upwards, it is seen that at a temperature of 29.8deg
+represented by the point H, it cuts the solubility curve of the
+[alpha]-tetrahydrate. This point is therefore a quadruple point at which
+the four phases hexahydrate, [alpha]-tetrahydrate, solution, and vapour can
+coexist. It is also the transition point for these two hydrates. Since, at
+temperatures above 29.8deg, the [alpha]-tetrahydrate is the stable form, it
+is evident from the data given before (p. 146), as also from Fig. 38, that
+the portion of the solubility curve of the hexahydrate lying above this
+temperature represents _metastable_ equilibria. The realization of the
+metastable melting point of the hexahydrate is, therefore, due to suspended
+transformation. At the transition point, 29.8deg, the solubility of the
+hexahydrate and [alpha]-tetrahydrate is 100.6 parts of CaCl_{2} in 100
+parts of water.
+
+The retroflex portion of the solubility curve of the hexahydrate extends to
+only 1deg below the melting point of the hydrate. At 29.2deg crystals of a
+new hydrate, [beta]-tetrahydrate, separate out, and the solution, which now
+contains 112.8 parts of CaCl_{2} to 100 parts of water, is saturated with
+respect to the two hydrates. Throughout its whole extent the solubility
+curve EDF of the [beta]-tetrahydrate represents metastable equilibria. The
+upper limit of the solubility curve of [beta]-tetrahydrate is reached at
+38.4deg (F), the point of intersection with the curve for the dihydrate.
+
+Above 29.8deg the stable hydrate is the [alpha]-tetrahydrate; and its
+solubility curve extends to 45.3deg (K), at which temperature it cuts the
+solubility curve of the dihydrate. The curve of the latter hydrate extends
+to 175.5deg (L), and is then succeeded by the curve for the monohydrate.
+The solubility curve of the anhydrous salt does not begin until a
+temperature of about 260deg. The whole diagram, therefore, shows a
+succession of stable hydrates, a metastable hydrate, a metastable melting
+point and retroflex solubility curve. {150}
+
+Pressure-Temperature Diagram.--The complete study of the equilibria between
+the two components calcium chloride and water would require the discussion
+of the vapour pressure of the different systems, and its variation with the
+temperature. For our present purpose, however, such a discussion would not
+be of great value, and will therefore be omitted here; in general, the same
+relationships would be found as in the case of sodium sulphate (p. 138),
+except that the rounded portion of the solubility curve of the hexahydrate
+would be represented by a similar rounded portion in the pressure
+curve.[227] As in the case of sodium sulphate, the transition points of the
+different hydrates would be indicated by breaks in the curve of pressures.
+Finally, mention may again be made of the difference of the pressure of
+dissociation of the hexahydrate according as it becomes dehydrated to the
+[alpha]- or the [beta]-tetrahydrate (p. 88).
+
+The Indifferent Point.--We have already seen that at 30.2deg calcium
+chloride hexahydrate melts congruently, and that, provided the pressure is
+maintained constant, addition or withdrawal of heat will cause the complete
+liquefaction or solidification, without the temperature of the system
+undergoing change. This behaviour, therefore, is similar to, but is not
+quite the same as the fusion of a simple substance such as ice; and the
+difference is due to the fact that in the case of the hexahydrate the
+emission of vapour by the liquid phase causes an alteration in the
+composition of the latter, owing to the non-volatility of the calcium
+chloride; whereas in the case of ice this is, of course, not so.
+
+Consider, however, for the present that the vapour phase is absent, and
+that we are dealing with the two-phase system solid--solution. Then, since
+there are two components, the system is bivariant. For any given value of
+the pressure, therefore, we should expect that the system could exist at
+different temperatures; which, indeed, is the case. It has, however,
+already been noted that when the composition of the liquid phase becomes
+the same as that of the solid, the system then behaves as a _univariant_
+system; for, at a given pressure, the system solid--solution can exist only
+at _one_ temperature, change of temperature producing complete
+transformation in {151} one or other direction. _The variability of the
+system has therefore been diminished._
+
+This behaviour will perhaps be more clearly understood when one reflects
+that since the composition of the two phases is the same, the system may be
+regarded as being formed of _one component_, just as the system NH_{4}Cl
+<--> NH_{3} + HCl was regarded as being composed of one component when the
+vapour had the same total composition as the solid (p. 13). One component
+in two phases, however, constitutes a univariant system, and we can
+therefore see that calcium chloride hexahydrate in contact with solution of
+the same composition will constitute a univariant system. The temperature
+of equilibrium will, however, vary with the pressure;[228] if the latter is
+constant, the temperature will also be constant.
+
+A point such as has just been referred to, which represents the special
+behaviour of a system of two (or more) components, in which the composition
+of two phases becomes identical, is known as an _indifferent point_,[229]
+and it has been shown[230] that at a given pressure the temperature in the
+indifferent point is the _maximum_ or _minimum_ temperature possible at the
+particular pressure[231] (cf. critical solution temperature). At such a
+point a system loses one degree of freedom, or behaves like a system of the
+next lower order.
+
+The Hydrates of Ferric Chloride.--A better illustration of the formation of
+compounds possessing a definite melting point, and of the existence of
+retroflex solubility curves, is afforded by the hydrates of ferric
+chloride, which not only possess definite points of fusion, but these
+melting points are stable. A very brief description of the relations met
+with will suffice.[232]
+
+{152}
+
+Ferric chloride can form no less than four stable hydrates, viz.
+Fe_{2}Cl_{6},12H_{2}O, Fe_{2}Cl_{6},7H_{2}O, Fe_{2}Cl_{6},5H_{2}O, and
+Fe_{2}Cl_{6},4H_{2}O, and each of these hydrates possesses a definite,
+stable melting point. On analogy with the behaviour of calcium chloride,
+therefore, we shall expect that the solubility curves of these different
+hydrates will exhibit a series of _temperature maxima_; the points of
+maximum temperature representing systems in which the composition of the
+solid and liquid phases is the same. A graphical representation of the
+solubility relations is given in Fig. 39, and the composition of the
+different saturated solutions which can be formed is given in the following
+tables, the composition being expressed in molecules of Fe_{2}Cl_{6} to 100
+molecules of water. The figures printed in thick type refer to transition
+and melting points.
+
+[Illustration: FIG. 39.]
+
+{153}
+
+  COMPOSITION OF THE SATURATED SOLUTIONS OF FERRIC CHLORIDE AND ITS
+      HYDRATES.
+
+  (_The name placed at the head of each table is the solid phase._)
+
+  ICE.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+  +/--55deg    |  +/-2.75
+     -40deg    |     2.37
+     -27.5deg  |     1.90
+     -20.5deg  |     1.64
+     -10deg    |     1.00
+       0deg    |     0
+  ---------------------------
+
+  Fe_{2}Cl_{6},12H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+    -55deg     |  +/-2.75
+    -41deg     |     2.81
+    -27deg     |     2.98
+      0deg     |     4.13
+     10deg     |     4.54
+     20deg     |     5.10
+     30deg     |     5.93
+     35deg     |     6.78
+     36.5deg   |     7.93
+     37deg     |     8.33
+     36deg     |     9.29
+     33deg     |    10.45
+     30deg     |    11.20
+     27.4deg   |    12.15
+     20deg     |    12.83
+     10deg     |    13.20
+      8deg     |    13.70
+  ---------------------------
+
+  Fe_{2}Cl_{6},7H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     20deg     |    11.35
+     27.4deg   |    12.15
+     32deg     |    13.55
+     32.5deg   |    14.29
+     30deg     |    15.12
+     25deg     |    15.54
+  ---------------------------
+
+  Fe_{2}Cl_{6},5H_{2}O.
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     20deg     |    11.35
+     12deg     |    12.87
+     20deg     |    13.95
+     27deg     |    14.85
+     30deg     |    15.12
+     35deg     |    15.64
+     50deg     |    17.50
+     55deg     |    19.15
+     56deg     |    20.00
+     55deg     |    20.32
+  ---------------------------
+
+  Fe_{2}Cl_{6},4H_{2}O
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     20deg     |    11.35
+     50deg     |    19.96
+     55deg     |    20.32
+     60deg     |    20.70
+     69deg     |    21.53
+     72.5deg   |    23.35
+     73.5deg   |    25.00
+     72.5deg   |    26.15
+     70deg     |    27.90
+     66deg     |    29.20
+  ---------------------------
+
+  Fe_{2}Cl_{6} (ANHYDROUS).
+  ---------------------------
+  Temperature. | Composition.
+  ---------------------------
+     20deg     |    11.35
+     66deg     |    29.20
+     70deg     |    29.42
+     75deg     |    28.92
+     80deg     |    29.20
+    100deg     |    29.75
+  ---------------------------
+
+The lowest portion of the curve, AB, represents the equilibria between ice
+and solutions containing ferric chloride. It represents, in other words,
+the lowering of the fusion point of ice by addition of ferric chloride. At
+the point B (-55deg), the cryohydric point (p. 117) is reached, at which
+the solution is in equilibrium with ice and ferric chloride dodecahydrate.
+As {154} has already been shown, such a point represents an invariant
+system; and the liquid phase will, therefore, solidify to a mixture of ice
+and hydrate without change of temperature. If heat is added, ice will melt
+and the system will pass to the curve BCDN, which is the solubility curve
+of the dodecahydrate. At C (37deg), the point of maximum temperature, the
+hydrate melts completely. The retroflex portion of this curve can be
+followed backwards to a temperature of 8deg, but below 27.4deg (D), the
+solutions are supersaturated with respect to the heptahydrate; point D is
+the eutectic point for dodecahydrate and heptahydrate. The curve DEF is the
+solubility curve of the heptahydrate, E being the melting point, 32.5deg.
+On further increasing the quantity of ferric chloride, the temperature of
+equilibrium is lowered until at F (30deg) another eutectic point is
+reached, at which the heptahydrate and pentahydrate can co-exist with
+solution. Then follow the solubility curves for the pentahydrate, the
+tetrahydrate, and the anhydrous salt; G (56deg) is the melting point of the
+former hydrate, J (73.5deg) the melting point of the latter. H and K, the
+points at which the curves intersect, represent eutectic points; the
+temperature of the former is 55deg, that of the latter 66deg. The dotted
+portions of the curves represent metastable equilibria.
+
+As is seen from the diagram, a remarkable series of solubility curves is
+obtained, each passing through a point of maximum temperature, the whole
+series of curves forming an undulating "festoon." To the right of the
+series of curves the diagram represents unsaturated solutions; to the left,
+supersaturated.
+
+If an unsaturated solution, the composition of which is represented by a
+point in the field to the right of the solubility curves, is cooled down,
+the result obtained will differ according as the composition of the
+solution is the same as that of a cryohydric point, or of a melting point,
+or has an intermediate value. Thus, if a solution represented by _x__{1} is
+cooled down, the composition will remain unchanged as indicated by the
+horizontal dotted line, until the point D is reached. At this point,
+dodecahydrate and heptahydrate will separate out, and the liquid will
+ultimately solidify completely to a mixture or "conglomerate" of these two
+hydrates; the temperature of {155} the system remaining constant until
+complete solidification has taken place. If, on the other hand, a solution
+of the composition _x__{3} is cooled down, ferric chloride dodecahydrate
+will be formed when the temperature has fallen to that represented by C,
+and the solution will completely solidify, without alteration of
+temperature, with formation of this hydrate. In both these cases,
+therefore, a point is reached at which complete solidification occurs
+without change of temperature.
+
+Somewhat different, however, is the result when the solution has an
+intermediate composition, as represented by _x__{2} or _x__{4}. In the
+former case the dodecahydrate will first of all separate out, but on
+further withdrawal of heat the temperature will fall, the solution will
+become relatively richer in ferric chloride, owing to separation of the
+hydrate, and ultimately the eutectic point D will be reached, at which
+complete solidification will occur. Similarly with the second solution.
+Ferric chloride dodecahydrate will first be formed, and the temperature
+will gradually fall, the composition of the solution following the curve CB
+until the cryohydric point B is reached, when the whole will solidify to a
+conglomerate of ice and dodecahydrate.
+
+Suspended Transformation.--Not only can the upper branch of the solubility
+curve of the dodecahydrate be followed backwards to a temperature of 8deg,
+or about 19deg below the temperature of transition to the heptahydrate; but
+suspended transformation has also been observed in the case of the
+heptahydrate and the pentahydrate. To such an extent is this the case that
+the solubility curve of the latter hydrate has been followed downwards to
+its point of intersection with the curve for the dodecahydrate. This point
+of intersection, represented in Fig. 39 by M, lies at a temperature of
+about 15deg; and at this temperature, therefore, it is possible for the two
+solid phases dodecahydrate and pentahydrate to coexist, so that M is a
+eutectic point for the dodecahydrate and the pentahydrate. It is, however,
+a metastable eutectic point, for it lies in the region of supersaturation
+with respect to the heptahydrate; and it can be realized only because of
+the fact that the latter hydrate is not readily formed.
+
+Evaporation of Solutions at Constant Temperature.--On {156} evaporating
+dilute solutions of ferric chloride at constant temperature, a remarkable
+series of changes is observed, which, however, will be understood with the
+help of Fig. 40. Suppose an unsaturated solution, the composition of which
+is represented by the point _x__{1}, is evaporated at a temperature of
+about 17deg - 18deg. As water passes off, the composition of the solution
+will follow the dotted line of constant temperature, until at the point
+where it cuts the curve BC the solid hydrate Fe_{2}Cl_{6},12H_{2}O
+separates out. As water continues to be removed, the hydrate must be
+deposited (in order that the solution shall remain saturated), until
+finally the solution dries up to the hydrate. As dehydration proceeds, the
+heptahydrate can be formed, and the dodecahydrate will finally pass into
+the heptahydrate; and this, in turn, into the pentahydrate.
+
+[Illustration: FIG. 40.]
+
+But the heptahydrate is not always formed by the dehydration of the
+dodecahydrate, and the behaviour on evaporation is therefore somewhat
+perplexing at first sight. After the solution has dried to the
+dodecahydrate, as explained above, further removal of water causes
+liquefaction, and the system is now represented by the point of
+intersection at _a_; at this point the solid hydrate is in equilibrium with
+a solution containing relatively more ferric chloride. If, therefore,
+evaporation is continued, the solid hydrate must _pass into solution_ in
+order that the composition of the latter may remain unchanged, so that
+ultimately a liquid will again be obtained. A very slight further
+dehydration will bring the solution into the state represented by _b_, at
+which the pentahydrate is formed, and the solution will at last disappear
+and leave this hydrate alone.
+
+Without the information to be obtained from the curves in Figs. 39 and 40,
+the phenomena which would be observed on carrying out the evaporation at a
+temperature of about 31 - 32deg {157} would be still more bewildering. The
+composition of the different solutions formed will be represented by the
+perpendicular line _x__{2}12345. Evaporation will first cause the
+separation of the dodecahydrate, and then total disappearance of the liquid
+phase. Then liquefaction will occur, and the system will now be represented
+by the point 2, in which condition it will remain until the solid hydrate
+has disappeared. Following this there will be deposition of the
+heptahydrate (point 3), with subsequent disappearance of the liquid phase.
+Further dehydration will again cause liquefaction, when the concentration
+of the solution will be represented by the point 4; the heptahydrate will
+ultimately disappear, and then will ensue the deposition of the
+pentahydrate, and complete solidification will result. On evaporating a
+solution, therefore, of the composition _x__{2}, the following series of
+phenomena will be observed: solidification to dodecahydrate; liquefaction;
+solidification to heptahydrate; liquefaction; solidification to
+pentahydrate.[233]
+
+Although ferric chloride and water form the largest and best-studied series
+of hydrates possessing definite melting points, examples of similar
+hydrates are not few in number; and more careful investigation is
+constantly adding to the list.[234] In all these cases the solubility curve
+will show a point of maximum temperature, at which the hydrate melts, and
+will end, above and below, in a cryohydric point. Conversely, if such a
+curve is found in a system of two components, we can argue that a definite
+compound of the components possessing a definite melting point is formed.
+
+Inevaporable Solutions.--If a saturated solution in contact with two
+hydrates, or with a hydrate and anhydrous salt is heated, the temperature
+and composition of the solution will, of course, remain unchanged so long
+as the two solid phases are present, for such a system is invariant. In
+addition to this, however, the _quantity_ of the solution will also remain
+unchanged, the water which evaporates being supplied by the higher hydrate.
+The same phenomenon is also observed in the case of cryohydric points when
+ice is a solid phase; so long as the latter is present, evaporation will be
+accompanied {158} by fusion of the ice, and the quantity of solution will
+remain constant. Such solutions are called _inevaporable_.[235]
+
+[Illustration: FIG. 41.]
+
+Illustration.--In order to illustrate the application of the principles of
+the Phase Rule to the study of systems formed by a volatile and a
+non-volatile component, a brief description may be given of the behaviour
+of sulphur dioxide and potassium iodide, which has formed the subject of a
+recent investigation. After it had been found[236] that liquid sulphur
+dioxide has the property of dissolving potassium iodide, and that the
+solutions thus obtained present certain peculiarities of behaviour, the
+question arose as to whether or not compounds are formed between the
+sulphur dioxide and the potassium iodide, and if so, what these compounds
+are. To find an answer to this question, Walden and Centnerszwer[237] made
+a complete investigation of the solubility curves (equilibrium curves) of
+these two components, the investigation extending from the freezing point
+to the critical point of sulphur dioxide. For convenience of reference, the
+results which they obtained are represented diagrammatically in Fig. 41.
+The freezing point (A) of pure sulphur dioxide was found to be -72.7deg.
+Addition of potassium iodide lowered the freezing point, but the maximum
+depression obtained was very small, and was reached when the concentration
+of the potassium iodide in the solution was only 0.336 mols. per cent.
+Beyond this point, an increase in the concentration of the iodide was
+accompanied by an elevation of the freezing point, the change of the
+freezing point with the concentration being represented by the curve BC.
+The solid {159} which separated from the solutions represented by BC was a
+bright _yellow_ crystalline substance. At the point C (-23.4deg) a
+temperature-maximum was reached; and as the concentration of the potassium
+iodide was continuously increased, the temperature of equilibrium first
+fell and then slowly rose, until at +0.26deg (E) a second
+temperature-maximum was registered. On passing the point D, the solid which
+was deposited from the solution was a _red_ crystalline substance. On
+withdrawing sulphur dioxide from the system, the solution became turbid,
+and the temperature remained constant. The investigation was not pursued
+farther at this point, the attention being then directed to the equilibria
+at higher temperatures.
+
+When a solution of potassium iodide in liquid sulphur dioxide containing
+1.49 per cent. of potassium iodide was heated, solid (potassium iodide) was
+deposited at a temperature of 96.4deg. Solutions containing more than about
+3 per cent. of the iodide separated, on being heated, into two layers, and
+the temperature at which the liquid became heterogeneous fell as the
+concentration was increased; a temperature-minimum being obtained with
+solutions containing 12 per cent. of potassium iodide. On the other hand,
+solutions containing 30.9 per cent. of the iodide, on being heated,
+deposited potassium iodide; while a solution containing 24.5 per cent. of
+the salt first separated into two layers at 89.3deg, and then, on cooling,
+solid was deposited and one of the liquid layers disappeared.
+
+Such are, in brief, the results of experiment; their interpretation in the
+light of the Phase Rule is the following:--
+
+The curve AB is the freezing-point curve of solid sulphur dioxide in
+contact with solutions of potassium iodide. BCD is the solubility curve of
+the yellow crystalline solid which is deposited from the solutions. C, the
+temperature-maximum, is the melting point of this _yellow_ solid, and the
+composition of the latter must be the same as that of the solution at this
+point (p. 145), which was found to be that represented by the formula
+KI,14SO_{2}. B is therefore the eutectic point, at which solid sulphur
+dioxide and the compound KI,14SO_{2} can exist together in equilibrium with
+solution and vapour. The curve DE is the solubility curve of the _red_
+crystalline solid, and the {160} point E, at which the composition of
+solution and solid is the same, is the melting point of the solid. The
+composition of this substance was found to be KI,4SO_{2}.[238] D is,
+therefore, the eutectic point at which the compounds KI,14SO_{2} and
+KI,4SO_{2} can coexist in equilibrium with solution and vapour. The curve
+DE does not exhibit a retroflex portion; on the contrary, on attempting to
+obtain more concentrated solutions in equilibrium with the compound
+KI,4SO_{2}, a new solid phase (probably potassium iodide) was formed. Since
+at this point there are four phases in equilibrium, viz. the compound
+KI,4SO_{2}, potassium iodide, solution, and vapour, the system is
+invariant. E is, therefore, the _transition point_ for KI,4SO_{2} and KI.
+
+Passing to higher temperatures, FG is the solubility curve of potassium
+iodide in sulphur dioxide; at G two liquid phases are formed, and the
+system therefore becomes invariant (cf. p. 121). The curve GHK is the
+solubility curve for two partially miscible liquids; and since complete
+miscibility occurs on _lowering_ the temperature, the curve is similar to
+that obtained with triethylamine and water (p. 101). K is also an invariant
+point at which potassium iodide is in equilibrium with two liquid phases
+and vapour.
+
+The complete investigation of the equilibria between sulphur dioxide and
+potassium iodide, therefore, shows that these two components form the
+compounds KI,14SO_{2} and KI,4SO_{2}; and that when solutions having a
+concentration between those represented by the points G and K are heated,
+separation into two layers occurs. The temperatures and concentrations of
+the different characteristic points are as follows:--
+
+  -------------------------------------------------------------
+                              |                | Composition of
+              Point.          |  Temperature.  |  the solution
+                              |                |  per cent. KI.
+  -------------------------------------------------------------
+  A (m.p. of SO_{2})          |       -72.7deg |       --
+  B (eutectic point)          |        --      |      0.86
+  C (m.p. of KI,14SO_{2})     |       -23.4deg |     17.63
+  E (m.p. of KI,4SO_{2})      |        +0.26deg|     39.33
+  G (KI + two liquid phases)  |(about) 88deg   |     24.0
+  H (critical solution point) |        77.3deg |     12
+  K (KI + two liquid phases)  |(about) 88deg   |      2.7
+  -------------------------------------------------------------
+
+       *       *       *       *       *
+
+
+{161}
+
+CHAPTER IX
+
+EQUILIBRIA BETWEEN TWO VOLATILE COMPONENTS
+
+General.--In the two preceding chapters certain restrictions were imposed
+on the discussion of the equilibria between two components; but in the
+present chapter the restriction that only one of the components is volatile
+will be allowed to fall, and the general behaviour of two volatile[239]
+components, each of which is capable of forming a liquid solution with the
+other, will be studied. As we shall see, however, the removal of the
+previous restriction produces no alteration in the general aspect of the
+equilibrium curves for concentration and temperature, but changes to some
+extent the appearance of the pressure-temperature diagram. The latter would
+become still more complicated if account were taken not only of the total
+pressure but also of the partial pressures of the two components in the
+vapour phase; this complication, however, will not be introduced in the
+present discussion.[240] In this chapter we shall consider the systems
+formed by the two components iodine and chlorine, and sulphur dioxide and
+water.
+
+Iodine and Chlorine.--The different systems furnished by iodine and
+chlorine, rendered classical by the studies of Stortenbeker,[241] form a
+very complete example of equilibria in a two-component system. We shall
+first of all consider the {162} relations between concentration and
+temperature, with the help of the accompanying diagram, Fig. 42.
+
+[Illustration: FIG. 42.]
+
+Concentration-Temperature Diagram.--In this diagram the temperatures are
+taken as the abscissae, and the composition of the solution, expressed in
+atoms of chlorine to one atom of iodine,[242] is represented by the
+ordinates. In the diagram, A represents the melting point of pure iodine,
+114deg. If chlorine is added to the system, a solution of chlorine in
+liquid iodine is obtained, and the temperature at which solid iodine is in
+equilibrium with the liquid solution will be all the lower the greater the
+concentration of the chlorine. We therefore obtain the curve ABF, which
+represents the composition of the solution {163} with which solid iodine is
+in equilibrium at different temperatures. This curve can be followed down
+to 0deg, but at temperatures below 7.9deg (B) it represents metastable
+equilibria. At B iodine monochloride can be formed, and if present the
+system becomes invariant; B is therefore a quadruple point at which the
+four phases, iodine, iodine monochloride, solution, and vapour, can
+coexist. Continued withdrawal of heat at this point will therefore lead to
+the complete solidification of the solution to a mixture or conglomerate of
+iodine and iodine monochloride, while the temperature remains constant
+during the process. B is the eutectic point for iodine and iodine
+monochloride.
+
+Just as we found in the case of aqueous salt solutions that at temperatures
+above the cryohydric or eutectic point, two different solutions could
+exist, one in equilibrium with ice, the other in equilibrium with the salt
+(or salt hydrate), so in the case of iodine and chlorine there can be two
+solutions above the eutectic point B, one containing a lower proportion of
+chlorine in equilibrium with iodine, the other containing a higher
+proportion of chlorine in equilibrium with iodine monochloride. The
+composition of the latter solution is represented by the curve BCD. As the
+concentration of chlorine is increased, the temperature at which there is
+equilibrium between iodine monochloride and solution rises until a point is
+reached at which the composition of the solution is the same as that of the
+solid. At this point (C), iodine monochloride melts. Addition of one of the
+components will lower the temperature of fusion, and a continuous
+curve,[243] exhibiting a retroflex portion as in the case of
+CaCl_{2},6H_{2}O, will be obtained. At temperatures below its melting
+point, therefore, iodine monochloride can be in equilibrium with two
+different solutions.
+
+The upper portion of this curve, CD, can be followed downwards to a
+temperature of 22.7deg. At this temperature iodine trichloride can separate
+out, and a second quadruple {164} point (D) is obtained. This is the
+eutectic point for iodine monochloride and iodine trichloride.
+
+By addition of heat and increase in the amount of chlorine, the iodine
+monochloride disappears, and the system passes along the curve DE, which
+represents the composition of the solutions in equilibrium with solid
+iodine trichloride. The concentration of chlorine in the solution increases
+as the temperature is raised, until at the point E, where the solution has
+the same composition as the solid, the maximum temperature is reached; the
+iodine trichloride melts. On increasing still further the concentration of
+chlorine in the solution, the temperature of equilibrium falls, and a
+continuous curve, similar to that for the monochloride, is obtained. The
+upper branch of this curve has been followed down to a temperature of
+30deg, the solution at this point containing 99.6 per cent. of
+chlorine.[244] The very rounded form of the curve is due to the trichloride
+being largely dissociated in the liquid state.
+
+One curve still remains to be considered. As has already been mentioned,
+iodine monochloride can exist in two crystalline forms, only one of which,
+however, is stable at temperatures below the melting point; the two forms
+are _monotropic_ (p. 44). The stable form which melts at 27.2deg, is called
+the [alpha]-form, while the less stable variety, melting at 13.9deg, is
+known as the [beta]-form. If, now, the presence of [alpha]-ICl is excluded,
+it is possible to obtain the [beta]-form, and to study the conditions of
+equilibrium between it and solutions of iodine and chlorine, from the
+eutectic point F to the melting point G. As the [beta]-ICl becomes less
+stable in presence of excess of chlorine, it has not been possible to study
+the retroflex portion of the curve represented by the dotted continuation
+of FG.
+
+The following table gives some of the numerical data from which Fig. 42 was
+constructed.[245]
+
+{165}
+
+  IODINE AND CHLORINE.
+
+  I. _Invariant systems._
+
+  -------------------------------------------------------------------------
+          |          |                  Phases present.
+   Temper-| Pressure.+--------------------+-----------------+--------------
+   ature. |          |      Solid.        |    Liquid.      |   Vapour.
+  --------+----------+--------------------+-----------------+--------------
+  7.9deg  | 11 mm.   | I_{2},[alpha]-ICl  | I[wavy]Cl_{0.66}| I + Cl_{0.92}
+  0.9deg  |   --     | I_{2},[beta]-ICl   | I[wavy]Cl_{0.72}|     --
+  22.7deg | 42 mm.   | [alpha]-ICl,ICl_{3}| I[wavy]Cl_{1.19}| I + Cl_{1.75}
+  [-102deg| <1 atm.  | ICl_{3},Cl_{2}     | I[wavy]Cl_{m}   | I + Cl_{n}]
+  --------+----------+--------------------+-----------------+--------------
+
+  II. _Melting points._
+
+  A. Iodine,[246] 114.15deg (pressure 89.8 mm.).
+  C. [alpha]-Iodine monochloride, 27.2deg (pressure 37 mm.).
+  E. Iodine trichloride, 101deg (pressure 16 atm.).
+  G. [beta]-Iodine monochloride, 13.9deg.
+
+Since the vapour pressure at the melting point of iodine trichloride
+amounts to 16 atm., the experiments must of course be carried out
+in closed vessels. At 63.7deg the vapour pressure of the system
+trichloride--solution--vapour is equal to 1 atm.
+
+Pressure-Temperature Diagram.--In this diagram there are represented the
+values of the vapour pressure of the saturated solutions of chlorine and
+iodine. To give a complete picture of the relations between pressure,
+temperature, and concentration, a solid model would be required, with three
+axes at right angles to one another along which could be measured the
+values of pressure, temperature, and concentration of the components in the
+solution. Instead of this, however, there may be employed the accompanying
+projection figure[247] (Fig. 43), the lower portion of which shows the
+projection of the equilibrium curve on the surface containing the
+concentration and temperature axes, while the upper portion is the
+projection on the plane containing the pressure and temperature axes. The
+lower portion is therefore a concentration-temperature diagram; {166} the
+upper portion, a pressure-temperature diagram. The corresponding points of
+the two diagrams are joined by dotted lines.
+
+[Illustration: FIG. 43.]
+
+Corresponding to the point C, the melting point of pure iodine, there is
+the point C_{1}, which represents the vapour pressure of iodine at its
+melting point. At this point three curves cut: 1, the sublimation curve of
+iodine; 2, the vaporization curve of fused iodine; 3, C_{1}B_{1}, the
+vapour-pressure curve of the saturated solutions in equilibrium with solid
+iodine. Starting, therefore, with the system solid iodine--liquid iodine,
+addition of chlorine will cause the temperature of equilibrium to fall
+continuously, while the vapour pressure will first increase, pass through a
+maximum and then fall continuously {167} until the eutectic point, B
+(B_{1}), is reached.[248] At this point the system is invariant, and the
+pressure will therefore remain constant until all the iodine has
+disappeared. As the concentration of the chlorine increases in the manner
+represented by the curve B_f_H, the pressure of the vapour also increases
+as represented by the curve B_{1}_f__{1}H_{1}. At H_{1}, the eutectic point
+for iodine monochloride and iodine trichloride, the pressure again remains
+constant until all the monochloride has disappeared. As the concentration
+of the solution passes along the curve HF, the pressure of the vapour
+increases as represented by the curve H_{1}F_{1}; F_{1} represents the
+pressure of the vapour at the melting point of iodine trichloride. If the
+concentration of the chlorine in the solution is continuously increased
+from this point, the vapour pressure first increases and then decreases,
+until the eutectic point for iodine trichloride and solid chlorine is
+reached (D_{1}). Curves Cl_{2} solid and Cl_{2} liquid represent the
+sublimation and vaporization curves of chlorine, the melting point of
+chlorine being -102deg.
+
+Although complete measurements of the vapour pressure of the different
+systems of pure iodine to pure chlorine have not been made, the
+experimental data are nevertheless sufficient to allow of the general form
+of the curves being indicated with certainty.
+
+Bivariant Systems.--To these, only a brief reference need be made. Since
+there are two components, two phases will form a bivariant system. The
+fields in which these systems can exist are shown in Fig. 43 and Fig. 44,
+which is a more diagrammatic representation of a portion of Fig. 43.
+
+    I. Iodine--vapour.
+   II. Solution--vapour.
+  III. Iodine trichloride--vapour.
+  IV. Iodine monochloride--vapour.
+
+[Illustration: FIG. 44.]
+
+The conditions for the existence of these systems will probably be best
+understood from Fig. 44. Since the curve B'A' {168} represents the
+pressures under which the system iodine--solution--vapour can exist,
+increase of volume (diminution of pressure) will cause the volatilization
+of the solution, and the system iodine--vapour will remain. If, therefore,
+we start with a system represented by _a_, diminution of pressure at
+constant temperature will lead to the condition represented by _x_. On the
+other hand, increase of pressure at _a_ will lead to the condensation of a
+portion of the vapour phase. Since, now, the concentration of chlorine in
+the vapour is greater than in the solution, condensation of vapour would
+increase the concentration of chlorine in the solution; a certain amount of
+iodine must therefore pass into solution in order that the composition of
+the latter shall remain unchanged.[249] If, therefore, the volume of vapour
+be sufficiently great, continued diminution of volume will ultimately lead
+to the disappearance of all the iodine, and there will remain only solution
+and vapour (field II.). As the diminution of volume is continued, the
+vapour pressure and the concentration of the chlorine in the solution will
+increase, until when the pressure has reached the value _b_, iodine
+monochloride can separate out. The system, therefore, again becomes
+univariant, and at constant temperature the pressure and composition of the
+phases must remain unchanged. Diminution of volume will therefore not
+effect an increase of pressure, but a condensation of the vapour; and since
+this is richer in chlorine than the {169} solution, solid iodine
+monochloride must separate out in order that the concentration of the
+solution remain unchanged.[250] As the result, therefore, we obtain the
+bivariant system iodine monochloride--vapour.
+
+A detailed discussion of the effect of a continued increase of pressure
+will not be necessary. From what has already been said and with the help of
+Fig. 44, it will readily be understood that this will lead successively to
+the univariant system (_c_), iodine monochloride--solution--vapour; the
+bivariant system solution--vapour (field II.); the univariant system (_d_),
+iodine trichloride--solution--vapour; and the bivariant system _x'_, iodine
+trichloride--vapour. If the temperature of the experiment is above the
+melting point of the monochloride, then the systems in which this compound
+occurs will not be formed.
+
+Sulphur Dioxide and Water.--In the case just studied we have seen that the
+components can combine to form definite compounds possessing stable melting
+points. The curves of equilibrium, therefore, resemble in their general
+aspect those of calcium chloride and water, or of ferric chloride and
+water. In the case of sulphur dioxide and water, however, the melting point
+of the compound formed cannot be realized, because transition to another
+system occurs; retroflex concentration-temperature curves are therefore not
+found here, but the curves exhibit breaks or sudden changes in direction at
+the transition points, as in the case of the systems formed by sodium
+sulphate and water. The case of sulphur dioxide and water is also of
+interest from the fact that two liquid phases can be formed.
+
+The phases which occur are--Solid: ice, sulphur dioxide hydrate,
+SO_{2},7H_{2}O. Liquid: two solutions, the one containing excess of sulphur
+dioxide, the other excess of water, and represented by the symbols SO_{2}
+[wavy] _x_H_{2}O (solution I.), and H_{2}O [wavy] _y_SO_{2} (solution II.).
+Vapour: a mixture of sulphur dioxide and water vapour in varying
+proportions. Since there are two components, sulphur dioxide and water, the
+number of {170} possible systems is considerable. Only the following,
+however, have been studied:--
+
+  I. _Invariant Systems: Four co-existing phases._
+    (_a_) Ice, hydrate, solution, vapour.
+    (_b_) Hydrate, solution I., solution II., vapour.
+
+  II. _Univariant Systems: Three co-existing phases._
+    (_a_) Hydrate, solution I., vapour.
+    (_b_) Hydrate, solution II., vapour.
+    (_c_) Solution I., solution II., vapour.
+    (_d_) Hydrate, solution I., solution II.
+    (_e_) Hydrate, ice, vapour.
+    (_f_) Ice, solution II., vapour.
+    (_g_) Ice, hydrate, solution II.
+
+  III. _Bivariant Systems: Two co-existing phases._
+    (_a_) Hydrate, solution I.
+    (_b_) Hydrate, solution II.
+    (_c_) Hydrate, vapour.
+    (_d_) Hydrate, ice.
+    (_e_) Solution I., solution II.
+    (_f_) Solution I., vapour.
+    (_g_) Solution I., ice.
+    (_h_) Solution II., vapour.
+    (_i_) Solution II., ice.
+    (_j_) Ice, vapour.
+
+[Illustration: FIG. 45.]
+
+Pressure-Temperature Diagram.[251]--If sulphur dioxide is passed into water
+at 0deg, a solution will be formed and the temperature at which ice can
+exist in equilibrium with this solution will fall more and more as the
+concentration of the sulphur dioxide increases. At -2.6deg, however, a
+cryohydric point is reached at which solid hydrate separates out,
+and the system becomes invariant. The curve AB (Fig. 45) therefore
+represents the pressure of the system ice--solution II.--vapour, and B
+represents the temperature and pressure at which the invariant system
+ice--hydrate--solution II.--vapour can exist. At this point the temperature
+is -2.6deg, and the pressure 21.1 cm. If heat is withdrawn from this
+system, the solution will ultimately {171} solidify to a mixture of
+ice and hydrate, and there will be obtained the univariant system
+ice--hydrate--vapour. The vapour pressure of this system has been
+determined down to a temperature of -9.5deg, at which temperature the
+pressure amounts to 15 cm. The pressures for this system are represented by
+the curve BC. If at the point B the volume is diminished, the pressure must
+remain constant, but the relative amounts of the different phases will
+undergo change. If suitable quantities of these are present, diminution of
+volume will ultimately lead to the total condensation of the vapour phase,
+and there will remain the univariant system ice--hydrate--solution. The
+temperature of equilibrium of this system will alter with the pressure,
+but, as in the case of the melting point of a simple substance, great
+differences of pressure will cause only comparatively small changes in the
+temperature of equilibrium. The change of the cryohydric point with the
+pressure is represented by the line BE; the actual values have not been
+determined, but the curve must slope towards the pressure axis because
+fusion is accompanied by diminution of volume, as in the case of pure ice.
+{172}
+
+A fourth univariant system can be formed at B. This is the system
+hydrate--solution II.--vapour. The conditions for the existence of
+this system are represented by the curve BF, which may therefore be
+regarded as the vapour-pressure curve of the saturated solution of
+sulphur dioxide heptahydrate in water. Unlike the curve for iodine
+trichloride--solution--vapour, this curve cannot be followed to the melting
+point of the hydrate. Before this point is reached, a second liquid phase
+appears, and an invariant system consisting of hydrate--solution
+I.--solution II.--vapour is formed. We have here, therefore, the phenomenon
+of melting under the solution as in the case of succinic nitrile and water
+(p. 122). This point is represented in the diagram by F; the temperature at
+this point is 12.1deg, and the pressure 177.3 cm. The range of stable
+existence of the hydrate is therefore from -2.6deg to 12.1deg;
+nevertheless, the curve FB has been followed down to a temperature of
+-6deg, at which point ice formed spontaneously.
+
+So long as the four phases hydrate, two liquid phases, and vapour are
+present, the condition of the system is perfectly defined. By altering the
+conditions, however, one of the phases can be made to disappear, and a
+univariant system will then be obtained. Thus, if the vapour phase is made
+to disappear, the univariant system solution I.--solution II.--hydrate,
+will be left, and the temperature at which this system is in equilibrium
+will vary with the pressure. This is represented by the curve FI; under a
+pressure of 225 atm. the temperature of equilibrium is 17.1deg. Increase of
+pressure, therefore, raises the temperature at which the three phases can
+coexist.
+
+Again, addition of heat to the invariant system at F will cause the
+disappearance of the solid phase, and there will be formed the univariant
+system solution I.--solution II.--vapour. In the case of this system the
+vapour pressure increases as the temperature rises, as represented by the
+curve FG. Such a system is analogous to the case of ether and water, or
+other two partially miscible liquids (p. 103). As the temperature changes,
+the composition of the two liquid phases will undergo change; but this
+system has not been studied fully.
+
+The fourth curve, which ends at the quadruple point F, is {173} that
+representing the vapour pressure of the system hydrate--solution I.--vapour
+(FH). This curve has been followed to a temperature of 0deg, the pressure
+at this point being 113 cm. The metastable prolongation of GF has also been
+determined. Although, theoretically, this curve must lie below FH, it was
+found that the difference in the pressure for the two curves was within the
+error of experiment.
+
+Bivariant Systems.--The different bivariant systems, consisting of two
+phases, which can exist within the range of temperature and pressure
+included in Fig. 45, were given on p. 170. The conditions under which these
+systems can exist are represented by the areas in the diagram, and the
+fields of the different bivariant systems are indicated by letters,
+corresponding to the letters on p. 170. Just as in the case of
+one-component systems (p. 29), we found that the field lying between any
+two curves gave the conditions of existence of that phase which was common
+to the two curves, so also in the case of two-component systems, a
+bivariant two-phase system occurs in the field enclosed[252] by the two
+curves to which the two phases are common. As can be seen, the same
+bivariant system can occur in more than one field.
+
+As is evident from Fig. 45, three different bivariant systems are capable
+of existing in the area HFI; which of these will be obtained will depend on
+the relative masses of the different phases in the univariant or invariant
+system. Thus, starting with a system represented by a point on the curve
+HF, diminution of volume at constant temperature will cause the
+condensation of a portion of the vapour, which is rich in sulphur dioxide;
+since this would increase the concentration of sulphur dioxide in the
+solution, it must be counteracted by the passage of a portion of the
+hydrate (which is relatively poor in sulphur dioxide) into the solution.
+If, therefore, the amount of hydrate present is relatively very small, the
+final result of the compression will be the production of the system _f_,
+solution I.--vapour. On the other hand, if the vapour is present in
+relatively small amount, it will be the first phase to disappear, {174} and
+the bivariant system _a_, hydrate--solution I., will be obtained. Finally,
+if we start with the invariant system at F, compression will cause the
+condensation of vapour, while the composition of the two solutions will
+remain unchanged. When all the vapour has disappeared, the univariant
+system hydrate--solution I.--solution II. will be left. If, now, the
+pressure is still further increased, while the temperature is kept below
+12deg, more and more hydrate must be formed at the expense of the two
+liquid phases (because 12deg is the lower limit for the coexistence of the
+two liquid phases), and if the amount of the solution I. (containing excess
+of sulphur dioxide) is relatively small, it will disappear before solution
+II., and there will be obtained the bivariant system hydrate--solution II.
+(bivariant system _b_).
+
+In a similar manner, account can be taken of the formation of the other
+bivariant systems.
+
+A behaviour similar to that of sulphur dioxide and water is shown by
+chlorine and water and by bromine and water, although these have not been
+so fully studied.[253] In the case of hydrogen bromide and water, and of
+hydrogen chloride and water, a hydrate, viz. HBr,2H_{2}O and HCl,2H_{2}O,
+is formed which possesses a definite melting point, as in the case of
+iodine trichloride. In these cases, therefore, a retroflex curve is
+obtained. Further, just as in the case of the chlorides of iodine the upper
+branch of the retroflex curve ended in a eutectic point, so also in the
+case of the hydrate HBr,2H_{2}O the upper branch of the curve ends in a
+eutectic point at which the system dihydrate--monohydrate--solution--vapour
+can exist. Before the melting point of the monohydrate is reached, two
+liquid phases are formed, as in the case of sulphur dioxide and water.
+
+       *       *       *       *       *
+
+
+{175}
+
+CHAPTER X
+
+SOLID SOLUTIONS. MIXED CRYSTALS
+
+General.--With the conception of gaseous and liquid solutions, every one is
+familiar. Gases can mix in all proportions to form homogeneous solutions.
+Gases can dissolve in or be "absorbed" by liquids; and solids, also, when
+brought in contact with liquids, "pass into solution" and yield a
+homogeneous liquid phase. On the other hand, the conception of a _solid
+solution_ is one which in many cases is found more difficult to appreciate;
+and the existence and behaviour of solid solutions, in spite of their not
+uncommon occurrence and importance, are in general comparatively little
+known.
+
+The reason of this is to be found, to some extent, no doubt, in the fact
+that the term "solid solution" was introduced at a comparatively recent
+date,[254] but it is probably also due in some measure to a somewhat hazy
+comprehension of the definition of the term "solution" itself. As has
+already been said (p. 92), a solution is a homogeneous phase, the
+composition of which can vary continuously within certain limits; the
+definition involves, therefore, no condition as to the physical state of
+the substances. Accordingly, solid solutions are homogeneous solid phases,
+the composition of which can undergo continuous variation within certain
+limits. Just as we saw that the range of variation of composition is more
+limited in the case of liquids than in the case of gases, so also we find
+that the limits of miscibility are in general still more restricted in the
+case of solids. Examples of complete miscibility are, however, not unknown
+even in the case of solid substances.
+
+Solid solutions have long been known, although, of course, {176} they were
+not defined as such. Thus, the phenomena of "occlusion" of gases by metals
+and other substances (occlusion of hydrogen by palladium; occlusion of
+hydrogen by iron) are due to the formation of solid solutions. The same is
+probably also true of the phenomena of "adsorption," as in the removal of
+organic colouring matter by charcoal, although, in this case, surface
+tension no doubt plays a considerable part.[255]
+
+As examples of the solution of gases in solids there may be cited (in
+addition to the phenomena of occlusion already mentioned), the hydrated
+silicates and the zeolites. During dehydration these crystalline substances
+remain clear and transparent, and the pressure of the water vapour which
+they emit varies with the degree of hydration or the concentration of water
+in the mineral.[256] As examples of the solution of solids in solids we
+have the cementation of iron by charcoal, the formation of glass, and the
+crystallization together of isomorphous substances.
+
+Although we have here spoken of the glasses as "solid solutions," it should
+be mentioned that the term "solid" is used in its popular sense. Strictly
+speaking, the glasses are to be regarded as supercooled liquids (see also
+p. 53, footnote).
+
+In discussing the equilibria in systems containing a solid solution, it is
+of essential importance to remember that a solid solution constitutes only
+_one_ phase, a phase of varying composition, as in the case of liquid
+solutions.
+
+Solution of Gases in Solids.--Comparatively little work has been done in
+this connection, the investigations being limited chiefly to the phenomena
+of occlusion or adsorption of gases by charcoal.[257] We shall, therefore,
+indicate only briefly {177} and in a general manner, the behaviour which
+the Phase Rule enables us to foresee.[258]
+
+In dealing with the systems formed by the two phases gas--solid, three
+chief cases call for mention:--
+
+I. _The gas is not absorbed by the solid, but when the pressure reaches a
+certain value, combination of the two components can result._
+
+[Illustration: Fig. 46.]
+
+The graphic representation of such a system is shown in Fig. 46, the
+ordinates being the pressures of the gas, and the abscissae the
+concentrations of the gaseous component in the solid phase. Since there is
+no formation of a solid solution, the concentration of gas in the solid
+phase remains zero until the pressure has increased to the point A. At this
+point combination can take place. There will now be three phases present,
+viz. solid component, compound, and vapour. The system is therefore
+univariant, and if the temperature is maintained constant, the vapour
+pressure will be constant, irrespective of the amount of compound formed,
+_i.e._ irrespective of the relative amounts of gas and solid. This is
+indicated by the line AB. When the solid component has entirely
+disappeared, the system ceases to be univariant, and if no absorption
+occurs, the pressure will increase again, as shown by BC. If a second
+compound can be formed, then a second _pc_-line will be obtained, similar
+to the preceding. To this group belong the salt hydrates (Chap. VII.).
+
+II. _The gas may be absorbed and may also form a compound._
+
+If absorption of gas occurs with formation of a solid solution, then, as
+the system consists of two phases, solution--vapour, it is bivariant. At
+constant temperature, therefore, the pressure will still vary with the
+concentration of the gaseous component in the solid phase. This is
+represented by the curve AB in Fig. 47. When, however, the pressure has
+reached a certain value, combination can take place; and since there are
+now three phases present, the system is {178} univariant, and at constant
+temperature the pressure is constant, as shown by the line BC.
+
+III. _Absorption of gas occurs, but at a certain concentration the solid
+solution can separate into two immiscible solid solutions._
+
+We have seen, in Chapter VI., that two liquids can form two immiscible
+solutions, and the same has also been found true of solid solutions, as we
+shall presently learn more fully. If, now, two immiscible solutions are
+formed, then the system will become univariant, and at constant temperature
+the _pc_-curve will be a straight line, as in the case of the formation of
+a compound (cf. p. 86). The behaviour of this system will, therefore, also
+be represented diagrammatically by Fig. 47.
+
+[Illustration: FIG. 47.]
+
+_Palladium and Hydrogen._--The phenomenon of the absorption of hydrogen by
+palladium, to which Graham gave the name "occlusion," is one that has
+claimed the attention of several investigators. Although Graham was not of
+opinion that a compound is formed, but rather that the gas undergoes very
+great condensation, acts as a quasi-metal (to which he gave the name
+hydrogenium), and forms a homogeneous alloy with the palladium, later
+investigations, especially those of Troost and Hautefeuille,[259] pointed
+to the formation of a definite chemical compound, having the formula
+Pd_{2}H. This conclusion has, however, not been confirmed by subsequent
+investigation.[260]
+
+Roozeboom and Hoitsema[261] sought to arrive at a final decision as to the
+nature of the phenomenon by an investigation of the equilibrium between
+hydrogen and palladium on the basis of the Phase Rule classification given
+above. If a compound is formed, diminution of volume would cause no
+increase of pressure, but only an increase in the amount of the compound.
+
+As this is the only case of gas absorption which has been {179} accurately
+studied from this point of view, a brief account of the results obtained
+will be given here, although these are not so clear and free from ambiguity
+as one would desire.
+
+The scientists just mentioned investigated the variation of the pressure of
+hydrogen with the amount absorbed by the metal at different temperatures,
+and a few of their results, typical of all, are represented graphically in
+Fig. 48; the curves indicating the variation of the gas pressure with the
+concentration of the hydrogen in the palladium at the temperatures 120deg,
+170deg, and 200deg. As can be seen, the curve consists of three parts, an
+ascending portion which passes gradually and continuously into an almost
+horizontal but slightly ascending middle part, which in turn passes without
+break into a second rapidly ascending curve. This, as Fig. 48 indicates, is
+the general form of the curve; but the length of the middle portion varies
+with the temperature, being shorter at higher than at lower temperatures.
+
+[Illustration: FIG. 48.]
+
+What is the interpretation to be put on these curves? With regard to the
+two end portions, these represent bivariant, two-phase systems, consisting
+of a solid solution and gas. They correspond, therefore, to curve AB in
+Fig. 47. If the middle portion were horizontal, it would indicate either
+the formation of a compound or of two immiscible solid solutions. If a
+compound Pd_{2}H were formed, then the middle portion would at all
+temperatures end at the same value of the concentration, viz. that
+corresponding to 0.5 atoms of hydrogen to 1 atom of palladium. As the
+figure shows, however, this is not the case; the higher the temperature,
+the lower is the concentration at which the middle passes into the terminal
+portion of the curve. {180} Such a behaviour would, however, agree with the
+assumption of the formation of two solid solutions, the "miscibility" of
+which increases with the temperature, as in the case of the liquid
+solutions of phenol and water (p. 97). Nevertheless, although the
+assumption of the formation of two solid solutions is more satisfactory
+than that of the formation of a compound, it does not entirely explain the
+facts. If two solid solutions are formed, the pressure curve should be
+horizontal, but this is not the case; and the deviation from the horizontal
+does not appear to be due to impurities either in the gas or in the metal,
+but is apparently a peculiarity of the system. Further, the gradual instead
+of abrupt passage of the three portions of the curve into one another
+remains unexplained. Hoitsema has expressed the opinion that the occlusion
+of hydrogen by palladium is a process of continuous absorption, the
+peculiar form of the curve--the flat middle portion--being possibly due to
+a condensation of the gas, even at temperatures far above the critical
+temperature of liquid hydrogen.
+
+While, therefore, the occlusion of hydrogen by palladium still presents
+some unexplained phenomena, the behaviour found by Hoitsema would appear to
+disprove conclusively the formation of a definite chemical compound.[262]
+
+SOLUTION OF SOLIDS IN SOLIDS. MIXED CRYSTALS.
+
+The introduction by van't Hoff of the term "solid solution" resulted from
+the discovery of a number of deviations from the Raoult-van't Hoff law for
+the depression of the freezing point by dissolved substances. In all cases,
+the depression was too small; in some instances, indeed, the freezing point
+may be raised. To explain these irregularities, van't Hoff assumed that the
+dissolved substance crystallized out along with the solid solvent; and he
+showed how this would account for the {181} deviations from the law of the
+depression of the freezing point, which had been developed on the
+assumption that only the pure solvent crystallized out from the
+solution.[263]
+
+The "mixed crystals" which were thus obtained, and which van't Hoff called
+dilute solid solutions, showed great resemblance in their behaviour to
+ordinary liquid solutions, and obeyed the laws applicable to these. These
+laws, however, can no longer be applied in the case of the concentrated
+solid solutions formed by the crystallization together of isomorphous
+substances, and known as isomorphous mixtures. Indeed, it has been
+contended[264] that these isomorphous mixtures should not be considered as
+solid solutions at all, although no sharp line of demarcation can be drawn
+between the two classes. The differences, however, in the behaviour of the
+two groups are of a quantitative rather than a qualitative nature; and
+since we are concerned at present only with the qualitative behaviour, we
+shall make no distinction between the crystalline solid solutions and the
+isomorphous mixtures, but shall study the behaviour of the two classes
+under the head of "mixed crystals."
+
+Mixed crystals can be formed either by sublimation[265] or from a liquid
+phase; and in the latter case the mixed crystals can be deposited either
+from solution in a common solvent or from a mixture of the fused
+components. In this method of formation, which alone will be discussed in
+the present chapter, we are dealing with the fusion curves of two
+substances, where, however, the liquid solution is in equilibrium not with
+one of the pure components, but with a solid solution or mixed crystal. The
+simple scheme (Fig. 29, p. 117) which was obtained in the case of two
+components which crystallize out in the pure state, is no longer sufficient
+in the case of the formation of mixed crystals. With the help of the Phase
+Rule, however, the different possible systems can be classified; and
+examples of the different cases predicted by the Phase Rule have also been
+obtained by experiment.
+
+{182}
+
+We shall now consider briefly the formation of mixed crystals by
+isomorphous substances; the consideration of the formation of mixed
+crystals of isodimorphous substances will, on account of the complexity of
+the relationships, not be undertaken here.[266]
+
+_Formation of Mixed Crystals of Isomorphous Substances._
+
+For the purpose of representing the relationships found here we shall
+employ a temperature-concentration diagram,[267] in which the ordinates
+represent the temperature and the abscissae the concentration of the
+components. Since there are two solutions, the liquid and the solid, and
+since the concentration of the components in these two phases is not, in
+general, the same, two curves will be required for each system, one
+relating to the liquid phase, the other relating to the solid. The
+temperature at which solid begins to be deposited from the liquid solution
+will be called the _freezing point_ of the mixture, and the temperature at
+which the solid solution just begins to liquefy will be called the _melting
+point_ of the solid solution. The temperature-concentration curve for the
+liquid phase will therefore be the freezing-point curve; that for the solid
+solution, the melting-point curve. The latter will be represented by a
+dotted line.[268]
+
+{183}
+
+I.--THE TWO COMPONENTS CAN FORM AN UNBROKEN SERIES OF MIXED CRYSTALS.
+
+Since, as has already been pointed out (p. 176), a mixed crystal (solid
+solution) constitutes only one phase, it is evident that if the two
+components are miscible with one another in all proportions in the solid
+state, there can never be more than one solid phase present, viz. the solid
+solution or mixed crystal. If the components are completely miscible in the
+solid state, they will also be completely miscible in the liquid state, and
+there can therefore be only one liquid phase. The system can at no point
+become invariant, because there can never be more than three phases
+present. When, therefore, the two components form a continuous series of
+mixed crystals, the equilibrium curve must also be continuous. Of these
+systems three types are found.
+
+[Illustration: FIG. 49.]
+
+(_a_) _The freezing points of all mixtures lie between the freezing points
+of the pure components_ (Curve I., Fig. 49).
+
+Examples.--This type of curve is represented by the mixed crystals of
+naphthalene and [beta]-naphthol.[269] The addition of [beta]-naphthol to
+naphthalene raises the freezing point of the latter, and the rise is
+directly proportional to the amount of naphthol added. The freezing point
+curve is therefore a straight line joining the melting points of the two
+components. This behaviour, however, is rather exceptional, the
+freezing-point curve lying generally above, sometimes also below, the
+straight line joining the melting points of the pure components. Thus the
+freezing-point curve of mixtures of [alpha]-monochlorocinnamic aldehyde and
+[alpha]-monobromocinnamic aldehyde[270] lies above the {184} straight line
+joining the melting points of the pure components (31.22deg and 69.56deg),
+as is evident from the following table:--
+
+  ----------------------------------------------------------------------
+  Molecules of bromo-        |                 |
+  cinnamic aldehyde in       | Freezing point. | Deviation from straight
+  100 mols. of mixture.      |                 | line.
+  ----------------------------------------------------------------------
+          0.00               |  31.22deg       |       --
+         10.48               |  37.28deg       |     2.04deg
+         21.91               |  43.12deg       |     3.50deg
+         30.07               |  46.80deg       |     4.05deg
+         45.04               |  52.94deg       |     4.45deg
+         62.16               |  58.82deg       |     3.77deg
+         82.98               |  65.07deg       |     2.03deg
+         93.50               |  67.91deg       |     0.84deg
+        100.00               |  69.56deg       |       --
+  ----------------------------------------------------------------------
+
+Melting-point Curve.--This curve, like the freezing-point curve, must also
+be continuous, and the melting points of the different solid solutions will
+lie between the melting points of the pure components. This is represented
+by the dotted line in Fig. 49, I. The relative position of the two curves,
+which can be deduced with the help of thermodynamics and also by
+experimental determination, is found in all cases to be in accordance with
+the following rule: At any given temperature, _the concentration of that
+component by the addition of which the freezing point is depressed, is
+greater in the liquid than in the solid phase_; or, conversely, _the
+concentration of that component by the addition of which the freezing point
+is raised, is greater in the solid than in the liquid phase_. An
+illustration of this rule is afforded by the two substances chloro- and
+bromo-cinnamic aldehyde already mentioned. As can be seen from the above
+table, the addition of chlorocinnamic aldehyde lowers the melting point of
+the bromo-compound. In accordance with the rule, therefore, the
+concentration of the chloro-compound in the liquid phase must be greater
+than in the solid phase; and this was found experimentally. At a
+temperature of 49.44deg, the liquid contained 58.52 per cent., the solid
+only 52.57 per cent. of the chlorocinnamic aldehyde.
+
+From this it will also be clear that on cooling a fused mixture of two
+substances capable of forming mixed crystals, {185} the temperature of
+solidification will not remain constant during the separation of the solid;
+nor, on the other hand, will the temperature of liquefaction of the solid
+solution be constant. Thus, for example, if a liquid solution of two
+components, A and B, having the composition represented by the point _x_
+(Fig. 50), is allowed to cool, the system will pass along the line _xx'_.
+At the temperature of the point _a_, mixed crystals will be deposited, the
+composition of which will be that represented by b. As the temperature
+continues to fall, more and more solid will be deposited; and since the
+solid phase is relatively rich in the component B, the liquid will become
+relatively poorer in this. The composition of the liquid solution will
+therefore pass along the curve _ad_, the composition of the solid solution
+at the same time passing along the curve _bc_; at the point _c_ the liquid
+will solidify completely.[271]
+
+[Illustration: FIG. 50.]
+
+Conversely, if mixed crystals of the composition and at the temperature
+_x'_ are heated, liquefaction will begin at the temperature _c_, yielding a
+liquid of the composition d. On continuing to add heat, the temperature of
+the mass will rise, more of the solid will melt, and the composition of the
+two phases will change as represented by the curves _da_ and _cb_. When the
+temperature has risen to _a_, complete liquefaction will have occurred. The
+process of solidification or of liquefaction is therefore extended over a
+temperature interval _ac_.
+
+Even when the freezing-point curve is a straight line joining {186} the
+melting points of the pure components, the melting-point curve will not
+necessarily coincide with the freezing-point curve, although it may
+approach very near to it; complete coincidence can take place only when the
+melting points of the two components are identical. An example of this will
+be given later (Chap. XII.).
+
+(_b_) _The freezing-point curve passes through a maximum_ (Curve II., Fig.
+49).
+
+[Illustration: FIG. 51.]
+
+This curve exhibits the greatest degree of contrast to the freezing-point
+curve which is obtained when the pure components crystallize out. For,
+since the curve passes through a maximum, it is evident that the freezing
+point of each of the components must be _raised_ by the addition of the
+other component.
+
+Example.--Very few cases belonging to this type are known. The best example
+is found in the freezing-point curve of mixtures of _d_- and
+_l_-carvoxime[272] (C_{10}H_{14}N.OH). The freezing points and melting
+points of the different mixtures of _d_- and _l_-carvoxime are given in the
+following table, and represented graphically in Fig. 51:--
+
+  ---------------+----------------+-----------------+-----------------
+  Per cent. of   | Per cent. of   | Freezing point. | Melting point.
+  _d_-carvoxime. | _l_-carvoxime. |                 |
+  ---------------+----------------+-----------------+-----------------
+     100         |      0         |     72.0deg     |    72.0deg
+      99         |      1         |     72.4deg     |      --
+      98         |      2         |     73.0deg     |      --
+      95         |      5         |     75.4deg     |    73.0deg
+      90         |     10         |     79.0deg     |    75.0deg
+      80         |     20         |     84.6deg     |    80.0deg
+      70         |     30         |     88.2deg     |    85.0deg
+      60         |     40         |     90.4deg     |      --
+      50         |     50         |     91.4deg     |    91.4deg
+      25         |     75         |     86.4deg     |    82.0deg
+       8         |     92         |     77.4deg     |      --
+       1         |     99         |     72.4deg     |      --
+       0         |    100         |     72.0deg     |    72.0deg
+  ---------------+----------------+-----------------+-----------------
+
+{187}
+
+In this figure, the melting-point curve, _i.e._ the
+temperature-concentration curve for the mixed crystals, is represented by
+the lower curve. Since the addition of the laevo-form to the dextro-form
+raises the melting point of the latter, the concentration of the laevo-form
+(on the right-hand branch of the curve) must, in accordance with the rule
+given, be greater in the solid phase than in the liquid. Similarly, since
+addition of the dextro-form raises the melting point of the laevo-form, the
+solid phase (on the left-hand branch of the curve) must be richer in
+dextro- than in laevo-carvoxime. At the maximum point, the melting-point
+and freezing-point curves touch; at this point, therefore, the composition
+of the solid and liquid phases must be identical. It is evident, therefore,
+that at the maximum point the liquid will solidify, or the solid will
+liquefy completely without change of temperature; and, accordingly, mixed
+crystals of the composition represented by the maximum point will exhibit a
+definite melting point, and will in this respect behave like a simple
+substance.
+
+(_c_) _The freezing-point curve passes through a minimum_ (Curve III., Fig.
+49).
+
+In this case, as in the case of those systems where the pure components are
+deposited, a minimum freezing point is obtained. In the latter case,
+however, there are two freezing-point curves which intersect at a eutectic
+point; in the case where mixed crystals are formed there is only one
+continuous curve. On one side of the minimum point the liquid phase
+contains relatively more, on the other side relatively less, of the one
+component than does the solid phase; while at the minimum point the
+composition of the two phases is the same. At this point, therefore,
+complete solidification or complete liquefaction will occur without change
+of temperature, and the mixed crystals will accordingly exhibit a definite
+melting point.
+
+[Illustration: FIG. 52.]
+
+{188}
+
+Example.--As an example of this there may be taken the mixed crystals of
+mercuric bromide and iodide.[273] Mercuric bromide melts at 236.5deg, and
+mercuric iodide at 255.4deg. The mixed crystal of definite constant melting
+point (minimum point) contains 59 mols. per cent. of mercuric bromide, the
+melting point being 216.1deg.
+
+The numerical data are contained in the following table, and represented
+graphically in Fig. 52:--
+
+  -----------------------------------------------------
+  Mols. per cent. of |                 |
+  HgBr_{2}.          | Freezing point. | Melting point.
+  -----------------------------------------------------
+        100          |    236.5deg     |     236deg
+         90          |    228.8deg     |     226deg
+         80          |    222.2deg     |     219deg
+         70          |    217.8deg     |     217deg
+         65          |    216.6deg     |     216deg
+         60          |    216.1deg     |     215.5deg
+         55          |    216.3deg     |     216deg
+         50          |    217.3deg     |     216deg
+         40          |    221.1deg     |     218deg
+         30          |    227.8deg     |     223deg
+         20          |    236.2deg     |     231deg
+         10          |    245.5deg     |     242deg
+          0          |    255.4deg     |     254deg
+  -----------------------------------------------------
+
+[Illustration: FIG. 53.]
+
+Fractional Crystallization of Mixed Crystals.--With the help of the
+diagrams already given it will be possible to predict what will be the
+result of the fractional crystallization of a fused mixture of two
+substances which can form mixed crystals. Suppose, for example, a fused
+mixture of the composition _x_ (Fig. 53) is cooled down; then, as we have
+already seen, when the temperature has fallen to _a_, mixed crystals of
+composition, _b_, are deposited. If the temperature is allowed to fall
+{189} to _x'_, and the solid then separated from the liquid, the mixed
+crystals so obtained will have the composition represented by e. If, now,
+the mixed crystals _e_ are completely fused and the fused mass allowed to
+cool, separation of solid will occur when the temperature has fallen to the
+point _f_. The mixed crystals which are deposited have now the composition
+represented by _g_, i.e. _they are richer in B than the original mixed
+crystals_. By repeating this process, the composition of the successive
+crops of mixed crystals which are obtained approximates more and more to
+that of the pure component B, while, on the other hand, the composition of
+the liquid phase produced tends to that of pure A. By a systematic and
+methodical repetition of the process of fractional crystallization,
+therefore, a _practically_ complete separation of the components can be
+effected; a perfect separation is theoretically impossible.
+
+From this it will be readily understood that in the case of substances the
+freezing point of which passes through a maximum, fractional
+crystallization will ultimately lead to mixed crystals having the
+composition of the maximum point, while the liquid phase will more and more
+assume the composition of either pure A or pure B, according as the initial
+composition was on the A side or the B side of the maximum point. In those
+cases, however, where the curves exhibit a minimum, the solid phase which
+separates out will ultimately be one of the pure components, while a liquid
+phase will finally be obtained which has the composition of the minimum
+point.
+
+II.--THE TWO COMPONENTS DO NOT FORM A CONTINUOUS SERIES OF MIXED CRYSTALS.
+
+This case corresponds to that of the partial miscibility of liquids. The
+solid component A can "dissolve" the component B until the concentration of
+the latter in the mixed crystal has reached a certain value. Addition of a
+further amount of B will not alter the composition of the mixed crystal,
+but there will be formed a second solid phase consisting {190} of a
+solution of A in B. At this point the four phases, mixed crystals
+containing excess of A, mixed crystals containing excess of B, liquid
+solution, vapour, can coexist; this will therefore be an invariant point.
+The temperature-concentration curves will therefore no longer be
+continuous, but will exhibit a break or discontinuity at the point at which
+the invariant system is formed.
+
+(_a_) _The freezing-point curve exhibits a transition point_ (Curve I.,
+Fig. 54).
+
+As is evident from the figure, addition of B raises the melting point of A,
+and, in accordance with the rule previously given, the concentration of B
+in the mixed crystals will be greater than in the solution. This is
+represented in the figure by the dotted curve AD. On the other hand,
+addition of A lowers the melting point of B, and the two curves BC and BE
+are obtained for the liquid and solid phases respectively. At the
+temperature of the line CDE the liquid solution of the composition
+represented by C is in equilibrium with the two different mixed crystals
+represented by D and E. At this temperature, therefore, the _tc_-curve for
+the solid phase exhibits a discontinuity; and, since the solid phase
+undergoes change at this point, the freezing-point curve must show a break
+(p. 111).
+
+[Illustration: FIG. 54.]
+
+Example.--Curves of the form given in Fig. 54 I. have been found
+experimentally in the case of silver nitrate and sodium nitrate.[274] The
+following table contains the numerical data, which are also represented
+graphically in Fig. 55:--
+
+{191}
+
+  -----------------------------------------------------
+  Molecules NaNO_{3} | Freezing point. | Melting point.
+      per cent.      |                 |
+  -----------------------------------------------------
+          0          |     208.6deg    |    208.6deg
+          8          |     211.4deg    |    210deg
+         15.06       |     215deg      |    212deg
+         19.46       |     217.2deg    |    214.8deg
+         21.9        |     222deg      |    215deg
+         26          |     228.4deg    |    216.5deg
+         29.7        |     234.8deg    |    217.5deg
+         36.2        |     244.4deg    |    217.5deg
+         47.3        |     259.4deg    |    237.6deg
+         58.9        |     272deg      |    257deg
+         72          |     284deg      |    274deg
+        100          |     308deg      |    308deg
+  -----------------------------------------------------
+
+The temperature of the transition point is 217.5deg; at this point the
+liquid contains 19.5, and the two conjugate solid solutions 26 and 38
+molecules of sodium nitrate per cent. respectively.
+
+[Illustration: FIG. 55.]
+
+[Illustration: FIG. 56.]
+
+(_b_) _The freezing-point curve exhibits a eutectic point_ (Curve II., Fig.
+54). {192}
+
+In this case the freezing point of each of the components is lowered by the
+addition of the other, until at last a point is reached at which the liquid
+solution solidifies to a mixture or conglomerate of two mixed crystals.
+
+Examples.--Curves belonging to this class have been obtained in the case of
+potassium and thallium nitrates[275] and of naphthalene and monochloracetic
+acid.[276] The data for the latter are given in the following table and
+represented in Fig. 56:--
+
+  -------------------------------------------------------------------------
+               |         Liquid solution.     |        Solid solution.
+               ------------------------------------------------------------
+  Temperature. |              |               |              |
+               |  Per cent.   |   Per cent.   |  Per cent.   |   Per cent.
+               | naphthalene. |     acid.     | naphthalene. |     acid.
+  -------------------------------------------------------------------------
+     62deg     |     --       |     100       |     --       |     100
+     60deg     |      4.0     |      96.0     |      1.7     |      98.3
+     55deg     |     21.0     |      79.0     |      2.1     |      97.9
+     53.5deg   |     29.4     |      70.0     |     --       |      --
+     55deg     |     31.3     |      68.7     |     59.6     |      40.4
+     60deg     |     42.4     |      57.6     |     80.3     |      19.7
+     65deg     |     53.3     |      46.7     |     89.2     |      10.8
+     70deg     |     69.7     |       2.3     |     95.4     |       4.6
+     75deg     |     84.4     |      15.6     |     96.6     |       3.4
+     79.9deg   |    100       |      --       |    100       |      --
+  -------------------------------------------------------------------------
+
+At the eutectic point the liquid solution is in equilibrium with two
+different mixed crystals the composition of which is represented by D and E
+respectively. If, therefore, a fused mixture containing the two components
+A and B in the proportions represented by C is cooled down, it will, when
+the temperature has reached the point C, solidify completely to a
+_conglomerate_ of mixed crystals, D and E.
+
+[Illustration: FIG. 57.]
+
+[Illustration: FIG. 58.]
+
+Changes in Mixed Crystals with the Temperature.--In the case of the
+different types of systems represented in Fig. 49, a homogeneous liquid
+solution of the two components will exist at temperatures above the
+freezing-point curve, a homogeneous mixed crystal at temperatures below the
+melting-point curve, while at any point between the freezing-point and
+melting-point {193} curves the mixture will separate into a solid phase and
+a liquid phase. In the case, however, of the two types shown in Fig. 54 the
+relationships are somewhat more complicated. As before, the area above the
+freezing-point curve gives the conditions under which homogeneous liquid
+solutions can exist; but below the melting-point curve two different mixed
+crystals can coexist. This will be best understood from Figs. 57 and 58. D
+and E represent, as we have seen, the composition of two mixed crystals
+which are in equilibrium with the liquid solution at the temperature of the
+point C. These two mixed crystals represent, in the one case, a saturated
+solution of B in A (point D), and the other a saturated solution of A in B
+(point E). Just as we saw that the mutual solubility of two liquids varied
+with the temperature, so also in the case of two solids; as the temperature
+alters, the solubility of the two solid components in one another will
+change. This alteration is indicated diagrammatically in Figs. 57 and 58 by
+the dotted curve similar to the solubility curves for two mutually soluble
+liquids (p. 101).
+
+Suppose, now, that a mixed crystal of the composition _x_ is cooled down,
+it will remain unchanged until, when the temperature has fallen to _t'_,
+the homogeneous mixed crystal breaks up into a conglomerate of two mixed
+crystals the composition of {194} which is represented by _x'_ and _x"_
+respectively. From this, then, it can be seen that in the case of
+substances which form two solid solutions, the mixed crystals which are
+desposited from the liquid fused mass need not remain unchanged in the
+solid state, but may at some lower temperature lose their homogeneity. This
+fact is of considerable importance for the formation of alloys.[277]
+
+A good example of this will soon be met with in the case of the iron and
+carbon alloys. The alloys of copper and tin also furnish examples of the
+great changes which may take place in the alloy between the temperature at
+which it separates out from the fused mass and the ordinary temperature.
+Thus, for example, one of the alloys of copper and tin which separates out
+from the liquid as a solid solution breaks up, on cooling, into the
+compound Cu_{3}Sn and liquid:[278] a striking example of a solid substance
+partially liquefying on being cooled.
+
+       *       *       *       *       *
+
+
+{195}
+
+CHAPTER XI
+
+EQUILIBRIUM BETWEEN DYNAMIC ISOMERIDES
+
+It has long been known that certain substances, _e.g._ acetoacetic ester,
+are capable when in solution or in the fused state, of reacting as if they
+possessed two different constitutions; and in order to explain this
+behaviour the view was advanced (by Laar) that in such cases a hydrogen
+atom oscillated between two positions in the molecule, being at one time
+attached to oxygen, at another time to carbon, as represented by the
+formula--
+
+  CH_{3}.C--CH.CO_{2}C_{2}H_{5}
+         .  ^
+         .  |
+         O<-H
+
+When the hydrogen is in one position, the substance will act as an
+hydroxy-compound; with hydrogen in the other position, as a ketone.
+Substances possessing this double function are called _tautomeric_.
+
+Doubt, however, arose as to the validity of the above explanation, and this
+doubt was confirmed by the isolation of the two isomerides in the solid
+state, and also by the fact that the velocity of change of the one
+isomeride into the other could in some cases be quantitatively measured.
+These and other observations then led to the view, in harmony with the laws
+of chemical dynamics, that tautomeric substances in the dissolved or fused
+state represent a _mixture_ of two isomeric forms, and that equilibrium is
+established not by _intra_- but by _inter_-molecular change, as expressed
+by the equation--
+
+  CH_{3}.CO.CH_{2}.CO_{2}C_{2}H_{5} <--> CH_{3}.C(OH):CH.CO_{2}C_{2}H_{5}
+
+{196} In the solid state, the one or other of the isomerides represents the
+stable form; but in the liquid state (solution or fusion) the stable
+condition is an equilibrium between the two forms.
+
+A similar behaviour is also found in the case of other isomeric substances
+where the isomerism is due to difference of structure, _i.e._ structure
+isomerism (_e.g._ in the case of the oximes
+
+  C_{6}H_{5}.C.H      C_{6}H_{5}.C.H
+             ||   and            ||  ),
+             N.OH             HO.N
+
+or to difference in configuration, _i.e._ stereoisomerism (_e.g._ optically
+active substances), or to polymerism (_e.g._ acetaldehyde and paraldehyde).
+In all such cases, although the different solid forms correspond to a
+single definite constitution, in the liquid state a condition of
+equilibrium between the two modifications is established. As a general name
+for these different classes of substances, the term "dynamic isomerides"
+has been introduced; and the different kinds of isomerism are classed
+together under the title "dynamic isomerism."[279]
+
+By reason of the importance of these phenomena in the study more especially
+of Organic Chemistry, a brief account of the equilibrium relations
+exhibited by systems composed of dynamic isomerides may be given here.[280]
+
+In studying the fusion and solidification of those substances which exhibit
+the relationships of dynamic isomerism, the phenomena observed will vary
+somewhat according as the reversible transformation of the one form into
+the other takes place with measurable velocity at temperatures in the
+neighbourhood of the melting points, or only at some higher temperature. If
+the transformation is very rapid, the system will behave like a
+one-component system, but if the isomeric change is comparatively slow, the
+behaviour will be that of a two-component system.
+
+Temperature-Concentration Diagram.--The relationships which are met with
+here will be most readily understood with {197} the help of Fig. 59.
+Suppose, in the first instance, that isomeric transformation does not take
+place at the temperature of the melting point, then the freezing point
+curve will have the simple form ACB; the formation of compounds being for
+the present excluded. This is the simplest type of curve, and gives the
+composition of the solutions in equilibrium with the one modification
+([alpha] modification) at different temperatures (curve AC); and of the
+solutions in equilibrium with the other modification ([beta] modification)
+at different temperatures (curve BC). C is the eutectic point at which the
+two solid isomerides can exist side by side in contact with the solution.
+
+[Illustration: FIG. 59.]
+
+Now, suppose that isomeric transformation takes place with measurable
+velocity. If the pure [alpha]-modification is heated to a temperature _t'_
+above its melting point, and the liquid maintained at that temperature
+until equilibrium has been established, a certain amount of the [beta]-form
+will be present in the liquid, the composition of which will be represented
+by the point _x'_. The same condition of equilibrium will also be reached
+by starting with pure [beta]. Similarly, if the temperature of the liquid
+is maintained at the temperature _t"_, equilibrium will be reached, we
+shall suppose, when the solution has the composition _x"_. The curve DE,
+therefore, which passes through all the different values of _x_
+corresponding to different values of _t_, will represent the change of
+equilibrium with the temperature. It will slope to the right (as in the
+figure) if the transformation of [alpha] into [beta] is accompanied by
+absorption of heat; to the left if the transformation is accompanied by
+evolution of heat, in accordance with van't Hoff's Law of movable
+equilibrium. If transformation occurs without heat effect, the equilibrium
+will be independent of the {198} temperature, and the equilibrium curve DE
+will therefore be perpendicular and parallel to the temperature axis.
+
+We must now find the meaning of the point D. Suppose the pure [alpha]- or
+pure [beta]-form heated to the temperature _t'_, and the temperature
+maintained constant until the liquid has the composition _x'_ corresponding
+to the equilibrium at that temperature. If the temperature is now allowed
+to fall sufficiently slowly so that the condition of equilibrium is
+continually readjusted as the temperature changes, the composition of the
+solution will gradually alter as represented by the curve _x'_D. Since D is
+on the freezing point curve of pure [alpha], this form will be deposited on
+cooling; and since D is also on the equilibrium curve of the liquid, D is
+the only point at which solid can exist in stable equilibrium with the
+liquid phase. (The vapour phase may be omitted from consideration, as we
+shall suppose the experiments carried out in open vessels.) All systems
+consisting of the two hylotropic[281] isomeric substances [alpha] and
+[beta] will, therefore, ultimately freeze at the point D, which is called
+the "natural" freezing point[282] of the system; provided, of course, that
+sufficient time is allowed for equilibrium to be established. From this it
+is apparent that _the stable modification at temperatures in the
+neighbourhood of the melting point is that which is in equilibrium with the
+liquid phase at the natural freezing point_.
+
+From what has been said, it will be easy to predict what will be the
+behaviour of the system under different conditions. If pure [alpha] is
+heated, a temperature will be reached at which it will melt, but this
+melting point will be sharp only if the velocity of isomeric transformation
+is comparatively slow; _i.e._ slow in comparison with the determination of
+the melting point. If the substance be maintained in the fused condition
+for some time, a certain amount of the [beta] modification will be formed,
+and on lowering the temperature the pure [alpha] form will be deposited,
+not at the temperature of the melting point, but at some lower temperature
+depending on the concentration of the [beta] modification in the liquid
+phase. If isomeric transformation {199} takes place slowly in comparison
+with the rate at which deposition of the solid occurs, the liquid will
+become increasingly rich in the [beta] modification, and the freezing point
+will, therefore, sink continuously. At the eutectic point, however, the
+[beta] modification will also be deposited, and the temperature will remain
+constant until all has become solid. If, on the other hand, the velocity of
+transformation is sufficiently rapid, then as quickly as the [alpha]
+modification is deposited, the equilibrium between the two isomeric forms
+in the liquid phase will continuously readjust itself, and the end-point of
+solidification will be the natural freezing point.
+
+Similarly, starting with the pure [beta] modification, the freezing point
+after fusion will gradually fall owing to the formation of the [alpha]
+modification; and the composition of the liquid phase will pass along the
+curve BC. If, now, the rate of cooling is not too great, or if the velocity
+of isomeric transformation is sufficiently rapid, complete solidification
+will not occur at the eutectic point; for at this temperature solid and
+liquid are not in stable equilibrium with one another. On the contrary, a
+further quantity of the [beta] modification will undergo isomeric change,
+the liquid phase will become richer in the [alpha] form, and the freezing
+point will _rise_; the solid phase in contact with the liquid being now the
+[alpha] modification. The freezing point will continue to rise until the
+point D is reached, at which complete solidification will take place
+without further change of temperature.
+
+The diagram also allows us to predict what will be the result of rapidly
+cooling a fused mixture of the two isomerides. Suppose that either the
+[alpha] or the [beta] modification has been maintained in the fused state
+at the temperature _t'_ sufficiently long for equilibrium to be
+established. The composition of the liquid phase will be represented by
+_x'_. If the liquid is now _rapidly_ cooled, the composition will remain
+unchanged as represented by the dotted line _x'_G. At the temperature of
+the point G solid [alpha] modification will be deposited. If the cooling is
+not carried below the point G, so as to cause complete solidification, the
+freezing point will be found to rise with time, owing to the conversion of
+some of the [beta] form into the [alpha] form {200} in the liquid phase;
+and this will continue until the composition of the liquid has reached the
+point D. From what has just been said, it can also be seen that if the
+freezing point curves can be obtained by actual determination of the
+freezing points of different synthetic mixtures of the two isomerides, it
+will be possible to determine the condition of equilibrium in the fused
+state at any given temperature without having recourse to analysis. All
+that is necessary is to rapidly cool the fused mass, after equilibrium has
+been established, and find the freezing point at which solid is deposited;
+that is, find the point at which the line of constant temperature cuts the
+freezing point curve. The composition corresponding to this temperature
+gives the composition of the equilibrium mixture at the given temperature.
+
+It will be evident, from what has gone before, that the degree of
+completeness with which the different curves can be realised will depend on
+the velocity with which isomeric change takes place, and on the rapidity
+with which the determinations of the freezing point can be carried out. As
+the two extremes we have, on the one hand, practically instantaneous
+transformation, and on the other, practically infinite slowness of
+transformation. In the former case, only one melting and freezing point
+will be found, viz. the natural freezing point; in the latter case, the two
+isomerides will behave as two perfectly independent components, and the
+equilibrium curve DE will not be realised.
+
+The diagram which is obtained when isomeric transformation does not occur
+within measurable time at the temperature of the melting point is somewhat
+different from that already given in Fig. 59. In this case, the two
+freezing point curves AC and BC (Fig. 60) can be readily realized, as no
+isomeric change occurs in the liquid phase. Suppose, however, that at a
+higher temperature, _t'_, reversible isomeric transformation can take
+place, the composition of the liquid phase will alter until at the point
+_x'_ a condition of equilibrium is reached; and the composition of the
+liquid at higher temperatures will be represented by the curve _x'_F. Below
+the temperature _t'_ the position of the equilibrium curve is hypothetical;
+but as the temperature {201} falls the velocity of transformation
+diminishes, and at last becomes _practically_ zero. The equilibrium curve
+can therefore be regarded as dividing into two branches _x'_G and _x'_H. At
+temperatures between G and _t'_ the [alpha] modification can undergo
+isomeric change leading to a point on the curve G_x'_; and the [beta]
+modification can undergo change leading to a point on the curve H_x'_. The
+same condition of equilibrium is therefore not reached from each side, and
+we are therefore dealing not with true but with false equilibrium (p. 5).
+Below the temperatures G and H, isomeric transformation does not occur in
+measurable time. We shall not, however, enter into a detailed discussion of
+the equilibria in such systems, more especially as they are not systems in
+true equilibrium, and as the temperature at which true equilibrium can be
+established with appreciable velocity alters under the influence of
+catalytic agents.[283] Examples of such systems will no doubt be found in
+the case of optically active substances, where both isomerides are
+apparently quite stable at the melting point. In the case of such
+substances, also, the action of catalytic agents in producing isomeric
+transformation (racemisation) is well known.
+
+[Illustration: FIG. 60.]
+
+Transformation of the Unstable into the Stable Form.--As has already been
+stated, the stable modification in the neighbourhood of the melting point
+is that one which is in equilibrium with the liquid phase at the natural
+freezing point. In the case of polymorphic substances, we have seen (p. 39)
+that that form which is stable in the neighbourhood of the melting point
+melts at the higher temperature. That was a {202} consequence of the fact
+that the two polymorphic forms on melting gave identical liquid phases. In
+the present case, however, the above rule does not apply, for the simple
+reason that the liquid phase obtained by the fusion of the one modification
+is not identical with that obtained by the fusion of the other. In the case
+of isomeric substances, therefore, the form of lower melting point _may_ be
+the more stable; and where this behaviour is found it is a sign that the
+two forms are isomeric (or polymeric) and not polymorphic.[284] An example
+of this is found in the case of the isomeric benzaldoximes (p. 203).
+
+Since in Fig. 59 the [alpha] modification has been represented as the
+stable form, the transformation of the [beta] into the [alpha] form will be
+possible at all temperatures down to the transition point. At temperatures
+below the eutectic point, transformation will occur without formation of a
+liquid phase; but at temperatures above the eutectic point liquefaction can
+take place. This will be more readily understood by drawing a line of
+constant temperature, HK, at some point between C and B. Then if the [beta]
+modification is maintained for a sufficiently long time at that
+temperature, a certain amount of the [alpha] modification will be formed;
+and when the composition of the mixture has reached the point H, fusion
+will occur. If the temperature is maintained constant, isomeric
+transformation will continue to take place in the liquid phase until the
+equilibrium point for that temperature is reached. If this temperature is
+higher than the natural melting point, the mixture will remain liquid all
+the time; but if it is below the natural melting point, then the [alpha]
+modification will be deposited when the system reaches the condition
+represented by the point on the curve AC corresponding to the particular
+temperature. As isomeric transformation continues, the freezing point of
+the system will rise until it reaches the natural freezing point D.
+Similarly, if the [alpha] modification is maintained at a temperature above
+that of the point D, liquefaction will ultimately occur, and the system
+will again reach the final state represented by D.[285]
+
+{203}
+
+Examples.--_Benzaldoximes._ The relationships which have just been
+discussed from the theoretical point of view will be rendered clearer by a
+brief description of cases which have been experimentally investigated. The
+first we shall consider is that of the two isomeric benzaldoximes:[286]--
+
+  C_{6}H_{5}.C.H            C_{6}H_{5}.C.H
+             ||                        ||
+          HO.N                         N.OH
+
+  Benzantialdoxime          Benzsynaldoxime
+  ([alpha]-modification).    ([beta]-modification).
+
+Fig. 61 gives a graphic representation of the results obtained.
+
+The melting point of the [alpha] modification is 34-35deg; the melting
+point of the unstable [beta]-modification being 130deg. The freezing curves
+AC and BC were obtained by determining the freezing points of different
+mixtures of known composition, and the numbers so obtained are given in the
+following table.
+
+{204}
+
+  ----------------------------------------------------
+  Grams of the [alpha] modification |
+    in 100 gm. of mixture.          | Freezing point.
+  ----------------------------------+-----------------
+               26.2                 |    101deg
+               49.2                 |     79deg
+               73.7                 |     46deg
+               91.7                 |     26.2deg
+               95.0                 |     28.6deg
+               96.0                 |     30.0deg
+  ----------------------------------------------------
+
+[Illustration: FIG. 61.]
+
+The eutectic point C was found to lie at 25-26deg, and the natural freezing
+point D was found to be 27.7deg. The equilibrium curve DE was determined by
+heating the liquid mixtures at different temperatures until equilibrium was
+attained, and then rapidly cooling the liquid. In all cases the freezing
+point was practically that of the point D. From this it is seen that the
+equilibrium curve must be a straight line parallel to the temperature axis;
+and, therefore, isomeric transformation in the case of the two
+benzaldoximes is not accompanied by any heat effect (p. 197). This
+behaviour has also been found in the case of acetaldoxime.[287]
+
+The isomeric benzaldoximes are also of interest from the fact that the
+stable modification has the _lower_ melting point (_v._ p. 202).
+
+_Acetaldehyde and Paraldehyde._--As a second example of the equilibria
+between two isomerides, we shall take the two isomeric (polymeric) forms of
+acetaldehyde, which have recently been exhaustively studied.[288]
+
+{205}
+
+In the case of these two substances the reaction
+
+  3CH_{3}.CHO <--> (CH_{3}.CHO)_{3}
+
+takes place at the ordinary temperature with very great slowness. For this
+reason it is possible to determine the freezing point curves of
+acetaldehyde and paraldehyde. The three chief points on these curves,
+represented graphically in Fig. 62, are:--
+
+  m.p. of acetaldehyde - 118.45deg
+  m.p. of paraldehyde  +  12.55deg
+  eutectic point       - 119.9deg
+
+[Illustration: FIG. 62.]
+
+In order to determine the position of the natural melting point, it was
+necessary, on account of the slowness of transformation, to employ a
+catalytic agent in order to increase the velocity with which the
+equilibrium was established. A drop of concentrated sulphuric acid served
+the purpose. In presence of a trace of this substance, isomeric
+transformation very speedily occurs, and leads to the condition of
+equilibrium. Starting in the one case with fused paraldehyde, and in the
+other case with acetaldehyde, the same freezing point, viz. 6.75deg, was
+obtained, the solid phase being paraldehyde. This temperature, 6.75deg, is
+therefore the natural freezing point, and paraldehyde, the solid in
+equilibrium with the liquid phase at this point, is the stable form.
+
+With regard to the change of equilibrium with the temperature, it was found
+that whereas the liquid phase contained 11.7 molecules per cent. of
+acetaldehyde at the natural freezing point, the liquid at the temperature
+of 41.6deg contains 46.6 molecules per cent. of acetaldehyde. As the
+temperature {206} rises, therefore, there is increased formation of
+acetaldehyde, or a decreasing amount of polymerisation. This is in harmony
+with the fact that the polymerisation of acetaldehyde is accompanied by
+evolution of heat.
+
+While speaking of these isomerides, it may be mentioned that at the
+temperature 41.6deg the equilibrium mixture has a vapour pressure equal to
+the atmospheric pressure. At this temperature, therefore, the equilibrium
+mixture (obtained quickly with the help of a trace of sulphuric acid)
+boils.[289]
+
+       *       *       *       *       *
+
+
+{207}
+
+CHAPTER XII
+
+SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF TWO
+COMPONENTS
+
+In this concluding chapter on two-component systems, it is proposed to
+indicate briefly how the Phase Rule has been applied to the elucidation of
+a number of problems connected with the equilibria between two components,
+and how it has been employed for the interpretation of the data obtained by
+experiment. It is hoped that the practical value of the Phase Rule may
+thereby become more apparent, and its application to other cases be
+rendered easier.
+
+The interest and importance of investigations into the conditions of
+equilibrium between two substances, lie in the determination not only of
+the conditions for the stable existence of the participating substances,
+but also of whether or not chemical action takes place between these two
+components; and if combination occurs, in the determination of the nature
+of the compounds formed and the range of their existence. In all such
+investigations, the Phase Rule becomes of conspicuous value on account of
+the fact that its principles afford, as it were, a touchstone by which the
+character of the system can be determined, and that from the form of the
+equilibrium curves obtained, conclusions can be drawn as to the nature of
+the interaction between the two substances. In order to exemplify the
+application of the principles of the Phase Rule more fully than has already
+been done, illustrations will be drawn from investigations on the
+interaction of organic compounds; on the equilibria between optically
+active compounds; and on alloys. {208}
+
+Summary of the Different Systems of Two Components.--Before passing to the
+consideration of the application of the Phase Rule to the investigation of
+particular problems, it will be well to collect together the different
+types of equilibrium curves with which we are already acquainted; to
+compare them with one another, in order that we may then employ these
+characteristic curves for the interpretation of the curves obtained as the
+result of experiment.
+
+In investigating the equilibria between two components, three chief classes
+of curves will be obtained according as--
+
+I. No combination takes place between the two components.
+
+II. The components can form definite compounds.
+
+III. The components separate out in the form of mixed crystals.
+
+The different types of curves which are obtained in these three cases are
+represented in Figs. 63, 64, 65. These different diagrams represent the
+whole series of equilibria, from the melting point of the one component (A)
+to that of the other component (B). The curves represent, in all cases, the
+composition of the solution, or phase of variable composition; the
+temperature being measured along one axis, and the composition along the
+other.
+
+We shall now recapitulate very briefly the characteristics of the different
+curves.
+
+[Illustration: FIG. 63.]
+
+If no compound is formed between the two components, {209} the general form
+of the equilibrium curve will be that of curve I. or II., Fig. 63. Type I.
+is the simplest form of curve found, and consists, as the diagram shows, of
+only two branches, AC and BC, meeting at the point C, _which lies below the
+melting point of either component_. The solid phase which is in equilibrium
+with the solutions AC is pure A; that in equilibrium with BC, pure B. C is
+the eutectic point. Although at the eutectic point the solution solidifies
+entirely without change of temperature, the solid which is deposited is not
+a homogeneous solid phase, but a mixture, or conglomerate of the two
+components. _The eutectic point, therefore, represents the melting or
+freezing point, not of a compound, but of a mixture_ (p. 119).
+
+Curve II., Fig. 63, is obtained when two liquid phases are formed. C is an
+eutectic point, D and F are transition points at which there can co-exist
+the four phases--solid, two liquid phases, vapour. DEF represents the
+change in the composition of the two liquid phases with rise of
+temperature; the curve might also have the reversed form with the critical
+solution point below the transition points D and F.
+
+[Illustration: FIG. 64.]
+
+In the second class of systems (Fig. 64), that in which combination between
+the components occurs, there are again two types according as the compound
+formed has a definite melting point (_i.e._ can exist in equilibrium with a
+solution of the same composition), or undergoes only partial fusion; that
+is, exhibits a transition point.
+
+If a compound possessing a definite melting point is formed, the
+equilibrium curve will have the general form shown by curve I., Fig. 64. A,
+B, and D are the melting points of pure A, pure B, and of the compound
+A_{x}B_{y} respectively. AC {210} is the freezing point curve of A in
+presence of B; BE that of B in presence of A; and DC and DE the freezing
+point curves of the compound in presence of a solution containing excess of
+one of the components. C and E are eutectic points at which mixtures of A
+and A_{x}B_{y}, or B and A_{x}B_{y} can co-exist in contact with solution.
+The curve CDE may be large or small, and the melting point of the compound,
+D, may lie above or below that of each of the components, or may have an
+intermediate position. If more than one compound can be formed, a series of
+curves similar to CDE will be obtained (_cf._ p. 152).
+
+On the other hand, if the compound undergoes transition to another solid
+phase at a temperature below its melting point, a curve of the form II.,
+Fig. 64, will be found. This corresponds to the case where a compound can
+exist only in contact with solutions containing excess of one of the
+components. The metastable continuation of the equilibrium curve for the
+compound is indicated by the dotted line, the summit of which would be the
+melting point of the compound. Before this temperature is reached, however,
+the solid compound ceases to be able to exist in contact with solution, and
+transition to a different solid phase occurs at the point E (_cf._ p. 134).
+This point, therefore, represents the limit of the existence of the
+compound AB. If a series of compounds can be formed none of which possess a
+definite melting point, then a series of curves will be obtained which do
+not exhibit a temperature-maximum, and there will be only one eutectic
+point. The limits of existence of each compound will be marked by a break
+in the curve (_cf._ p. 143).
+
+[Illustration: FIG. 65.]
+
+Turning, lastly, to the third class of systems, in which formation of mixed
+crystals can occur, five different types of curves can be obtained, as
+shown in Fig. 65. With regard to the first three types, curves I., II., and
+III., {211} these differ entirely from those of the previous classes, in
+that they are continuous; they exhibit no eutectic point, and no transition
+point. Curve II. bears some resemblance to the melting-point curve of a
+compound (_e.g._ CDE, Fig. 64, I.), but differs markedly from it in not
+ending in eutectic points.
+
+Further, in the case of the formation of a compound, the composition of the
+solid phase remains unchanged throughout the whole curve between the
+eutectic points; whereas, when mixed crystals are produced, the composition
+of the solid phase varies with the composition of the liquid solution. On
+passing through the maximum, the relative proportions of A and B in the
+solid and the liquid phase undergo change; on the one side of the maximum,
+the solid phase contains relatively more A, and on the other side of the
+maximum, relatively more B than the liquid phase. Lastly, when mixed
+crystals are formed, the temperature at which complete solidification
+occurs changes as the composition of the solution changes, whereas in the
+case of the formation of compounds, the temperature of complete
+solidification for all solutions is a eutectic point.
+
+The third type of curve, Fig. 65, can be distinguished in a similar manner
+from the ordinary eutectic curve, Fig. 63, I., to which it bears a certain
+resemblance. Whereas in the case of the latter, the eutectic point is the
+temperature of complete solidification of all solutions, the point of
+minimum temperature in the case of the formation of mixed crystals, is the
+solidification point only of solutions having one particular composition;
+that, namely, of the minimum point. For all other solutions, the
+temperature of complete solidification is different. Whereas, also, in the
+case of the simple eutectic curve, the solid which separates out from the
+solutions represented by either curve remains the same throughout the whole
+extent of that curve, the composition of the mixed crystal varies with
+variation of the composition of the liquid phase, and the relative
+proportions of the two components in the solid and the liquid phase are
+reversed on passing through the minimum.[290]
+
+In a similar manner, type IV., Fig. 65, can be distinguished from type II.,
+Fig. 64, by the fact that it does not exhibit a {212} eutectic point, and
+that the composition of the solid phase undergoes continuous variation with
+variation of the liquid phase on either side of the transition point.
+Lastly, type V., which does exhibit a eutectic point, differs from the
+eutectic curve of Fig. 63, in that the eutectic point does not constitute
+the point of complete solidification for all solutions, and that the
+composition of the solid phase varies with the composition of the liquid
+phase.
+
+Such, then, are the chief general types of equilibrium curves for
+two-components; they are the pattern curves with which other curves,
+experimentally determined, can be compared; and from the comparison it will
+be possible to draw conclusions as to the nature of the equilibria between
+the two components under investigation.
+
+1. _Organic Compounds._
+
+[Illustration: FIG. 66.]
+
+The principles of the Phase Rule have been applied to the investigation of
+the equilibria between organic compounds, and Figs. 66-69 reproduce some of
+the results which have been obtained.[291]
+
+{213}
+
+Fig. 66, the freezing point curve (curve of equilibrium) for
+_o_-nitrophenol and _p_-toluidine, shows a curve of the simplest type[292]
+(type I., Fig. 63), in which two branches meet at an eutectic point. The
+solid phase in equilibrium with solutions represented by the left-hand
+branch of the curve was _o_-nitrophenol (m.p. 44.1deg); that in equilibrium
+with the solutions represented by the right-hand branch, was _p_-toluidine
+(m.p. 43.3deg). At the eutectic point (15.6deg), these two solid phases
+could co-exist with the liquid phase. This equilibrium curve, therefore,
+shows that _o_-nitrophenol and _p_-toluidine do not combine with one
+another.
+
+In connection with this curve, attention may be called to the interesting
+fact that although the solid produced by cooling the liquid phase at the
+eutectic point has a composition approximating to that of a compound of
+equimolecular proportions of the phenol and toluidine, and a constant
+melting point, it is nevertheless a _mixture_. Although, as a rule, the
+constituents of the eutectic mixture are not present in simple molecular
+proportions, there is no reason why they should not be so; and it is
+therefore necessary to beware of assuming the formation of compounds in
+such cases.[293]
+
+Fig. 67, on the other hand, indicates with perfect certainty the formation
+of a compound between phenol and [alpha]-naphthylamine.[294] (_Cf._ curve
+I., Fig. 64.)
+
+Phenol freezes at 40.4deg, but the addition of [alpha]-naphthylamine lowers
+the freezing point as represented by the curve AC. At C (16.0deg) the
+compound C_{6}H_{5}OH,C_{10}H_{7}NH_{2} is formed, and the system becomes
+invariant. On increasing the amount of the amine, the temperature of
+equilibrium rises, the solid phase now being the compound. At D, the curve
+passes through a maximum (28.8deg), at which the solid and liquid phases
+have the same composition. This is the melting point of the compound.
+Further addition of the amine lowers the temperature of equilibrium, until
+at E solid [alpha]-naphthylamine separates out, and a second eutectic point
+(24.0deg) is obtained. BE is the {214} freezing-point curve of
+[alpha]-naphthylamine in presence of phenol, the freezing point of the pure
+amine being 48.3deg.
+
+On account of the great sluggishness with which the compound of phenol and
+[alpha]-naphthylamine crystallizes, it was found possible to follow the
+freezing point curves of phenol and the amine to temperatures considerably
+below the eutectic points, as shown by the curves CF and EG.
+
+[Illustration: FIG. 67.]
+
+Phenol can also combine with _p_-toluidine in equimolecular proportions;
+and this compound is of interest, from the fact that it exists in two
+crystalline forms melting at 28.5deg and 30deg. Each of these forms now
+must have its own equilibrium curve, and it was found that the intermediate
+portion of the freezing point curve was duplicated, as shown in Fig.
+68.[295]
+
+{215}
+
+[Illustration: FIG. 68.]
+
+[Illustration: FIG. 69.]
+
+{216}
+
+Lastly, a curve is given, Fig. 69,[296] which corresponds with curve II.,
+Fig. 64. Picric acid and benzene can form a compound, which, however, can
+exist only in contact with solutions _containing excess of benzene_. When
+the temperature is raised, a point (K) is reached at which the compound
+melts with separation of solid picric acid. The point, K, is, therefore, a
+_transition point_; analysis, however, showed that the composition of the
+solution at this point is very nearly that of the compound
+C_{6}H_{2}(NO_{2})_{3}OH,C_{6}H_{6}, so that the melting point of the
+compound can almost be reached. The fusion of the compound of benzene and
+picric acid with separation of the latter is analogous to the (partial)
+fusion of Glauber's salt with separation of anhydrous sodium sulphate.
+
+2. _Optically Active Substances._
+
+The question as to whether a resolvable inactive body is a mixture of the
+two oppositely active constituents (a _dl_-mixture), or a racemic compound,
+is one which has given rise to considerable discussion during the past
+decade; and several investigators have endeavoured to establish general
+rules by which the question could be decided. In the case of inactive
+liquids it is a matter of great difficulty to arrive at a certain
+conclusion as to whether one is dealing with a mixture or a compound, for
+in this case the usual physical methods give but a dubious answer; and
+although the existence of a racemate in the liquid state (in the case of
+conine) has been asserted,[297] most chemists incline to the belief that
+such a thing is improbable.
+
+Even in the case of crystalline substances, where the differences between
+the various forms is greater, it was not always easy to discriminate
+between the _dl_-mixture and the racemic compound. The occurrence of
+hemihedral faces was considered by Pasteur to be a sufficient criterion for
+an optically active substance. It has, however, been found that hemihedry
+in crystals, although a frequent accompaniment of {217} optical activity,
+is by no means a necessary or constant expression of this property. Other
+rules, also, which were given, although in some cases reliable, were in
+other cases insufficient; and all were in so far unsatisfactory that they
+lacked a theoretical basis.
+
+With the help of the Phase Rule, however, it is possible from a study of
+the solubility or fusion curves of the optically active and inactive
+substances, to decide the nature of the inactive substance, at least under
+certain conditions. On account of the interest and importance which these
+compounds possess, a brief description of the application of the Phase Rule
+to the study of such substances will be given here;[298] the two optical
+antipodes being regarded as the two components.
+
+In the present chapter we shall consider only the fusion curves, the
+solubility curves being discussed in the next section on three-component
+systems. The rules which are hereby obtained, have reference only to the
+nature of the inactive substance in the neighbourhood of the melting
+points.
+
+I. _The inactive substance is a _dl_-mixture._
+
+In this case the fusion curves will have the simple form shown in type I,
+Fig. 63. A and B are the melting points of the two optical isomerides, and
+C the eutectic point at which the inactive mixture consisting of equal
+amounts of d- and l-form melts. Owing to the similar effect of the one form
+on the freezing point of the other, the figure is symmetrical. No example
+of this simple case has been investigated.
+
+II. _The two components form a racemic compound._
+
+In this case there will be three melting point curves as in Fig. 64, type
+I. In this case also the figure must be symmetrical.
+
+Examples.--As examples of this, may be taken dimethyl tartrate and mandelic
+acid, the freezing point curves of which are given in Figs. 70 and 71.[299]
+As can be seen, the curve for the racemic tartrate occupies a large part of
+the diagram, {218} while that for racemic mandelic acid is much smaller. In
+the case of dimethyldiacetyl tartrate, this middle portion is still less.
+
+[Illustration: FIG. 70.]
+
+[Illustration: FIG. 71.]
+
+[Illustration: FIG. 72.]
+
+Active dimethyl tartrate melts at 43.3deg; racemic dimethyl tartrate at
+89.4deg. Active mandelic acid melts at 132.8deg; the racemic acid at
+118.0deg. In the one case, therefore, the racemic compound has a higher, in
+the other a lower melting point than the active forms. {219}
+
+In the case of partially racemic compounds (_i.e._ the compound of a
+racemate with an optically active substance) the type of curve will be the
+same, but the figure will no longer be symmetrical. Such a curve has been
+found in the case of the l-menthyl esters of d- and l-mandelic acid (Fig.
+72).[300] The freezing point of l-menthyl d-mandelate is 97.2deg, of
+l-menthyl l-mandelate 77.6deg, and of l-menthyl r-mandelate 83.7.deg It
+will be observed that the summit of the curve for the partially racemic
+mandelate is very flat, indicating that the compound is largely dissociated
+into its components at the temperature of fusion.
+
+III. _The inactive substance is a pseudo-racemic mixed crystal._
+
+In cases where the active components can form mixed crystals, the
+freezing-point curve will exhibit one of the forms given in Fig. 65. The
+inactive mixed crystal containing 50 per cent. of the dextro and laevo
+compound, is known as a pseudo-racemic mixed crystal.[301] So far, only
+curves of the types I. and II. have been obtained.
+
+Examples.--The two active camphor oximes are of interest from the fact that
+they form a continuous series of mixed crystals, _all of which have the
+same melting point_. The curve which is obtained in this case is,
+therefore, a straight line joining the melting points of the pure active
+components; the melting point of the active isomerides and of the whole
+series of mixed crystals being 118.8deg.
+
+[Illustration: FIG. 73.]
+
+In the case of the carvoximes mixed crystals are also formed, but the
+equilibrium curve in this case exhibits a maximum (Fig. 73). At this
+maximum point the composition of the solid and of the liquid solution is
+the same. Since the curve must be symmetrical, this maximum point must
+occur in the case of the solution containing 50 per cent. {220} of each
+component, which will therefore be inactive. Further, this inactive mixed
+crystal will melt and solidify at the same temperature, and behave,
+therefore, like a chemical compound (p. 187). The melting point of the
+active compounds is 72deg; that of the inactive pseudo-racemic mixed
+crystal is 91.4deg.
+
+Transformations.--As has already been remarked, the conclusions which can
+be drawn from the fusion curves regarding the nature of the inactive
+substances formed hold only for temperatures in the neighbourhood of the
+melting points. At temperatures below the melting point transformation may
+occur; _e.g._ a racemate may break up into a _dl_-mixture, or a
+pseudo-racemic mixed crystal may form a racemic compound. We shall at a
+later point meet with examples of a racemic compound changing into a
+_dl_-mixture at a definite transition point; and the pseudo-racemic mixed
+crystal of camphoroxime is an example of the second transformation.
+Although at temperatures in the neighbourhood of the melting point the two
+active camphoroximes form only mixed crystals but no compound, a racemic
+compound is formed at temperatures below 103deg. At this temperature the
+inactive pseudo-racemic mixed crystal changes into a racemic compound; and
+in the case of the other mixed crystals transformation to racemate and
+(excess of) active component also occurs, although at a lower temperature
+than in the case of the inactive mixed crystal. Although this behaviour is
+one of considerable importance, this brief reference to it must suffice
+here.[302]
+
+3. _Alloys._
+
+One of the most important classes of substances in the study of which the
+Phase Rule has been of very considerable importance, is that formed by the
+mixtures or compounds of metals with one another known as alloys. Although
+in the investigation of the nature of these bodies various methods are
+employed, one of the most important is the determination of the character
+of the freezing-point curve; for from the form of this, valuable
+information can, as we have already learned, be {221} obtained regarding
+the nature of the solid substances which separate out from the molten
+mixture.
+
+Although it is impossible here to discuss fully the experimental results
+and the oftentimes very complicated relationships which the study of the
+alloys has brought to light, a brief reference to these bodies will be
+advisable on account both of the scientific interest and of the industrial
+importance attaching to them.[303]
+
+We have already seen that there are three chief types of freezing-point
+curves in systems of two components, viz. those obtained when (1) the pure
+components crystallize out from the molten mass; (2) the components form
+one or more compounds; (3) the components form mixed crystals. In the case
+of the metals, representatives of these three classes are also found.
+
+1. _The components separate out in the pure state._
+
+In this case the freezing-point curve is of the simple type, Fig. 63, I.
+Such curves have been obtained in the case of a number of pairs of metals,
+_e.g._ zinc--cadmium, zinc--aluminium, copper--silver (Heycock and
+Neville), tin--zinc, bismuth--lead (Gautier), and in other cases. From
+molten mixtures represented by one branch of the freezing-point curve one
+of the metals will be deposited; while from mixtures represented by the
+other branch, the other metal will separate out. At the eutectic point the
+molten mass will solidify to a _heterogeneous mixture_ of the two metals,
+forming what is known as the _eutectic alloy_. Such an alloy, therefore,
+will melt at a definite temperature lower than the melting point of either
+of the pure metals.
+
+{222}
+
+In the following table are given the temperature and the composition of the
+liquid at the eutectic point, for three pairs of metals:--
+
+  -------------------------------------------------------------------
+                  | Temperature. |      Composition of liquid.
+  -------------------------------------------------------------------
+  Zinc--cadmium   |   264.5deg   | 73.5 atoms per cent. of cadmium.
+  Zinc--aluminium |   380.5deg   | 11     "      "         aluminium.
+  Copper--silver  |   778deg     | 40     "      "         copper.
+  -------------------------------------------------------------------
+
+The melting points of the pure metals are, zinc, 419deg; cadmium, 322deg;
+silver, 960deg; copper, 1081deg; aluminium, 650deg.
+
+2. _The two metals can form one or more compounds._
+
+In this case there will be obtained not only the freezing-point curves of
+the pure metals, but each compound formed will have its own freezing-point
+curve, exhibiting a point of maximum temperature, and ending on either side
+in an eutectic point. The simplest curve of this type will be obtained when
+only one compound is formed, as is the case with mercury and thallium.[304]
+This curve is represented in Fig. 74, where the summit of the intermediate
+curve corresponds with a composition TlHg_{2}. Similar curves are also
+given by nickel and tin, by aluminium and silver, and by other metals, the
+formation of definite compounds between these pairs of metals being thereby
+indicated.[305]
+
+[Illustration: FIG. 74.]
+
+{223}
+
+A curve belonging to the same type, but more complicated, is obtained with
+gold and aluminium;[306] in this case, several compounds are formed, some
+of which have a definite melting point, while others exhibit only a
+transition point. The chief compound is AuAl_{2}, which has practically the
+same melting point as pure gold.
+
+3. _The two metals form mixed crystals (solid solutions)._
+
+The simplest case in which the metals crystallize out together is found in
+silver and gold.[307] The freezing-point curve in this case is an almost
+straight line joining the freezing points of the pure metals (_cf._ curve
+I., Fig. 65, p. 210). These two metals, therefore, can form an unbroken
+series of mixed crystals.
+
+In some cases, however, the two metals do not form an unbroken series of
+mixed crystals. In the case of zinc and silver,[308] for example, the
+addition of silver _raises_ the freezing point of the mixture, until a
+transition point is reached. This corresponds with curve IV., Fig. 65.
+Silver and copper, and gold and copper, on the other hand, do not form
+unbroken series of mixed crystals, but the freezing-point curve exhibits an
+eutectic point, as in curve V., Fig. 65.
+
+Not only may there be these three different types of curves, but there may
+also be combinations of these. Thus the two metals may not only form
+compounds, but one of the metals may not separate out in the pure state at
+all, but form mixed crystals. In this case the freezing point may rise (as
+in the case of silver and zinc), and one of the eutectic points will be
+absent.
+
+Iron-Carbon Alloys.--Of all the different binary alloys, probably the most
+important are those formed by iron and carbon: alloys consisting not of two
+metals, but of a metal and a non-metal. On account of the importance of
+these alloys, an attempt will be made to describe in brief some of the most
+important relationships met with.
+
+Before proceeding to discuss the applications of the Phase Rule to the
+study of the iron-carbon alloys, however, the main {224} facts with which
+we have to deal may be stated very briefly. With regard to the metal
+itself, it is known to exist in three different allotropic modifications,
+called [alpha]-, [beta]-, and [gamma]-ferrite respectively. Like the two
+modifications of sulphur and of tin, these different forms exhibit
+transition points at which the relative stability of the forms changes.
+Thus the transition point for [alpha]- and [beta]-ferrite is about 780deg;
+and below this temperature the [alpha]- form, above it the [beta]- form is
+stable. For [beta]- and [gamma]-ferrite, the transition point is about
+870deg, the [gamma]- form being the stable modification above this
+temperature.
+
+The different modifications of iron also possess different properties.
+Thus, [alpha]-ferrite is magnetic, but does not possess the power of
+dissolving carbon; [beta]-ferrite is non-magnetic, and likewise does not
+dissolve carbon; [gamma]-ferrite is also non-magnetic, but possesses the
+power of dissolving carbon, and of thus giving rise to solid solutions of
+carbon in iron.
+
+Various alloys of iron and carbon, also, have to be distinguished. First of
+all, there is _hard steel_, which contains varying amounts of carbon up to
+2 per cent. Microscopic examination shows that these mixtures are all
+homogeneous; and they are therefore to be regarded as solid solutions of
+carbon in iron ([gamma]-ferrite). To these solutions the name _martensite_
+has been given. _Pearlite_ contains about 0.8 per cent. of carbon, and, on
+microscopic examination, is found to be a heterogeneous mixture. If heated
+above 670deg, pearlite becomes homogeneous, and forms martensite. Lastly,
+there is a definite compound of iron and carbon, iron carbide or
+_cementite_, having the formula Fe_{3}C.
+
+A short description may now be given of the application of the Phase Rule
+to the two-component system iron--carbon; and of the diagram showing how
+the different systems are related, and with the help of which the behaviour
+of the different mixtures under given conditions can be predicted.
+Although, with regard to the main features of this diagram, the different
+areas to be mapped and the position of the frontier lines, there is general
+agreement; a final decision has not yet been reached with regard to the
+interpretation to be put on all the curves.
+
+[Illustration: FIG. 75.]
+
+The chief relationships met with in the case of the {225} iron-carbon
+alloys are represented graphically in Fig. 75.[309] The curve AC is the
+freezing-point curve for iron,[310] BC the unknown freezing-point curve for
+graphite. C is an eutectic point. Suppose, now, that we start with a
+mixture of iron and carbon, represented by the point _x_. On lowering the
+temperature, a point, _y_, will be reached at which solid begins to
+separate out. This solid phase, however, is not pure iron, but a solid
+solution of carbon in iron, having the composition represented by _y'_ (cf.
+p. 185). As the temperature continues to fall, the {226} composition of the
+liquid phase changes in the direction of _y_C, while the composition of the
+solid which separates out changes in the direction _y'_D; and, finally,
+when the composition of the molten mass is that of the point C (4.3 per
+cent. of carbon), the whole mass solidifies to a heterogeneous mixture of
+two solid solutions, one of which is represented by D (containing 2 per
+cent. of carbon), while the other will consist practically of pure
+graphite, and is not shown in the figure. The temperature of the eutectic
+point is 1130deg.
+
+Even below the solidification point, however, changes can take place. As
+has been said, the solid phase which finally separates out from the molten
+mass is a solid solution represented by the point D; and the curve DE
+represents the change in the composition of this solid solution with the
+temperature. As indicated in the figure, DE forms a part of a curve
+representing the mutual solubility of graphite in iron and iron in
+graphite; the latter solutions, however, not being shown, as they would lie
+far outside the diagram. As the temperature falls below 1130deg, more and
+more graphite separates out, until at E, when the temperature is 1000deg,
+the solid solution contains only 1.8 per cent. of carbon. At this
+temperature cementite also begins to be formed, so that as the temperature
+continues to fall, separation of cementite (represented by the line E'F')
+occurs, and the composition of the solid solution undergoes alteration, as
+represented by the curve EF. Below the temperature of the point F (670deg)
+the martensite becomes heterogeneous, and forms pearlite.
+
+From the above description, therefore, it follows that if we start with a
+molten mixture of iron and carbon, the composition of which is represented
+by any point between D and C (from 2 to 4.3 per cent. of carbon), we shall
+obtain, on cooling the mass, first of all solid solutions, the composition
+of which will be represented by points on the line AD; that then, after the
+mass has completely solidified at 1130deg, further cooling will lead to a
+separation of graphite and a change in the composition of the martensite
+(from 2 to 1.8 per cent. of carbon). On cooling below 1000deg, however, the
+martensite and graphite will give rise to cementite and solid solutions
+{227} containing less carbon than before, until, at temperatures below
+670deg, we are left with a mixture of pearlite and cementite.
+
+We have already said that iron consists in three allotropic modifications,
+the regions of stability of which are separated by definite transition
+points. The transition point for [alpha]- and [beta]-ferrite (780deg) is
+represented in Fig. 75 by the point H; and the transition point for [beta]-
+and [gamma]-ferrite (870deg) by the point I. Since neither the [alpha]- nor
+the [beta]-ferrite dissolves carbon, the transition point will be
+unaffected by addition of carbon, and we therefore obtain the horizontal
+transition curve HG. In the case of the [beta]- and [gamma]-ferrite,
+however, the latter dissolves carbon, and the transition point is
+consequently affected by the amount of carbon present. This is shown by the
+line IG.
+
+If a martensite containing less carbon than that represented by the point G
+is cooled down from a temperature of, say, 900deg, then when the
+temperature has fallen to that, represented by a point on the curve IG,
+[beta]-ferrite will separate out, and, as the temperature falls, the
+composition of the solid solution will alter as represented by IG. On
+passing below the temperature of HG, the [beta]-ferrite will be converted
+into [alpha]-ferrite, and, as the temperature falls, the latter will
+separate out more and more, while the composition of the solid solution
+alters in the direction GF. On passing to still lower temperatures, the
+solid solution at F (0.8 per cent. of carbon) breaks up into pearlite. If
+the percentage of carbon in the original solid solution was between that
+represented by the points G and F, then, on cooling down, no
+[beta]-ferrite, but only [alpha]-ferrite would separate out.
+
+We see, therefore, that when martensite is allowed to cool _slowly_, it
+yields a heterogeneous mixture either of ferrite and pearlite (when the
+original mixture contained up to 0.8 per cent. of carbon), or pearlite and
+cementite (when the original mixture contained between 0.8 and 2 per cent.
+of carbon). These heterogeneous mixtures constitute soft steels, or, when
+the carbon content is low, wrought iron.
+
+The case, however, is different if the solid solution of carbon in iron is
+_rapidly_ cooled (quenched) from a temperature above the curve IGFE to a
+temperature below this {228} curve. In this case, the rapid cooling does
+not allow time for the various changes which have been described to take
+place; so that the homogeneous solid solution, on being rapidly cooled,
+remains homogeneous. In this way hard steel is obtained. By varying the
+rapidity of cooling, as is done in the tempering of steel, varying degrees
+of hardness can be obtained.
+
+The interpretation of the curves given above is that due essentially to
+Roozeboom, who concluded from the experimental data that at temperatures
+below 1000deg the stable systems are martensite and cementite, or ferrite
+and cementite, graphite being labile. It has, however, been pointed out,
+more especially by E. Heyn,[311] that this is not in harmony with the facts
+of metallurgy, which show that graphite is undoubtedly formed on slow
+cooling, and more especially when small quantities of silicon are present
+in the iron.[312] While, therefore, the relationships represented by Fig.
+75 are obtained under certain conditions (especially when manganese is
+present), Heyn considers that all the curves in that figure, except ACB,
+represent _metastable_ systems--systems, therefore, akin to supercooled
+liquids. Rapid cooling will favour the production of the metastable systems
+containing cementite, and therefore give rise to relationships represented
+by Fig. 75; whereas slow cooling will lead to the stable system ferrite and
+graphite. Presence of silicon tends to prevent, presence of manganese tends
+to assist, the production of the metastable systems.
+
+Although this view put forward by Heyn has not been conclusively proved, it
+must be said that there is much evidence in its favour. Further
+investigation is, however, required before a final decision as to the
+interpretation of the curves can be reached.
+
+Determination of the Composition of Compounds, without Analysis.--Since the
+equilibrium between a solid and a liquid phase depends not only on the
+composition of the liquid (solution) but also on that of the solid, it is
+necessary {229} to determine the composition of the latter. In some cases
+this is easily effected by separating the solid from the liquid phase and
+analyzing it. In other cases, however, this method is inapplicable, or is
+accompanied by difficulties, due either to the fact that the solid phase
+undergoes decomposition (_e.g._ when it contains a volatile constituent),
+or to the difficulty of completely separating the mother liquor; as, for
+example, in the case of alloys. In all such cases, therefore, recourse must
+be had to other methods.
+
+In the first place, synthetic methods may be employed.[313] In this case we
+start with a solution of the two components, to which a third substance is
+added, which, however, does not enter into the solid phase.[314] We will
+assume that the initial solution contains _x_ gm. of A and _y_ gm. of B to
+1 gm. of C. After the solution has been cooled down to such a temperature
+that solid substance separates out, a portion of the liquid phase is
+removed with a pipette and analyzed. If, now, the composition of the
+solution is such that there are _x'_ gm. of A and _y'_ gm. of B to 1 gm. of
+C., then the composition of the solid phase is _x_ - _x'_ gm. of A and _y_
+- _y'_ gm. of B. When _x_ = _x'_, the solid phase is pure B; when _y_ =
+_y'_, the solid phase is pure A.
+
+We have assumed here that there is only one solid phase present, containing
+A and B. To make sure that the solid phase is not a solid solution in which
+A and B are present in the same ratio as in the liquid solution, a second
+determination of the composition must be made, with different initial and
+end concentrations. If the solid phase is a solid solution, the composition
+will now be found different from that found previously.
+
+The composition of the solid phase can, however, be determined in another
+manner, viz. by studying the fusion curve and the curve of cooling. From
+the form of the fusion curve alone, it is possible to decide whether the
+two components {230} form a compound or not; and if the compounds which may
+be formed have a definite melting point, the position of the latter gives
+at once the composition of the compounds (cf. p. 231).
+
+This method, however, cannot be applied when the compounds undergo
+decomposition before the melting point is reached. In such cases, however,
+the form of the cooling curve enables one to decide the composition of the
+solid phase.[315] If a solution is allowed to cool slowly, and the
+temperature noted at definite times, the graphic representation of the rate
+of cooling will give a continuous curve; _e.g._ _ab_ in Fig. 76. So soon,
+however, as a solid phase begins to be formed, the rate of cooling alters
+abruptly, and the cooling curve then exhibits a break, or change in
+direction (point _b_). When the eutectic point is reached, the temperature
+remains constant, until all the liquid has solidified. This is represented
+by the line _cd_. When complete solidification has occurred, the fall of
+temperature again becomes uniform (_de_).
+
+[Illustration: FIG. 76.]
+
+[Illustration: FIG. 77.]
+
+[Illustration: FIG. 78.]
+
+The length of time during which the temperature remains constant at the
+point _c_, depends, of course, on the eutectic solution. If, therefore, we
+take equal amounts of solution having a different initial composition, the
+period of constant temperature in the cooling curve will evidently be
+greatest in the case of the solution having the composition of the eutectic
+point; and the period will become less and less as we increase the amount
+of one of the components. The relationship between initial composition of
+solution and the duration of constant temperature at the eutectic point is
+represented by the curve _a'c'b'_ (Fig. 77). When a compound possessing a
+definite melting point is formed, it behaves as a pure substance. If,
+therefore, the initial composition of the {231} solution is the same as
+that of the compound, no eutectic solution will be obtained; and therefore
+no line of constant temperature, such as _cd_ (Fig. 76). In such a case, if
+we represent graphically the relation between the initial composition of
+the solution and the duration of constant temperature, a diagram is
+obtained such as shown in Fig. 78. The two maxima on the time-composition
+curve represent eutectic points, and the minima, _a'_, _b'_, _e'_, pure
+substances. The position of _e'_ gives the composition of the compound.
+When a series of compounds is formed, then for each compound a minimum is
+found on the time-composition curve.
+
+[Illustration: FIG. 79.]
+
+If the compound formed has no definite melting point, the diagram obtained
+is like that shown in Fig. 79. If we start with a solution, the composition
+of which is represented by a point between _d_ and _b_, then, on cooling,
+_b_ will separate out first, and the temperature will fall until the point
+_d_ is reached. The temperature then remains constant until the component
+_b_, which has separated out, is converted into the compound. After this
+the temperature again falls, until it again remains constant at the
+eutectic point c. In the case of the first halt, the period of constant
+temperature is greatest when the initial composition of the solution is the
+same as that of the compound; and it becomes shorter and shorter with {232}
+increase in the amount of either component. In this way we obtain the
+time-composition curve _b'e"d'_, of which the maximum point _e"_ gives the
+composition of the compound.
+
+On the other hand, the period of constant temperature for the eutectic
+point _c_ is greatest in the case of solutions having the same initial
+_composition_ as that corresponding with the eutectic point; and it
+decreases the more the initial composition approaches that of the pure
+component _a_ or the component e. In this way we obtain the
+time-composition curve _a'c'e'_. Here also the point _e'_ represents the
+composition of the compound. We see, therefore, that from the graphic
+representation of the freezing-point curve, and from the duration of the
+temperature-arrests on the cooling curve, for solutions of different
+initial composition, it is possible, without having recourse to analysis,
+to decide what solid phases are formed, and what is their composition.
+
+Formation of Minerals.--Important and interesting as is the application of
+the Phase Rule to the study of alloys, its application to the study of the
+conditions regulating the formation of minerals is no less so; and although
+we do not propose to consider different cases in detail here, still
+attention must be drawn to certain points connected with this interesting
+subject.
+
+In the first place, it will be evident from what has already been said,
+that that mineral which first crystallizes out from a molten magma is not
+necessarily the one with the highest melting point. The _composition_ of
+the fused mass must be taken into account. When the system consists of two
+components which do not form a compound, one or other of these will
+separate out in a pure state, according as the composition of the molten
+mass lies on one or other side of the eutectic composition; and the
+separation of the one component will continue until the composition of the
+eutectic point is reached. Further cooling will then lead to the
+simultaneous separation of the two components.
+
+If, however, the two components form a stable compound (_e.g._ orthoclase,
+from a fused mixture of silica and potassium aluminate), then the
+freezing-point curve will resemble that {233} shown in Fig. 64; _i.e._
+there will be a middle curve possessing a dystectic point, and ending on
+either side at a eutectic point. This curve would represent the conditions
+under which orthoclase is in equilibrium with the molten magma. If the
+initial composition of the magma is represented by a point between the two
+eutectic points, orthoclase will separate first. The composition of the
+magma will thereby change, and the mass will finally solidify to a mixture
+of orthoclase and silica, or orthoclase and potassium aluminate, according
+to the initial composition.
+
+What has just been said holds, however, only for stable equilibria, and it
+must not be forgotten that complications can arise owing to suspended
+transformation (when, for example, the magma is rapidly cooled) and the
+production of metastable equilibria. These conditions occur very frequently
+in nature.
+
+The study of the formation of minerals from the point of view of the Phase
+Rule is still in its initial stages, but the results which have already
+been obtained give promise of a rich harvest in the future.[316]
+
+       *       *       *       *       *
+
+
+{234}
+
+CHAPTER XIII
+
+SYSTEMS OF THREE COMPONENTS
+
+General.--It has already been made evident that an increase in the number
+of the components from one to two gives rise to a considerable increase in
+the possible number of systems, and introduces not a few complications into
+the equilibrium relations of these. No less is this the case when the
+number of components increases from two to three; and although examples of
+all the possible types of systems of three components have not been
+investigated, nor, indeed, any one type fully, nevertheless, among the
+systems which have been studied experimentally, cases occur which not only
+possess a high scientific interest, but are also of great industrial
+importance. On account not only of the number, but more especially of the
+complexity of the systems constituted of three components, no attempt will
+be made to give a full account, or, indeed, even a survey of all the cases
+which have been subjected to a more or less complete experimental
+investigation; on the contrary, only a few of the more important classes
+will be selected, and the most important points in connection with the
+behaviour of these described.
+
+On applying the Phase Rule
+
+  P + F = C + 2
+
+to the systems of three components, we see that in order that the system
+shall be invariant, no fewer than five phases must be present together, and
+an invariant system will therefore exist at a _quintuple_ point. Since the
+number of liquid phases can never exceed the number of the components, and
+since there can be only one vapour phase, it is evident that in this case,
+{235} as in others, there must always be at least one solid phase present
+at the quintuple point. As the number of phases diminishes, the variability
+of the system can increase from one to four, so that in the last case the
+condition of the system will not be completely defined until not only the
+temperature and the total pressure of the system, but also the
+concentrations of two of the components have been fixed. Or, instead of the
+concentrations, the partial pressures of the components may also be taken
+as independent variables.
+
+Graphic Representation.--Hitherto the concentrations of the components have
+been represented by means of rectangular co-ordinates, although the
+numerical relationships have been expressed in two different ways. In the
+one case, the concentration of the one component was expressed in terms of
+a fixed amount of the other component. Thus, the solubility of a salt was
+expressed by the number of grams of salt dissolved by 100 grams of water or
+other solvent; and the numbers so obtained were measured along one of the
+co-ordinates. The second co-ordinate was then employed to indicate the
+change of another independent variable, _e.g._ temperature. In the other
+case, the combined weights of the two components A and B were put equal to
+unity, and the concentration of the one expressed as a fraction of the
+whole amount. This method allows of the representation of the complete
+series of concentrations, from pure A to pure B, and was employed, for
+example, in the graphic representation of the freezing point curves.
+
+Even in the case of three components rectangular co-ordinates can also be
+employed, and, indeed, are the most convenient in those cases where the
+behaviour of two of the components to one another is very different from
+their behaviour to the third component; as, for example, in the case of two
+salts and water. In these cases, the composition of the system can be
+represented by measuring the amounts of each of the two components in a
+given weight of the third, along two co-ordinates at right angles to one
+another; and the change of the system with the temperature can then be
+represented by a third axis at right angles to the first two. In those
+cases, {236} however, where the three components behave in much the same
+manner towards one another, the rectangular co-ordinates are not at all
+suitable, and instead of these a _triangular diagram_ is employed. Various
+methods have been proposed for the graphic representation of systems of
+three components by means of a triangle, but only two of these have been
+employed to any considerable extent; and a short description of these two
+methods will therefore suffice.[317]
+
+[Illustration: FIG. 80.]
+
+In the method proposed by Gibbs an equilateral triangle of unit height is
+used (Fig 80).[318] The quantities of the different components are
+expressed as fractional parts of the whole, and the sum of their
+concentrations is therefore equal to unity, and can be represented by the
+height of the triangle. The corners {237} of the triangle represent the
+pure substances A, B, and C respectively. A point on one of the sides of
+the triangle will give the composition of a mixture in which only two
+components are present, while a point within the triangle will represent
+the composition of a ternary mixture. Since every point within the triangle
+has the property that the sum of the perpendiculars from that point on the
+sides of the triangle is equal to unity (the height of the triangle), it is
+evident that the composition of a ternary mixture can be represented by
+fixing a point within the triangle such that the lengths of the
+_perpendiculars_ from the point to the sides of the triangle are equal
+respectively to the fractional amounts of the three components present; the
+fractional amount of A, B, or C being represented by the perpendicular
+distance from the side of the triangle _opposite_ the corners A, B, and C
+respectively.
+
+The location of this point is simplified by dividing the normals from each
+of the corners on the opposite side into ten or one hundred parts, and
+drawing through these divisions lines at right angles to the normal and
+parallel to the side of the triangle. A network of rhombohedra is thus
+obtained, and the position of any point can be read off in practically the
+same manner as in the case of rectangular co-ordinates. Thus the point P in
+Fig. 80 represents a ternary mixture of the composition A = 0.5, B = 0.3, C
+= 0.2; the perpendiculars P_a_, P_b_, and P_c_ being equal respectively to
+0.5, 0.2, and 0.3 of the height of the triangle.
+
+Another method of representation, due to Roozeboom, consists in employing
+an equilateral triangle, the length of whose _side_ is made equal to unity,
+or one hundred; the sum of the fractional or percentage amounts of the
+three components being represented therefore by a side of the triangle. In
+this case the composition of a ternary mixture is obtained by determining,
+not the _perpendicular_ distance of a point P from the three sides of the
+triangle, but the distance in a direction _parallel_ to the sides of the
+triangle (Fig. 81). Conversely, in order to represent a mixture consisting
+of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one
+side of the triangle, say AB, is first of all divided into ten or one {238}
+hundred parts; a portion, B_x_ = _a_, is then measured off, and represents
+the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off
+and represents the fractional amount of B, while the remainder, _xx'_ =
+_c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel
+to the sides of the triangle, and the point of intersection, P, represents
+the composition of the ternary mixture of given composition; for, as is
+evident from the figure, the distance of the point P from the three sides
+of the triangle, when measured in directions _parallel_ to the sides, is
+equal to _a_, _b_, and _c_ respectively. From the division marks on the
+side AB, it is seen that the point P in this figure also represents a
+mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives
+exactly the same result as the previous method. The employment of a
+right-angled isosceles triangle has also been suggested,[319] but is not in
+general use.
+
+[Illustration: FIG. 81.]
+
+In employing the triangular diagram, it will be of use to note a property
+of the equilateral triangle. A line drawn from one corner of the triangle
+to the opposite side, represents the composition of all mixtures in which
+the _relative_ amounts of two of the components remain unchanged. Thus, as
+Fig. 82 shows, if the component C is added to a mixture x, in which A and B
+are present in the proportions of _a_ : _b_, a mixture _x'_, which is
+thereby obtained, also contains A and B in the ratio _a_ : b. For the two
+triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and
+KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_
+and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ :
+E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore,
+the ratio of A to B is the same.
+
+[Illustration: FIG. 82.]
+
+[Illustration: FIG. 83.]
+
+If it is desired to represent at the same time the change of another
+independent variable, _e.g._ temperature, this can be done by measuring the
+latter along axes drawn perpendicular to the corners of the triangle. In
+this way a right prism (Fig. 83) is obtained, and each section of this cut
+parallel to the base represents therefore an _isothermal surface_.
+
+       *       *       *       *       *
+
+
+{240}
+
+CHAPTER XIV
+
+SOLUTIONS OF LIQUIDS IN LIQUIDS
+
+We have already seen (p. 95) that when two liquids are brought together,
+they may mix in all proportions and form one homogeneous liquid phase; or,
+only partial miscibility may occur, and two phases be formed consisting of
+two mutually saturated solutions. In the latter case, the concentration of
+the components in either phase and also the vapour pressure of the system
+had, at a given temperature, perfectly definite values. In the case of
+three liquid components, a similar behaviour may be found, although
+complete miscibility of three components with the formation of only one
+liquid phase is of much rarer occurrence than in the case of two
+components. When only partial miscibility occurs, various cases are met
+with according as the three components form one, two, or three pairs of
+partially miscible liquids. Further, when two of the components are only
+partially miscible, the addition of the third may cause either an increase
+or a diminution in the mutual solubility of these. An increase in the
+mutual solubility is generally found when the third component dissolves
+readily in each of the other two; but when the third component dissolves
+only sparingly in the other two, its addition diminishes the mutual
+solubility of the latter.
+
+We shall consider here only a few examples illustrating the three chief
+cases which can occur, viz. (1) A and B, and also B and C are miscible in
+all proportions, while A and C are only partially miscible. (2) A and B are
+miscible in all proportions, but A and C and B and C are only partially
+miscible. (3) A and B, B and C, and A and C are only partially miscible. A,
+B, and C here represent the three components.
+
+1.--_The three components form only one pair of partially miscible
+liquids._ {241}
+
+An example of this is found in the three substances: chloroform, water, and
+acetic acid.[320] Chloroform and acetic acid, and water and acetic acid,
+are miscible with one another in all proportions, but chloroform and water
+are only partially miscible with one another. If, therefore, chloroform is
+shaken with a larger quantity of water than it can dissolve, two layers
+will be formed consisting one of a saturated solution of water in
+chloroform, the other of a saturated solution of chloroform in water. The
+composition of these two solutions at a temperature of about 18deg, will be
+represented by the points _a_ and _b_ in Fig. 84; _a_ representing a
+solution of the composition: chloroform, 99 per cent.; water, 1 per cent.;
+and _b_ a solution of the composition: chloroform, 0.8 per cent.; water,
+99.2 per cent. When acetic acid is added, it distributes itself between the
+two liquid layers, and two conjugate _ternary_ solutions, consisting of
+chloroform, water, and acetic acid are thereby produced which are in
+equilibrium with one another, and the composition of which will be
+represented by two points inside the triangle. In this way a series of
+pairs of ternary solutions will be obtained by the addition of acetic acid
+to the mixture of chloroform and water. By this addition, also, not only do
+the two liquid phases become increasingly rich in acetic acid, but the
+mutual solubility of the chloroform and water increases; so that the layer
+_a_ becomes relatively richer in water, and layer _b_ relatively richer in
+chloroform. This is seen from the following table, which gives the
+percentage composition of different conjugate ternary solutions at 18deg.
+
+  -------------------------------------------------------------------------
+             Heavier layer.           |            Lighter layer.
+  -------------------------------------------------------------------------
+  Chloroform. | Water. | Acetic acid. | Chloroform. | Water. | Acetic acid.
+  -------------------------------------------------------------------------
+     99.01    |  0.99  |     0        |     0.84    | 99.16  |     0
+     91.85    |  1.38  |     6.77     |     1.21    | 73.69  |    25.10
+     80.00    |  2.28  |    17.72     |     7.30    | 48.58  |    44.12
+     70.13    |  4.12  |    25.75     |    15.11    | 34.71  |    50.18
+     67.15    |  5.20  |    27.65     |    18.33    | 31.11  |    50.56
+     59.99    |  7.93  |    32.08     |    25.20    | 25.39  |    49.41
+     55.81    |  9.58  |    34.61     |    28.85    | 23.28  |    47.87
+  -------------------------------------------------------------------------
+
+{242}
+
+By the continued addition of acetic acid, the composition of the successive
+conjugate solutions in equilibrium with one another becomes, as the table
+shows, more nearly the same, and a point is at length reached at which the
+two solutions become identical. This will therefore be a _critical point_
+(p. 98). Increased addition of acetic acid beyond this point will lead to a
+single homogeneous solution.
+
+These relationships are represented graphically by the curve _a_K_b_, Fig.
+84. The points on the branch _a_K represent the composition of the
+solutions relatively rich in chloroform (heavier layer), those on the curve
+_b_K the composition of solutions relatively rich in water (lighter layer);
+and the points on these two branches representing conjugate solutions are
+joined together by "tie-lines." Thus, the points _a'b'_ represent conjugate
+solutions, and the line _a'b'_ is a tie-line.
+
+[Illustration: FIG. 84.]
+
+Since, now, acetic acid when added to a heterogeneous mixture of chloroform
+and water does not enter in equal amounts into the two layers, but in
+amounts depending on its coefficient of distribution between chloroform and
+water,[321] the {243} tie-lines will not be parallel to AB, but will be
+inclined at an angle. As the solutions become more nearly the same, the
+tie-lines diminish in length, and at last, when the conjugate solutions
+become identical, shrink to a point. For the reason that the tie-lines are,
+in general, not parallel to the side of the triangle, the critical point at
+which the tie-line vanishes will not be at the summit of the curve, but
+somewhere below this, as represented by the point K.
+
+The curve _a_K_b_, further, forms the boundary between the heterogeneous
+and homogeneous systems. A mixture of chloroform, water, and acetic acid
+represented by any point outside the curve _a_K_b_, will form only one
+homogeneous phase; while any mixture represented by a point within the
+curve, will separate into two layers having the composition represented by
+the ends of the tie-line passing through that point. Thus, a mixture of the
+total composition _x_, will separate into two layers having the composition
+_a'_ and _b'_ respectively.
+
+Since three components existing in three phases (two liquid and a vapour
+phase) constitute a bivariant system, the final result, _i.e._ the
+composition of the two layers and the total vapour pressure, will not
+depend merely on the temperature, as in the case of two-component systems
+(p. 102), but also on the composition of the mixture with which we start.
+At constant temperature, however, all mixtures, the composition of which is
+represented by a point on one and the same tie-line, will separate into the
+same two liquid phases, although the relative _amounts_ of the two phases
+will vary. If we omit the vapour phase, the condition of the system will
+depend on the pressure as well as on the temperature and composition of the
+initial mixture. By keeping the pressure constant, _e.g._ at atmospheric
+pressure (by working with open vessels), the system again becomes
+bivariant. We see, therefore, that the position of the curve _a_K_b_, or,
+in other words, the composition of the different conjugate ternary
+solutions, will vary with the temperature, and only with the temperature,
+if we assume either constancy of pressure or the presence of the vapour
+phase. Since at the critical point the condition is imposed that the two
+liquid phases become identical, one degree of freedom is thereby {244}
+lost, and therefore only one degree of freedom remains. The critical point,
+therefore, depends on the temperature, and only on the temperature; always
+on the assumption, of course, that the pressure is constant, or that a
+vapour phase is present. Fig. 84, therefore, represents an isothermal
+(p. 239).
+
+It is of importance to note that the composition of the different ternary
+solutions obtained by the addition of acetic acid to a heterogeneous
+mixture of chloroform and water, will depend not only on the amount of
+acetic acid added, but also on the relative amounts of chloroform and water
+at the commencement. Suppose, for example, that we start with chloroform
+and water in the proportions represented by the point _c'_ (Fig. 84). On
+mixing these, two liquid layers having the composition _a_ and _b_
+respectively will be formed. Since by the addition of acetic acid the
+relative amounts of these two substances in the system as a whole cannot
+undergo alteration, the total composition of the different ternary systems
+which will be obtained must be represented by a point on the line C_c'_
+(p. 238). Thus, for example, by the addition of acetic acid a system may be
+obtained, the total composition of which is represented by the point _c"_.
+Such a system, however, will separate into two conjugate ternary solutions,
+the composition of which will be represented by the ends of the tie-line
+passing through the point _c"_. So long as the total composition of the
+system lies below the point S, _i.e._ the point of intersection of the line
+C_c'_ with the boundary curve, two liquid layers will be formed; while all
+systems having a total composition represented by a point on the line
+C_c'_, above S, will form only one homogeneous solution.
+
+From the figure, also, it is evident that as the amount of acetic acid is
+increased, the relative amounts of the two liquid layers formed differ more
+and more until at S a limiting position is reached, when the amount of the
+one liquid layer dwindles to nought, and only one solution remains.
+
+The same reasoning can be carried through for different initial amounts of
+chloroform and water, but it would be fruitless to discuss all the
+different systems which can be obtained. The reason for the preceding
+discussion was to show that {245} although the addition of acetic acid to a
+mixture of chloroform and water will, in all cases, lead ultimately to a
+limiting system, beyond which homogeneity occurs, that point is not
+necessarily the critical point. On the contrary, in order that addition of
+acetic acid shall lead to the critical mixture, it is necessary to start
+with a binary mixture of chloroform and water in the proportions
+represented by the point _c'_. In this case, addition of acetic acid will
+give rise to a series of conjugate ternary solutions, the composition of
+which will gradually approach to one another, and at last become identical.
+
+From the foregoing it will be evident that the amount of acetic acid
+required to produce a homogenous solution, will depend on the relative
+amounts of chloroform and water from which we start, and can be ascertained
+by joining the corner C with the point on the line AB representing the
+total composition of the initial binary system. The point where this line
+intersects the boundary curve _a_K_b_ will indicate the minimum amount of
+acetic acid which, under these particular conditions, is necessary to give
+one homogeneous solution.
+
+Retrograde Solubility.--As a consequence of the fact that acetic acid
+distributes itself unequally between chloroform and water, and the critical
+point K, therefore, does not lie at the summit of the curve, it is possible
+to start with a homogeneous solution in which the percentage amount of
+acetic acid is greater than at the critical point, and to pass from this
+first to a heterogenous and then again to a homogenous system merely by
+altering the relative amounts of chloroform and water. This phenomenon, to
+which the term _retrograde solubility_ is applied, will be observed not
+only in the case of chloroform, water, and acetic acid, but in all other
+systems in which the critical point lies below the highest point of the
+boundary curve for heterogeneous systems. This will be seen from the
+diagram, Fig. 85. Starting with the homogeneous system represented by _x_,
+in which, therefore, the concentration of C is greater than in the critical
+mixture (K), if the relative amounts of A and B are altered in the
+direction _xx'_, while the amount of C is maintained constant, the system
+will become heterogeneous when the composition reaches the point _y_, and
+will remain {246} heterogeneous with changing composition until the point
+_y'_ is passed, when it will again become homogeneous. If the relative
+concentration of C is increased above that represented by the line SS, this
+phenomenon will, of course, no longer be observed.
+
+[Illustration: FIG. 85.]
+
+Relationships similar to those described for chloroform, water, and acetic
+acid are also found in the case of a number of other trios, _e.g._ ether,
+water, and alcohol; chloroform, water, and alcohol.[322] They have also
+been observed in the case of a considerable number of molten metals.[323]
+Thus, molten lead and silver, as well as molten zinc and silver, mix in all
+proportions; but molten lead and zinc are only partially miscible with one
+another. When melted together, therefore, the last two metals will separate
+into two liquid layers, one rich in lead, the other rich in zinc. If silver
+is now added, and the temperature maintained above the freezing point of
+the mixture, the silver passes for the most part, in accordance with the
+law of distribution, into the upper layer, which is rich in zinc; silver
+being more soluble in molten zinc than in molten lead. This is clearly
+shown by the following figures:--[324]
+
+{247}
+
+  --------------------------------------------------
+       Heavier alloy.     |     Lighter alloy.
+  --------------------------------------------------
+    Percentage amount of  |  Percentage amount of
+  Silver. | Lead. | Zinc. | Silver. | Lead. | Zinc.
+  --------------------------------------------------
+    1.25  | 96.69 |  2.06 |  38.91  |  3.12 | 57.97
+    1.71  | 96.43 |  1.86 |  45.01  |  3.37 | 51.62
+    5.55  | 93.16 |  1.29 |  54.93  |  4.21 | 40.86
+  --------------------------------------------------
+
+The numbers in the same horizontal row give the composition of the
+conjugate alloys, and it is evident that the upper layer consists almost
+entirely of silver and zinc. On allowing the mixture to cool slightly, the
+upper layer solidifies first, and can be separated from the still molten
+lead layer. It is on this behaviour of silver towards a mixture of molten
+lead and zinc that the Parkes's method for the desilverization of lead
+depends.[325] If aluminium is also added, a still larger proportion of
+silver passes into the lighter layer, and the desilverization of the lead
+is more complete.[326]
+
+[Illustration: FIG. 86.]
+
+[Illustration: FIG. 87.]
+
+The Influence of Temperature.--As has already been said, a ternary system
+existing in three phases possesses two degrees of freedom; and the state of
+the system is therefore dependent not only on the relative concentration of
+the components, but also on the temperature. As the temperature changes,
+therefore, the boundary curve of the heterogeneous system will also alter;
+and in order to represent this alteration we shall make use of the right
+prism, in which the temperature is measured upwards. In this way the
+boundary curve passes into a boundary surface (called a dineric surface),
+as shown in Fig. 86. In this figure the curve _akb_ is the isothermal for
+the ternary system; the curve _a_K_b_ shows the change in the _binary_
+system AB with the temperature, with {248} a critical point at K. This
+curve has the same meaning as those given in Chapter VI. The curve _k_K is
+a critical curve joining together the critical points of the different
+isothermals. In such a case as is shown in Fig. 86, there does not exist
+any real critical temperature for the ternary system, for as the
+temperature is raised, the amount of C in the "critical" solution becomes
+less and less, and at K only two components, A and B, are present. In the
+case, however, represented in Fig. 87, a real ternary critical point is
+found. In this figure _ak'b_ is an isothermal, _ak"_ is the curve for the
+binary system, and K is the ternary critical point. All points outside the
+helmet-shaped boundary surface represent homogeneous ternary solutions,
+while all points within the surface belong to heterogeneous systems. Above
+the temperature of the point K, the three components are miscible in all
+proportions. An example of a ternary system yielding such a boundary
+surface is that consisting of phenol, water, and acetone.[327] In this case
+the critical temperature K is 92deg, and the composition at this ternary
+critical point is--
+
+  Water   59 per cent.
+  Acetone 12    "
+  Phenol  29    "
+
+[Illustration: FIG. 88.]
+
+The difference between the two classes of systems just mentioned, is seen
+very clearly by a glance at the Figs. 88 and 89, which show the projection
+of the isothermals on the base of the prism. In Fig. 88, the projections
+yield paraboloid curves, the two branches of which are cut by one side of
+the triangle; and the critical point is represented by a point on {249}
+this side. In the second case (Fig. 89), however, the projections of the
+isothermals form ellipsoidal curves surrounding the supreme critical point,
+which now lies _inside the triangle_. At lower temperatures, these
+isothermal boundary curves are cut by a side of the triangle; at the
+critical temperature, _k"_, of the binary system AB, the boundary curve
+_touches_ the side AB, while at still higher temperatures the boundary
+curve comes to lie entirely within the triangle. At any given temperature,
+therefore, between the critical point of the binary system (_k"_), and the
+supreme critical point of the ternary system (K), each pair of the three
+components are miscible with one another in all proportions; for the region
+of heterogeneous systems is now bounded by a closed curve lying entirely
+within the triangle. Outside this curve only homogeneous systems are found.
+Binary mixtures, therefore, represented by any point on one of the sides of
+the triangle must be homogeneous, for they all lie outside the boundary
+curve for heterogeneous states.
+
+[Illustration: FIG. 89.]
+
+2. _The three components can form two pairs of partially miscible liquids._
+
+In the case of the three components water, alcohol, and succinic nitrile,
+water and alcohol are miscible in all proportions, but not so water and
+succinic nitrile, or alcohol and succinic nitrile.
+
+[Illustration: FIG. 90.]
+
+[Illustration: FIG. 91.]
+
+As we have already seen (p. 122), water and succinic nitrile can form two
+liquid layers between the temperatures 18.5deg and 55.5deg; while alcohol
+and nitrile can form two liquid layers between 13deg and 31deg. If, then,
+between these two temperature limits, alcohol is added to a heterogeneous
+mixture of water and nitrile, or water is added to a mixture of alcohol and
+nitrile, two heterogeneous ternary systems will be formed, {250} and two
+boundary curves will be obtained in the triangular diagram, as shown in
+Fig. 90.[328] On changing the temperature, the boundary curves will also
+undergo alteration, in a manner similar to that just discussed. As the
+temperature falls, the two curves will spread out more and more into the
+centre of the triangle, and might at last meet one another; while at still
+lower temperatures we may imagine the curves still further expanding so
+that the two heterogeneous regions flow into one another and form a _band_
+on the triangular diagram (Fig. 91). This, certainly, has not been realized
+in the case of the three components mentioned, because at a temperature
+higher than that at which the two heterogeneous regions could fuse
+together, solid separates out.
+
+[Illustration: FIG. 92.]
+
+The gradual expansion of a paraboloid into a band-like area of
+heterogeneous ternary systems, has, however, been observed in the case of
+water, phenol, and aniline.[329] In Fig. 92 are shown three isothermals,
+viz. those for 148deg, 95deg, and 50deg. At 148deg, water and aniline form
+two layers having the composition--
+
+  Water,   83.5 per cent. }     { water,   20 per cent.
+                          } and {
+  Aniline, 16.5    "      }     { aniline, 80    "
+
+{251}
+
+and the critical point _k'_ has the composition--
+
+  Water, 65; phenol, 13.2; aniline, 21.8 per cent.
+
+At 95deg, the composition of the two binary solutions is--
+
+  Water,  93 per cent. }     { water     8 per cent.
+                       } and {
+  Aniline, 7    "      }     { aniline, 92    "
+
+while the point _k"_ has the composition
+
+  Water, 69.9; phenol, 26.6; aniline, 3.5 per cent.
+
+At 50deg, the region of heterogeneous states now forms a band, and the two
+layers formed by water and aniline have the composition--
+
+  Water,  96.5 per cent. }     { water,    5.5 per cent.
+                         } and {
+  Aniline, 3.5    "      }     { aniline, 94.5    "
+
+while the two layers formed by water and phenol have the composition--
+
+  Water,  89 per cent.}     { water,  38 per cent.
+                      } and {
+  Phenol, 11    "     }     { phenol, 62    "
+
+All mixtures of water, phenol, and aniline, therefore, the composition of
+which is represented by any point within the band _abcd_, will form two
+ternary solutions; while if the composition is represented by a point
+outside the band, only one homogeneous solution will be produced.
+
+3. _The three components form three pairs of partially miscible liquids._
+
+[Illustration: FIG. 93.]
+
+The third chief case which can occur is that no two of the components are
+completely miscible with one another. In this case, therefore, we shall
+obtain three paraboloid boundary curves, as shown in Fig. 93. If, now, we
+imagine these three curves to expand in towards the centre of the triangle,
+as might happen, for example, by lowering the temperature, a point will
+{252} be reached at which the curves partly overlap, and we shall get the
+appearance shown in Fig. 94.
+
+The points _a_, _b_, and _c_ represent the points where the three curves
+cut, and the triangle _abc_ is a region where the curves overlap. From this
+diagram we can see that any mixture having a composition represented by a
+point in one of the clear spaces at the corners of the larger triangle,
+will form a homogeneous solution; if the composition corresponds to any
+point lying in one of the quadrilateral regions _x__{1}, _x__{2} or
+_x__{3}, two ternary solutions will be formed; while, if the composition is
+represented by any point in the inner triangle, separation into three
+layers will occur.
+
+[Illustration: FIG. 94.]
+
+Since in the clear regions at the corners of the triangle we have three
+components in two phases, liquid and vapour, the systems have three degrees
+of freedom. At constant temperature, therefore, the condition of the system
+is not defined until the concentrations of two of the components are fixed.
+A system belonging to one of the quadrilateral spaces has, as we have seen,
+two degrees of freedom; besides the temperature, one concentration must be
+fixed. Lastly, a system the composition of which falls within the inner
+triangle _abc_, will form three layers, and will therefore possess only one
+degree of freedom. If the temperature is fixed, the composition of the
+three layers is also determined, viz. that of the points _a_, _b_, and _c_
+respectively; and a change in the composition of the original mixture can
+lead only to a difference in the relative amounts of the three layers, not
+to a difference in their composition.
+
+An example of a system which can form three liquid phases is found in
+water, ether, and succinic nitrile.[330]
+
+       *       *       *       *       *
+
+
+{253}
+
+CHAPTER XV
+
+PRESENCE OF SOLID PHASES
+
+A. The Ternary Eutectic Point.--In passing to the consideration of those
+ternary systems in which one or more solid phases can exist together with
+one liquid phase, we shall first discuss not the solubility curves, as in
+the case of two-component systems, but the simpler relationships met with
+at the freezing point. That is, we shall first of all examine the freezing
+point curves of ternary systems.
+
+[Illustration: FIG. 95.]
+
+Since it is necessary to take into account not only the changing
+composition of the liquid phase, but also the variation of the temperature,
+we shall employ the right prism for the graphic representation of the
+systems, as shown in Fig. 95. A, B, and C in this figure, therefore, denote
+the melting points of the pure components. If we start with the component A
+at its melting point, and add B, which is capable of dissolving in liquid
+A, the freezing point of A will be lowered; and, similarly, the freezing
+point of B by addition of A. In this way we get the freezing point curve
+A_k__{1}B for the binary system; _k__{1}; being an eutectic point. This
+curve will of course lie in the plane formed by one face of the prism. In a
+similar manner we obtain the freezing point curves A_k__{2}C and B_k__{3}C.
+These curves give the composition of the binary liquid phases in
+equilibrium {254} with one of the pure components, or at the eutectic
+points, with a mixture of two solid components. If, now, to the system
+represented say by the point _k__{1}, a small quantity of the third
+component, C, is added, the temperature at which the two solid phases A and
+B can exist in equilibrium with the liquid phase is lowered; and this
+depression of the eutectic point is all the greater the larger the addition
+of C. In this way we obtain the curve _k__{1}K, which slopes inwards and
+downwards, and indicates the varying composition of the ternary liquid
+phase with which a mixture of solid A and B are in equilibrium. Similarly,
+the curves _k__{2}K and _k__{3}K are the corresponding eutectic curves for
+A and C, and B and C in equilibrium with ternary solutions. At the point K,
+the three solid components are in equilibrium with the liquid phase; and
+this point, therefore, represents _the lowest temperature attainable with
+the three components given_. Each of the ternary eutectic curves, as they
+may be called, is produced by the intersection of two surfaces, while at
+the ternary eutectic point, three surfaces, viz. A_k__{1}K_k__{2},
+B_k__{1}K_k__{3}, and C_k__{1}K_k__{3} intersect. Any point on one of these
+surfaces represents a ternary solution in equilibrium with only one
+component in the solid state; the lines or curves of intersection of these
+represent equilibria with two solid phases, while at the point K, the
+ternary eutectic point, there are three solid phases in equilibrium with a
+liquid and a vapour phase. The surfaces just mentioned represent bivariant
+systems. One component in the solid state can exist in equilibrium with a
+ternary liquid phase under varying conditions of temperature and
+concentration of the components in the solution; and before the state of
+the system is defined, these two variables, temperature and composition of
+the liquid phase, must be fixed. On the other hand, the curves formed by
+the intersection of these planes represent univariant systems; at a given
+temperature two solid phases can exist in equilibrium with a ternary
+solution, only when the latter has a definite composition. Lastly, the
+ternary eutectic point, K, represents an invariant system; three solid
+phases can exist in equilibrium with a ternary solution, only when the
+latter has one fixed composition and when the temperature has a definite
+value. This eutectic point, therefore, {255} has a perfectly definite
+position, depending only on the nature of the three components.
+
+Instead of employing the prism, the change in the composition of the
+ternary solutions can also be indicated by means of the _projections_ of
+the curves _k__{1}K, _k__{2}K, and _k__{3}K on the base of the prism, the
+particular temperature being written beside the different eutectic points
+and curves. This is shown in Fig. 96.
+
+[Illustration: FIG. 96.]
+
+The numbers which are given in this diagram refer to the eutectic points
+for the system bismuth--lead--tin, the data for which are as
+follows:--[331]
+
+  --------------------------------------------------------------------
+  Melting point of | Percentage composition of | Temperature of binary
+     pure  metal.  |  binary eutectic mixture. |    eutectic point.
+  --------------------------------------------------------------------
+                   |      Bi    Pb    Sn       |
+   Bismuth, 268deg |      55    45    --       |     Bi--Pb, 127deg
+   Lead, 325deg    |      58    --    42       |     Bi--Sn, 133deg
+   Tin, 232deg     |      --    37    63       |     Pb--Sn, 182deg
+  --------------------------------------------------------------------
+
+  --------------------------------------------------
+  Percentage composition of | Temperature of ternary
+  ternary eutectic mixture. |    eutectic point.
+  --------------------------------------------------
+       Bi    Pb    Sn       |
+       52    32    16       |         96deg
+  --------------------------------------------------
+
+Formation of Compounds.--In the case just discussed, the components
+crystallized out from solution in the pure state. If, however, combination
+can take place between two of the components, the relationships will be
+somewhat different; the curves which are obtained in such a case being
+represented in Fig. 97. From the figure, we see that the two components B
+{256} and C form a compound, and the freezing point curve of the binary
+system has therefore the form shown in Fig. 64 (p. 209). Further, there are
+two _ternary_ eutectic points, K_{1} and K_{2}, the solid phases present
+being A, B, and compound, and A, C, and compound respectively.
+
+[Illustration: FIG. 97.]
+
+The particular point, now, to which it is desired to draw attention is
+this. Suppose the ternary eutectic curves projected on a plane parallel to
+the face of the prism containing B and C, _i.e._ suppose the concentrations
+of the two components B and C, between which interaction can occur,
+expressed in terms of a constant amount of the third component A,[332]
+curves will then be obtained which are in every respect analogous to the
+freezing point curves of binary systems. Thus, suppose the eutectic curves
+_k__{1}K and _k__{2}K in Fig. 95 projected on the face BC of the prism,
+then evidently a curve will be obtained consisting of two branches
+meeting in an eutectic point. On the other hand, the projection of the
+ternary eutectic curves in Fig. 97 on the face BC of the prism, will
+give a curve consisting of three portions, as shown by the outline
+_k__{1}K_{1}K_{2}_k__{2} in Fig. 97.
+
+Various examples of this have been studied, and the following table
+contains some of the data for the system ethylene bromide (A), picric acid
+(B), and [beta]-naphthol (C), obtained by Bruni.[333]
+
+{257}
+
+  -------------------------------------------------------------------------
+                     | Temperature |            Solid phases present.
+  -------------------------------------------------------------------------
+  Point _k__{1}      |   9.41deg   | Ethylene bromide, picric acid.
+  Curve _k__{1}K_{1} |     --      |         "               "
+  Point K_{1}        |   9.32deg   | Ethylene bromide, picric acid, and
+                     |             |  [beta]-naphthol picrate.
+  Curve K_{1}D'K_{2} |     --      | Ethylene bromide,
+                     |             |               [beta]-naphthol picrate.
+  Point D'           |   9.75deg   |     "       "         "        "
+  Point K_{2}        |   8.89deg   |     "       "     [beta]-naphthol,
+                     |             |                           and picrate.
+  Curve K_{2}_k__{2} |     --      |     "       "     [beta]-naphthol.
+  Point _k__{2}      |   9.04deg   |     "       "               "
+  -------------------------------------------------------------------------
+
+From what has been said, it will be apparent that if the ternary eutectic
+curve of a three-component system (in which one of the components is
+present in constant amount) is determined, it will be possible to state,
+from the form of curve obtained, whether or not the two components present
+in varying amount crystallize out pure or combine with one another to form
+a compound. It may be left to the reader to work out the curves for the
+other possible systems; but it will be apparent, that the projections of
+the ternary eutectic curves in the manner given will yield a series of
+curves alike in all points to the binary curves given in Figs. 63-65,
+pp. 208-210.
+
+Since, from the method of investigation, the temperatures of the eutectic
+curves will depend on the melting point of the third component (A), it is
+possible, by employing substances with widely differing melting points, to
+investigate the interaction of the two components (_e.g._ two optical
+antipodes) B and C over a range of temperature; and thus determine the
+range of stability of the compound, if one is formed. Since, in some cases,
+two substances which at one temperature form mixed crystals combine at
+another temperature to form a definite compound, the relationships which
+have just been described can be employed, and indeed, have been employed,
+to determine the temperature at which this change occurs.[334] By means of
+this method, Adriani found that below 103deg _i_-camphoroxime exists as a
+racemic compound, while above {258} that temperature it occurs as a racemic
+mixed crystal[335] (_cf._ p. 219).
+
+B. Equilibria at Higher Temperatures. Formation of Double Salts.--After
+having studied the relationships which are found in the neighbourhood of
+the freezing points of the components, we now pass to the discussion of the
+equilibria which are met with at higher temperatures. In this connection we
+shall confine the discussion entirely to the systems formed of two salts
+and water, dealing more particularly with those cases in which the water is
+present in relatively large amount and acts as solvent. Further, in
+studying these systems, one restriction must be made, viz. that the single
+salts are salts either of the same base or of the same acid; or are, in
+other words, capable of yielding a common ion in solution. Such a
+restriction is necessary, because otherwise the system would be one not of
+three but of four components.[336]
+
+Transition Point.--As is very well known, there exist a number of hydrated
+salts which, on being heated, undergo apparent partial fusion; and in
+Chapter V. the behaviour of such hydrates was more fully studied in the
+light of the Phase Rule. Glauber's salt, or sodium sulphate decahydrate,
+for example, on being heated to a temperature of about 32.5deg, partially
+liquefies, owing to the fact that the water of crystallization is split off
+and anhydrous sodium sulphate formed, as shown by the equation--
+
+  Na_{2}SO_{4},10H_{2}O = Na_{2}SO_{4} + 10H_{2}O
+
+The temperature of 32.5deg, it was learned, constituted a _transition
+point_ for the decahydrate and anhydrous salt plus water; decomposition of
+the hydrated salt occurring above this temperature, combination of the
+anhydrous salt and water below it.
+
+Analogous phenomena are met with in systems constituted of two salts and
+water in which the formation of double salts can take place. Thus, for
+example, if _d_-sodium potassium {259} tartrate is heated to above 55deg,
+apparent partial fusion occurs, and the two single salts, _d_-sodium
+tartrate and _d_-potassium tartrate, are deposited, the change which occurs
+being represented by the equation--
+
+  4NaKC_{4}O_{6}H_{4},4H_{2}O = 2Na_{2}C_{4}O_{6}H_{4},2H_{2}O
+                              + 2K_{2}C_{4}O_{6}H_{4},1/2H_{2}O + 11H_{2}O
+
+On the other hand, if sodium and potassium tartrates are mixed with water
+in the proportions shown on the right side of the equation, the system will
+remain partially liquid so long as the temperature is maintained above
+55deg (in a closed vessel to prevent loss of water), but on allowing the
+temperature to fall below this point, complete solidification will ensue,
+owing to the formation of the hydrated double salt. Below 55deg, therefore,
+the hydrated double salt is the stable system, while above this temperature
+the two single salts plus saturated solution are stable.[337]
+
+A similar behaviour is found in the case of the double salt copper
+dipotassium chloride (CuCl_{2},2KCl,2H_{2}O or CuK_{2}Cl_{4},2H_{2}O).[338]
+When this salt is heated to 92deg, partial liquefaction occurs, and the
+original blue plate-shaped crystals give place to brown crystalline needles
+and white cubes; while on allowing the temperature to fall, re-formation of
+the blue double salt ensues. The temperature 92deg is, therefore, a
+transition point at which the reversible reaction--
+
+  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O
+
+takes place.
+
+The decomposition of sodium potassium tartrate, or of copper dipotassium
+chloride, differs in so far from that of Glauber's salt that _two_ new
+solid phases are formed; and in the case of copper dipotassium chloride,
+one of the decomposition products is itself a double salt.
+
+In the two examples of double salt decomposition which have just been
+mentioned, sufficient water was yielded to cause a partial liquefaction;
+but other cases are known where this is not so. Thus, when copper calcium
+acetate is heated to a {260} temperature of 75deg, although decomposition
+of the double salt into the two single salts occurs as represented by the
+equation[339]--
+
+  CuCa(C_{2}H_{3}O_{2})_{4},8H_{2}O = Cu(C_{2}H_{3}O_{2})_{2},H_{2}O
+                                    + Ca(C_{2}H_{3}O_{2})_{2},H_{2}O
+                                    + 6H_{2}O
+
+the amount of water split off is insufficient to give the appearance of
+partial fusion, and, therefore, only a change in the crystals is observed.
+
+The preceding examples, in which decomposition of the double salt was
+effected by a rise of temperature, were chosen for first consideration as
+being more analogous to the case of Glauber's salt; but not a few examples
+are known where the reverse change takes place, formation of the double
+salt occurring _above_ the transition point, and decomposition into the
+constituent salts below it. Instances of this behaviour are found in the
+case of the formation of astracanite from sodium and magnesium sulphates,
+and of sodium ammonium racemate from the two sodium ammonium tartrates, to
+which reference will be made later. Between these various systems, however,
+there is no essential difference; and whether decomposition or formation of
+the double salt occurs at temperatures above the transition point, will of
+course depend on the heat of change at that point. For, in accordance with
+van't Hoff's law of movable equilibrium (p. 58), that change will take
+place at the higher temperature which is accompanied by an absorption of
+heat. If, therefore, the formation of the double salt from the single salts
+is accompanied by an absorption of heat, the double salt will be formed
+from the single salts on raising the temperature; but if the reverse is the
+case, then the double salt on being heated will decompose into the
+constituent salts.[340]
+
+In those cases, now, which have so far been studied, the change at the
+transition point is accompanied by a taking up or a splitting off of water;
+and _in such cases the general rule can be given, that if the water of
+crystallization of the two constituent {261} salts together is greater than
+that of the double salt, the latter will be produced from the former on
+raising the temperature_ (_e.g._ astracanite from sodium and magnesium
+sulphates); _but if the double salt contains more water of crystallization
+than the two single salts, increase of temperature will effect the
+decomposition of the double salt_. When we seek for the connection between
+this rule and the law of van't Hoff, it is found in the fact that the heat
+effect involved in the hydration or dehydration of the salts is much
+greater than that of the other changes which occur, and determines,
+therefore, the sign of the total heat effect.[341]
+
+Vapour Pressure. Quintuple Point.--In the case of Glauber's salt, we saw
+that at a certain temperature the vapour pressure curve of the hydrated
+salt cut that of the saturated solution of anhydrous sodium sulphate. That
+point, it will be remembered, was a quadruple point at which the four
+phases sodium sulphate decahydrate, anhydrous sodium sulphate, solution,
+and vapour, could co-exist; and was also the point of intersection of the
+curves for four univariant systems. In the case of the formation of double
+salts, similar relationships are met with; and also certain differences,
+due to the fact that we are now dealing with systems of three components.
+Two cases will be chosen here for brief description, one in which
+formation, the other in which decomposition of the double salt occurs with
+rise of temperature.
+
+On heating a mixture of sodium sulphate decahydrate and magnesium sulphate
+heptahydrate, it is found that at 22deg partial liquefaction occurs with
+formation of astracanite. At this temperature, therefore, there can coexist
+the five phases--
+
+  Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O;
+  solution; vapour.
+
+This constitutes, therefore, a _quintuple point_; and since there are three
+components present in five phases, the system is invariant. This point,
+also, will be the point of intersection of curves for five univariant
+systems, which, in this case, must each be composed of four phases. These
+systems are--
+
+{262}
+
+    I. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
+    Na_{2}Mg(SO_{4})_{2},4H_{2}O; vapour.
+
+    II. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; solution; vapour.
+
+    III. MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution; vapour.
+
+    IV. Na_{2}SO_{4},10H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution;
+    vapour.
+
+    V. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
+    Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution.
+
+[Illustration: FIG. 98.]
+
+On representing the vapour pressures of these different systems
+graphically, a diagram is obtained such as is shown in Fig. 98,[342] the
+curves being numbered in accordance with the above list. When the system I.
+is heated, the vapour pressure increases until at the quintuple point the
+liquid phase (solution) is formed, and it will then depend on the relative
+amounts of the different phases whether on further heating there is formed
+system III., IV., or V. If either of the first two is produced, we shall
+obtain the vapour pressure of the solutions saturated with respect to both
+double salt and one of the single salts; while if the vapour phase
+disappears, there will be obtained the pressure of the condensed systems
+formed of double salt, two single salts and solution. This curve,
+therefore, indicates the _change of the transition point with pressure_;
+and since in the ordinary determinations of the transition point in open
+vessels, we are in reality dealing with condensed systems under the
+pressure of 1 atm., it will be evident that the transition point does not
+accurately coincide with the quintuple point (at which the system is under
+the pressure of its own vapour). As in the case of other condensed systems,
+however, pressure has only a slight influence on the temperature of the
+transition point. Whether or not pressure raises or lowers the transition
+point will depend on whether transformation is accompanied by an increase
+or {263} diminution of volume (theorem of Le Chatelier, p. 58). In the case
+of the formation of astracanite, expansion occurs, and the transition point
+will therefore be raised by increase of pressure. Although measurements
+have not been made in the case of this system, the existence of such a
+curve has been experimentally verified in the case of copper and calcium
+acetates and water (v. _infra_).[343]
+
+[Illustration: FIG. 99.]
+
+The vapour pressure diagram in the case of copper calcium acetate and water
+(Fig. 99), is almost the reverse of that already discussed. In this case,
+the double salt decomposes on heating, and the decomposition is accompanied
+by a contraction. Curve I. is the vapour pressure curve for double salt,
+two single salts (p. 260), and vapour; curves II. and III. give the vapour
+pressures of solutions saturated with respect to double salt and one of the
+single salts; curve IV. is the curve of pressures for the solutions
+saturated with respect to the two single salts; while curve V. again
+represents the change of the transition point with pressure. On examining
+this diagram, it is seen that whereas {264} astracanite could exist both
+above and below the quintuple point, copper calcium acetate can exist only
+_below_ the quintuple point. This behaviour is found only in those cases in
+which the double salt is decomposed by rise of temperature, and where the
+decomposition is accompanied by a diminution of volume.[344]
+
+As already mentioned, the decomposition of copper calcium acetate into the
+single salts and saturated solution is accompanied by a contraction, and it
+was therefore to be expected that increase of pressure would _lower_ the
+transition point. This expectation of theory was confirmed by experiment,
+for van't Hoff and Spring found that although the transition point under
+atmospheric pressure is about 75deg, decomposition of the double salt took
+place even at the ordinary temperature when the pressure was increased to
+6000 atm.[345]
+
+Solubility Curves at the Transition Point.--At the transition point, as has
+already been shown, the double salt and the two constituent salts can exist
+in equilibrium with the same solution. The transition point, therefore,
+must be the point of intersection of two solubility curves; the solubility
+curve of the double salt and the solubility curve of the mixtures of the
+two constituent salts. It should be noted here that we are not dealing with
+the solubility curves of the single salts separately, for since the systems
+are composed of three components, a single solid phase can, at a given
+temperature, be in equilibrium with solutions of different composition, and
+two solid phases in contact with solution (and vapour) are therefore
+necessary to give an univariant system. The same applies, of course, to the
+solubility of the double salt; for a double salt also constitutes a single
+phase, and can therefore exist in equilibrium with solutions of varying
+composition. If, however, we make the restriction (which we do for the
+present) that the double salt is not decomposed by water, then the solution
+will contain the constituent salts in the same relative proportions as they
+are contained in the double salt, and the system may therefore be regarded
+as one of _two_ components, viz. double salt and water. In this case one
+solid phase is sufficient, with solution and {265} vapour, to give an
+univariant system; and at a given temperature, therefore, the solubility
+will have a perfectly definite value.
+
+Since in almost all cases the solubility is determined in open vessels, we
+shall in the following discussion consider that the vapour phase is absent,
+and that the system is under a constant pressure, that of the atmosphere.
+With this restriction, therefore, four phases will constitute an invariant
+system, three phases an univariant, and two phases a bivariant system.
+
+It has already been learned that in the case of sodium sulphate and water,
+the solubility curve of the salt undergoes a sudden change in direction at
+the transition point, and that this is accompanied by a change in the solid
+phase in equilibrium with the solution. The same behaviour is also found in
+the case of double salts. To illustrate this, we shall briefly discuss the
+solubility relations of a few double salts, beginning with one of the
+simplest cases, that of the formation of rubidium racemate from rubidium
+_d_- and _l_-tartrates. The solubilities are represented diagrammatically
+in Fig. 100, the numerical data being contained in the following table, in
+which the solubility is expressed as the number of gram-molecules
+Rb_{2}C_{4}H_{4}O_{6} in 100 gm.-molecules of water.[346]
+
+  ---------------------------------------------------------------
+  Temperature. | Solubility of tartrate | Solubility of racemate.
+               |        mixture.        |
+  ---------------------------------------------------------------
+     25deg     |         13.03          |          10.91
+     35deg     |          --            |          12.63
+     40.4deg   |          --            |          13.48
+     40.7deg   |         13.46          |           --
+     54deg     |         13.83          |           --
+  ---------------------------------------------------------------
+
+In Fig. 100 the curve AB represents the solubility of the racemate, while
+A'BC represents the solubility of the mixed tartrates. Below the transition
+point, therefore, the solubility of the racemate is less than that of the
+mixed tartrates. The solution, saturated with respect to the latter, will
+be supersaturated with respect to the racemate; and if a nucleus of this is
+present, racemate will be deposited, and the mixed tartrates, if present in
+equimolecular amounts, will ultimately {266} entirely disappear, and only
+racemate will be left as solid phase. The solution will then have the
+composition represented by a point on the curve AB. Conversely, above the
+transition point, the saturated solution of the racemate would be
+supersaturated with respect to the two tartrates, and transformation into
+the latter would ensue. If, therefore, a solution of equimolecular
+proportions of rubidium _d_- and _l_-tartrates is allowed to evaporate at a
+temperature above 40deg, a mixture of the two tartrates will be deposited;
+while at temperatures below 40deg the racemate will separate out.
+
+[Illustration: FIG. 100.]
+
+Similar relationships are met with in the case of sodium ammonium _d_- and
+_l_-tartrate and sodium ammonium racemate; but in this case the racemate is
+the stable form in contact with solution above the transition point
+(27deg).[347] Below the transition point, therefore, the solubility curve
+of the mixed tartrates will lie below the solubility curve of the racemate.
+Below the transition point, therefore, sodium ammonium racemate will break
+up in contact with solution into a mixture of sodium ammonium _d_- and
+_l_-tartrates. At a higher temperature, 35deg, sodium ammonium racemate
+undergoes decomposition into sodium racemate and ammonium racemate.[348]
+
+The behaviour of sodium ammonium racemate is of interest from the fact that
+it was the first racemic substance to be resolved into its optically active
+forms by a process of crystallization. On neutralizing a solution of
+racemic tartaric acid, half with soda and half with ammonia, and allowing
+the solution to evaporate, Pasteur[349] obtained a mixture of sodium
+ammonium {267} _d_- and _l_-tartrates. Since Pasteur was unaware of the
+existence of a transition point, the success of his experiment was due to
+the happy chance that he allowed the solution to evaporate at a temperature
+below 27deg; for had he employed a temperature above this, separation of
+the racemate into the two enantiomorphous forms would not have occurred.
+For this reason the attempt of Staedel to perform the same resolution met
+only with failure.[350]
+
+Decomposition of the Double Salt by Water.--In the two cases just
+described, the solubility relationships at the transition point are of a
+simpler character than in the case of most double salts. If, at a
+temperature above the transition point, a mixture of rubidium _d_- and
+_l_-tartrates in equimolecular proportions is brought in contact with water
+a solution will be obtained, which is saturated with respect to both
+enantiomorphous forms; and since the solubility of the two optical
+antipodes is identical, and the effect of one on the solubility of the
+other also the same, the solution will contain equimolecular amounts of the
+_d_- and _l_-salt. If, now, the solution is cooled down in contact with the
+solid salts to just below the transition point, it becomes supersaturated
+with respect to the racemate, and this will be deposited. The solution
+thereby becomes unsaturated with respect to the mixture of the active
+salts, and these must therefore pass into solution. As the latter are
+equally soluble, equal amounts of each will dissolve, and a further
+quantity of the racemate will be deposited. These processes of solution and
+deposition will continue until the single tartrates have completely
+disappeared, and only racemate is left as solid phase. As a consequence of
+the identical solubility of the two tartrates, therefore, no excess of
+either form will be left on passing through the transition point. From this
+it will be evident that the racemate can exist as single solid phase in
+contact with its saturated solution at the transition point; or, in other
+words, the racemate is not decomposed by water at the transition point. The
+same behaviour will evidently be exhibited by sodium ammonium racemate at
+27deg, for the two enantiomorphous sodium ammonium tartrates have also
+identical solubility.
+
+{268}
+
+Very different, however, is the behaviour of, say, astracanite, or of the
+majority of double salts; for the solubility of the constituent salts is
+now no longer the same. If, for example, excess of a mixture of sodium
+sulphate and magnesium sulphate, in equimolecular proportions, is brought
+in contact with water below the transition point (22deg), more magnesium
+sulphate than sodium sulphate will dissolve, the solubility of these two
+salts in a common solution being given by the following figures, which
+express number of molecules of the salt in 100 molecules of water.[351]
+
+  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
+  Na_{2}SO_{4},10H_{2}O AND MgSO_{4},7H_{2}O.
+
+  ----------------------------------------
+  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
+  ----------------------------------------
+     18.5deg   |      2.16     |   4.57
+     24.5deg   |      3.43     |   4.68
+  ----------------------------------------
+
+At the transition point, then, it is evident that the solution contains
+more magnesium sulphate than sodium sulphate: and this must still be the
+case when astracanite, which contains sodium sulphate and magnesium
+sulphate in equimolecular proportions, separates out. If, therefore, the
+temperature is raised slightly above the transition point, magnesium
+sulphate and sodium sulphate will pass into solution, the former, however,
+in larger quantities than the latter, and astracanite will be deposited;
+and this will go on until all the magnesium sulphate has disappeared, and a
+mixture of astracanite and sodium sulphate decahydrate is left as solid
+phases. Since there are now three phases present, the system is univariant
+(by reason of the restriction previously made that the vapour phase is
+absent), and at a given temperature the solution will have a definite
+composition; as given in the following table:--
+
+  COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
+  Na_{2}Mg(SO_{4})_{2},4H_{2}O AND Na_{2}SO_{4},10H_{2}O.
+
+  ----------------------------------------
+  Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
+  ----------------------------------------
+     22deg     |      2.95     |   4.70
+     24.5deg   |      3.45     |   3.62
+  ----------------------------------------
+
+{269}
+
+From the above figures, therefore, it will be seen that at a temperature
+just above the transition point a solution in contact with the two solid
+phases, astracanite and Glauber's salt, contains a relatively smaller
+amount of sodium sulphate than a pure solution of astracanite would; for in
+this case there would be equal molecular amounts of Na_{2}SO_{4} and
+MgSO_{4}. A solution which is saturated with respect to astracanite alone,
+will contain more sodium sulphate than the solution saturated with respect
+to astracanite plus Glauber's salt, and the latter will therefore be
+deposited. From this, therefore, it is clear that if astracanite is brought
+in contact with water at about the transition point, it will undergo
+decomposition with separation of Glauber's salt (supersaturation being
+excluded).
+
+[Illustration: FIG. 101.]
+
+This will perhaps be made clearer by considering Fig. 101. In this diagram
+the ordinates represent the ratio of sodium sulphate to magnesium sulphate
+in the solutions, and the abscissae represent the temperatures. The line AB
+represents solutions saturated with respect to a mixture of the single
+salts (p. 268); BC refers to solutions in equilibrium with astracanite and
+magnesium sulphate; while BX represents the composition of solutions in
+contact with the solid phases astracanite and Glauber's salt. The values of
+the solubility are contained in the following table, and in that on p. 268,
+and are, as before, expressed in gm.-molecules of salt in 100 gm.-molecules
+of water.[352]
+
+{270}
+
+  -------------------------------------------------------------------------
+               | Astracanite                | Astracanite
+  Temperature. |        + sodium sulphate.  |        + magnesium sulphate.
+               |----------------------------|------------------------------
+               | Na_{2}SO_{4}. |  MgSO_{4}. | Na_{2}SO_{4}. |  MgSO_{4}.
+  -------------------------------------------------------------------------
+     18.5deg   |       --      |      --    |      3.41     |    4.27
+     22deg     |      2.95     |     4.70   |      2.85     |    4.63
+     24.5deg   |      3.45     |     3.62   |      2.68     |    4.76
+     30deg     |      4.58     |     2.91   |      2.30     |    5.31
+     35deg     |      4.30     |     2.76   |      1.73     |    5.88
+  -------------------------------------------------------------------------
+
+At the transition point the ratio of sodium sulphate to magnesium sulphate
+is approximately 1 : 1.6. In the case of solutions saturated with respect
+to both astracanite and Glauber's salt, the relative amount of sodium
+sulphate increases as the temperature rises, while in the solutions
+saturated for astracanite and magnesium sulphate, the ratio of sodium
+sulphate to magnesium sulphate decreases.
+
+If, now, we consider only the temperatures above the transition point, we
+see from the figure that solutions represented by points above the line BX
+contain relatively more sodium sulphate than solutions in contact with
+astracanite and Glauber's salt; and solutions lying below the line BC
+contain relatively more magnesium sulphate than solutions saturated with
+this salt and astracanite. These solutions will therefore not be stable,
+but will deposit in the one case, astracanite and Glauber's salt, and in
+the other case, astracanite and magnesium sulphate, until a point on BX or
+BC is reached. All solutions, however, lying to the right of CBX, will be
+_unsaturated_ with respect to these two pairs of salts, and only the
+solutions represented by the line XY (and which contain equimolecular
+amounts of sodium and magnesium sulphates) will be saturated with respect
+to the pure double salt.
+
+Transition Interval.--Fig. 101 will also render intelligible a point of
+great importance in connection with astracanite, and of double salts
+generally. At temperatures between those represented by the points B and X,
+the double salt when brought in contact with water will be decomposed with
+separation of sodium sulphate. Above the temperature of the point {271} X,
+however, the solution of the pure double salt is stable, because it can
+still take up a little of either of the components. At temperatures, then,
+above that at which the solution in contact with the double salt and the
+less soluble single salt, contains the single salts in the ratio in which
+they are present in the double salt, solution of the latter will take place
+without decomposition. _The range of temperature between that at which
+double salt can begin to be formed (the transition point) and that at which
+it ceases to be decomposed by water is called the transition
+interval._[353] If the two single salts have identical solubility at the
+transition point, the transition interval diminishes to nought.
+
+In those cases where the double salt is the stable form below the
+transition point, the transition interval will extend downwards to a lower
+temperature. Fig. 101 will then have the reverse form.
+
+Summary.--With regard to double salts we have learned that their formation
+from and their decomposition into the single salts, is connected with a
+definite temperature, the _transition temperature_. At this transition
+temperature two vapour pressure curves cut, viz. a curve of dehydration of
+a mixture of the single salts and the solubility curve of the double salt;
+or the dehydration curve of the double salt and the solubility curve of the
+mixed single salts. The solubility curves, also, of these two systems
+intersect at the transition point, but although the formation of the double
+salt commences at the transition point, complete stability in contact with
+water may not be attained till some temperature above (or below) that
+point. _Only when the temperature is beyond the transition interval, will a
+double salt dissolve in water without decomposition (_e.g._ the alums)._
+
+       *       *       *       *       *
+
+
+{272}
+
+CHAPTER XVI
+
+ISOTHERMAL CURVES AND THE SPACE MODEL
+
+In the preceding chapter we considered the changes in the solubility of
+double salts and of mixtures of their constituent salts with the
+temperature; noting, more especially, the relationships between the two
+systems at the transition point. It is now proposed to conclude the study
+of the three-component systems by discussing very briefly the solubility
+relations at constant temperature, or the isothermal solubility curves. In
+this way fresh light will be thrown on the change in the solubility of one
+component by the addition of another component, and also on the conditions
+of formation and stable existence of double salts in solution. With the
+help of these isothermal curves, also, the phenomena of crystallization at
+constant temperature--phenomena which have not only a scientific interest
+but also an important bearing on the industrial preparation of double
+salts--will be more clearly understood.[354]
+
+A brief description will also be given of the method of representing the
+variation of the concentration of the two salts in the solution with the
+temperature.
+
+Non-formation of Double Salts.--In Fig. 102 are shown the solubility curves
+of two salts, A and B, which at the given temperature do not form a double
+salt.[355] The ordinates represent the amount of A, the abscissae the
+amount of B in a _constant amount_ of the third component, the solvent. The
+{273} point A, therefore, represents the solubility of the salt A at the
+given temperature; and similarly, point B represents the solubility of B.
+Since we are dealing with a three-component system, one solid phase in
+contact with solution will constitute a bivariant system (in the absence of
+the vapour phase and under a constant pressure). At any given temperature,
+therefore, the concentration of the solution in equilibrium with the solid
+can undergo change. If, now, to a pure solution of A a small quantity of B
+is added, the solubility of A will in general be altered; as a rule it is
+diminished, but sometimes it is increased.[356] The curve AC represents the
+varying composition of the solution in equilibrium with the solid component
+A. Similarly, the curve BC represents the composition of the solutions in
+contact with pure B as solid phase. At the point, C, where these two curves
+intersect, there are two solid phases, viz. pure A and pure B, in
+equilibrium with solution, and the system becomes invariant. At this point
+the solution is saturated with respect to both A and B, and at a given
+temperature must have a perfectly definite composition. To take an example,
+if we suppose A to represent sodium sulphate decahydrate, and B, magnesium
+sulphate heptahydrate, and the temperature to be 18.5deg (_i.e._ below the
+transition point), the point C would represent a solution containing 2.16
+gm.-molecules Na_{2}SO_{4} and 4.57 gm.-molecules MgSO_{4} per 100
+gm.-molecules of water (p. 268). The curve ACB is the boundary curve for
+saturated solutions; solutions lying outside this curve are supersaturated,
+those lying within the area ACBO, are unsaturated.
+
+[Illustration: FIG. 102.]
+
+[Illustration: FIG. 103.]
+
+[Illustration: FIG. 104.]
+
+Formation of Double Salt.--We have already learned in the preceding chapter
+that if the temperature is outside[357] the {274} transition interval, it
+is possible to prepare a pure saturated solution of the double salt. If,
+now, we suppose the double salt to contain the two constituent salts in
+equimolecular proportions, its saturated solution must be represented by a
+point lying on the line which bisects the angle AOB; _e.g._ point D, Fig.
+103. But a double salt constitutes only a single phase, and can exist,
+therefore, in contact with solutions of varying concentration, as
+represented by EDF.
+
+Let us compare, now, the relations between the solubility curve for the
+double salt, and those for the two constituent salts. We shall suppose that
+the double salt is formed from the single salts when the temperature is
+raised above a certain point (as in the formation of astracanite). At a
+temperature below the transition point, as we have already seen, the
+solubility of the double salt is greater than that of a mixture of the
+single salts. The curve EDF, therefore, must lie above the point C, in the
+region representing solutions supersaturated with respect to the single
+salts (Fig. 104). Such a solution, however, would be metastable, and on
+being brought in contact with the single salts would deposit these and
+yield a solution represented by the point C. At this particular
+temperature, therefore, the isothermal solubility curve will consist of
+only two branches.
+
+[Illustration: FIG. 105.]
+
+Suppose, now, that the temperature is that of the transition point. At this
+point, the double salt can exist together with the single salts in contact
+with solution. The solubility curve {275} of the double salt must,
+therefore, pass through the point C, as shown in Fig. 105.
+
+From this figure, now, it is seen that a solution saturated with respect to
+double salt alone (point D), is supersaturated with respect to the
+component A. If, then, at the temperature of the transition point, excess
+of the double salt is brought in contact with water,[358] and if
+supersaturation is excluded, _the double salt will undergo decomposition
+and the component A will be deposited_. The relative concentration of the
+component B in the solution will, therefore, increase, and the composition
+of the solution will be thereby altered in the direction DC. When the
+solution has the composition of C, the single salt ceases to be deposited,
+for at this point the solution is saturated for both double and single
+salt; and the system becomes invariant.
+
+This diagram explains very clearly the phenomenon of the decomposition of a
+double salt at the transition point. As is evident, this decomposition will
+occur when the solution which is saturated at the temperature of the
+transition point, with respect to the two single salts (point C), does not
+contain these salts in the same ratio in which they are present in the
+double salt. If point C lay on the dotted line bisecting the right angle,
+then the pure saturated solution of the double salt would not be
+supersaturated with respect to either of the single salts, and the double
+salt would, therefore, not be decomposed by water. As has already been
+mentioned, this behaviour is found in the case of optically active
+isomerides, the solubilities of which are identical.
+
+At the transition point, therefore, the isothermal curve also consists of
+two branches; but the point of intersection of the two branches now
+represents a solution which is saturated not {276} only with respect to the
+single salts, but also for the double salt in presence of the single salts.
+
+We have just seen that by a change of temperature the two solubility
+curves, that for the two single salts and that for the double salt, were
+made to approach one another (_cf._ Figs. 104 and 105). In the previous
+chapter, however, we found that on passing the transition point to the
+region of stability for the double salt, the solution which is saturated
+for a mixture of the two constituent salts, is supersaturated for the
+double salt. In this case, therefore, point C must lie above the solubility
+curve of the pure double salt (Fig. 106), and a solution of the composition
+C, if brought in contact with double salt, will deposit the latter. If the
+single salts were also present, then as the double salt separated out, the
+single salts would pass into solution, because so long as the two single
+salts are present, the composition of the solution must remain unaltered.
+If one of the single salts disappear before the other, there will be left
+double salt plus A or double salt plus B, according to which was in excess;
+and the composition of the solution will be either that represented by D
+(saturated for double salt plus A), or that of the point F (saturated for
+double salt plus B).
+
+[Illustration: FIG. 106.]
+
+In connection with the isothermal represented in Fig. 106, it should be
+noted that at this particular temperature a solution saturated with respect
+to the pure double salt is no longer supersaturated for one of the single
+salts (point D); so that at the temperature of this isothermal the double
+salt is not decomposed by water. At this temperature, further, the boundary
+curve consists of three branches AD, DF, and FB, which give the composition
+of the solutions in equilibrium with pure A, double salt, and pure B
+respectively; while the points D and F represent solutions saturated for
+double salt plus A and double salt plus B.
+
+On continuing to alter the temperature in the same direction {277} as
+before, the relative shifting of the solubility curves becomes more marked,
+as shown in Fig. 107. At the temperature of this isothermal, the solution
+saturated for the double salt now lies in a region of distinct unsaturation
+with respect to the single salts; and the double salt can now exist as
+solid phase in contact with solutions containing both relatively more of A
+(curve ED), and relatively more of B (curve DF), than is contained in the
+double salt itself.
+
+[Illustration: FIG. 107.]
+
+Transition Interval.--From what has been said, and from an examination of
+the isothermal diagrams, Figs. 104-107, it will be seen that by a variation
+of the temperature we can pass from a condition where the double salt is
+quite incapable of existing in contact with solution (supersaturation being
+excluded), to a condition where the existence of the double salt in
+presence of solution becomes possible; only in the presence, however, of
+one of the single salts (_transition point_, Fig. 105). A further change of
+temperature leads to a condition where the stable existence of the pure
+double salt in contact with solution just becomes possible (Fig. 106); and
+from this point onwards, pure saturated solutions of the double salt can be
+obtained (Fig. 107). _At any temperature, therefore, between that
+represented by Fig. 105, and that represented by Fig. 106, the double salt
+undergoes partial decomposition, with deposition of one of the constituent
+salts._ The temperature range between the transition point and the
+temperature at which a stable saturated solution of the pure double salt
+just begins to be possible, is known as the _transition interval_ (p. 270).
+As the figures show, the transition interval is limited on the one side by
+the transition temperature, and on the other by the temperature at which
+the solution saturated for double salt and the less soluble of the single
+salts, contains the component salts in the same ratio as they are present
+in the double salt. The greater the difference in the solubility of the
+single salts, the larger will be the transition interval. {278}
+
+Isothermal Evaporation.--The isothermal solubility curves are of great
+importance for obtaining an insight into the behaviour of a solution when
+subjected to isothermal evaporation. To simplify the discussion of the
+relationships found here, we shall still suppose that the double salt
+contains the single salts in equimolecular proportions; and we shall, in
+the first instance, suppose that the unsaturated solution with which we
+commence, also contains the single salts in the same ratio. The composition
+of the solution must, therefore, be represented by some point lying on the
+line OD, the bisectrix of the right angle.
+
+From what has been said, it is evident that when the formation of a double
+salt can occur, three temperature intervals can be distinguished, viz. the
+single-salt interval, the transition interval, and the double-salt
+interval.[359] When the temperature lies in the first interval, evaporation
+leads first of all to the crystallization of one of the single salts, and
+then to the separation of both the single salts together. In the second
+temperature interval, evaporation again leads, in the first place, to the
+deposition of one of the single salts, and afterwards to the
+crystallization of the double salt. In the third temperature interval, only
+the double salt crystallizes out. This will become clearer from what
+follows.
+
+[Illustration: FIG. 108.]
+
+[Illustration: FIG. 109.]
+
+If an unsaturated solution of the two single salts in equimolecular
+proportion (_e.g._ point _x_, Fig. 108) is evaporated at a temperature at
+which the formation of double salt is impossible, the component A, the
+solubility curve of which is {279} cut by the line OD, will first separate
+out; the solution will thereby become richer in B. On continued
+evaporation, more A will be deposited, and the composition of the solution
+will change until it attains the composition represented by the point C,
+when both A and B will be deposited, and the composition of the solution
+will remain unchanged. The result of evaporation will therefore be a
+mixture of the two components.
+
+If the formation of double salt is possible, but if the temperature lies
+within the transition interval, the relations will be represented by a
+diagram like Fig. 109. Isothermal evaporation of the solution X will lead
+to the deposition of the component A, and the composition of the solution
+will alter in the direction DE; at the latter point the double salt will be
+formed, and the composition of the solution will remain unchanged so long
+as the two solid phases are present. As can be seen from the diagram,
+however, the solution in E contains less of component A than is contained
+in the double salt. Deposition of the double salt at E, therefore, would
+lead to a relative decrease in the concentration of A in the solution, and
+to counterbalance this, _the salt which separated out at the commencement
+must redissolve_.
+
+Since the salts were originally present in equimolecular proportions, the
+final result of evaporation will be the pure double salt. If when the
+solution has reached the point E the salt A which had separated out is
+removed, double salt only will be left as solid phase. At a given
+temperature, however, a single solid phase can exist in equilibrium with
+solutions of different composition. If, therefore, isothermal evaporation
+is continued after the removal of the salt A, double salt will be
+deposited, and the composition of the solution will change in the direction
+EF. At the point F the salt B will separate out, and on evaporation both
+double salt and the salt B will be deposited. In the former case (when the
+salt A disappears on evaporation) we are dealing with an _incongruently
+saturated solution_; but in the latter case, where both solid phases
+continue to be deposited, the solution is said to be _congruently
+saturated_.[360]
+
+A "congruently saturated solution" is one from which the {280} solid phases
+are continuously deposited during isothermal evaporation to dryness,
+whereas in the case of "incongruently saturated solutions," at least one of
+the solid phases disappears during the process of evaporation.
+
+[Illustration: FIG. 110.]
+
+Lastly, if the temperature lies outside the transition interval, isothermal
+evaporation of an unsaturated solution of the composition X (Fig. 110) will
+lead to the deposition of pure double salt from beginning to end. If a
+solution of the composition Y is evaporated, the component A will first be
+deposited and the composition of the solution will alter in the direction
+of E, at which point double salt will separate out. Since the solution at
+this point contains relatively more of A than is present in the double
+salt, both the double salt and the single salt A will be deposited on
+continued evaporation, in order that the composition of the solution shall
+remain unchanged. In the case of solution Z, first component B and
+afterwards the double salt will be deposited. The result will, therefore,
+be a mixture of double salt and the salt B (congruently saturated
+solutions),
+
+It may be stated here that the same relationships as have been explained
+above for double salts are also found in the resolution of racemic
+compounds by means of optically active substances (third method of
+Pasteur). In this case the single salts are doubly active substances
+(_e.g._ strychnine-_d_-tartrate and strychnine-_l_-tartrate), and the
+double salt is a partially racemic compound.[361]
+
+Crystallization of Double Salt from Solutions containing Excess of One
+Component.--One more case of isothermal crystallization may be discussed.
+It is well known that a double salt which is decomposed by pure water can
+nevertheless be obtained pure by crystallization from a solution containing
+excess of one of the single salts (_e.g._ in the case of carnallite). Since
+the double salt is partially decomposed by water, the temperature of the
+experiment must be within the transition {281} interval, and the relations
+will, therefore, be represented by a diagram like Fig. 109. If, now,
+instead of starting with an unsaturated solution containing the single
+salts in equimolecular proportions, we commence with one in which excess of
+one of the salts is present, as represented by the point Y, isothermal
+evaporation will cause the composition to alter in the direction YD', the
+relative amounts of the single salts remaining the same throughout. When
+the composition of the solution reaches the point D', pure double salt will
+be deposited. The separation of double salt will, however, cause a relative
+decrease in the concentration of the salt A, and the composition of the
+solution will, therefore, alter in the direction D'F. If the evaporation is
+discontinued before the solution has attained the composition F, only
+double salt will have separated out. Even within the transition interval,
+therefore, pure double salt can be obtained by crystallization, provided
+the original solution has a composition represented by a point lying
+between the two lines OE and OF. Since, as already shown, the composition
+of the solution alters on evaporation in the direction EF, it will be best
+to employ a solution having a composition near to the line OE.
+
+Formation of Mixed Crystals.--If the two single salts A and B do not
+crystallize out pure from solution, but form an unbroken series of mixed
+crystals, it is evident that an invariant system cannot be produced. The
+solubility curve will therefore be continuous from A to B; the liquid
+solutions of varying composition being in equilibrium with solid solutions
+also of varying composition. If, however, the series of mixed crystals is
+not continuous, there will be a break in the solubility curve at which two
+solid solutions of different composition will be in equilibrium with liquid
+solution. This, of course, will constitute an invariant system, and the
+point will correspond to the point C in Fig. 108. A full discussion of
+these systems would, however, lead us too far, and the above indication of
+the behaviour must suffice.[362]
+
+{282}
+
+Application to the Characterization of Racemates.--The form of the
+isothermal solubility curves is also of great value for determining whether
+an inactive substance is a racemic compound or a conglomerate of equal
+proportions of the optical antipodes.[363]
+
+As has already been pointed out, the formation of racemic compounds from
+the two enantiomorphous isomerides, is analogous to the formation of double
+salts. The isothermal solubility curves, also, have a similar form. In the
+case of the latter, indeed, the relationships are simplified by the fact
+that the two enantiomorphous forms have identical solubility, and the
+solubility curves are therefore symmetrical to the line bisecting the angle
+of the co-ordinates. Further, with the exception of the partially racemic
+compounds to be mentioned later, there is no transition interval.
+
+In Fig. 111, are given diagrammatically two isothermal solubility curves
+for optically active substances. From what has been said in the immediately
+preceding pages, the figure ought really to explain itself. The upper
+isothermal _acb_ represents the solubility relations when the formation of
+a racemic compound is excluded, as, _e.g._ in the case of rubidium _d_- and
+_l_-tartrates above the transition point (p. 265). The solution at the
+point _c_ is, of course, inactive, and _is unaffected by addition of either
+the _d_- or _l_- form_. The lower isothermal, on the other hand, would be
+obtained at a temperature at which the racemic compound could be formed.
+The curve _a'e_ is the solubility curve for the _l_- form; _b'f_, that for
+the _d_- form; and _edf_, that for the racemic compound in presence of
+solutions of varying concentration. The point _d_ corresponds to saturation
+for the pure racemic compound.
+
+[Illustration: FIG. 111.]
+
+From these curves now, it will be evident that it will be possible, in any
+given case, to decide whether or not an inactive body is a mixture or a
+racemic compound. For this purpose, {283} two solubility determinations are
+made, first with the inactive material alone (in excess), and then with the
+inactive material plus excess of one of the optically active forms. If we
+are dealing with a mixture, the two solutions thus obtained will be
+identical; both will have the composition corresponding to the point _c_,
+and will be inactive. If, however, the inactive material is a racemic
+compound, then two different solutions will be obtained; namely, an
+inactive solution corresponding to the point _d_ (Fig. 111), and an
+_active_ solution corresponding either to _e_ or to _f_, according to which
+enantiomorphous form was added.
+
+_Partially racemic compounds._[364] In this case we are no longer dealing
+with enantiomorphous forms, and the solubility of the two oppositely active
+isomerides is no longer the same. The symmetry of the solubility curves
+therefore disappears, and a figure is obtained which is identical in its
+general form with that found in the case of ordinary double salts (Fig.
+112). In this case there is a transition interval.
+
+[Illustration: FIG. 112.]
+
+The curves _acb_ belong to a temperature at which the partially racemic
+compound cannot be formed; _a'dfb'_, to the temperature at which the
+compound just begins to be stable in contact with water, and _a"ed'f'b"_
+belongs to a temperature at which the partially racemic compound is quite
+stable in contact with water. Suppose now solubility determinations, made
+in the first case with the original material alone, and then with the
+original body plus each of the two compounds, formed from the
+enantiomorphous substances separately, then if the original body was a
+mixture, identical solutions will be obtained in all three cases (point
+_c_); if it was a partially racemic compound, three different solutions
+(_e_, _d'_, and _f'_) will be obtained if the temperature was outside the
+transition interval, and two solutions, _d_ and _f_, if the temperature
+belonged to the transition interval.
+
+{284}
+
+_Representation in Space._
+
+Space Model for Carnallite.--Interesting and important as the isothermal
+solubility curves are, they are insufficient for the purpose of obtaining a
+clear insight into the complete behaviour of the systems of two salts and
+water. A short description will, therefore, be given here of the
+representation in space of the solubility relations of potassium and
+magnesium chlorides, and of the double salt which they form,
+carnallite.[365]
+
+[Illustration: FIG. 113.]
+
+Fig. 113 is a diagrammatic sketch of the model for carnallite looked at
+sideways from above. Along the X-axis is measured the concentration of
+magnesium chloride in the {285} solution; along the Y-axis, the
+concentration of potassium chloride; while along the T-axis is measured the
+temperature. The three axes are at right angles to one another. The
+XT-plane, therefore, contains the solubility curve of magnesium chloride;
+the YT-plane, the solubility curve of potassium chloride, and in the space
+between the two planes, there are represented the composition of solutions
+containing both magnesium and potassium chlorides. Any _surface_ between
+the two planes will represent the various solutions in equilibrium with
+only one solid phase, and will therefore indicate the area or field of
+existence of bivariant ternary systems. A _line_ or _curve_ formed by the
+intersection of two surfaces will represent solutions in equilibrium with
+two solid phases (viz. those belonging to the intersecting surfaces), and
+will show the conditions for the existence of univariant systems. Lastly,
+_points_ formed by the intersection of three surfaces will represent
+invariant systems, in which a solution can exist in equilibrium with three
+solid phases (viz. those belonging to the three surfaces).
+
+We shall first consider the solubility relations of the single salts. The
+complete equilibrium curve for magnesium chloride and water is represented
+in Fig. 113 by the series of curves ABF_{1} G_{1} H_{1} J_{1} L_{1} N_{1}.
+AB is the freezing-point curve of ice in contact with solutions containing
+magnesium chloride, and B is the cryohydric point at which the solid phases
+ice and MgCl_{2},12H_{2}O can co-exist with solution. BFG is the solubility
+curve of magnesium chloride dodecahydrate. This curve shows a point of
+maximum temperature at F_{1}, and a retroflex portion F_{1}G_{1}. The curve
+is therefore of the form exhibited by calcium chloride hexahydrate, or the
+hydrates of ferric chloride (Chapter VIII.). G_{1} is a transition point at
+which the solid phase changes from dodecahydrate to octahydrate, the
+solubility of which is represented by the curve G_{1}H_{1}. At H_{1} the
+octahydrate gives place to the hexahydrate, which is the solid phase in
+equilibrium with the solutions represented by the curve H_{1}J_{1}. J_{1}
+and L_{1} are also transition points at which the solid phase undergoes
+change, in the former case from hexahydrate to tetrahydrate; and in the
+latter case, {286} from tetrahydrate to dihydrate. The complete curve of
+equilibrium for magnesium chloride and water is, therefore, somewhat
+complicated, and is a good example of the solubility curves obtained with
+salts capable of forming several hydrates.
+
+The solubility curve of potassium chloride is of the simplest form,
+consisting only of the two branches AC, the freezing-point curve of ice,
+and CO, the solubility curve of the salt. C is the cryohydric point. This
+point and the two curves lie in the YT-plane.
+
+On passing to the ternary systems, the composition of the solutions must be
+represented by points or curves situated _between_ the two planes. We shall
+now turn to the consideration of these. BD and CD are ternary eutectic
+curves (p. 284). They give the composition of solutions in equilibrium with
+ice and magnesium chloride dodecahydrate (BD), and with ice and potassium
+chloride (CD). D is a _ternary cryohydric point_. If the temperature is
+raised and the ice allowed to disappear, we shall pass to the solubility
+curve for MgCl_{2},12H_{2}O + KCl (curve DE). At E carnallite is formed and
+the potassium chloride disappears; EFG is then the solubility curve for
+MgCl_{2},12H_{2}O + carnallite (KMgCl_{3},6H_{2}O). This curve also shows a
+point of maximum temperature (F) and a retroflex portion. GH and HJ
+represent the solubility curves of carnallite + MgCl_{2},8H_{2}O and
+carnallite + MgCl_{2},6H_{2}O, G and H being transition points. JK is the
+solubility curve for carnallite + MgCl_{2},4H_{2}O. At the point K we have
+the _highest temperature at which carnallite can exist with magnesium
+chloride in contact with solution_. Above this temperature decomposition
+takes place and potassium chloride separates out.
+
+If at the point E, at which the two single salts and the double salt are
+present, excess of potassium chloride is added, the magnesium chloride will
+all disappear owing to the formation of carnallite, and there will be left
+carnallite and potassium chloride. The solubility curve for a mixture of
+these two salts is represented by EMK; a simple curve exhibiting, however,
+a temperature maximum at M. This maximum point corresponds with the fact
+that dry carnallite melts at this temperature with separation of potassium
+chloride. _At all temperatures {287} above this point, the formation of
+double salt is impossible_. The retroflex portion of the curve represents
+solutions in equilibrium with carnallite and potassium chloride, but in
+which the ratio MgCl_{2} : KCl is greater than in the double salt.
+
+Throughout its whole course, _the curve EMK represents solutions in which
+the ratio of MgCl_{2} : KCl is greater than in the double salt_. As this is
+a point of some importance, it will be well, perhaps, to make it clearer by
+giving one of the isothermal curves, _e.g._ the curve for 10deg, which is
+represented diagrammatically in Fig. 114. E and F here represent solutions
+saturated for carnallite plus magnesium chloride hydrate, and for
+carnallite plus potassium chloride. As is evident, the point F lies above
+the line representing equimolecular proportions of the salts (OD).
+
+[Illustration: FIG. 114.]
+
+Summary and Numerical Data.--We may now sum up the different systems which
+can be formed, and give the numerical data from which the model is
+constructed.[366]
+
+  I. _Bivariant Systems._
+
+  --------------------------------------
+     Solid phase.   | Area of existence.
+  --------------------------------------
+  Ice               | ABDC
+  KCl               | CDEMKLNO
+  Carnallite        | EFGHJKM
+  MgCl_{2},12H_{2}O | BF_{1}G_{1}GFED
+  MgCl_{2},8H_{2}O  | G_{1}H_{1}HG
+  MgCl_{2},6H_{2}O  | H_{1}I_{1}IH
+  MgCl_{2},4H_{2}O  | I_{1}L_{1}LKI
+  MgCl_{2},2H_{2}O  | L_{1}N_{1}NL
+  --------------------------------------
+
+II. _Univariant Systems._--The different univariant systems have already
+been described. The course of the curves will be sufficiently indicated if
+the temperature and composition of the solutions for the different
+invariant systems are given.
+
+{288}
+
+  III.--_Invariant Systems--Binary and Ternary._
+
+  -------------------------------------------------------------------------
+         |                           |          | Composition of solution.
+  Point. |      Solid Phases.        | Temper-  | Gram-molecules of salt
+         |                           |  ature.  | per 1000 gram-mol. water.
+  -------------------------------------------------------------------------
+  A      | Ice                       |   0deg   | --
+         |                           |          |
+  B      | Ice; MgCl_{2},12H_{2}O    | -33.6deg | 49.2 MgCl_{2}
+         |                           |          |
+  C      | Ice; KCl                  | -11.1deg | 59.4 KCl
+         |                           |          |
+  D      |{ Ice; MgCl_{2},12H_{2}O; }| -34.3deg | 43 MgCl_{2}; 3 KCl
+         |{ KCl                     }|          |
+         |                           |          |
+  E      |{ MgCl_{2},12H_{2}O; KCl; }| -21deg   | 66.1 MgCl_{2}; 4.9 KCl
+         |{ carnallite              }|          |
+         |                           |          |
+  F_{1}  | MgCl_{2},12H_{2}O         | -16.4deg | 83.33 MgCl_{2}
+         |                           |          |
+  F      |{ MgCl_{2},12H_{2}O;      }| -16.6deg |{ Almost same as F_{1};
+         |{ carnallite              }|          |{ contains small amount
+         |                           |          |{ of KCl
+         |                           |          |
+  G_{1}  |{ MgCl_{2},12H_{2}O;      }| -16.8deg | 87.5 MgCl_{2}
+         |{ MgCl_{2},8H_{2}O        }|          |
+         |                           |          |
+  G      |{ MgCl_{2},12H_{2}O;      }| -16.9deg |{ Almost same as G_{1},
+         |{ MgCl_{2},8H_{2}O;       }|          |{ but contains small
+         |{ carnallite              }|          |{ quantity of KCl
+         |                           |          |
+  H_{1}  |{ MgCl_{2},8H_{2}O;       }|  -3.4deg | 99 MgCl_{2}
+         |{ MgCl_{2},6H_{2}O        }|          |
+         |                           |          |
+  H      |{ MgCl_{2},8H_{2}O;       }|ca.-3.4deg|{ Almost same as H_{1},
+         |{ MgCl_{2},6H_{2}O;       }|          |{ but contains small
+         |{ carnallite              }|          |{ amount of KCl
+         |                           |          |
+  J_{1}  |{ MgCl_{2},6H_{2}O;       }| 116.67deg| 161.8 MgCl_{2}
+         |{ MgCl_{2},4H_{2}O        }|          |
+         |                           |          |
+  J      |{ MgCl_{2},6H_{2}O;       }| 115.7deg | 162 MgCl_{2}; 4 KCl
+         |{ MgCl_{2},4H_{2}O;       }|          |
+         |{ carnallite              }|          |
+         |                           |          |
+  K      |{ MgCl_{2},4H_{2}O; KCl;  }| 152.5deg | 200 MgCl_{2}; 24 KCl
+         |{ carnallite              }|          |
+         |                           |          |
+  L_{1}  |{ MgCl_{2},4H_{2}O;       }| 181deg   | 238.1 MgCl_{2}
+         |{ MgCl_{2},2H_{2}O        }|          |
+         |                           |          |
+  L      |{ MgCl_{2},4H_{2}O;       }| 176deg   | 240 MgCl_{2}; 41 KCl
+         |{ MgCl_{2},2H_{2}O; KCl   }|          |
+         |                           |          |
+  M      | Carnallite; KCl           | 167.5deg | 166.7 MgCl_{2}; 41.7 KCl
+         |                           |          |
+  [N_{1} | MgCl_{2},2H_{2}O          | 186deg   | ca. 241 MgCl_{2}]
+         |                           |          |
+  N      | MgCl_{2},2H_{2}O; KCl     | 186deg   | 240 MgCl_{2}; 63 KCl
+         |                           |          |
+  [O     | KCl                       | 186deg   | 195.6 KCl]
+  -------------------------------------------------------------------------
+
+With the help of the data in the preceding table and of the solid model it
+will be possible to state in any given case what will be the behaviour of a
+system composed of magnesium chloride, potassium chloride and water. One or
+two different cases will be very briefly described; and the reader should
+have no difficulty in working out the behaviour under other conditions with
+the help of the model and the numerical data just given. {289}
+
+In the first place it may be again noted that at a temperature above
+167.5deg (point M) carnallite cannot exist. If, therefore, a solution of
+magnesium and potassium chlorides is evaporated at a temperature above this
+point, the result will be a mixture of potassium chloride and either
+magnesium chloride tetrahydrate or magnesium chloride dihydrate, according
+as the temperature is below or above 176deg. The isothermal curve here
+consists of only two branches.
+
+Further, reference has already been made to the fact that all points of the
+carnallite area correspond to solutions in equilibrium with carnallite, but
+in which the ratio of MgCl_{2} to KCl is greater than in the double salt. A
+solution which is saturated with respect to double salt alone will be
+supersaturated with respect to potassium chloride. At all temperatures,
+therefore, carnallite is decomposed by water with separation of potassium
+chloride; hence all solutions obtained by adding excess of carnallite to
+water will lie on the curve EM. _A pure saturated solution of carnallite
+cannot be obtained._
+
+If an unsaturated solution of the two salts in equimolecular amounts is
+evaporated, potassium chloride will first be deposited, because the plane
+bisecting the right angle formed by the X and Y axes cuts the area for that
+salt. Deposition of potassium chloride will lead to a relative increase in
+the concentration of magnesium chloride in the solution; and on continued
+evaporation a point (on the curve EM) will be reached at which carnallite
+will separate out. So long as the two solid phases are present, the
+composition of the solution must remain unchanged. Since the separation of
+carnallite causes a decrease in the relative concentration of the potassium
+chloride in the solution, the portion of this salt which was deposited at
+the commencement must _redissolve_, and carnallite will be left on
+evaporating to dryness. (_Incongruently saturated solution._)
+
+Although carnallite is decomposed by pure water, it will be possible to
+crystallize it from a solution having a composition represented by any
+point in the carnallite area. Since during the separation of the double
+salt the relative amount of magnesium chloride increases, it is most
+advantageous to {290} commence with a solution the composition of which is
+represented by a point lying just above the curve EM (cf. p. 281).
+
+From the above description of the behaviour of carnallite in solution, the
+processes usually employed for obtaining potassium chloride will be readily
+intelligible.[367]
+
+Ferric Chloride--Hydrogen Chloride--Water.--In the case of another system
+of three components which we shall now describe, the relationships are
+considerably more complicated than in those already discussed. They deserve
+discussion, however, on account of the fact that they exhibit a number of
+new phenomena.
+
+In the system formed by the three components, ferric chloride, hydrogen
+chloride, and water, not only can various compounds of ferric chloride and
+water (p. 152), and of hydrogen chloride and water be formed, each of which
+possesses a definite melting point, but various ternary compounds are also
+known. Thus we have the following solid phases:--
+
+  2FeCl_{3},12H_{2}O   HCl,3H_{2}O   2FeCl_{3},2HCl,12H_{2}O
+  2FeCl_{3},7H_{2}O    HCl,2H_{2}O   2FeCl_{3},2HCl,8H_{2}O
+  2FeCl_{3},5H_{2}O    HCl,H_{2}O    2FeCl_{3},2HCl,4H_{2}O
+  2FeCl_{3},4H_{2}O
+  FeCl_{3}
+
+From this it will be readily understood that the complete study of the
+conditions of temperature and concentration under which solutions can
+exist, either with one solid phase or with two or three solid phases, are
+exceedingly complicated; and, as a matter of fact, only a few of the
+possible equilibria have been investigated. We shall attempt here only a
+brief description of the most important of these.[368]
+
+If we again employ rectangular co-ordinates for the graphic {291}
+representation of the results, we have the two planes XOT and YOT (Fig.
+115): the concentration of ferric chloride being measured along the X-axis,
+the concentration of hydrogen chloride along the Y-axis, and the
+temperature along the T-axis. The curve ABCDEFGHJK is, therefore, the
+solubility curve of ferric chloride in water (p. 152), and the curve
+A'B'C'D'E'F' the solubility curve of hydrogen chloride and its hydrates. B'
+and D' are the melting points of the hydrates HCl,3H_{2}O and HCl,2H_{2}O.
+In the space between these two planes are represented those systems in
+which all three components are present. As already stated, only a few of
+the possible ternary systems have been investigated, and these are
+represented in Fig. 116. The figure shows the model resting on the
+XOT-plane, so that the lower edge represents the solubility curve of ferric
+chloride, the concentration increasing from right to left. The
+concentration of hydrogen chloride is measured upwards, and the temperature
+forwards. The further end of the model represents the isothermal surface
+for -30deg. The surface of the model on the left does not correspond with
+the plane YOT in Fig. 115, but with a parallel plane which cuts the
+concentration axis for ferric chloride at a point representing 65
+gm.-molecules FeCl_{3} in 100 gm.-molecules of water. The upper surface
+corresponds with a plane parallel to the axis XOT, at a distance
+corresponding with the concentration of 50 gm.-molecules HCl in 100
+gm.-molecules of water.
+
+[Illustration: FIG. 115.]
+
+Ternary Systems.--We pass over the binary system FeCl_{3}--H_{2}O, which
+has already been discussed (p. 152), and the similar system HCl--H_{2}O
+(see Fig. 115), and turn to the discussion of some of the ternary systems
+represented by {292} points on the surface of the model between the planes
+XOT and YOT. As in the case of carnallite, a plane represents the
+conditions of concentration of solution and temperature under which a
+ternary solution can be in equilibrium with a _single_ solid phase
+(bivariant systems), a line represents the conditions for the coexistence
+of a solution with two solid phases (univariant systems), and a point the
+conditions for equilibrium with three solid phases (invariant systems).
+
+[Illustration: FIG. 116.]
+
+In the case of a binary system, in which 2FeCl_{3},12H_{2}O is in
+equilibrium with a solution of the same composition, addition of hydrogen
+chloride must evidently lower the temperature at which equilibrium can
+exist; and the same holds, of course, {293} for all other binary solutions
+in equilibrium with this solid phase. In this way we obtain the surface I.,
+which represents the temperatures and concentrations of solutions in which
+2FeCl_{3},12H_{2}O can be in equilibrium with a ternary solution containing
+ferric chloride, hydrogen chloride, and water. This surface is analogous to
+the curved surface K_{1}K_{2}_k__{4}_k__{3} in Fig. 97 (p. 256). Similarly,
+the surfaces II., III., IV., and V. represent the conditions for
+equilibrium between the solid phases 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O;
+2FeCl_{3},4H_{2}O; FeCl_{3} and ternary solutions respectively. The lines
+CL, EM, GN, and IO on the model represent univariant systems in which a
+ternary solution is in equilibrium with two solid phases, viz. with those
+represented by the adjoining fields. These lines correspond with the
+ternary eutectic curves _k__{3}K_{1} and _k__{4}K_{2} in Fig. 97. Besides
+the surfaces already mentioned, there are still three others, VI., VII.,
+and VIII., which also represent the conditions for equilibrium between one
+solid phase and a ternary solution; but in these cases, the solid phase is
+not a binary compound or an anhydrous salt, but a ternary compound
+containing all three components. The solid phases which are in equilibrium
+with the ternary solutions represented by the surfaces VI., VII., and
+VIII., are 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
+2FeCl_{3},2HCl,12H_{2}O respectively.
+
+The model for FeCl_{3}--HCl--H_{2}O exhibits certain other peculiarities
+not found in the case of MgCl_{2}--KCl--H_{2}O. On examining the model more
+closely, it is found that the field of the ternary compound
+2FeCl_{3},2HCl,8H_{2}O (VII.) resembles the surface of a sugar cone, and
+has a projecting point, the end of which corresponds with a higher
+temperature than does any other point of the surface. At the point of
+maximum temperature the composition of the liquid phase is the same as that
+of the solid. This point, therefore, represents the melting point of the
+double salt of the above composition.
+
+The curves representing univariant systems are of two kinds. In the one
+case, the two solid phases present are both binary compounds; or one is a
+binary compound and the other is one of the components. In the other case,
+either one or both solid phases are ternary compounds. Curves belonging
+{294} to the former class (so-called _border curves_) start from binary
+eutectic points, and their course is always towards lower temperatures,
+_e.g._ CL, EM, GN, IO. Curves belonging to the latter class (so-called
+_medial curves_) would, in a triangular diagram, lie entirely within the
+triangle. Such curves are YV, WV, VL, LM, MV, NS, ST, SO, OZ. These curves
+do not always run from higher to lower temperatures, but may even exhibit a
+point of maximum temperature. Such maxima are found, for example, at U
+(Fig. 116), and also on the curves ST and LV.
+
+Finally, whereas all the other ternary univariant curves run in valleys
+between the adjoining surfaces, we find at the point X a similar appearance
+to that found in the case of carnallite, as the univariant curve here rises
+above the surrounding surface. The point X, therefore, does not correspond
+with a eutectic point, but with a transition point. At this point the
+ternary compound 2FeCl_{3},2HCl,12H_{2}O melts with separation of
+2FeCl_{3},12H_{2}O, just as carnallite melts at 168deg with separation of
+potassium chloride.
+
+The Isothermal Curves.--A deeper insight into the behaviour of the system
+FeCl_{3}--HCl--H_{2}O is obtained from a study of the isothermal curves,
+the complete series of which, so far as they have been studied, is given in
+Fig. 117.[369] In this figure the lightly drawn curves represent isothermal
+solubility curves, the particular temperature being printed beside the
+curve.[370] The dark lines give the composition of the univariant systems
+at different temperatures. The point of intersection of a dark with a light
+curve gives the composition of the univariant solution at the temperature
+represented by the light curve; and the point of intersection of two dark
+lines gives the composition of the invariant solution in equilibrium with
+three solid phases. The dotted lines represent metastable systems, and the
+points P, Q, and R represent solutions of {295} the composition of the
+ternary salts, 2FeCl_{3},2HCl,4H_{2}O; 2FeCl_{3},2HCl,8H_{2}O; and
+2FeCl_{3},2HCl,12H_{2}O.
+
+[Illustration: FIG. 117.]
+
+The farther end of the model (Fig. 116) corresponds, as already mentioned,
+to the temperature -30deg, so that the outline evidently represents the
+isothermal curve for that temperature. Fig. 117 does not show this.
+We can, however, follow the isothermal for -20deg, which is the extreme
+curve on the right in Fig. 117. Point A represents the solubility
+of 2FeCl_{3},12H_{2}O in water. If hydrogen chloride is added, the
+concentration of ferric chloride in the solution first decreases
+and then increases, until at point 34 the ternary double salt
+2FeCl_{3},2HCl,12H_{2}O is formed. If the addition of hydrogen chloride is
+continued, the ferric chloride disappears ultimately, and only the ternary
+double salt remains. This salt can coexist with solutions of the
+composition represented by the curve which passes through the points 173,
+174, 175. At the last-mentioned point, the ternary salt with 8H_{2}O is
+formed. The composition of the solutions with which this salt is in
+equilibrium at -20deg is represented by the curve which passes through a
+point of maximal concentration with respect to HCl, and cuts the curve SN
+at the point 112, at which the solution is in equilibrium with the two
+solid phases 2FeCl_{3},4H_{2}O and 2FeCl_{3},2HCl,8H_{2}O. The succeeding
+portion of the isotherm represents the solubility curve at -20deg of
+2FeCl_{3},4H_{2}O, which cuts the dark line OS at point 113, at which the
+solution is in equilibrium with the two solid phases 2FeCl_{3},4H_{2}O and
+2FeCl_{3},2HCl,4H_{2}O. Thereafter comes the solubility curve of the latter
+compound.
+
+The other isothermal curves can be followed in a similar manner. If the
+temperature is raised, the region of existence of the ternary double salts
+becomes smaller and smaller, and at temperatures above 30deg the ternary
+salts with 12H_{2}O and 8H_{2}O are no longer capable of existing. If the
+temperature is raised above 46deg, only the binary compounds of ferric
+chloride and water and the anhydrous salt can exist as solid phases.
+The isothermal curve for 0deg represents the solubility curve for
+2FeCl_{3},12H_{2}O; 2FeCl_{3},7H_{2}O; 2FeCl_{3},5H_{2}O; and
+2FeCl_{3},4H_{2}O. {296}
+
+Finally, in the case of the system FeCl_{3}--HCl--H_{2}O, we find
+_closed_ isothermal curves. Since, as already stated, the salt
+2FeCl_{3},2HCl,8H_{2}O has a definite melting point, the temperature
+of which is therefore higher than that at which this compound is in
+equilibrium with solutions of other composition, it follows that the line
+of intersection of an isothermal plane corresponding with a temperature
+immediately below the melting point of the salt with the cone-shaped
+surface of its region of existence, will form a closed curve. This is shown
+by the isotherm for -4.5deg, which surrounds the point Q, the melting point
+of the ternary salt.
+
+The following table gives some of the numerical data from which the curves
+and the model have been constructed:--
+
+  -------------------------------------------------------------------------
+         |                             |          | Composition of the sol-
+         |                             |          | ution in gm.-mols. salt
+  Point. |         Solid phases.       | Temper-  | to 100 gm.-mols. water.
+         |                             |  ature.  |------------------------
+         |                             |          |   HCl   |  FeCl_{3}
+  -------------------------------------------------------------------------
+  A      | 2FeCl_{3},12H_{2}O          | -20deg   |   --    |   6.56
+         |                             |          |         |
+  C      |{ 2FeCl_{3},12H_{2}O;       }|  27.4deg |   --    |  24.30
+         |{ 2FeCl_{3},7H_{2}O         }|          |         |
+         |                             |          |         |
+  E      |{ 2FeCl_{3},7H_{2}O;        }|  30deg   |   --    |  30.24
+         |{ 2FeCl_{3},5H_{2}O         }|          |         |
+         |                             |          |         |
+  G      |{ 2FeCl_{3},5H_{2}O;        }|  55deg   |   --    |  40.64
+         |{ 2FeCl_{3},4H_{2}O         }|          |         |
+         |                             |          |         |
+  J      | 2FeCl_{3},4H_{2}O; FeCl_{3} |  66deg   |   --    |  58.40
+         |                             |          |         |
+         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
+  L      |{ 2FeCl_{3},7H_{2}O;        }|  -7.5deg |  19.22  |  23.72
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},7H_{2}O;        }|          |         |
+  M      |{ 2FeCl_{3},5H_{2}O;        }|  -7.3deg |  23.08  |  28.55
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},5H_{2}O;        }|          |         |
+  N      |{ 2FeCl_{3},4H_{2}O;        }| -16deg   |  28.40  |  31.89
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
+  S      |{ 2FeCl_{3},2HCl,8H_{2}O;   }| -27.5deg |  32.33  |  34.21
+         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},4H_{2}O;        }|          |         |
+  O      |{ FeCl_{3};                 }|  29deg   |  33.71  |  49.84
+         |{ 2FeCl_{3},2HCl,4H_{2}O    }|          |         |
+         |                             |          |         |
+  U      |{ 2FeCl_{3},7H_{2}O;        }|  -4.5deg |  20.66  |  25.74
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+         |{ 2FeCl_{3},12H_{2}O;       }|          |         |
+  V      |{ 2FeCl_{3},2HCl,12H_{2}O;  }| -13deg   |  22.40  |  18.00
+         |{ 2FeCl_{3},2HCl,8H_{2}O    }|          |         |
+         |                             |          |         |
+  X      |{ 2FeCl_{3},12H_{2}O;       }| -12.5deg |  22.14  |  16.69
+         |{ 2FeCl_{3},2HCl,12H_{2}O   }|          |         |
+         |                             |          |         |
+  Q      | 2FeCl_{3},2HCl,8H_{2}O      |  -3deg (melting point)
+  -------------------------------------------------------------------------
+
+Basic Salts.--Another class of systems in the study of {297} which the
+Phase Rule has performed exceptional service, is that of the basic salts.
+In many cases it is impossible, by the ordinary methods of analysis, to
+decide whether one is dealing with a definite chemical individual or with a
+mixture. The question whether a solid phase is a chemical individual can,
+however, be answered, in most cases, with the help of the principles which
+we have already learnt. Let us consider, for example, the formation of
+basic salts from bismuth nitrate, and water. In this case we can choose as
+components Bi_{2}O_{3}, N_{2}O_{5}, and H_{2}O; since all the systems
+consist of these in varying amounts. If we are dealing with a condition of
+equilibrium at constant temperature between liquid and solid phases, three
+cases can be distinguished,[371] viz.--
+
+1. The solutions in different experiments have the same composition, but
+the composition of the precipitate alters. In this case there must be two
+solid phases.
+
+2. The solutions in different experiments can have varying composition,
+while the composition of the precipitate remains unchanged. In this case
+only one solid phase exists, a definite compound.
+
+3. The composition both of the solution and of the precipitate varies. In
+this case the solid phase is a solid solution or a mixed crystal.
+
+In order, therefore, to decide what is the nature of a precipitate produced
+by the hydrolysis of a normal salt, it is only necessary to ascertain
+whether and how the composition of the precipitate alters with alteration
+in the composition of the solution. If the composition of the solution is
+represented by abscissae, and the composition of the precipitate by
+ordinates, the form of the curves obtained would enable us to answer our
+question; for vertical lines would indicate the presence of two solid
+phases (1st case), horizontal lines the presence of only one solid phase
+(2nd case), and slanting lines the presence of mixed crystals (3rd case).
+This method of representation cannot, however, be carried out in most
+cases. It is, however, {298} generally possible to find one pair or several
+pairs of components, the _relative amounts_ of which in the solution or in
+the precipitate undergo change when, and only when, the composition of the
+solution or of the precipitate changes. Thus, in the case of bismuth,
+nitrate, and water, we can represent the ratio of Bi_{2}O_{3} : N_{2}O_{5}
+in the precipitate as ordinates, and N_{2}O_{5} : H_{2}O in the solution as
+abscissae. A horizontal line then indicates a single solid phase, and a
+vertical line two solid phases. An example of this is given in Fig.
+118.[372]
+
+[Illustration: FIG. 118.]
+
+Bi_{2}O_{3}--N_{2}O_{5}--H_{2}O.--Although various systems have been
+studied in which there is formation of basic salts,[373] we shall content
+ourselves here with the description of some of the conditions for the
+formation of basic salts of bismuth nitrate, and for their equilibrium in
+contact with solutions.[374]
+
+Three normal salts of bismuth oxide and nitric acid are known, viz.
+Bi_{2}O_{3},3N_{2}O_{5},10H_{2}O(S_{10});
+Bi_{2}O_{3},3N_{2}O_{5},4H_{2}O(S_{4}); and
+Bi_{2}O_{3},3N_{2}O_{5},3H_{2}O(S_{3}). Besides these normal salts, there
+are the following basic salts:--
+
+{299}
+
+  Bi_{2}O_{3},N_{2}O_{5},2H_{2}O   (represented by B_{1-1-2})
+  Bi_{2}O_{3},N_{2}O_{5},H_{2}O    (     "      "  B_{1-1-1})
+  6Bi_{2}O_{3},5N_{2}O_{5},9H_{2}O (     "      "  B_{6-5-9})
+  2Bi_{2}O_{3},N_{2}O_{5},H_{2}O   (     "      "  B_{2-1-1})
+
+Probably some others also exist. The problem now is to find the conditions
+under which these different normal and basic salts can be in equilibrium
+with solutions of varying concentration of the three components. Having
+determined the equilibrium conditions for the different salts, it is then
+possible to construct a model similar to that for MgCl_{2}--KCl--H_{2}O or
+for FeCl_{3}--HCl--H_{2}O, from which it will be possible to determine the
+limits of stability of the different salts, and to predict what will occur
+when we bring the salts in contact with solutions of nitric acid of
+different concentrations and at different temperatures.
+
+For our present purpose it is sufficient to pick out only some of the
+equilibria which have been studied, and which are represented in the model
+(Fig. 119). In this case use has been made of the triangular method of
+representation, so that the surface of the model lies within the prism.
+
+[Illustration: FIG. 119.]
+
+This model shows the three surfaces, A, B, and C, which represent the
+conditions for the stable existence of the salts B_{1-1-1}, S_{10}, and
+S_{3} in contact with solution at different {300} temperatures. The front
+surface of the model represents the temperature 9deg, and the farther end
+the temperature 75.5deg. The dotted curve represents the isotherm for
+20deg. The prominences between the surfaces represent, of course, solutions
+which are saturated in respect of two solid phases. Thus, for example,
+_pabc_ represents solutions in equilibrium with B_{1-1-1} and S_{10}; and
+the ridge _qdc_, solutions in equilibrium with S_{10} and S_{3}. The point
+_b_, which lies at 75.5deg, is the point of maximum temperature for S_{10}.
+If the temperature is raised above this point, S_{10} decomposes into the
+basic salt B_{1-1-1} and solution. This point is therefore analogous to the
+point M in the carnallite model, at which this salt decomposes into
+potassium chloride and solution (p. 284); or to the point at which the salt
+2FeCl_{3},2HCl,12H_{2}O decomposes into 2FeCl_{3},12H_{2}O and solution
+(p. 294). The curve _pab_ has been followed to the temperature of 72deg
+(point _c_). The end of the model is incomplete, but it is probable that in
+the neighbourhood of the point _c_ there exists a quintuple point at which
+the basic salt B_{1-2-2} appears. In the neighbourhood of _e_ also there
+probably exists another quintuple point at which S_{4} is formed. These
+systems have, however, not been studied.
+
+The following tables give some of the numerical data:--
+
+  ISOTHERM FOR 20deg.
+
+  ----------------------------------------------------------------------
+                               | Composition of the solution. Gram-mols.
+                               | in 1000 gm.-mols. of water.
+  Solid phase.                 |----------------------------------------
+                               |    Bi_{2}O_{3}  |      N_{2}O_{5}
+  -----------------------------|-----------------|----------------------
+  B_{1-1-1}                    |      10.50      |       38.65
+  --                           |      27.20      |       83.84
+  B_{1-1-1}; S_{10}            |      30.15      |       97.97
+  S_{10}                       |      29.70      |       96.57
+  --                           |      19.65      |       98.76
+  --                           |      10.51      |      162.58
+  --                           |      33.51      |      355.87
+  S_{10}; S_{3}                |      51.00      |      403.0
+  S_{3}                        |      14.35      |      492.0
+  --                           |       7.45      |      592.9
+  ----------------------------------------------------------------------
+
+SYSTEMS IN EQUILIBRIUM WITH B_{1-1-1} AND S_{10} (CURVE _pabc_).
+
+  ------------------------------------------------------------
+                     | Composition of the solution. Gram-mols.
+                     |  in 1000 gm.-mols. of water.
+  Temperature.       |----------------------------------------
+                     |  Bi_{2}O_{3}  |  N_{2}O_{5}
+  -------------------|---------------|------------------------
+  9deg               |    26.7       |     88.2
+  20deg (point _a_)  |    30.15      |     97.97
+  30deg              |    33.6       |    112.3
+  50deg              |    41.8       |    148.4
+  65deg              |    57.21      |    190.8
+  75.5deg (point _b_)|    87.9       |    288.4
+  72deg (point _c_)  |    96.0       |    327.0
+  ------------------------------------------------------------
+
+  SYSTEMS IN EQUILIBRIUM WITH S_{10} AND S_{3} (CURVE _qde_).
+
+  --------------------------------------------------------
+                | Composition of the solution. Gram-mols.
+                |  in 1000 gm.-mols. of water.
+  Temperature.  |-----------------------------------------
+                |  Bi_{2}O_{3}  |  N_{2}O_{5}
+  --------------|---------------|-------------------------
+  11.5deg       |    44.5       |    396
+  20deg         |    51.0       |    405.4
+  50deg         |    66.5       |    444.2
+  65deg         |    80.0       |    454.4
+  --------------------------------------------------------
+
+Basic Mercury Salts.--The Phase Rule has also been applied by A. J.
+Cox[375] in an investigation of the basic salts of mercury, the result of
+which has been to show that, of the salts mentioned in text-books, quite a
+number are incorrectly stated to be chemical compounds or chemical
+individuals (p. 92). The investigation, which was carried out essentially
+in the manner described above, included the salts mentioned in the
+following table; and of the basic salts said to be derived from them, only
+those mentioned really exist. In the following table, the numbers in the
+second column give the minimum values of the concentration of the acid,
+expressed in equivalent normality, necessary for the existence of the {301}
+corresponding salts in contact with solution at the temperature given in
+the third column:--
+
+  -------------------------------------------------------------
+               Salt.             | Normality of  | Temperature.
+                                 |     acid.     |
+  -------------------------------------------------------------
+  HgCrO_{4}                      |     1.41      |    50deg
+  3HgO.CrO_{3}                   | 2.6 x 10^{-4} |    50deg
+                                 |               |
+  Hg(NO_{3})_{2}.H_{2}O          |    18.72      |    25deg
+  3HgO.N_{2}O_{5}                |     0.159     |    25deg
+                                 |               |
+  HgSO_{4}                       |     6.87      |    25deg
+  3HgO.SO_{3}                    | 1.3 x 10^{-3} |    25deg
+                                 |               |
+  HgF_{2}                        |     1.14      |    25deg
+                                 |               |
+  HgNO_{3}.H_{2}O                |     2.95      |    25deg
+  5Hg_{2}O.3N_{2}O_{5}.2H_{2}O   | ca. 0.293     |    25deg
+  2Hg_{2}O.N_{2}O_{5}(?)         |     0.110     |    25deg
+  3Hg_{2}O.N_{2}O_{5}.2H_{2}O(?) | 1.7 x 10^{-3} |    25deg
+                                 |               |
+  Hg_{2}SO_{4}                   | 4.2 x 10^{-3} |    25deg
+  2Hg_{2}O.SO_{3}.H_{2}O         | 5.6 x 10^{-4} |    25deg
+  -------------------------------------------------------------
+
+Mercuric fluoride does not form any basic salt.
+
+Since two succeeding members of a series can coexist only in contact with a
+solution of definite concentration, we can prepare acid solutions of
+definite concentration by bringing an excess of two such salts in contact
+with water.
+
+Indirect Determination of the Composition of the Solid Phase.--It has
+already been shown (p. 228) how the composition of the solid phase in a
+system of two components can be determined without analysis, and we shall
+now describe how this can be done in a system of three components.[376]
+
+We shall assume that we are dealing with the aqueous solution of two salts
+which can give rise to a double salt, in which case we can represent the
+solubility relations in a system of rectangular co-ordinates. In this case
+we should obtain, as before (Fig. 120), the isotherm _adcb_, if we express
+the {302} composition of the solution in gram-molecules of A or of B to 100
+gram-molecules of water.
+
+[Illustration: FIG. 120.]
+
+Let us suppose, now, that the double salt is in equilibrium with the
+solution at a definite temperature, and that the composition of the
+solution is represented by the point e. The greater part of the solution is
+now separated from the solid phase, and the latter, _together with the
+adhering mother liquor_, is analyzed. The composition (expressed, as
+before, in gram-molecules of A and B to 100 gram molecules of water) will
+be represented by a point (_e.g._ _f_) on the line _e_S, where S represents
+the composition of the double salt. That this is so will be evident when
+one considers that the composition of the whole mass must lie between the
+composition of the solution and that of the double salt, no matter what the
+relative amounts of the solid phase and the mother liquor.
+
+If, in a similar manner, we analyze a solution of a different composition
+in equilibrium with the same double salt (not necessarily at the same
+temperature as before), and also the mixture of solid phase and solution,
+we shall obtain two other points, as, for example, _g_ and _h_, and the
+line joining these must likewise pass through S. The method of finding the
+{303} composition of an unknown double salt consists, therefore, in
+finding, in the manner just described, the position of two lines such as
+_ef_ and _gh_. The point of intersection of these lines then gives the
+composition of the double salt.
+
+If the double salt is anhydrous, the point S lies at infinity, and the
+lines _ef_ and _gh_ are parallel to each other.
+
+The same result is arrived at by means of the triangular method of
+representation.[377] If we start with the three components in known
+amounts, and represent the initial composition of the whole by a point in
+the triangle, and then ascertain the final composition of the solution in
+equilibrium with the solid phase at a definite temperature, the line
+joining the points representing the initial and end concentration passes
+through the point representing the composition of the solid phase. If two
+determinations are made with solutions having different initial and final
+concentrations in equilibrium with the same solid phase, then the point of
+intersection of the two lines so obtained gives the composition of the
+solid phase.
+
+       *       *       *       *       *
+
+
+{304}
+
+CHAPTER XVII
+
+ABSENCE OF A LIQUID PHASE
+
+In the preceding chapters dealing with equilibria in three-component
+systems, our attention was directed only to those cases in which liquid
+solutions formed one or more phases. Mention must, however, be made of
+certain systems which contain no liquid phase, and in which only solids and
+gases are in equilibrium. Since, in all cases, there can be but one gas
+phase, four solid phases will be necessary in order to form an invariant
+system. When only three solid phases are present, the system is univariant;
+and when only two solid phases coexist with gas, it is bivariant. If,
+however, we make the restriction that the gas pressure is constant, we
+diminish the variability by one.
+
+On account of their great industrial importance, we shall describe briefly
+some of the systems belonging to this class.
+
+Iron, Carbon Monoxide, Carbon Dioxide.--Some of the most important systems
+of three components in which equilibrium exists between solid and gas
+phases are those formed by the three components--iron, carbon monoxide, and
+carbon dioxide--and they are of importance especially for the study of the
+processes occurring in the blast furnace.
+
+If carbon monoxide is passed over reduced iron powder at a temperature of
+about 600deg, the iron is oxidized and the carbon monoxide reduced with
+separation of carbon in accordance with the equation
+
+  Fe + CO = FeO + C
+
+This reaction is succeeded by the two reactions
+
+  FeO + CO = Fe + CO_{2}
+    CO_{2} + C = 2CO
+
+{305}
+
+[Illustration: FIG. 121.]
+
+The former of these reactions is not complete, but leads to a definite
+equilibrium. The result of the different reactions is therefore an
+equilibrium between the three solid phases, carbon, iron, and ferrous
+oxide, and the gas phase consisting of carbon monoxide and dioxide. We have
+here four phases; and if the total pressure is maintained constant,
+equilibrium can occur only at a definite temperature.
+
+Since, under certain conditions, we can also have the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+{306} a second series of equilibria can be obtained of a character similar
+to the former. These various equilibria have been investigated by Baur and
+Glaessner,[378] and the following is a short account of the results of
+their work.
+
+Mixtures of the solid phases in equilibrium with carbon monoxide and
+dioxide were heated in a porcelain tube at a definite temperature until
+equilibrium was produced, and the gas was then pumped off and analyzed. The
+results which were obtained are given in the following tables, and
+represented graphically in Fig. 121.
+
+  SOLID PHASES: Fe_{3}O_{4}; FeO.
+
+  -----------------------------------------------------------------
+      |             | Duration of    |              | Percentage of
+  No. | Tube filled | the experiment | Temperature. |
+      |    with     | in hours.      |              | CO_{2} | CO
+  -----------------------------------------------------------------
+    1 |    CO       |       14       |    600deg    |  59.3  | 40.7
+    2 |    CO       |       15       |    590deg    |  54.7  | 45.3
+    3 |    CO_{2}   |       16       |    590deg    |  64.6  | 35.4
+    4 |    CO       |       24       |    590deg    |  58.4  | 41.6
+    5 |    CO       |       22       |    730deg    |  67.7  | 32.3
+    6 |    CO_{2}   |       22       |    730deg    |  86.1  | 31.9
+    7 |    CO       |       22       |    750deg    |  68.4  | 31.6
+    8 |    CO_{2}   |       22       |    610deg    |  64.9  | 35.1
+    9 |    CO       |       23       |    420deg    |  56.0  | 44.0
+   10 |    CO       |       47       |    350deg    |  65.6  | 34.4
+   11 |    CO_{2}   |       46       |    350deg    |  72.8  | 27.2
+   12 |    CO       |       53       |    350deg    |  64.0  | 36.0
+   13 |    CO       |       18       |    570deg    |  53.4  | 46.6
+   14 |    CO       |       19       |    680deg    |  60.5  | 39.5
+   15 |    CO_{2}   |       24       |    540deg    |  55.5  | 44.5
+   16 |    CO       |       21       |    630deg    |  57.5  | 42.5
+   17 |    CO_{2}   |       17       |    690deg    |  65.5  | 34.5
+   18 |    CO_{2}   |       17       |    670deg    |  67.0  | 33.0
+   19 |    CO_{2}   |       24       |    410deg    |  58.5  | 41.5
+   20 |    CO       |       24       |    490deg    |  51.7  | 48.8
+   21 |    CO_{2}   |       23       |    590deg    |  54.4  | 45.6
+   22 |    CO_{2}   |        4       |    950deg    |  77.0  | 23.0
+   23 |    CO_{2}   |       15       |    850deg    |  73.4  | 26.6
+   24 |    CO       |        8       |    800deg    |  71.2  | 28.8
+   25 |    CO_{2}   |       24       |    540deg    |  56.7  | 43.3
+  -----------------------------------------------------------------
+
+{307}
+
+  SOLID PHASES: FeO; Fe.
+
+  ------+-------------+-------------+--------------+---------------
+        |             | Duration of |              | Percentage of
+   No.  | Tube filled | experiment  | Temperature. |
+        | with        | in hours.   |              | CO_{2} |  CO
+  ------+-------------+-------------+--------------+--------+------
+  I.    |    CO       |      15     |    800deg    |  35.2  | 64.8
+  II.   |    CO       |      18     |    530deg    |  29.1  | 70.9
+  III.  |    CO       |      13     |    880deg    |  30.2  | 69.6
+  IV.   |    CO_{2}   |      24     |    870deg    |  32.3  | 67.7
+  V.    |    CO       |      18     |    760deg    |  36.9  | 63.1
+  VI.   |    CO_{2}   |      16     |    820deg    |  34.7  | 65.3
+  VII.  |    CO_{2}   |      18     |    730deg    |  41.1  | 58.9
+  VIII. |    CO       |      18     |    630deg    |  34.9  | 65.1
+  IX.   |    CO_{2}   |      17     |    630deg    |  61.6  | 58.4
+  X.    |    CO       |      18     |    540deg    |  25.0  | 75.0
+  XI.   |    CO_{2}   |      25     |    540deg    |  36.5  | 63.5
+  ------+-------------+-------------+--------------+--------+------
+
+As is evident from the above tables and from the curves in Fig. 121, the
+curve of equilibrium in the case of the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+exhibits a maximum for the ratio CO : CO_{2}, at 490deg, while, for the
+reaction
+
+  FeO + CO = Fe + CO_{2}
+
+this ratio has a minimum value at 680deg. From these curves can be derived
+the conditions under which the different solid phases can exist in contact
+with gas. Thus, for example, at a temperature of 690deg, FeO and
+Fe_{3}O_{4} can coexist with a mixture of 65.5 per cent. of CO_{2} and 34.5
+per cent. of CO. If the partial pressure of CO_{2} is increased, there
+occurs the reaction
+
+  3FeO + CO_{2} = Fe_{3}O_{4} + CO
+
+and if carbon dioxide is added in sufficient amount, the ferrous oxide
+finally disappears completely. If, on the other hand, the partial pressure
+of CO is increased, there occurs the reaction
+
+  Fe_{3}O_{4} + CO = 3FeO + CO_{2}
+
+and all the ferric oxide can be made to disappear. We see, therefore, that
+Fe_{3}O_{4} can only exist at temperatures and in {308} contact with
+mixtures of carbon monoxide and dioxide, represented by the area which lies
+below the under curve in Fig. 121. Similarly, the region of existence of
+FeO is that represented by the area between the two curves; while metallic
+iron can exist under the conditions of temperature and composition of gas
+phase represented by the area above the upper curve in Fig. 121. If,
+therefore, ferric oxide or metallic iron is heated for a sufficiently long
+time at temperatures above 700deg (to the right of the dotted line; _vide
+infra_), complete transformation to ferrous oxide finally occurs.
+
+In another series of equilibria which can be obtained, carbon is one of the
+solid phases. In Fig. 121 the equilibria between carbon, carbon monoxide,
+and carbon dioxide under pressures of one and of a quarter atmosphere, are
+represented by dotted lines.[379]
+
+If we consider only the dotted line on the right, representing the
+equilibria under atmospheric pressure, we see that the points in which the
+dotted line cuts the other two curves must represent systems in which
+carbon monoxide and carbon dioxide are in equilibrium with FeO +
+Fe_{3}O_{4} + C, on the one hand, and with Fe + FeO + C on the other. These
+systems can only exist at one definite temperature, if we make the
+restriction that the pressure is maintained constant (atmospheric
+pressure). Starting, therefore, with the equilibrium FeO + Fe_{3}O_{4} + CO
++ CO_{2} at a temperature of about 670deg, and then add carbon to the
+system, the reaction
+
+  C + CO_{2} = 2CO
+
+will occur, because the concentration of CO_{2} is greater than what
+corresponds with the system FeO + Fe_{3}O_{4} + C in equilibrium with
+carbon monoxide and dioxide. In consequence of this reaction, the
+equilibrium between FeO + Fe_{3}O_{4} and the gas phase is disturbed, and
+the change in the composition of the gas phase is opposed by the reaction
+Fe_{3}O_{4} + CO = 3FeO + CO_{2}, which continues until either all the
+carbon {309} or all the ferric oxide is used up. If the ferric oxide first
+disappears, the equilibrium corresponds with a point on the dotted line in
+the middle area of Fig. 121, which represents equilibria between FeO + C as
+solid phases, and a mixture of carbon monoxide and dioxide as gas phase. If
+the temperature is higher than 685deg, at which temperature the curve for
+C--CO--CO_{2} cuts that for Fe--FeO--CO--CO_{2}; then, when all the ferric
+oxide has disappeared, the concentration of CO_{2} is still too great for
+the coexistence of FeO and C. Consequently, there occurs the reaction C +
+CO_{2} = 2CO, and the composition of the gas phase alters until a point on
+the upper curve is reached. A further increase in the concentration of CO
+is opposed by the reaction FeO + CO = Fe + CO_{2}, and the pressure remains
+constant until all the ferrous oxide is reduced and only iron and carbon
+remain in equilibrium with gas. If the quantities of the substances have
+been rightly chosen, we ultimately reach a point on the dotted curve in the
+upper part of Fig. 121.
+
+Fig. 121 shows us, also, what are the conditions under which the reduction
+of ferric to ferrous oxide by carbon can occur. Let us suppose, for
+example, that we start with a mixture of carbon monoxide and dioxide at
+about 600deg (the lowest point on the dotted line), and maintain the total
+pressure constant and equal to one atmosphere. If the temperature is
+increased, the concentration of the carbon dioxide will diminish, owing to
+the reaction C + CO_{2} = 2CO, but the ferric oxide will undergo no change
+until the temperature reaches 647deg, the point of intersection of the
+dotted curve with the curve for FeO and Fe_{3}O_{4}. At this point further
+increase in the concentration of carbon monoxide is opposed by the
+reduction of ferric oxide in accordance with the equation Fe_{3}O_{4} + CO
+= 3FeO + CO_{2}. The pressure, therefore, remains constant until all the
+ferric oxide has disappeared. If the temperature is still further raised,
+we again obtain a univariant system, FeO + C, in equilibrium with gas
+(univariant because the total pressure is constant); and if the temperature
+is raised the composition of the gas must undergo change. This is effected
+by the reaction C + CO_{2} = 2CO. When the {310} temperature rises to
+685deg, at which the dotted curve cuts the curve for Fe--FeO, further
+change is prevented by the reaction FeO + CO = Fe + CO_{2}. When all the
+ferrous oxide is used up, we obtain the system Fe + C in equilibrium with
+gas. If the temperature is now raised, the composition of the gas undergoes
+change, as shown by the dotted line. The two temperatures, 647deg and
+685deg, give, evidently, the limits within which ferric or ferrous oxide
+can be reduced directly by carbon.
+
+It is further evident that at any temperature to the right of the dotted
+line, carbon is unstable in presence of iron or its oxides; while at
+temperatures lower than those represented by the dotted line, it is stable.
+In the blast furnace, therefore, separation of carbon can occur only at
+lower temperatures, and the carbon must disappear on raising the
+temperature.
+
+Finally, it may be remarked that the equilibrium curves show that ferrous
+oxide is most easily reduced at 680deg, since the concentration of the
+carbon monoxide required at this temperature is a minimum. On the other
+hand, ferric oxide is reduced with greatest difficulty at 490deg, since at
+this temperature the requisite concentration of carbon monoxide is a
+maximum.
+
+Other equilibria between solid and gas phases are: Equilibrium between
+iron, ferric oxide, water vapour, and hydrogen,[380] and the equilibria
+between carbon, carbon monoxide, carbon dioxide, water vapour, and
+hydrogen,[381] which is of importance for the manufacture of water gas.
+
+       *       *       *       *       *
+
+
+{311}
+
+CHAPTER XVIII
+
+SYSTEMS OF FOUR COMPONENTS
+
+In the systems which have so far been studied, we have met with cases where
+two or three components could enter into combination; but in no case did we
+find double decomposition occurring. The reason of this is that in the
+systems previously studied, in which double decomposition might have been
+possible, namely in those systems in which two salts acted as components,
+the restriction was imposed that either the basic or the acid constituent
+of these salts must be the same; a restriction imposed, indeed, for the
+very purpose of excluding double decomposition. Now, however, we shall
+allow this restriction to fall, thereby extending the range of study.
+
+Hitherto, in connection with four-component systems, the attention has been
+directed solely to the study of aqueous solutions of salts, and more
+especially of the salts which occur in sea-water, _i.e._ chiefly, the
+sulphates and chlorides of magnesium, potassium, and sodium. The importance
+of these investigations will be recognized when one recollects that by the
+evaporation of sea-water there have been formed the enormous salt-beds at
+Stassfurt, which constitute at present the chief source of the sulphates
+and chlorides of magnesium and potassium. The investigations, therefore,
+are not only of great geological interest as tending to elucidate the
+conditions under which these salt-beds have been formed, but are of no less
+importance for the industrial working of the deposits.
+
+It is, however, not the intention to enter here into any detailed
+description of the different systems which have so far been studied, and of
+the sometimes very complex relationships {312} met with, but merely to
+refer briefly to some points of more general import in connection with
+these systems.[382]
+
+Reciprocal Salt-Pairs. Choice of Components.--When two salts undergo double
+decomposition, the interaction can be expressed by an equation such as
+
+  NH_{4}Cl + NaNO_{3} = NaCl + NH_{4}NO_{3}
+
+Since one pair of salts--NaCl + NH_{4}NO_{3}--is formed from the other
+pair--NH_{4}Cl + NaNO_{3}--by double decomposition, the two pairs of salts
+are known as _reciprocal salt-pairs_.[383] It is with systems in which the
+component salts form reciprocal salt-pairs that we have to deal here.
+
+It must be noted, however, that the four salts formed by two reciprocal
+salt-pairs do not constitute a system of four, but only of _three_
+components. This will be understood if it is recalled that only so many
+constituents are taken as components as are necessary to _express_ the
+composition of all the phases present (p. 12). It will be seen, now, that
+the composition of each of the four salts which can be present together can
+be expressed in terms of three of them. Thus, for example, in the case of
+NH_{4}Cl, NaNO_{3}, NH_{4}NO_{3}, NaCl, we can express the composition of
+NH_{4}Cl by NH_{4}NO_{3} + NaCl - NaNO_{3}; or of NaNO_{3} by NH_{4}NO_{3}
++ NaCl - NH_{4}Cl. In all these cases it will be seen that negative
+quantities of one of the components must be employed; but that we have seen
+to be quite permissible (p. 12). The number of components is, therefore,
+three; but any three of the four salts can be chosen.
+
+Since, then, two reciprocal salt-pairs constitute only three {313}
+components or independently variable constituents, another component is
+necessary in order to obtain a four-component system. As such, we shall
+choose water.
+
+Transition Point.--In the case of the formation of double salts from two
+single salts, we saw that there was a point--the _quintuple point_--at
+which five phases could coexist. This point we also saw to be a transition
+point, on one side of which the double salt, on the other side the two
+single salts in contact with solution, were found to be the stable system.
+A similar behaviour is found in the case of reciprocal salt-pairs. The
+four-component system, two reciprocal salt-pairs and water, can give rise
+to an invariant system in which the six phases, four salts, solution,
+vapour, can coexist; the temperature at which this is possible constitutes
+a _sextuple point_. Now, this sextuple point is also a transition point, on
+the one side of which the one salt-pair, on the other side the reciprocal
+salt-pair, is stable in contact with solution.
+
+The sextuple point is the point of intersection of the curves of six
+univariant systems, viz. four solubility curves with three solid phases
+each, a vapour-pressure curve for the system: two reciprocal
+salt-pairs--vapour; and a transition curve for the condensed system: two
+reciprocal salt-pairs--solution. If we omit the vapour phase and work under
+atmospheric pressure (in open vessels), we find that the transition point
+is the point of intersection of four solubility curves.
+
+Just as in the case of three-component systems we saw that the presence of
+one of the single salts along with the double salt was necessary in order
+to give a univariant system, so in the four-component systems the presence
+of a third salt is necessary as solid phase along with one of the
+salt-pairs. In the case of the reciprocal salt-pairs mentioned above, the
+transition point would be the point of intersection of the solubility
+curves of the systems with the following groups of salts as solid phases:
+Below the transition point: NH_{4}Cl + NaNO_{3} + NaCl; NH_{4}Cl + NaNO_{3}
++ NH_{4}NO_{3}; above the transition point: NaCl + NH_{4}NO_{3} + NaNO_{3};
+NaCl + NH_{4}NO_{3} + NH_{4}Cl. From this we see that the two salts
+NH_{4}Cl and NaNO_{3} would be able to exist together with solution below
+the transition point, but not above it. This transition point has not been
+determined. {314}
+
+Formation of Double Salts.--In all cases of four-component systems so far
+studied, the transition points have not been points at which one salt-pair
+passed into its reciprocal, but at which a double salt was formed. Thus, at
+4.4deg Glauber's salt and potassium chloride form glaserite and sodium
+chloride, according to the equation
+
+  2Na_{2}SO_{4},10H_{2}O + 3KCl = K_{3}Na(SO_{4})_{2} + 3NaCl + 20H_{2}O
+
+Above the transition point, therefore, there would be K_{3}Na(SO_{4})_{2},
+NaCl and KCl; and it may be considered that at a higher temperature the
+double salt would interact with the potassium chloride according to the
+equation
+
+  K_{3}Na(SO_{4})_{2} + KCl = 2K_{2}SO_{4} + NaCl
+
+thus giving the reciprocal of the original salt-pair. This point has,
+however, not been experimentally realized.[384]
+
+Transition Interval.--A double salt, we learned (p. 277), when brought in
+contact with water at the transition point undergoes partial decomposition
+with separation of one of the constituent salts; and only after a certain
+range of temperature (transition interval) has been passed, can a pure
+saturated solution be obtained. A similar behaviour is also found in the
+case of reciprocal salt-pairs. If one of the salt-pairs is brought in
+contact with water at the transition point, interaction will occur and one
+of the salts of the reciprocal salt-pair will be deposited; and this will
+be the case throughout a certain range of temperature, after which it will
+be possible to prepare a solution saturated only for the one salt-pair. In
+the case of ammonium chloride and sodium nitrate the lower limit of the
+transition interval is 5.5deg, so that above this temperature and up to
+that of the transition point (unknown), ammonium chloride and sodium
+nitrate in contact with water would give rise to a third salt by double
+decomposition, in this case to sodium chloride.[385]
+
+{315}
+
+Graphic Representation.--For the graphic representation of systems of four
+components, four axes may be chosen intersecting at a point like the edges
+of a regular octahedron (Fig. 122).[386] Along these different axes the
+equivalent molecular amounts of the different salts are measured.
+
+[Illustration: FIG. 122.]
+
+[Illustration: FIG. 123.]
+
+To represent a given system consisting of _x_B, _y_C, and _z_D in a given
+amount of water (where B, C, and D represent equivalent molecular amounts
+of the salts), measure off on OB and OC lengths equal to _x_ and _y_
+respectively. The point of intersection _a_ (Fig. 122) represents a
+solution containing _x_B and _y_C (_ab_ = _x_; _ac_ = _y_). From _a_ a line
+_a_P is drawn parallel to OD and equal to _z_. P then represents the
+solution of the above composition.
+
+It is usual, however, not to employ the three-dimensional figure, but its
+horizontal and vertical projections. Fig. 122, if projected on the base of
+the octahedron, would yield a diagram such as is shown in Fig. 123. The
+projection of the edges of the octahedron form two axes at right angles and
+give rise to four quadrants similar to those employed for the
+representation of ternary solutions (p. 273). Here, the point _a_
+represents a ternary solution saturated with respect to B and C; and _a_P,
+quaternary solutions in equilibrium with the same two salts as solid
+phases. Such a diagram represents the conditions of equilibrium only for
+one definite temperature, and corresponds, therefore, to the isothermal
+diagrams for ternary systems (p. 273). In such a diagram, since the
+temperature and {316} pressure are constant (vessels open to the air), a
+surface will represent a solution in equilibrium with only one solid phase;
+a line, a solution with two solid phases, and a point, one in equilibrium
+with three solid phases.
+
+[Illustration: FIG. 124.]
+
+Example.--As an example of the complete isothermal diagram, there may be
+given one representing the equilibria in the system composed of water and
+the reciprocal salt-pair sodium sulphate--potassium chloride for the
+temperature 0deg (Fig. 124).[387] The amounts of the different salts are
+measured along the four axes, and the composition of the solution is {317}
+expressed in equivalent gram-molecules per 1000 gram-molecules of
+water.[388]
+
+The outline of this figure represents four ternary solutions in which the
+component salts have a common acid or basic constituent; viz. sodium
+chloride--sodium sulphate, sodium sulphate--potassium sulphate, potassium
+sulphate--potassium chloride, potassium chloride--sodium chloride. These
+four sets of curves are therefore similar to those discussed in the
+previous chapter. In the case of sodium and potassium sulphate, a double
+salt, _glaserite_ [K_{3}Na(SO_{4})_{2}] is formed. Whether glaserite is
+really a definite compound or not is still a matter of doubt, since
+isomorphic mixtures of Na_{2}SO_{4} and K_{2}SO_{4} have been obtained.
+According to van't Hoff and Barscholl,[389] glaserite is an isomorphous
+mixture; but Gossner[390] considers it to be a definite compound having the
+formula K_{3}Na(SO_{4})_{2}. Points VIII. and IX. represent solutions
+saturated with respect to glaserite and sodium sulphate, and glaserite and
+potassium sulphate respectively.
+
+The lines which pass inwards from these boundary curves represent solutions
+containing three salts, but in contact with only two solid phases; and the
+points where three lines meet, or where three fields meet, represent
+solutions in equilibrium with three solid phases; with the phases, namely,
+belonging to the three concurrent fields.
+
+If it is desired to represent a solution containing the salts say in the
+proportions, 51Na_{2}Cl_{2}, 9.5K_{2}Cl_{2}, 3.5K_{2}SO_{4}, the difficulty
+is met with that two of the salts, sodium chloride and potassium sulphate,
+lie on opposite axes. To overcome this difficulty the difference 51 - 3.5 =
+47.5 is taken and measured off along the sodium chloride axis; and the
+solution is therefore represented by the point 47.5Na_{2}Cl_{2},
+9.5K_{2}Cl_{2}. In order, therefore, to find the amount of potassium
+sulphate present {318} from such a diagram, it is necessary to know the
+total number of salt molecules in the solution. When this is known, it is
+only necessary to subtract from it the sum of the molecules of sodium and
+potassium chloride, and the result is equal to twice the number of
+potassium sulphate molecules. Thus, in the above example, the total number
+of salt molecules is 64. The number of molecules of sodium and potassium
+chloride is 57; 64 - 57 = 7, and therefore the number of potassium sulphate
+molecules is 3.5.
+
+Another method of representation employed is to indicate the amounts of
+only two of the salts in a plane diagram, and to measure off the total
+number of molecules along a vertical axis. In this way a solid model is
+obtained.
+
+The numerical data from which Fig. 124 was constructed are contained in the
+following table, which gives the composition of the different solutions at
+0deg:--[391]
+
+  ----------------------------------------
+         |                               |
+         |                               |
+  Point. |       Solid phases.           |
+         |                               |
+  ----------------------------------------
+      I. | NaCl                          |
+         |                               |
+     II. | KCl                           |
+         |                               |
+    III. | Na_{2}SO_{4},10H_{2}O         |
+         |                               |
+     IV. | K_{2}SO_{4}                   |
+         |                               |
+      V. | NaCl; KCl                     |
+         |                               |
+     VI. | NaCl; Na_{2}SO_{4},10H_{2}O   |
+         |                               |
+    VII. | KCl; K_{2}SO_{4}              |
+         |                               |
+   VIII. |{ Glaserite;                  }|
+         |{ Na_{2}SO_{4},10H_{2}O       }|
+         |                               |
+     IX. |  Glaserite; K_{2}SO_{4}       |
+         |                               |
+      X. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
+         |{ NaCl                        }|
+         |                               |
+     XI. |{ Na_{2}SO_{4},10H_{2}O; KCl; }|
+         |{ glaserite                   }|
+         |                               |
+    XII. |  K_{2}SO_{4}; KCl; glaserite  |
+  ----------------------------------------
+  [Transcriber's note: table continued below...]
+  -------------------------------------------------------------------------
+   Composition of solution in gram-mols.                       | Total
+   per 1000 gram-mols. water.                                  | number
+  -------------------------------------------------------------| of salt
+   Na_{2}Cl_{2}. | K_{2}Cl_{2}. | Na_{2}SO_{4}. | K_{2}SO_{4}. | molecules.
+  -------------------------------------------------------------------------
+        55       |      --      |      --       |      --      |   55
+                 |              |               |              |
+         --      |     34.5     |      --       |      --      |   34.5
+                 |              |               |              |
+         --      |      --      |      6        |      --      |    6
+                 |              |               |              |
+         --      |      --      |      --       |      9       |    9
+                 |              |               |              |
+        46.5     |     12.5     |      --       |      --      |   59
+                 |              |               |              |
+        47.5     |      --      |      8        |      --      |   55.5
+                 |              |               |              |
+         --      |     34.5     |      --       |      1       |   35.5
+                 |              |               |              |
+         --      |      --      |     10        |     10       |   20
+                 |              |               |              |
+                 |              |               |              |
+         --      |      --      |      7.5      |     10       |   17.5
+                 |              |               |              |
+        51       |      9.5     |      --       |      3.5     |   64
+                 |              |               |              |
+                 |              |               |              |
+        40.5     |     13       |      --       |      3.5     |   57
+                 |              |               |              |
+                 |              |               |              |
+        18       |     23       |      --       |      3       |   44
+  -------------------------------------------------------------------------
+
+From the aspect of these diagrams the conditions under which the salts can
+coexist can be read at a glance. Thus, {319} for example, Fig. 124 shows
+that at 0deg Glauber's salt and potassium chloride can exist together with
+solution; namely, in contact with solutions having the composition X--XI.
+This temperature must therefore be below the transition point of this
+salt-pair (p. 314). On raising the temperature to 4.4deg, it is found that
+the curve VIII.--XI. moves so that the point XI. coincides with point X. At
+this point, therefore, there will be _four_ concurrent fields, viz.
+Glauber's salt, potassium chloride, glaserite, and sodium chloride. But
+these four salts can coexist with solution only at the transition point; so
+that 4.4deg is the transition temperature of the salt-pair: Glauber's
+salt--potassium chloride. At higher temperatures the line VIII.--XI. moves
+still further to the left, so that the field for Glauber's salt becomes
+entirely separated from the field for potassium chloride. This shows that
+at temperatures above the transition point the salt-pair Glauber's
+salt--potassium chloride cannot coexist in presence of solution.
+
+[Illustration: FIG. 125.]
+
+If it is only desired to indicate the mutual relationships of the different
+components and the conditions for their coexistence (_paragenesis_), a
+simpler diagram than Fig. 124 can be employed. Thus if the boundary curves
+of Fig. 124 are so drawn that they cut one another at right angles, a
+figure such as Fig. 125 is obtained, the Roman numerals here corresponding
+with those in Fig. 124.
+
+Ammonia-Soda Process.--One of the most important applications of the Phase
+Rule to systems of four components with reciprocal salt-pairs has recently
+been made by Fedotieff[392] in his investigations of the conditions for the
+formation of sodium carbonate by the so-called ammonia-soda (Solvay) {320}
+process.[393] This process consists, as is well known, in passing carbon
+dioxide through a solution of common salt saturated with ammonia.
+
+Whatever differences of detail there may be in the process as carried out
+in different manufactories, the reaction which forms the basis of the
+process is that represented by the equation
+
+  NaCl + NH_{4}HCO_{3} = NaHCO_{3} + NH_{4}Cl
+
+We are dealing here, therefore, with reciprocal salt-pairs, the behaviour
+of which has just been discussed in the preceding pages. The present case
+is, however, simpler than that of the salt-pair Na_{2}SO_{4}.10H_{2}O +
+KCl, inasmuch as under the conditions of experiment neither hydrates nor
+double salts are formed. Since the study of the reaction is rendered more
+difficult on account of the fact that ammonium bicarbonate in solution,
+when under atmospheric pressure, undergoes decomposition at temperatures
+above 15deg, this temperature was the one chosen for the detailed
+investigation of the conditions of equilibrium. Since, further, it has been
+shown by Bodlaender[394] that the bicarbonates possess a definite
+solubility only when the pressure of carbon dioxide in the solution has a
+definite value, the measurements were carried out in solutions saturated
+with this gas. This, however, does not constitute another component,
+because we have made the restriction that the sum of the partial pressures
+of carbon dioxide and water vapour is equal to 1 atmosphere. The
+concentration of the carbon dioxide is, therefore, not independently
+variable (p. 10).
+
+[Illustration: FIG. 126.]
+
+In order to obtain the data necessary for a discussion of the conditions of
+soda formation by the ammonia-soda process, solubility determinations with
+the four salts, NaCl, NH_{4}Cl, NH_{4}HCO_{3}, and NaHCO_{3} were made,
+first with the single salts and then {321} with the salts in pairs. The
+results obtained are represented graphically in Fig. 126, which is an
+isothermal diagram similar to that given by Fig. 124. The points I., II.,
+III., IV., represent the composition of solutions in equilibrium with two
+solid salts. We have, however, seen (p. 314) that the transition point,
+when the experiment is carried out under constant pressure (atmospheric
+pressure), is the point of intersection of four solubility curves, each of
+which represents the composition of solutions in equilibrium with three
+salts, viz. one of the reciprocal salt-pairs along with a third salt.
+Since, now, it was found that the stable salt-pair at temperatures between
+0deg and 30deg is sodium bicarbonate and ammonium chloride, determinations
+were made of the composition of solutions in equilibrium with NaHCO_{3} +
+NH_{4}Cl + NH_{4}HCO_{3} and with NaHCO_{3} + NH_{4}Cl + NaCl as solid
+phases. Under the {322} conditions of experiment (temperature = 15deg)
+sodium chloride and ammonium bicarbonate cannot coexist in contact with
+solution. These determinations gave the data necessary for the construction
+of the complete isothermal diagram (Fig. 127). The most important of these
+data are given in the following table (temperature, 15deg):--
+
+  -------------------------------------------------------------------------
+         |                  | Composition of the solution in gram-molecules
+         |                  | to 1000 gram-molecules
+  Point. |  Solid phases.   | of water.
+         |                  |----------------------------------------------
+         |                  | NaHCO_{3} | NaCl | NH_{4}HCO_{3} | NH_{4}Cl
+  -------------------------------------------------------------------------
+   --    | NaHCO_{3}        |    1.08   |  --  |       --      |     --
+   --    | NaCl             |     --    | 6.12 |       --      |     --
+   --    | NH_{4}HCO_{3}    |     --    |  --  |      2.36     |     --
+   --    | NH_{4}Cl         |     --    |  --  |       --      |    6.64
+   I.    | NaHCO_{3}; NaCl  |    0.12   | 6.06 |       --      |     --
+   II.   | NaCl; NH_{4}Cl   |     --    | 4.55 |       --      |    3.72
+   III.  | NH_{4}Cl;        |     --    |  --  |      0.81     |    6.40
+         | NH_{4}HCO_{3}    |           |      |               |
+   IV.   | NaHCO_{3};       |    0.71   |  --  |      2.16     |     --
+         | NH_{4}HCO_{3}    |           |      |               |
+   P_{1} | NaHCO_{3};       |    0.93   | 0.51 |       --      |    6.28
+         | NH_{4}HCO_{3};   |           |      |               |
+         | NH_{4}Cl         |           |      |               |
+   P_{2} | NaHCO_{3};       |    0.18   | 4.44 |       --      |    3.73
+         | NaCl; NH_{4}Cl   |           |      |               |
+  -------------------------------------------------------------------------
+
+With reference to the solution represented by the point P_{1}, it may be
+remarked that it is an incongruently saturated solution (p. 279). If sodium
+chloride is added to this solution, the composition of the latter undergoes
+change; and if a sufficient amount of the salt is added, the solution P_{2}
+is obtained.
+
+Turning now to the practical application of the data so obtained, consider
+first what is the influence of concentration on the yield of soda. Since
+the reaction consists essentially in a double decomposition between sodium
+chloride and ammonium bicarbonate, then, after the deposition of the sodium
+bicarbonate, we obtain a solution containing sodium chloride, ammonium
+chloride, and sodium bicarbonate. In order to ascertain to what extent the
+sodium chloride has been converted into solid sodium bicarbonate, it is
+necessary to examine the composition of the solution which is obtained
+{323} with definite amounts of sodium chloride and ammonium bicarbonate.
+
+[Illustration: FIG. 127.]
+
+Consider, in the first place, the solutions represented by the curve
+P_{2}P_{1}. With the help of this curve we can state the conditions under
+which a solution, saturated for ammonium chloride, is obtained, after
+deposition of sodium bicarbonate. In the following table the composition of
+the solutions is given which are obtained with different initial amounts of
+sodium chloride and ammonium bicarbonate. The last two columns give the
+percentage amount of the sodium used, which is deposited as solid sodium
+bicarbonate (U_{Na}); and likewise the percentage amount of ammonium
+bicarbonate which is usefully converted into sodium bicarbonate, that is to
+say, the amount of the radical HCO_{3} deposited (U_{NH_{4}}):-- {324}
+
+  ------+---------------------+
+        |Initial composition  |
+        |of the solutions:    |
+        |grams of salt to 1000|
+  Point.|grams of water.      |
+        +------+--------------+
+        | NaCl | NH_{4}HCO_{3}|
+  ------+------+--------------+
+  P_{2} | 479  |     295      |
+  --    | 448  |     360      |
+  --    | 417  |     431      |
+  P_{1} | 397  |     496      |
+  ------+------+--------------+
+  [Transcriber's note: table continued below...]
+  +----------------------------------+---------+----------
+  |                                  |         |
+  |Composition of solutions obtained:|         |
+  |gram-equivalents per 1000 grams   |U_{Na}   |U_{NH_{4}}
+  |of water.                         |per cent.|per cent.
+  +----------+------+------+---------+         |
+  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |         |
+  +----------+------+------+---------+---------+----------
+  |   0.18   | 8.17 | 4.62 |  3.73   |  43.4   |   95.1
+  |   0.31   | 7.65 | 3.39 |  4.56   |  55.7   |   93.4
+  |   0.51   | 7.13 | 2.19 |  5.45   |  69.2   |   90.5
+  |   0.92   | 6.79 | 1.44 |  6.28   |  78.8   |   85.1
+  +----------+------+------+---------+---------+----------
+
+This table shows that the greater the excess of sodium chloride, the
+greater is the percentage utilization of ammonia (Point P_{2}); and the
+more the amount of sodium chloride decreases, the greater is the percentage
+amount of sodium chloride converted into bicarbonate. In the latter case,
+however, the percentage utilization of the ammonium bicarbonate decreases;
+that is to say, less sodium bicarbonate is deposited, or more of it remains
+in solution.
+
+Consider, in the same manner, the relations for solutions represented by
+the curve P_{2}IV, which gives the composition of solutions saturated with
+respect to sodium bicarbonate and ammonium bicarbonate. In this case we
+obtain the following results:--
+
+  ------+---------------------+
+        |Initial composition  |
+        |of the solutions:    |
+        |grams of salt to 1000|
+  Point.|grams of water.      |
+        +------+--------------+
+        | NaCl | NH_{4}HCO_{3}|
+  ------+------+--------------+
+  P_{1} | 397  |     496      |
+  --    | 351  |     446      |
+  --    | 316  |     412      |
+  --    | 294  |     389      |
+  --    | 234  |     327      |
+  ------+------+--------------+
+  [Transcriber's note: table continued below...]
+  +----------------------------------+------+----------
+  |                                  |      |
+  |Composition of solutions obtained:|      |
+  |in gram-equivalents per 1000 grams|U_{Na}|U_{NH_{4}}
+  |of water.                         |      |
+  +----------+------+------+---------+      |
+  |  HCO_{3} |  Cl  |  Na  | NH_{4}  |      |
+  +----------+------+------+---------+------+----------
+  |   0.92   | 6.79 | 1.44 |  6.28   | 78.8 |   85.1
+  |   0.99   | 6.00 | 1.34 |  5.65   | 77.7 |   82.5
+  |   1.07   | 5.41 | 1.27 |  5.21   | 76.4 |   79.5
+  |   1.12   | 5.03 | 1.23 |  4.92   | 75.5 |   75.1
+  |   1.30   | 4.00 | 1.16 |  4.14   | 71.0 |   68.6
+  +----------+------+------+---------+------+----------
+
+As is evident from this table, diminution in the relative amount of sodium
+chloride exercises only a slight influence {325} on the utilization of this
+salt, but is accompanied by a rapid diminution of the effective
+transformation of the ammonium bicarbonate. So far as the efficient
+conversion of the sodium is concerned, we see that it reaches its maximum
+at the point P_{1}, and that it decreases both with increase and with
+decrease of the relative amount of sodium chloride employed; and faster,
+indeed, in the former than in the latter case. On the other hand, the
+effective transformation of the ammonium bicarbonate reaches its maximum at
+the point P_{2}, and diminishes with increase in the relative amount of
+ammonium bicarbonate employed. Since sodium chloride is, in comparison with
+ammonia--even when this is regenerated--a cheap material, it is evidently
+more advantageous to work with solutions which are relatively rich in
+sodium chloride (solutions represented by the curve P_{1}P_{2}). This fact
+has also been established empirically.
+
+When, as is the case in industrial practice, we are dealing with solutions
+which are saturated not for two salts but only for sodium bicarbonate, it
+is evident that we have then to do with solutions the composition of which
+is represented by points in the area P_{1}P_{2}I,IV. Since in the
+commercial manufacture, the aim must be to obtain as complete a utilization
+of the materials as possible, the solutions employed industrially must lie
+in the neighbourhood of the curves P_{2}P_{1}IV, as is indicated by the
+shaded portion in Fig. 127. The best results, from the manufacturer's
+standpoint, will be obtained, as already stated, when the composition of
+the solutions approaches that given by a point on the curve P_{2}P_{1}.
+Considered from the chemical standpoint, the results of the experiments
+lead to the conclusion that the Solvay process, _i.e._ passage of carbon
+dioxide through a solution of sodium chloride saturated with ammonia, is
+not so good as the newer method of Schloesing, which consists in bringing
+together sodium chloride and ammonium bicarbonate with water.[395]
+
+{326}
+
+Preparation of Barium Nitrite.--Mention may also be made here of the
+preparation of barium nitrite by double decomposition of barium chloride
+and sodium nitrite.[396]
+
+The reaction with which we are dealing here is represented by the equation
+
+  BaCl_{2} + 2NaNO_{2} = 2NaCl + Ba(NO_{2})_{2}
+
+It was found that at the ordinary temperature NaCl and Ba(NO_{2})_{2} form
+the stable salt-pair. If, therefore, barium chloride and sodium nitrite are
+brought together with an amount of water insufficient for complete
+solution, transformation to the stable salt-pair occurs, and sodium
+chloride and barium nitrite are deposited. When, however, a stable
+salt-pair is in its transition interval (p. 315), a third salt--in this
+case barium chloride--will be deposited, as we have already learned. On
+bringing barium chloride and sodium nitrite together with water, therefore,
+three solid phases are obtained, viz. BaCl_{2}, NaCl, Ba(NO_{2})_{2}. These
+three phases, together with solution and vapour, constitute a univariant
+system, so that at each temperature the composition of the solution must be
+constant.
+
+Witt and Ludwig found that the presence of solid barium chloride can be
+prevented by adding an excess of sodium nitrite, as can be readily foreseen
+from what has been said. Since the solution in presence of the three solid
+phases must have a definite composition at a definite temperature, the
+addition of sodium nitrite to the solution must have, as its consequence,
+the solution of an equivalent amount of barium chloride, and the deposition
+of an equivalent amount of sodium chloride and barium nitrite. By
+sufficient addition of sodium nitrite, the complete disappearance of the
+solid barium chloride can be effected, and there will remain only the
+stable salt-pair sodium chloride and barium nitrite. As was pointed out by
+Meyerhoffer, however, the disappearance of the barium chloride is effected,
+not by a change in the {327} composition of the solution, but by the
+necessity for the composition of the solution remaining constant.
+
+[Illustration: FIG. 128.]
+
+Barium Carbonate and Potassium Sulphate.--As has been found by
+Meyerhoffer,[397] these two salts form the stable pair, not only at the
+ordinary temperature, but also at the melting point. For the ordinary
+temperatures this was proved in the following manner: A solution with the
+solid phases K_{2}SO_{4} and K_{2}CO_{3}.2H_{2}O in excess can only coexist
+in contact either with BaCO_{3} or with BaSO_{4}, since, evidently, in one
+of the two groups the stable system must be present. Two solutions were
+prepared, each with excess of K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O, {328} and
+to one was added BaCO_{3} and to the other BaSO_{4}. After stirring for a
+few days, the barium sulphate was completely transformed to BaCO_{3},
+whereas the barium carbonate remained unchanged. Consequently, BaCO_{3} +
+K_{2}SO_{4} + K_{2}CO_{3}.2H_{2}O is stable, and, therefore, so also is
+BaCO_{3} + K_{2}SO_{4}. That BaCO_{3} + K_{2}SO_{4} is the stable pair also
+at the melting point was proved by a special analytical method which allows
+of the detection of K_{2}CO_{3} in a mixture of the four solid salts. This
+analysis showed that a mixture of BaCO_{3} + K_{2}SO_{4}, after being fused
+and allowed to solidify, contains only small amounts of K_{2}CO_{3}; and
+this is due entirely to the fact that BaCO_{3} + K_{2}SO_{4} on fusion
+deposits a little BaSO_{4}, thereby giving rise at the same time to the
+separation of an equivalent amount of K_{2}CO_{3}.
+
+The different solubilities are shown in Fig. 128. In this diagram the
+solubility of the two barium salts has been neglected. A is the solubility
+of K_{2}CO_{3}.2H_{2}O; addition of BaCO_{3} does not alter this. B is the
+solubility of K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} + BaCO_{3}. A and B almost
+coincide, since the potassium sulphate is very slightly soluble in the
+concentrated solution of potassium carbonate. D gives the concentration of
+the solution in equilibrium with K_{2}SO_{4} + BaSO_{4}. The most
+interesting point is C. This solution is obtained by adding a small
+quantity of water to BaCO_{3} + K_{2}SO_{4}, whereupon, being in the
+transition interval, BaSO_{4} separates out and an equivalent amount of
+K_{2}CO_{3} goes into solution. C is the end point of the curve CO, which
+is called the Guldberg-Waage curve, because these investigators determined
+several points on it.
+
+In their experiments, Guldberg and Waage found the ratio K_{2}CO_{3} :
+K_{2}SO_{4} in solution to be constant and equal to 4. This result is,
+however, not exact, for the curve CO is not a straight line, as it should
+be if the above ratio were constant; but it is concave to the abscissa
+axis, and more so at lower than at higher temperatures.
+
+The following table refers to the temperature of 25deg. The Roman numbers
+in the first column refer to the points in Fig. 128. The numbers in the
+column [Sigma]_k__{2} give the amount, {329} in gram-molecules, of
+K_{2}CO_{3} + K_{2}SO_{4} contained in 1000 gram-molecules of water:--
+
+  SOLUBILITY DETERMINATIONS AT 25deg.
+
+  -----+-------------------------------------+-----------------------+
+       |                                     | 100 gms. of the       |
+       |                                     | solution contain,     |
+   No. |           Solid phases.             | in grams,             |
+       |                                     |           |           |
+       |                                     |K_{2}CO_{3}|K_{2}SO_{4}|
+  -----+-------------------------------------+-----------+-----------+
+  I.   | K_{2}CO_{3}.2H_{2}O + BaCO_{3}      |    53.2   |     --    |
+       |                                     |           |           |
+  II.  |{ K_{2}CO_{3}.2H_{2}O + K_{2}SO_{4} }|    53.0   |    0.023  |
+       |{ + BaCO_{3}                        }|           |           |
+       |                                     |           |           |
+  III.}| K_{2}SO_{4} + BaCO_{3}              |  { 28.5   |    0.886  |
+  IV. }|                                     |  { 22.1   |    1.72   |
+       |                                     |           |           |
+  V.   | BaCO_{3} + K_{2}SO_{4} + BaSO_{4}   |    17.81  |    2.485  |
+       |                                     |           |           |
+  VI. }| K_{2}SO_{4} + BaSO_{4}              |  { 12.6   |    3.92   |
+  VII.}|                                     |  {  5.85  |    6.76   |
+       |                                     |           |           |
+  VIII.| K_{2}SO_{4}                         |     --    |   10.76   |
+       |                                     |           |           |
+  IX. }| BaCO_{3} + BaSO_{4}                 |  {  7.35  |    0.602  |
+  X.  }|                                     |  {  2.85  |    0.173  |
+  -----+-------------------------------------+-----------+-----------+
+  [Transcriber's note: table continued below...]
+  -----+-----------------------+-----------------+-----------
+       | 1000 moles            |                 |
+       | of water contain,     |                 | K_{2}CO_{3}
+   No. | in moles,             |[Sigma]_k__{2}   | -----------
+       |           |           |                 | K_{2}SO_{4}
+       |K_{2}CO_{3}|K_{2}SO_{4}|                 |
+  -----+-----------+-----------+-----------------+-----------
+  I.   |   147.9   |     --    |        --       |    --
+       |           |           |                 |
+  II.  |   147.8   |    0.051  |        --       |    --
+       |           |           |                 |
+       |           |           |                 |
+  III.}|    52.58  |    1.296  |        --       |    --
+  IV. }|    37.79  |    2.333  |        --       |    --
+       |           |           |                 |
+  V.   |    29.11  |    3.220  |       32.32     |   9.03
+       |           |           |                 |
+  VI. }|    19.66  |    4.853  |        --       |    --
+  VII.}|     8.724 |    7.995  |        --       |    --
+       |           |           |                 |
+  VIII.|     --    |   12.47   |        --       |    --
+       |           |           |                 |
+  IX. }|    10.43  |    0.676  |       11.11     |  15.0
+  X.  }|     3.828 |    0.184  |        4.0      |  21.0
+  -----+-----------+-----------+-----------------+-----------
+
+The Guldberg-Waage curve at 100deg was also determined, and it was found
+that the ratio K_{2}CO_{3}: K_{2}SO_{4} is also not constant, although the
+variations are not so great as at 25deg.
+
+  GULDBERG-WAAGE CURVE AT 100deg.
+
+  ----------------------+-----------------------+-----------------+-------
+                        |100 moles of water     |                 | K2CO3
+       Solid phases.    |contain, in moles,     | [Sigma]_k__{2}  | -----
+                        |           |           |                 | K2SO4
+                        |K_{2}CO_{3}|K_{2}SO_{4}|                 |
+  ----------------------+-----------+-----------+-----------------+-------
+  BaCO_{3} + K_{2}SO_{4}|   23.9    |   12.65   |      35.65      |  1.82
+         + BaSO_{4}     |           |           |                 |
+  BaCO_{3} + BaSO_{4}   |    6.28   |    2.02   |       8.3       |  3.1
+     "           "      |    3.17   |    0.851  |       4.025     |  3.7
+  ----------------------+-----------+-----------+-----------------+-------
+
+       *       *       *       *       *
+
+
+{330}
+
+APPENDIX
+
+EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT
+
+For the purpose of determining the transition temperature, a number of
+methods have been employed, and the most important of these will be briefly
+described here. In any given case it is sometimes possible to employ more
+than one method, but all are not equally suitable, and the values of the
+transition point obtained by the different methods are not always
+identical. Indeed, a difference of several degrees in the value found may
+quite well occur.[398] In each case, therefore, some care must be taken to
+select the method most suitable for the purpose.
+
+I. The Dilatometric Method.--Since, in the majority of cases,
+transformation at the transition point is accompanied by an appreciable
+change of volume, it is only necessary to ascertain the temperature at
+which this change of volume occurs, in order to determine the transition
+point. For this purpose the _dilatometer_ is employed, an apparatus which
+consists of a bulb with capillary tube attached, and which constitutes a
+sort of large thermometer (Fig. 129). Some of the substance to be examined
+is passed into the bulb A through the tube B, which is then sealed off. The
+rest of the bulb and a small portion of the capillary tube is then filled
+with some liquid, which, of course, must be without chemical action on the
+substance under investigation. A liquid, however, may be employed which
+dissolves the substance, for, as we have seen (p. 70), the transformation
+at the transition point is, as a rule, accelerated by the presence of a
+solvent. On the other hand, the liquid must not dissolve in the substance
+under examination, for the temperature of transformation would be thereby
+altered.
+
+{331}
+
+In using the dilatometer, two methods of procedure may be followed.
+According to the first method, the dilatometer containing the form stable
+at lower temperatures is placed in a thermostat, maintained at a constant
+temperature, until it has taken the temperature of the bath. The height of
+the meniscus is then read on a millimetre scale attached to the capillary.
+The temperature of the thermostat is then raised degree by degree, and the
+height of the meniscus at each point ascertained. If, now, no change takes
+place in the solid, the expansion will be practically uniform, or the rise
+in the level of the meniscus per degree of temperature will be practically
+the same at the different temperatures, as represented diagrammatically by
+the line AB in Fig. 130. On passing through the transition point, however,
+there will be a more or less sudden increase in the rise of the meniscus
+per degree (line BC) if the specific volume of the form stable at higher
+temperatures is greater than that of the original modification; thereafter,
+the expansion will again be uniform (line CD). Similarly, on cooling,
+contraction will at first be uniform and then at the transition point there
+will be a relatively large diminution of volume.
+
+[Illustration: FIG. 129.]
+
+[Illustration: FIG. 130.]
+
+If, now, transformation occurred immediately the transition point was
+reached, the sudden expansion and contraction would take place at the same
+temperature. It is, however, generally found that there is a lag, and that
+with rising temperature the relatively large expansion does not take place
+until a temperature somewhat higher than the transition point; and with
+falling temperature the contraction occurs at a temperature somewhat below
+the transition point. This is represented in Fig. 130 by the lines BC and
+EF. The amount of lag will vary from case to case, and will {332} also
+depend on the length of time during which the dilatometer is maintained at
+constant temperature.
+
+As an example, there may be given the results obtained in the determination
+of the transition point at which sodium sulphate and magnesium sulphate
+form astracanite (p. 268).[399] The dilatometer was charged with a mixture
+of the two sulphates.
+
+  --------------------------------------------------------
+  Temperature. | Level of oil in capillary. | Rise per 1deg.
+  --------------------------------------------------------
+    15.6deg    |            134             |
+    16.6deg    |            141             |       7
+    17.6deg    |            148             |       7
+    18.6deg    |            154             |       6
+    19.6deg    |            161             |       7
+    20.6deg    |            168             |       7
+    21.6deg    |            241             |      73
+    22.6deg    |            243             |       2
+    23.6deg    |            251             |       8
+    24.6deg    |            259             |       8
+  --------------------------------------------------------
+
+The transition point, therefore, lies about 21.6deg (p. 268).
+
+The second method of manipulation depends on the fact that, while above or
+below the transition point transformation of one form into the other can
+take place, at the transition point the two forms undergo no change. The
+bulb of the dilatometer is, therefore, charged with a mixture of the stable
+and metastable forms and a suitable liquid, and is then immersed in a bath
+at constant temperature. After the temperature of the bath has been
+acquired, readings of the height of the meniscus are made from time to time
+to ascertain whether expansion or contraction occurs. If expansion is
+found, the temperature of the thermostat is altered until a temperature is
+obtained at which a gradual contraction takes place. The transition point
+must then lie between these two temperatures; and by repeating the
+determinations it will be possible to reduce the difference between the
+temperatures at which expansion and contraction take place to, say, 1deg,
+and to fix the temperature of the transition point, therefore, to within
+half a degree. By this method the transition point, for example, of sulphur
+was found to be 95.6deg under a pressure of 4 atm.[400] The following are
+the figures obtained by Reicher, who used a mixture {333} of 1 part of
+carbon disulphide (solvent for sulphur) and 5 parts of turpentine as the
+measuring liquid.
+
+  TEMPERATURE 95.1deg.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      343.5
+         30        |      340.5
+         55        |      335.75
+         65        |      333
+  -----------------------------------
+
+  TEMPERATURE 96.1deg.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      342.75
+         30        |      354.75
+         55        |      360.5
+         60        |      361.5
+  -----------------------------------
+
+  TEMPERATURE 95.6deg.
+
+  -----------------------------------
+  Time in minutes. | Level of liquid.
+  -----------------------------------
+          5        |      368.75
+        100        |      368
+        110        |      368.75
+  -----------------------------------
+
+At a temperature of 95.1deg there is a contraction, _i.e._ monoclinic
+sulphur passes into the rhombic, the specific volume of the former being
+greater than that of the latter. At 96.1deg, however, there is expansion,
+showing that at this temperature rhombic sulphur passes into monoclinic;
+while at 95.6deg there is neither expansion nor contraction. This is,
+therefore, the transition temperature; and since the dilatometer was sealed
+up to prevent evaporation of the liquid, the pressure within it was 4 atm.
+
+II. Measurement of the Vapour Pressure.--In the preceding pages it has been
+seen repeatedly that the vapour pressures of the two systems undergoing
+reciprocal transformation become identical at the transition point (more
+strictly, at the triple or {334} multiple point), and the latter can
+therefore be determined by ascertaining the temperature at which this
+identity of vapour pressure is established. The apparatus usually employed
+for this purpose is the Bremer-Frowein tensimeter (p. 91).
+
+Although this method has not as yet been applied to systems of one
+component, it has been used to a considerable extent in the case of systems
+containing water or other volatile component. An example of this has
+already been given in Glauber's salt (p. 139).
+
+III. Solubility Measurements.--The temperature of the transition point can
+also be fixed by means of solubility measurements, for at that point the
+solubility of the two systems becomes identical. Reference has already been
+made to several cases in which this method was employed, _e.g._ ammonium
+nitrate (p. 112), Glauber's salt (p. 134), astracanite and sodium and
+magnesium sulphates (p. 268).
+
+The determinations of the solubility can be carried out in various ways.
+One of the simplest methods, which also gives sufficiently accurate results
+when the temperature is not high or when the solvent is not very volatile,
+can be carried out in the following manner. The solid substance is finely
+powdered (in order to accelerate the process of solution), and placed in
+sufficient quantity along with the solvent in a tube carefully closed by a
+glass stopper; the latter is protected by a rubber cap, such as a rubber
+finger-stall. The tube is then rotated in a thermostat, the temperature of
+which does not vary more than one or two tenths of a degree, until
+saturation is produced. The solution is withdrawn by means of a pipette to
+which a small glass tube, filled with cotton wool to act as a filter, is
+attached. The solution is then run into a weighing bottle, and weighed;
+after which the amount of solid in solution is determined in a suitable
+manner.
+
+For more accurate determinations of the solubility, especially when the
+solvent is appreciably volatile at the temperature of experiment, other
+methods are preferable. In Fig. 131 is shown the apparatus employed by H.
+Goldschmidt,[401] and used to a considerable extent in the laboratory of
+van't Hoff. This consists essentially of three parts: _a_, a tube in which
+the solvent and salt are placed; this is closed at the foot by an
+india-rubber stopper. Through this stopper there passes the bent tube _cb_,
+which connects the tube _a_ with the weighing-tube d. At _c_ there is a
+plug of cotton wool. Tube _e_ is open to the air. The wider portion of the
+tube _cb_, which passes through the rubber stopper in _a_, can be closed by
+a plug {335} attached to a glass rod _ff_, which passes up through a hollow
+Witt stirrer, _g_. After being fitted together, the whole apparatus is
+immersed in the thermostat. After the solution has become saturated, the
+stopper of the bent tube is raised by means of the rod _ff_ and a
+suction-pump attached to the end of e. The solution is thereby drawn into
+the weighing-tube _d_, the undissolved salt being retained by the plug at
+c. The apparatus is then removed from the thermostat, tube _d_ detached and
+immediately closed by a ground stopper. It is then carefully dried and
+weighed.
+
+[Illustration: FIG. 131.]
+
+Another form of solubility vessel, due to Meyerhoffer and Saunders, is
+shown in Fig. 132.[402] This consists of a single tube, and the stirring is
+effected by means of a glass screw.
+
+[Illustration: FIG. 132.]
+
+The progress of the solution towards saturation can be very well tested by
+determining the density of the solution from time to {336} time. This is
+most conveniently carried out by means of the pipette shown in Fig.
+133.[403] With this pipette the solution can not only be removed for
+weighing, but the volume can be determined at the same time. It consists of
+the wide tube _a_, to which the graduated capillary _b_, furnished with a
+cap _c_, is attached. To the lower end of the pipette the tube _e_, with
+plug of cotton wool, can be fixed. After the pipette has been filled by
+sucking at the end of _b_, the stop-cock _d_ is closed and the cap _c_
+placed on the capillary. The apparatus can then be weighed, and the volume
+of the solution be ascertained by means of the graduations.
+
+As has already been insisted, particular care must be paid to the
+characterization of the solid in contact with the solution.
+
+[Illustration: FIG. 133.]
+
+IV. Thermometric Method.--If a substance is heated, its temperature will
+gradually rise until the melting point is reached, and the temperature will
+then remain constant until all the solid has passed into liquid. Similarly,
+if a substance which can undergo transformation is heated, the temperature
+will rise until the transition point is reached, and will then remain
+constant until complete transformation has taken place.
+
+This method, it will be remembered, was employed by Richards for the
+determination of the transition point of sodium sulphate decahydrate
+(p. 136). The following figures give the results obtained by Meyerhoffer in
+the case of the transformation:--
+
+  CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O
+
+the temperature being noted from minute to minute: 95deg, 93deg, 91.8deg,
+91.7deg, 92deg, 92.3deg, 92.4deg, 92.2deg, 92.2deg, 92deg, 90.5deg, 89deg,
+and then a rapid fall in the temperature. From this we see that the
+transition point is about 92.2deg. It is also evident that a slight
+supercooling took place (91.7deg), owing to a delay in the transformation,
+but that then the temperature rose to the transition point. This is
+analogous to the supercooling of a liquid.
+
+A similar halt in the temperature would be observed on passing from lower
+to higher temperatures; but owing to a lag in the transformation, the same
+temperature is not always obtained.
+
+{337}
+
+V. Optical Method.--The transition point can sometimes be determined by
+noting the temperature at which some alteration in the appearance of the
+substance occurs, such as a change of colour or of the crystalline form.
+Thus mercuric iodide changes colour from red to yellow, and the blue
+quadratic crystals of copper calcium acetate change, on passing the
+transition point, into green rhombs of copper acetate and white needles of
+calcium acetate (p. 260). Or again, changes in the double refraction of the
+crystals may be also employed to ascertain the temperature of the
+transition point. These changes are best observed by means of a microscope.
+
+For the purpose of regulating the temperature of the substance a small
+copper air-bath is employed.[404]
+
+VI. Electrical Methods.--Electrical methods for the determination of the
+transition point are of two kinds, based on measurements of conductivity or
+of electromotive force. Both methods are restricted in their application,
+but where applicable give very exact results.
+
+The former method, which has been employed in several cases, need not be
+described here. The second method, however, is of considerable interest and
+importance, and calls for special reference.[405]
+
+If two pieces, say, of zinc, connected together by a conducting wire, are
+placed in a solution of a zinc salt, _e.g._ zinc sulphate, the potential of
+the two electrodes will be the same, and no current will be produced in the
+connecting wire. If, however, the zinc electrodes are immersed in two
+solutions of _different_ concentration contained in separate vessels, but
+placed in connection with one another by means of a bent tube filled with a
+conducting solution, the potentials at the electrodes will no longer be the
+same, and a current will now flow through the connecting wire. The
+direction of this current _in the cell_ will be from the weaker to the more
+concentrated solution.
+
+The greater the difference in the concentration of the solutions with
+respect to zinc, the greater will be the difference of the potential at the
+two electrodes, or the greater will be the E.M.F. of the cell. When the
+concentration of the two solutions becomes the same, the E.M.F. will become
+zero, and no current will pass.
+
+It will be understood now how this method can be made use of {338} for
+determining the transition point of a salt, when we bear in mind that at
+the transition point the solubility of the two forms becomes identical.
+Thus, for example, the transition point of zinc sulphate heptahydrate into
+hexahydrate could be determined in the following manner. Tube A (Fig. 134)
+contains, say, a saturated solution of the heptahydrate along with some of
+the solid salt; tube B, a saturated solution of the hexahydrate along with
+the solid salt. The tube C is a connecting tube bent downwards so as to
+prevent the mixing of the solutions by convection currents. ZZ are two zinc
+electrodes immersed in the solution; the cell is placed in a thermostat and
+the zinc electrodes connected with a galvanometer. Since, now, at
+temperatures below the transition point the solubility of the hexahydrate
+(the metastable form) is greater than that of the heptahydrate, a current
+will be produced, flowing in the cell from heptahydrate to hexahydrate. As
+the temperature is raised towards the transition point, the solubilities of
+the two hydrates also approach, and the current produced will therefore
+become weaker, because the E.M.F. of the cell becomes less; and when the
+transition point is attained, the E.M.F. becomes zero, and the current
+ceases. If the temperature is raised above this, the solubility of the
+heptahydrate becomes greater than that of the hexahydrate, and a current
+will again be produced, but in the opposite direction. By noting the
+temperature, therefore, at which the current ceases, or the E.M.F. becomes
+zero, the transition temperature can be ascertained.[406]
+
+[Illustration: FIG. 134.]
+
+In the case just described, the electrodes consisted of the same metal as
+was contained in the salt. But in some cases, _e.g._ sodium sulphate,
+electrodes of the metal contained in the salt cannot be employed.
+Nevertheless, the above electrical method can be used {339} even in those
+cases, if a suitable non-polarizable mercury electrode is employed.[407]
+
+Although, as we saw, no current was produced when two pieces of zinc were
+immersed in the same solution of zinc salt, a current will be obtained if
+two different metals, or even two different modifications of the same
+metal, are employed. Thus an E.M.F. will be established when electrodes of
+grey and of white tin are immersed in the same solution of zinc salt, but
+at the transition point this E.M.F. will become zero. By this method Cohen
+determined the transition point of grey and white tin (p. 42).
+
+       *       *       *       *       *
+
+
+{340}
+
+NAME INDEX
+
+          A
+  Abegg, 52
+  Adriani, 186, 217, 220
+  Alexejeff, 97, 125
+  Allan, 298
+  Allen, L. E., 109
+  Allen, R. W., 63
+  Ampolla, 213
+  Andreae, 109
+  Aristotle, 41
+  Armstrong, E. F., 313
+  Armstrong, H. E., 196
+  Arzruni, 33
+  Aten, 147, 163
+  Auerbach, 326
+
+          B
+  Babo, 126
+  Bancroft, 102, 104, 161, 176, 196, 202, 229, 246, 260, 261, 272, 281, 302
+  Barnes, 331, 339
+  Barschall, 318
+  Barus, 67
+  Battelli, 23
+  Baur, 233, 307
+  Beckmann, 49
+  Bell, 229
+  Berthollet, 7
+  Bodlaender, 181, 247, 311, 321
+  Bogojawlenski, 72
+  Boudouard, 309, 311
+  Braun, 107
+  Brauns, 40, 51, 74
+  Bredig, 52
+  Bremer, 91
+  Brodie, 34, 47
+  Bruner, 126
+  Bruni, 181, 182, 256, 257
+  Bunsen, 67
+
+          C
+  Cady, 192
+  Calvert, 130
+  Cameron, 203
+  Carnelley, 47
+  Carpenter, 225
+  Carveth, 204, 255
+  Centnerszwer, 158
+  Chapman, 47
+  Chappuis, 51, 176
+  Charpy, 255
+  Churchill, 140
+  Coehn, 52
+  Cohen, 41, 72, 136, 139, 140
+  Cooke, 331, 339
+  Cox, 301
+
+          D
+  Dawson, 263
+  Debray, 74, 81, 139
+  Deville, 49, 74
+  Dewar, 26, 51, 178
+  Dietz, 157
+  {341}
+  Doelter, 233
+  Donnan, 8, 18
+  Dreyer, 73
+  Duhem, 56, 151
+  Dutoit, 204
+
+          E
+  Etard, 115, 135
+
+          F
+  Fahrenheit, 30
+  Faraday, 82, 89
+  Fath, 204
+  Fedotieff, 315, 320
+  Findlay, 111, 204, 206, 219
+  Foote, 69
+  Friedlaender, 72
+  Fritsche, 41
+  Frowein, 91
+  Fuechtbauer, 75
+  Fyffe, 143
+
+          G
+  Gattermann, 51, 52
+  Gautier, 222, 223
+  Gay-Lussac, 135
+  Gernez, 72
+  Gibbs, 7, 8, 151, 236
+  Glaessner, 307
+  Goldschmidt, E., 41
+  Goldschmidt, H., 335
+  Goldschmidt, V., 32
+  Goossens, 26
+  Gossner, 318
+  Graham, 178
+  Guertler, 73
+  Guldberg, 7
+  Guthrie, 97, 104, 117, 118, 119, 233
+
+          H
+  Haber, 311
+  Hahn, 309, 311
+  Hallock, 35
+  Hammerl, 145
+  Hautefeuille, 46, 49, 50, 51, 178
+  Heller, 311
+  Henry, 94
+  Herold, 321
+  Hertz, 49
+  Heycock, 194, 221, 223
+  Heyn, 225, 228
+  Hickmans, 219
+  Hiorns, 228
+  Hissink, 115, 190
+  Hoitsema, 14, 90, 177, 178, 298
+  Hollmann, 204
+  Holsboer, 110
+  Horstmann, 8, 83, 89
+  Hudson, 102
+  Hulett, 10, 48, 52, 54, 67, 109
+
+          I
+  Isaac, 114
+  Isambert, 80, 82, 84
+
+          J
+  Jaffe, 74, 114
+  Joulin, 176
+  Juhlin, 23, 24, 30
+  von Jueptner, 225
+
+          K
+  Kastle, 71
+  Kaufler, 49
+  Kaufmann, 112
+  Kayser, 176
+  Keeling, 225
+  Kelvin, 25
+  Kenrick, 263, 297
+  Kipping, 219
+  Kirchhoff, 32
+  Knorr, 203
+  de Kock, 53, 182, 194
+  Konowaloff, 102, 103, 104
+  Krasnicki, 144
+  Kremann, 147, 212
+  Kuenen, 105
+  Kultascheff, 233
+  {342}
+  Kuriloff, 216
+  Kurnakoff, 221, 222, 230
+  Kuester, 72, 181, 183
+
+          L
+  Laar, 195
+  Labenburg, 216
+  Lattey, 101
+  Le Chatelier, 58, 81, 233
+  Lehfeldt, 338, 340
+  Lehmann, 33, 52, 53
+  Lidbury, 147
+  Loewel, 134, 135
+  Loewenherz, 134, 316
+  Lowry, 196, 198
+  Ludwig, 327
+  Lumsden, 80, 109, 110
+  Lussana, 68
+  Luther, 22
+
+          M
+  Mack, 67
+  Magnus, 22
+  Mathews, 221
+  Mellor, 80
+  Meusser, 142
+  Meyer, J., 71
+  Meyer, V., 47
+  Meyerhoffer, 158, 233, 259, 268, 271, 278, 279, 280, 284, 313, 315, 317,
+      319, 327, 328, 336, 337
+  Middelberg, 116
+  Miers, 114
+  Miller, 297
+  Mitscherlich, 33, 49
+  Mond, 178
+  Moore, 72
+  Moss, 66
+  Mueller, 112, 265
+  Mylius, 109, 142, 157
+
+          N
+  Naumann, 49
+  Neville, 194, 221, 223
+
+          O
+  Offer, 119
+  Ostwald, 8, 10, 13, 16, 22, 44, 58, 68, 70, 74, 85, 88, 92, 102, 110,
+      117, 125, 127, 130, 141, 198
+
+          P
+  Padoa, 73, 181
+  Parsons, 298
+  Pasteur, 266
+  Paterno, 213
+  Payen, 74
+  Pedler, 47
+  Pfaundler, 119
+  Philip, 213, 214
+  von Pickardt, 73
+  Planck, 68
+  Pope, 219
+  Poynting, 68
+  Preuner, 311
+  Puschin, 222
+
+          Q
+  Quincke, 52
+
+          R
+  Rabe, 113
+  Ramsay, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 90, 165, 178
+  Raoult, 180
+  Reed, 71
+  Regnault, 22
+  Reicher, 36, 37, 110, 260, 333
+  Reinders, 71, 185, 188
+  Reinitzer, 51, 52
+  Richards, 136, 140
+  Riddle, 47
+  Riecke, 48, 55
+  Roberts-Austen, 63, 194, 221, 223, 225
+  Roloff, 117
+  Roozeboom, 10, 38, 45, 47, 49, 50, 51, 54, 56, 57, 62, 63, 68, 88, 103,
+      126, 145, 147, 150, 151, 157, 162, 170, 174, 178, 182, 196, 201, 211,
+      217, 220, 225, 236, 238, 262, 264, 269, 272, 273, 281, 282, 290, 331
+  {343}
+  Rose, 223
+  Rotarski, 52
+  Rothmund, 97, 98, 100
+  Rutten, 298
+
+          S
+  Saposchnikoff, 212
+  Saunders, 313, 317, 319, 336, 337
+  Saurel, 151
+  Schaum, 49, 75
+  Scheel, 22, 23, 30
+  Schenck, 49, 52, 54, 311
+  Schneider, 52
+  Schoenbeck, 75
+  Schreinemakers, 122, 126, 246, 248, 250, 252, 290, 302
+  Schroetter, 46
+  Schukowsky, 52
+  Schwarz, 331
+  Seitz, 52
+  Shenstone, 109, 115, 135
+  Shepherd, 221, 255
+  Shields, 178
+  Skirrow, 130
+  Spring, 63
+  von Stackelberg, 107, 110
+  Staedel, 267
+  Stansfield, 194, 221
+  Stokes, 236
+  Stortenbeker, 44, 147, 161, 164, 281
+
+          T
+  Taber, 229
+  Tammann, 26, 32, 33, 37, 38, 39, 48, 52, 65, 67, 68, 72, 73, 140, 151,
+      176, 221, 230
+  Thiesen, 22, 23, 30
+  Thomson, J., 25, 28, 32
+  Thomson, W., 25
+  Tilden, 109, 115, 135
+  Trevor, 16
+  Troost, 46, 49, 50, 51
+  Tumlirz, 72
+
+          V
+  Van Bemmelen, 180
+  Van Deventer, 110, 139, 266, 267, 333
+  Van Eyk, 41, 63, 192, 338
+  Van't Hoff, 36, 38, 58, 70, 90, 92, 108, 127, 139, 140, 165, 175, 225,
+      258, 260, 263, 265, 266, 267, 272, 284, 290, 313, 318, 333, 340
+  Van Leeuwen, 259
+  Van Wyk, 185
+  Vogt, 5, 233
+
+          W
+  Waage, 7
+  Wald, 92
+  Walden, 158
+  Walker, 80, 105, 122, 126, 143
+  Wegscheider, 10, 49, 202
+  Wells, 136
+  Wenzel, 7
+  Wiebe, 22
+  Witt, 327
+  Wright, 241, 246, 247
+  von Wrochem, 109, 142
+
+          Y
+  Young, 3, 22, 23, 24, 30, 32, 63, 64, 66, 79, 105, 165
+
+          Z
+  Zacharias, 180
+  Zawidski, 63
+  Zenghelis, 35
+  Zimmermann, 311
+  Zincke, 44
+  Ziz, 141
+
+       *       *       *       *       *
+
+
+{344}
+
+SUBJECT INDEX
+
+          A
+  Acetaldehyde and paraldehyde, 204
+  Acetic acid, chloroform, water, 241
+  Acetone, phenol, water, 248
+  Adsorption, 176
+  Alcohol, chloroform, water, 246
+    ----, ether, water, 246
+  Alloys, equilibrium curves of, 221
+    ---- of copper and tin, liquefaction of, by cooling, 194
+    ---- of iron and carbon, 223
+    ---- of thallium and mercury, 222
+    ----, ternary, 246
+  Ammonia compounds of metal chlorides, 82
+  Ammonia silver chlorides, 82
+    ---- ---- ----, dissociation pressures of, 84
+  Ammonia-soda process, 320
+  Ammonium chloride, dissociation of, 3, 79
+    ---- cyanide, dissociation of, 80
+    ---- hydrosulphide, dissociation of, 80
+    ---- nitrate, solubility of, 113
+  Aniline, phenol, water, 250
+  Astracanite, 260, 261, 268, 274
+
+          B
+  Babo, law of, 126
+  Barium acetate, solubility of, 143
+  Barium carbonate and potassium sulphate, 328
+    ---- nitrite, preparation of, 327
+  Basic salts, 296
+  Benzaldoximes, 203
+  Benzene and picric acid, 216
+  Bismuth, effect of pressure on the melting point of, 67
+    ----, lead, tin, 255
+    ---- nitrates, basic, 298
+  Bivariant systems, 16
+  Bromocinnamic aldehyde and chlorocinnamic aldehyde, 183
+
+          C
+  Calcium carbonate, dissociation of, 3, 11, 81
+    ---- chloride hexahydrate, solubility of, 146
+    ---- ----, solubility of hydrates of, 148
+    ---- ----, vapour-pressure of hydrates of, 88
+  Camphor oximes, 219, 257
+  Carnallite, 284
+  Carvoximes, 186, 219
+  Cementite, 224
+  Chlorine and iodine, 161
+  Chlorocinnamic aldehyde and bromocinnamic aldehyde, 183
+  Chloroform, acetic acid, water, 241
+    ----, alcohol, water, 246
+  {345}
+  Classification of systems, 17
+  Component, 8, 10, 12
+    ----, systems of one, 21, 55
+  Components, choice of, 12, 13, 14, 76, 313
+    ----, determination of number of, 13
+    ----, systems of four, 312
+    ----, ---- of three, 234
+    ----, ---- of two, 76, 207
+    ----, variation in number of, 11, 14
+  Composition, determination of, without analysis, 228, 302
+  Concentration-temperature curve for two liquids, 101
+  Condensed systems, 36
+  Constituent, 10
+  Cooling curve, 230
+  Copper calcium acetate, 260
+    ---- chloride, heat of solution of, 110
+    ---- dipotassium chloride, 259
+    ---- sulphate, 85
+  Critical concentration, 98, 242
+    ---- pressure of water, 23
+    ---- solution temperature, 98
+    ---- temperature of water, 23
+  Cryohydrates, 117, 118
+  Cryohydric point, 117
+    ---- ----, changes at the, 119
+    ---- ---- for silver nitrate and ice, 116
+  Crystals, liquid, 51
+    ----, ----, equilibria of, 53
+    ----, ----, list of, 54
+    ----, ----, nature of, 52
+    ----, mixed, 180
+  Crystallization, velocity of, 72, 74
+    ----, spontaneous, 114
+
+          D
+  Deliquescence, 130
+  Devitrification, 73
+  Diethylamine and water, solubility of, 101
+  Dilatometer, determination of transition points by, 331
+  Dineric surface, 247
+  Dissociation equilibrium, effect of addition of dissociation products on,
+      4
+    ---- of ammonia compounds of metal chlorides, 82, 84
+    ---- of ammonium chloride, 3, 79
+    ---- ---- cyanide, 80
+    ---- ---- hydrosulphide, 80
+    ---- of calcium carbonate, 3, 81
+    ---- of compounds, degree of, 147
+    ---- of phosphonium bromide, 80
+    ---- of salt hydrates, 85
+    ----, phenomena of, 79
+  Dissociation pressure, 81
+  Distillation of supercooled liquid to solid, 32, 50
+  Double salt interval, 278
+    ---- salts, crystallization from solution, 280
+    ---- ----, decomposition by water, 267
+    ---- ----, formation of, 258, 273, 315
+
+          E
+  Efflorescence, 86
+  Electrical methods of determining transition points, 338
+  Enantiotropy, 44, 51
+  Equilibria, Gibbs's theory of, 8
+    ----, metastable, 69
+  Equilibrium apparent (false), 5, 6
+    ---- between ice and solution, 116
+    ---- between ice and water, 25
+    ---- between ice, water, vapour, 27
+    ---- between water and vapour, 21
+    ----, chemical, 3, 16
+    ----, heterogeneous, 5
+    ----, homogeneous, 5
+    ----, independence of, on amounts of phases, 9
+    ----, law of movable, 58
+  {346}
+    ----, physical, 3, 16
+    ---- real (true), 5, 6
+  Ether, alcohol, water, 246
+    ----, succinic nitrile, water, 252
+  Ethylene bromide, picric acid, [beta]-naphthol, 256
+  Eutectic mixtures, 117, 191, 209, 255, 257
+    ---- point, 117, 209, 213, 253
+
+          F
+  Ferric chloride, evaporation of solutions of, 155
+    ---- ----, hydrates of, 151, 153
+    ---- ----, hydrogen chloride and water, systems of, 290
+  Ferrite, modifications of, 224
+  Freedom, degree of, 14
+  Freezing mixtures, 120
+    ---- point, natural, 198
+  Fusion curve, 66
+    ---- ---- of ice, 25
+    ---- of ice, influence of pressure on, 26
+    ----, partial, 139
+
+          G
+  Glaserite, 315, 317
+  Glasses, 176
+  Glauber's salt, 13, 134
+    ---- ----, transition curve of, 68, 140
+  Graphic representation in space, 77, 284
+
+          H
+  Hydrates, range of existence of, 89
+    ---- chloride and water, 174
+  Hydrogen bromide and water, 174
+  Hylotropic substances, 198
+
+          I
+  Ice I., 32
+    ---- II., 32
+    ---- III., 32
+    ----, equilibrium between water and, 25
+    ----, influence of pressure on melting point of, 25, 26
+    ----, sublimation curve of, 24
+    ----, vapour pressure of, 25, 31
+  Indifferent point, 150
+  Individual, chemical, 92
+  Inversion temperature, 36
+  Iodine and chlorine, 161
+  Iron--carbon alloys, 223
+    ----, carbon monoxide and carbon dioxide, 305
+  Isomerides, dynamic, 195, 196
+    ----, ----, equilibrium between, 195, 196
+    ----, ----, equilibrium point of, 198
+    ----, transformation of unstable into stable, 201
+  Isomerism, dynamic, 196
+  Isothermal evaporation, 278
+    ---- solubility curves, 272
+
+          L
+  Lead, bismuth, tin, 255
+    ----, desilverization of, 247
+    ----, silver, zinc, 246
+  Le Chatelier, theorem of, 57
+  Lime, burning of, 3
+  Liquidus curve, 182
+
+          M
+  Mandelic acid, 217
+  Martensite, 224
+  Mass action, law of, 7
+  Melting point, influence of pressure on, 66
+  {347}
+    ---- ----, congruent, 146
+    ---- ----, incongruent, 139
+    ---- under the solvent, 122
+  Menthyl mandelates, 219
+  Mercuric bromide and iodide, 188
+  Mercury salts, basic, 301
+  Metastable equilibria, 69
+    ---- region, 30
+    ---- state, 30
+  Methylethyl ketone and water, 100
+  Minerals, formation of, 232
+  Miscibility of liquids, complete, 95, 104, 114
+    ---- ----, partial, 95, 96, 121
+  Mixed crystals, 180, 281
+    ---- ----, changes in, with temperature, 192
+    ---- ----, examples of, 183, 186, 187, 190, 192, 219, 223
+    ---- ----, formation of, 181, 182
+    ---- ----, fractional crystallization of, 188
+    ---- ----, freezing points of, 182
+    ---- ----, melting points of, 182, 184
+    ---- ----, pseudoracemic, 219
+  Mixtures, isomorphous, 181
+    ---- of constant boiling point, 105
+    ---- of constant melting point, 117, 186, 187, 192, 209, 255, 257
+  Monotropy, 44, 51
+  Multivariant systems, 16
+
+          N
+  Naphthalene and monochloracetic acid, 192
+    ---- and [beta]-naphthol, mixed crystals of, 183
+  [beta]-Naphthol, ethylene bromide, picric acid, 256
+  [alpha]-Naphthylamine and phenol, 213
+  Nickel iodate, solubility of, 142
+  _o_-Nitrophenol and _p_-toluidine, 213
+
+          O
+  Occlusion of gases, 176
+  Optical method of determining transition points, 338
+  Optically active substances, freezing-point curves of, 216
+  Order of a system, 13
+  Organic compounds, application of Phase Rule to, 212
+
+          P
+  Palladium and hydrogen, 90, 178
+  Paragenesis, 320
+  Paraldehyde and acetaldehyde, 204
+  Partial pressures of two components, 102
+  Pearlite, 224
+  Phase, 8
+    ---- Rule, 8, 16
+    ---- ----, deduction of, 18
+    ---- ----, scope of, 1
+  Phases, formation of new, 69
+    ----, number of, 9
+  Phenol, acetone, water, 248
+    ----, aniline, water, 250
+    ---- and [alpha]-naphthylamine, 213
+    ---- and _p_-toluidine, 214
+    ---- and water, solubility of, 97
+  Phosphonium bromide, dissociation of, 80
+    ---- chloride, 65
+  Phosphorus, 46
+    ----, distillation of white to red, 50
+    ----, melting point of red, 47
+    ----, ---- ---- of white, 48
+    ----, solubility of white and red, 47
+    ----, vapour pressure of white and red, 46
+  Picric acid and benzene, 216
+    ---- ----, ethylene bromide, and [beta]-naphthol, 256
+  Polymorphic forms, solubility of, 112
+  {348}
+    ---- substances, list of, 63
+  Polymorphism, 33
+  Potassium nitrate and thallium nitrate, 192
+  Potential, chemical, 19
+  Pressure-concentration diagram for two liquids, 102
+  Pressure-temperature diagram for solutions, 126
+  Pseudomonotropy, 45
+  Pseudo-racemic mixed crystals, 21
+  Pyridine and methyl iodide, 147
+  Pyrometer, registering, 230
+
+          Q
+  Quadruple point, 116
+  Quintuple point, 234, 261
+
+          R
+  Racemates, characterization of, 217, 282
+  Reactions, law of successive, 73
+  Reciprocal salt-pairs, 313
+    ---- ----, transition point of, 314
+  Rubidium tartrates, 265
+
+          S
+  Salt hydrates, 85
+    ---- ----, indefiniteness of vapour pressure of, 87
+    ---- ---- with definite melting point, 145
+  Separation of salt on evaporation, 130
+  Silicates, hydrated, 176
+  Silver, lead, zinc, 246
+  Silver nitrate, solubility of, 114
+    ---- ---- and sodium nitrate, 190
+  Single salt interval, 278
+  Sodium ammonium tartrates, 266
+    ---- nitrate and silver nitrate, 190
+    ---- sulphate and water, equilibria between, 134
+  Sodium sulphate and water, vapour pressures of, 138, 140
+    ---- ----, anhydrous, dehydration by, 138
+    ---- ----, solubility of, 135
+    ---- ---- decahydrate, solubility of, 134
+    ---- ---- ----, transition point of, 136, 139
+    ---- ---- heptahydrate, solubility of, 136
+    ---- ---- ----, transition point of, 137
+  Solidus curve, 182
+  Solubility curve at higher temperatures, 114
+    ---- ----, form of, 108
+    ---- ---- of anhydrous salts, 111
+    ---- ----, retroflex, 146, 151, 162
+    ---- curves, interpolation and extrapolation of, 111
+    ---- ---- of three component systems, 264
+    ----, determination of transition points by, 335
+    ----, influence of pressure on, 107
+    ----, ---- of subdivision on, 10
+    ----, ---- of temperature on, 109
+    ---- of metastable forms, 47, 112, 137
+  Solubility of polymorphic forms, 112
+    ---- of salt hydrates, 133, 145
+    ---- of supercooled liquids, 125
+    ----, retrograde, 245
+  Solute, 93
+  Solution, definition of, 92
+    ----, heat of, 109, 110
+    ----, saturated, 106, 108
+    ----, supersaturated, 108
+    ---- temperature, critical, 98
+    ----, unsaturated, 108
+  Solutions, bivariant systems, 129
+    ----, congruently saturated, 279
+    ---- conjugate, 97, 241
+  {349}
+    ----, incongruently saturated, 279, 289
+    ----, inevaporable, 157
+    ---- of gases in liquids, 93
+    ---- ---- in solids, 176
+    ---- of liquids in liquids (binary), 95
+    ---- ---- ---- (ternary), 240
+    ---- ----, influence of temperature on, 247
+    ---- of solids in liquids, 106
+    ---- ---- in solids, 180
+    ----, solid, 175, 180
+    ----, univariant systems, 127
+  Space model for carnallite, 284
+  Stability limit, 202
+  Steel, formation of, 223
+  Sublimation curve, 63
+    ---- ---- of ice, 24
+    ---- without fusion, 65
+  Succinic nitrile and water, 122
+    ---- ether, water, 252
+  Sulphur, 33, 34
+    ---- dioxide and water, 169
+    ---- ---- and potassium iodide, 158
+    ----, transition point of rhombic and monoclinic, 36
+  Supersaturation, 113, 114, 124
+    ----, limits of, 114
+  Systems, condensed, 36
+    ---- of one component, 21
+    ---- of two components, 76, 77, 207
+
+          T
+  Tachydrite, influence of pressure on the transition point of, 263
+  Tartrate, dimethyl, 217
+    ----, sodium potassium, 259
+  Tautomeric substances, 195
+  Tensimeter, 91
+  Thallium nitrate and potassium nitrate, 192
+  Theorem of van't Hoff and Le Chatelier, 57
+  Thermometric  determination of transition point, 337
+  Tin, 41
+    ----, lead, bismuth, 255
+    ---- plague, 43
+    ----, transition point of white and grey, 41
+  _p_-Toluidine and _o_-nitrophenol, 213
+    ---- and phenol, 214
+  Transformation of optically active substances, 220
+    ----, suspended, 37, 69, 89, 113, 137, 155
+    ----, velocity of, 70
+  Transition curve, 66
+    ---- ---- of Glauber's salt, 68, 140
+    ---- ---- of rhombic and monoclinic sulphur, 37
+    ---- interval, 270, 277, 315
+    ---- point, 34
+    ---- ---- for double salts, 258
+    ---- ----, influence of pressure on the, 68
+    ---- points, as fixed points in thermometry, 140
+    ---- ----, methods of determining, 331
+    ---- ---- of polymorphic substances, 63
+  Triangle, graphic representation by, 235
+  Triethylamine and water, 101
+  Triple point, 27, 55
+    ---- ----, arrangement of curves round, 56
+    ---- ----, changes at, 58
+    ---- ----, ice, water, vapour, 27
+    ---- ----, ice II., ice III., and water, 33
+    ---- ----, metastable, 38
+    ---- ----, monoclinic sulphur, liquid, vapour, 38
+    ---- ----, monoclinic and rhombic sulphur, liquid, 38
+    ---- ----, monoclinic and rhombic sulphur, vapour, 34
+  {350}
+    ---- ----, red phosphorus, liquid, vapour, 47
+    ---- ----, rhombic sulphur, liquid, vapour, 38
+    ---- ---- solid, solid, vapour, 62
+    ---- ----, white phosphorus, liquid, vapour, 48
+
+          U
+  Univariant systems, 16
+
+          V
+  Van't Hoff, theorem of, 57
+  Vaporization curve, 63
+    ---- ----, interpolation and extrapolation of, 66
+    ---- ---- of water, 21, 23
+  Vapour pressure, constancy of, and formation of compounds, 90
+    ---- ----, dependence of, on solid phase, 88
+    ---- ----, influence of surface tension on, 2
+    ---- ---- in three-component systems, 261
+    ---- ----, measurement of, 91, 334
+    ---- ---- of calcium chloride solutions, 150
+    ---- ---- of ice, 25, 31
+    ---- ---- of small drops, 10
+    ---- ---- of sodium sulphate and water, 138
+  Vapour pressure of solid, solution, vapour, 126
+    ---- ---- of water, 21, 31
+  Variability of a system, 14, 16
+  Variance of a system, 16
+  Volatile components, two, 161
+
+          W
+  Water, 21
+    ----, acetic acid, chloroform, 241
+    ----, acetone, phenol, 248
+    ----, alcohol, ether, 246
+    ----, ----, chloroform, 246
+    ----, aniline, phenol, 250
+    ----, bivariant systems of, 29
+    ----, critical pressure of, 23
+    ----, critical temperature of, 23
+    ----, equilibrium between ice and, 25
+    ----, ---- between vapour and, 21
+    ----, ether, succinic nitrile, 252
+    ----, supercooled, 30
+    ----, ----, vapour pressure of, 31
+    ----, vaporization curve of, 21
+    ----, vapour pressure of, 23
+
+          Z
+  Zeolites, 176
+  Zinc, lead, silver, 246
+  ---- chloride in water, solubility of, 157
+
+THE END
+
+PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES.
+
+       *       *       *       *       *
+
+
+NOTES
+
+[1] Except when the volume of the liquid becomes exceedingly small, in
+which case the surface tension exerts an influence on the vapour pressure.
+
+[2] For reasons which will appear later (Chap. IV.), the volume of the
+vapour is supposed to be large in comparison with that of the solid and
+liquid.
+
+[3] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.
+
+[4] See, more especially, Vogt, _Die Silikatschmelzloesungen_.
+(Christiania, 1903, 1904.)
+
+[5] _Trans. Connecticut Acad._, 1874-1878.
+
+[6] Lehre von der chemischen Verwandtschaft der Koerper, 1777.
+
+[7] See Ostwald's _Klassiker_, No. 74.
+
+[8] Etudes sur les affinites chimiques, 1867; Ostwald's _Klassiker_, No.
+104.
+
+[9] Died April, 1903.
+
+[10] For a mathematical treatment of the Phase Rule the reader is referred
+to the volume in this series on Thermodynamics, by F. G. Donnan.
+
+[11] Liebig's _Annalen_, 1873, 170, 192; Ostwald, _Lehrbuch_, II. 2. 111.
+
+[12] The action of gravity and other forces being excluded (see p. 5).
+
+[13] It may seem as if this were a contradiction to what was said on p. 4
+as to the effect of the addition of ammonia or hydrogen chloride to the
+system constituted by solid ammonium chloride in contact with its products
+of dissociation. There is, however, no contradiction, because in the case
+of ammonium chloride the gaseous phase consists of ammonia and hydrogen
+chloride in equal proportions, and in adding ammonia or hydrogen chloride
+alone we are not adding the gaseous phase, but only a constituent of it.
+Addition of ammonia and hydrogen chloride together in the proportions in
+which they are combined to form ammonium chloride would cause no change in
+the equilibrium.
+
+[14] The vapour pressure of water in small drops is greater than that of
+water in mass, and the solubility of a solid is greater when in a state of
+fine subdivision than when in large pieces (_cf._ Hulett, _Zeitschr.
+physikal. Chem._, 1901, 37. 385).
+
+[15] See Ostwald, _Lehrbuch_, II. 2. 476, 934; Roozeboom, _Zeitschr.
+physikal. Chem._, 1894, 15. 150; _Heterogene Gleichgewichte_, I. p. 16;
+Wegscheider, _Zeitschr. physikal. Chem._, 1903, 43. 89.
+
+[16] Ostwald, _Lehrbuch_, II. 2. 478.
+
+[17] See also Hoitsema, _Zeitschr. physikal. Chem._ 1895, 17. 651.
+
+[18] The term "degree of freedom" employed here must not be confused with
+the same term used to denote the various movements of a gas molecule
+according to the kinetic theory.
+
+[19] Trevor, _Jour. Physical Chem._, 1902, 6. 136.
+
+[20] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
+Findlay, 2nd edit., p. 7. (Macmillan, 1904.)
+
+[21] See the volume in this series on _Thermodynamics_ by F. G. Donnan.
+
+[22] _Pogg. Annalen_, 1844, 61. 225.
+
+[23] _Memoires de l'Acad._, 26. 751.
+
+[24] _Phil. Trans._ 1884, 175. 461; 1892, A, 183. 107.
+
+[25] _Bihang Svenska Akad. Handl._ 1891, 17. I. 1.
+
+[26] Abh_andl. physikal.-tech. Reichsanstalt_, 1900, 3. 71.
+
+[27] Ostwald-Luther, _Physiko-chemische Messungen_, 2nd edit., p. 156.
+
+[28] _Annales chim. et phys._, 1892 [6], 26. 425.
+
+[29] The vapour pressure of water at 0deg has recently been very accurately
+determined by Thiesen and Scheel (_loc. cit._), and found to be 4.579 +/-
+0.001 mm. of mercury (at 0deg), or equal to 0.006025 atm.
+
+[30] Juhlin, _Bihang Svenska Akad. Handl._, 1891, 17. I. 58. See also
+Ramsay and Young, _loc. cit._
+
+[31] _Trans. Roy. Soc. Edin._, 1849, 16. 575.
+
+[32] _Proc. Roy. Soc. Edin._, 1850, 2, 267.
+
+[33] _Annalen der Physik_, 1899 [3], 68. 564; 1900 [4], 2. 1, 424. See
+also Dewar, _Proc. Roy. Soc._, 1880, 30. 533.
+
+[34] The pressure of 1 atmosphere is equal to 1.033 kilogm. per sq. cm.; or
+the pressure of 1 kilogm. per sq. cm. is equal to 0.968 atm.
+
+[35] Tammann, _loc. cit._, 1900, 2. 1, 424; cf. Goossens, _Arch. neerland_,
+1886, 20. 449.
+
+[36] J. Thomson, _Proc. Roy. Soc._, 1874, 22. 28.
+
+[37] A field is "enclosed" by two curves when these cut at an angle less
+than two right angles. It may be useful to remember that an invariant
+system is represented by a _point_, a univariant system by a _line_, and a
+bivariant system by an _area_.
+
+[38] _Phil. Trans._, 1724, 39. 78.
+
+[39] Juhlin, _loc. cit._, p. 61; cf. Ramsay and Young, _loc. cit._: Thiesen
+and Scheel, _loc. cit._
+
+[40] This small difference is due to experimental errors in the
+determination of the vapour pressures; a differential method betrayed no
+difference between the vapour pressure of ice and of water at 0deg.
+
+[41] _Phil. Mag._, 1874 [4], 47. 447; _Proc. Roy. Soc._, 1873, 22. 27.
+
+[42] _Pogg. Annalen_, 1858, 103, 206.
+
+[43] See _Phil. Trans._, 1884, 175, 461.
+
+[44] This phenomenon of distillation from the supercooled liquid to the
+solid has been very clearly observed in the case of furfuraldoxime (V.
+Goldschmidt, _Zeitschr. f. Krystallographie_, 1897, 28. 169).
+
+[45] _Annalen der Physik_, 1900 [4], 2. 1, 424.
+
+[46] A similar triple point has been determined by Tammann in the case of
+phenol (_Annalen der Physik_, 1902 [4], 9. 249).
+
+[47] _Annales chim. et phys._, 1821, 19. 414.
+
+[48] Lehmann, _Molekularphysik_, I. 153.; Arzruni, _Physikalische Chemie
+der Krystalle_. (Graham-Otto, _Lehrbuch der Chemie_, I. 3.)
+
+[49] Brodie, _Proc. Roy. Soc._, 1855, 7. 24.
+
+[50] That solid sulphur does possess a certain vapour pressure has been
+shown by Hallock, who observed the formation at the ordinary temperature of
+copper sulphide in a tube containing copper and sulphur (_Amer. Jour.
+Sci._, 1889 [3], 37. 405). See also Zenghelis, _Zeitschr. physikal.
+Chem._, 1904, 50. 219.
+
+[51] _Zeitschr. fuer Krystallographie_, 1884, 8. 593.
+
+[52] Van't Hoff, _Studies on Chemical Dynamics_, p. 163.
+
+[53] Reicher, _loc. cit._ See also Tammann, _Annalen der Physik_, 1899
+[3], 68. 663.
+
+[54] Tammann, _Annalen der Physik_, 1899 [3], 68. 633.
+
+[55] Rec. Trav. _Chim. Pays-Bas_, 1887, 6. 314.
+
+[56] Cf. van't Hoff, _Lectures on Physical Chemistry_, I., p. 27 (Arnold).
+
+[57] _Annalen der Physik_, 1899 [3], 68. 663.
+
+[58] Brauns, _Jahrbuch fuer Mineralogie_, 1899-1901, 13. Beilage, p. 39.
+
+[59] Fritsche, _Ber._, 1869, 2. 112, 540.
+
+[60] _De mirabilibus Auscultationibus_, Cap. 51 (_v._ Cohen, _Zeitschr.
+physikal. Chem._, 1901, 36. 513).
+
+[61] E. Cohen and C. van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 601;
+Cohen, _ibid._, 1900, 33. 59; 35. 588; 1901, 36. 513; Cohen and E.
+Goldschmidt, _ibid._, 1904, 50. 225.
+
+[62] _Zeitschr. physikal. Chem._, 1900, 33, 58.
+
+[63] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. Trav.
+Chim. Pays-Bas_, 1888, 7. 152.
+
+[64] Zincke, _Ber._, 1871, 4. 576.
+
+[65] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 313.
+
+[66] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.
+
+[67] Roozeboom, _ibid._, p. 179.
+
+[68] Schroetter, _Pogg. Annalen_, 1850, 81. 276; Troost and Hautefeuille,
+_Annales de Chim. et Phys._ 1874 [5], 2. 153; _Ann. Scient. Ecole Norm._
+1868 [2], II. 266.
+
+[69] Pedler, _Trans. Chem. Soc._, 1890, 57. 599.
+
+[70] Brodie, _Trans. Chem. Soc._, 1853, 5, 289.
+
+[71] This is a familiar fact in the case of the solubility in carbon
+disulphide.
+
+[72] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 170.
+
+[73] _Trans. Chem. Soc._, 1899, 57. 734.
+
+[74] Carnelley, _Trans. Chem. Soc._, 1876, 29. 489; 1878, 33. 275. V. Meyer
+and Riddle, _Ber._, 1893, 26. 2443.
+
+[75] Riecke, _Zeitschr. physikal. Chem._, 1890, 6. 411.
+
+[76] _Annalen der Physik._, 1898 [3], 66. 492.
+
+[77] _Zeitschr. physikal. Chem._, 1899, 28. 666.
+
+[78] See Naumann, _Ber._, 1872, 4. 646; Troost and Hautefeuille, _Compt.
+rend._, 1868, 66. 795; 1868, 67. 1345; Roozeboom, _Das Heterogene
+Gleichgewicht_, I. pp. 62, 171.
+
+[79] Mitscherlich, _Lieb. Annalen_, 1834, 12. 137; Deville and Troost,
+_Compt. rend._, 1863, 56. 891.
+
+[80] Beckmann, _Zeitschr. physikal. Chem._, 1890, 5. 79; Hertz, _ibid._, 6.
+358.
+
+[81] _Ber._, 1902, 35. 351. _Cf._ also, K. Schaum, _Annalen der Chem._,
+1898, 300. 221; R. Wegscheider and Kaufler, _Sitzungsber. kaiserl. Akad.
+Wissensch. in Wien_, 1901, 110, II. 606.
+
+[82] See also Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 177.
+
+[83] _Annales de Chim. et Phys._, 1874 [5], 2. 154.
+
+[84] _Compt. rend._, 1887, 104. 1505.
+
+[85] _Compt. rend._, 1868, 66. 795.
+
+[86] _Phil. Mag._, 1884 [5], 18. 210. See also Roozeboom, _Das Heterogene
+Gleichgewicht_, I. p. 177.
+
+[87] Brauns, _Neues Jahrbuch fuer Mineralogie_, 1900, 13. Beilage-Band, p.
+39; Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 181.
+
+[88] _Monatshefte_, 1888, 9. 435.
+
+[89] Gattermann, _Ber._, 1890, 53. 1738.
+
+[90] _Zeitschr. physikal. Chem._, 1889, 4. 468; _Annalen der Physik_, 1900
+[4], 2. 649.
+
+[91] Quincke, _Annalen der Physik_, 1894 [3], 53. 613; Tammann, _Annalen
+der Physik_, 1901 [4], 4. 524; 1902, 8. 103; Rotarski, _ibid._, 4. 528.
+
+[92] _Annalen der Physik_, 1900 [4], 2. 649.
+
+[93] _Annalen der Physik_, 1902 [4], 8. 911.
+
+[94] See, more especially, O. Lehmann, _Annalen der Physik_, 1900 [4], 2.
+649; Reinitzer, _Sitzungsber. kaiserl. Akad. zu Wien._, 1888, 94. (2), 719;
+97. (1), 167; Gattermann, _loc. cit._; Schenck, _Zeitschr. physikal.
+Chem._, 1897, 23. 703; 1898, 25. 337; 27. 170; 1899, 28. 280; Schenck and
+Schneider, _ibid._, 1899, 29. 546; Abegg and Seitz, _ibid._, 1899, 29. 491;
+Hulett, _ibid._, 1899, 28. 629; Coehn, _Zeitschr. Elektrochem._, 1904, 10.
+856: Bredig and Schukowsky, _ibid._, 3419. For a full account of the
+subject, the reader is referred to the work by Lehmann, _Fluessige
+Kristalle_ (Engelmann, 1904), or the smaller monograph by Schenck,
+_Kristallinische Fluessigkeiten und fluessige Kristalle_ (Engelmann, 1905).
+
+[95] A. C. de Kock, _Zeitschr. physikal. Chem._, 1904, 48. 129.
+
+[96] On account of the fact that all grades of rigidity have been realized
+between the ordinary solid and the liquid state, in the case both of
+crystalline and amorphous substances, it has been proposed to abandon the
+terms "solid" and "liquid," and to class bodies as "crystalline" or
+"amorphous," the passage from the one condition to the other being
+discontinuous; crystalline bodies possess a certain regular orientation of
+their molecules and a directive force, while in amorphous bodies these are
+wanting (see Lehmann, _Annalen der Physik_, 1900 [4], 2. 696).
+
+[97] Hulett, _loc. cit._
+
+[98] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 144. See also
+Schenck, _Kristallinische Fluessigkeiten und fluessige Kristalle_, p. 8
+(Engelmann, 1904).
+
+[99] The possible number of triple points in a one-component system is
+given by the expression (_n_(_n_ - 1)(_n_ - 2))/1.2.3, where _n_ is the
+number of phases (Riecke, _Zeitschr. physikal. Chem._, 1890, 6, 411). The
+number of triple points, therefore, increases very rapidly as the number of
+possible phases increases.
+
+[100] Duhem, _Zeitschr. physikal. Chem._, 1891, 8. 371. _Cf._ Roozeboom,
+_Das Heterogene Gleichgewicht_, p. 94 ff.
+
+[101] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 99.
+
+[102] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.
+
+[103] These changes can be predicted quantitatively by means of the
+thermodynamic equation, _dp_/_dt_ = Q/(T(_v_{2}_ - _v_{1}_)), provided the
+specific volumes of the phases are known, and the heat effect which
+accompanies the transformation of one phase into the other.
+
+[104] _Studies on Chemical Dynamics_, translated by Ewan, p. 218.
+
+[105] Le Chatelier, _Compt. rend._, 1884, 99. 786.
+
+[106] See _Principles of Inorganic Chemistry_, translated by Findlay, 2nd
+edit., p. 133. (Macmillan, 1904.)
+
+[107] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 474.
+
+[108] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 189.
+
+[109] Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 125. See also
+Zawidski, _Zeitschr. physikal. Chem._, 1904, 47. 727; van Eyk, _ibid._,
+1905, 51. 720.
+
+[110] Roberts-Austen, _Proc. Roy. Soc._, 63. 454; Spring, _Zeitschr.
+physikal. Chem._, 1894, 15. 65. See also p. 35.
+
+[111] Ramsay and Young, _Phil. Trans._, 1884, 175. 461; Allen, _Trans.
+Chem. Soc._, 1900, 77. 413.
+
+[112] Ramsay and Young, _Phil. Trans._ 1886, 177. 87.
+
+[113] This is exemplified in the well-known experiment with the cryophorus.
+
+[114] Tammann has, however, found that the fusion curve (solid in contact
+with liquid) of phosphonium chloride can be followed up to temperatures
+above the critical point (_Arch. neer._, 1901 [2], 6. 244).
+
+[115] _Phil. Mag._, 1886, 21. 33. See also S. A. Moss, _Physical Review_,
+1903, 16. 356.
+
+[116] This is found also in the case of bismuth. See Tammann, _Zeitschr.
+anorgan. Chem._, 1904, 40. 54.
+
+[117] See p. 57, footnote.
+
+[118] _Pogg. Annalen_, 1850, 81. 562.
+
+[119] Barus, _Amer. Jour. Sci._, 1892, 42. 125; Mack, _Compt. rend._, 1898,
+127. 361; Hulett, _Zeitschr. physikal. Chem._, 1899, 38. 629.
+
+[120] _Annalen der Physik_, 1899 [3], 68. 553, 629; 1900 [4], 1. 275; 2.
+1; 3. 161. See also Tammann, _Kristallisieren und Schmelzen_ (Leipzig,
+1903).
+
+[121] Ostwald, _Lehrbuch_, II. 2. 373; Poynting, _Phil. Mag._, 1881 [5],
+12. 2; Planck, _Wied. Annalen_, 1882, 15. 446.
+
+[122] Bakhuis Roozeboom, _Das Heterogene Gleichgewicht_, I. p. 91.
+
+[123] Lussana, _Il nuovo Cimento_, 1895 [4], 1. 105.
+
+[124] Tammann, _Zeitschr. physikal. Chem._, 1903, 46. 818.
+
+[125] Foote, _Zeitschr. physikal. Chem._, 1900, 33. 740.
+
+[126] Ostwald, _Zeitschr. physikal. Chem._, 1897, 22. 289.
+
+[127] Van't Hoff, _Arch, neer._, 1901, 6. 471.
+
+[128] See, for example, the determinations of the solubility of rhombic and
+monoclinic sulphur, by J. Meyer, _Zeitschr. anorg. Chem._, 1902, 33. 140.
+
+[129] _Zeitschr. physikal. Chem._, 1899, 32. 506.
+
+[130] Kastle and Reed, _Amer. Chem. Jour._, 1902, 27. 209.
+
+[131] _Zeitschr. physikal. Chem._, 1900, 35. 581.
+
+[132] _Compt. rend._, 1882, 95. 1278; 1884, 97. 1298, 1366, 1433.
+
+[133] _Zeitschr. physikal. Chem._, 1893, 12. 545.
+
+[134] _Sitzungsber. Wiener Akad._, 1894, 103. IIa. 226.
+
+[135] _Zeitschr. physikal. Chem._, 23-29. See also Kuester, _ibid._, 25-28.
+
+[136] _Zeitschr. physikal. Chem._, 1897, 24. 152.
+
+[137] _Ibid._, 1898, 27. 585.
+
+[138] See W. Guertler, _Zeitschr. anorgan. Chem._, 1904, 40. 268; Tammann,
+_Zeitschr. Elektrochem._, 1904, 10. 532.
+
+[139] E. von Pickardt, _Zeitschr. physikal. Chem._, 1902, 42. 17.
+
+[140] _Zeitschr. physikal. Chem._, 1904, 48. 467.
+
+[141] M. Padoa, _Accad. Lincei, Atti_, 1904, 13. 329.
+
+[142] Deville, _Compt. rend._, 1852, 34. 561; Payen, _ibid._, 1852, 34.
+508; Debray, _ibid._, 1858, 46. 576. It has also been found by Jaffe
+(_Zeitschr. physikal. Chem._, 1903, 43. 465) that when spontaneous
+crystallization from solution occurs, the less stable form always separates
+first when purification has been carried sufficiently far.
+
+[143] Brauns, _Neues Jahrbuch fuer Mineralogie_, 1899, 13. (Beilage Band)
+84.
+
+[144] _Lehrbuch_, II. 2. 445. See also _Principles of Inorganic Chemistry_,
+2nd edit., p. 210 ff.
+
+[145] Schaum and Schoenbeck, _Annalen der Physik_, 1902 [4], 8. 652. See
+also Chr. Fuechtbauer, _Zeitschr. physikal. Chem._, 1904, 48. 549.
+
+[146] Ramsay and Young, _Phil. Trans._, 1886, 177. 87.
+
+[147] See volume in this series on _Chemical Dynamics_, by Dr. J. W.
+Mellor.
+
+[148] Isambert, _Compt. rend._, 1881, 92. 919; 1882, 94. 958; 1883, 96.
+643. Walker and Lumsden, _Jour. Chem. Soc._, 1897, 71. 428.
+
+[149] _Compt. rend._, 1867, 64. 603.
+
+[150] _Compt. rend._, 1883, 102. 1243.
+
+[151] _Compt. rend._, 1868, 66, 1259.
+
+[152] Horstmann, _Ber._, 1876, 9. 749.
+
+[153] _Loc. cit._
+
+[154] For the reasons for choosing anhydrous salt and water instead of salt
+hydrate and water as components, see p. 14.
+
+[155] See Ostwald, _Lehrbuch_, II. 2. 527.
+
+[156] Ostwald, _Lehrbuch_, II. 2. 538.
+
+[157] _Zeitschr. physikal. Chem._, 1889, 4. 43.
+
+[158] _Ber._, 1876, 9. 749.
+
+[159] See, for example, van't Hoff, _Lectures on Theoretical and Physical
+Chemistry_, I. p. 62 (Arnold).
+
+[160] _Jour. Chem. Soc._, 1877, 32. 395.
+
+[161] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.
+
+[162] _Zeitschr. physikal. Chem._, 1887, 1. 5; 1895, 17. 52.
+
+[163] It is important to powder the salt, since otherwise the dehydration
+of the hydrate and the production of equilibrium occurs with comparatively
+great tardiness.
+
+[164] A chemical individual is a substance which persists as a phase of
+constant composition when the conditions of temperature, pressure, and
+composition of the other phases present, undergo continuous alteration
+within certain limits--the limits of existence of the substance (Wald,
+_Zeitschr. physikal. Chem._, 1897, 24. 648).
+
+[165] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323; Ostwald,
+_Lehrbuch_, I. 606.
+
+[166] That mercury does dissolve in water can be argued from analogy, say,
+with mercury and bromonaphthalene. At the ordinary temperature these two
+liquids appear to be quite insoluble in one another, but at a temperature
+of 280deg the mercury dissolves in appreciable quantity; for on heating a
+tube containing bromonaphthalene over mercury the latter sublimes _through_
+the liquid bromonaphthalene and condenses on the upper surface of the tube.
+
+[167] _Phil. Mag._, 1884, [5], 18. 22; 495.
+
+[168] _Wied. Annalen_, 1886, 28. 305.
+
+[169] _Zeitschr. physikal. Chem._, 1898, 26. 433.
+
+[170] Rothmund, _loc. cit._
+
+[171] Rothmund, _loc. cit._
+
+[172] A similar behaviour is found in the case of diethylamine and water
+(R. T. Lattey, _Phil. Mag._, 1905, [6], 10, 397).
+
+[173] C. S. Hudson, _Zeitschr. physikal. Chem._, 1904, 47. 113.
+
+[174] Konowaloff, _Wied. Annalen_, 1881, 14. 219. Ostwald, _Lehrbuch_, II.
+2. 687. Bancroft, _Phase Rule_, p. 96.
+
+[175] Konowaloff, _loc. cit._
+
+[176] Roozeboom, _Zeitschr. physikal. Chem._, 1891, 8. 526; _Rec. Trav.
+Chim. Pays-Bas_, 1884, 3. 38.
+
+[177] Konowaloff, _loc. cit._ Cf. Bancroft, _Phase Rule_, p. 100.
+
+[178] _Phil. Mag._, 1884 [5], 18. 503.
+
+[179] See, for example, Walker, _Introduction to Physical Chemistry_, 3rd
+edit., p. 86 (Macmillan, 1903). Consult also Young, _Fractional
+Distillation_ (Macmillan, 1903), or Kuenen, _Verdampfung und Verfluessigung
+von Gemischen_ (Barth, 1906), where the subject is fully treated.
+
+[180] Since this is the only phase of variable composition present.
+
+[181] E. von Stackelberg, _Zeitschr. physikal. Chem._, 1896, 20. 337. If
+the change of volume which accompanies solution, and the heat effect are
+known, the quantitative change of the solubility with the pressure can be
+calculated (Braun, _Zeitschr. physikal. Chem._, 1887, 1. 259).
+
+[182] Van't Hoff, _Arch. neerland._ 1901 [2], 6. 471.
+
+[183] Tilden and Shenstone, _Phil. Trans._ 1884, 175. 23; Hulett and Allen,
+_Jour. Amer. Chem. Soc._ 1902, 24. 667; Andreae, _Jour. prak. Chem._ 137.
+474; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Mylius and v. Wrochem,
+_Ber._ 1900, 33. 3689.
+
+[184] E. von Stackelberg, _Zeitschr. physikal. Chem._ 1896, 20. 159; 1898,
+26. 533; Lumsden, _Jour. Chem. Soc._, 1902, 81. 350; Holsboer, _Zeitschr.
+physikal. Chem._, 1902, 39. 691.
+
+[185] Reicher and van Deventer, _Zeitschr. physikal. Chem._ 1890, 5. 559;
+cf. Ostwald, _Lehrbuch_, II. 2. 803.
+
+[186] It has been shown that the formula of Ramsay and Young (p. 66) can be
+applied (with certain restrictions) to the interpolation and extrapolation
+of the solubility curve of a substance provided two (or three) points on
+the curve are known. In this case T, T_{1}, etc., refer to the temperatures
+at which the two substances--one the solubility curve of which is known,
+the other the solubility curve of which is to be calculated--have equal
+solubilities, instead of, as in the previous case, equal vapour pressures.
+(Findlay, _Proc. Roy. Soc._, 1902, 69. 471; _Zeitschr. physikal. Chem._,
+1903, 42. 110.)
+
+[187] W. Mueller and P. Kaufmann, _Zeitschr. physikal. Chem._ 1903, 42.
+497.
+
+[188] W. O. Rabe, _Zeitschr. physikal. Chem._, 1901, 38. 175.
+
+[189] With regard to the limits of supersaturation and the spontaneous
+crystallization of the solute from supersaturated solutions, see Jaffe,
+_Zeitschr. physikal. Chem._, 1903, 43. 565, and the very interesting paper
+by Miers and Isaac, _Trans. Chem. Soc._, 1906, 89. 413.
+
+[190] _Annales chim. phys._, 1894 [7], 2. 524.
+
+[191] _Phil. Trans._, 1884, 175. 23.
+
+[192] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 543.
+
+[193] _Zeitschr. physikal. Chem._, 1903, 43. 313.
+
+[194] Guthrie, _Phil. Mag._, 1875, [4], 49. 1; 1884, [5], 17. 462.
+
+[195] See Roloff, _Zeitschr. physikal. Chem._, 1895, 17. 325; Guthrie,
+_loc. cit._
+
+[196] Guthrie, _Phil. Mag._, _loc. cit._ Cf. Ostwald, _Lehrbuch_, II. 2.
+843.
+
+[197] Guthrie, _Phil. Mag._, 1875 [4], 49. 269.
+
+[198] _Ber._, 1877, 20. 2223.
+
+[199] _Silz-Ber. Wien. Akad._, 1880, 81. II. 1058.
+
+[200] Guthrie, _Phil. Mag._, 1875 [4], 49. 206.
+
+[201] If in the neighbourhood of the cryohydric point solution should be
+accompanied by an evolution of heat, then as the solubility would in that
+case increase with fall of temperature, salt would pass into solution.
+
+[202] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193.
+
+[203] _Zeitschr. physikal. Chem._, 1897, 23. 418.
+
+[204] Provided the solid nitrile is not present in too great excess.
+
+[205] _Wied. Annalen_, 1886, 28. 328. Cf. Ostwald, _Lehrbuch_, II. 2. 872.
+
+[206] Walker, _Zeitschr. physikal. Chem._, 1890, 5. 193. Schreinemakers,
+_ibid._, 1897, 23. 417. Roozeboom, _Rec. trav. chim. Pays-Bays_, 1889, 8.
+257. Bruner, _Zeitschr. physikal. Chem._, 1897, 23. 542.
+
+[207] Van't Hoff, _Lectures on Theoretical Chemistry_, I. p. 42. Ostwald,
+_Lehrbuch_, II. 2. 824.
+
+[208] Ostwald, _Principles of Inorganic Chemistry_, translated by A.
+Findlay, 2nd edit., p. 453 (Macmillan, 1904); Skirrow and Calvert,
+_Zeitschr. physikal. Chem._, 1901, 37. 217.
+
+[209] _Vide_ Loewel, _Annales chim. phys._, 1857 [3], 49. 32. Cf.
+Loewenherz, _Zeitschr. physikal. Chem._, 1895, 18. 82.
+
+[210] Loewel, _loc. cit._ Gay-Lussac, _Annales chim. phys._, 1819, 11. 296.
+For the solubility at higher temperatures, see Tilden and Shenstone, _Phil.
+Trans._, 1884, 175. 23. Etard, _Annales chim. phys._, 1894 [7], 2. 548.
+
+[211] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690; Richards and
+Wells, _ibid._, 1903, 43. 465. This temperature is not quite the same as
+that of the _quadruple point_ anhydrous salt--hydrated
+salt--solution--vapour, because the latter is the temperature at which the
+system is under the pressure of its own vapour. Since, however, the
+influence of pressure on the solubility is very slight (p. 107), the
+position of the two points will not be greatly different. The quadruple
+point was found by Cohen (_Zeitschr. physikal. Chem._, 1894, 14. 90) to be
+32.6deg and 30.8 mm. of mercury.
+
+[212] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+185. Cf. Cohen, _ibid._, 1894, 14. 88.
+
+[213] Debray, _Compt. rend._, 1868, 66. 194.
+
+[214] Richards, _Zeitschr. physikal. Chem._, 1898, 26. 690. A number of
+other salt hydrates, having transition-points ranging from 20deg to 78deg,
+which might be used for the same purpose, have been given by Richards and
+Churchill, _ibid._, 1899, 28. 313.
+
+[215] _Zeitschr. physikal. Chem._, 1903, 46. 818.
+
+[216] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 67.
+
+[217] Cohen, _Zeitschr. physikal. Chem._, 1894, 14. 90.
+
+[218] Ziz, _Schweigger's Journal_, 1815, 15. 166. See Ostwald, _Lehrbuch_,
+II. 2. 717.
+
+[219] See, for example, the solubility determinations published in
+_Wissenschaftliche Abhandl. der physikalisch-technischen Reichsanstalt_,
+Vol. III., or in the _Berichte_, for the years 1897-1901.
+
+[220] Meusser, _Ber._, 1901, 34. 2440.
+
+[221] Mylius and von Wrochem, _Ber._, 1900, 33. 3693.
+
+[222] Walker and Fyffe, _Jour. Chem. Soc._, 1903, 83. 180.
+
+[223] _Monatshefte_, 1887, 8. 601.
+
+[224] The equilibria between calcium chloride and water have been most
+completely studied by Roozeboom (_Zeitschr. physikal. Chem._, 1889, 4. 31).
+
+[225] Hammerl, _Sitzungsber. Wien. Akad._, 2^{te} Abteil, 1878, 78. 59.
+Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.
+
+[226] Lidbury, _Zeitschr. physikal. Chem._, 1902, 39. 453. The curvature at
+the melting point is all the greater the more the compound is dissociated
+into its components in the liquid state. If the compound is _completely
+undissociated_, even in the vapour phase, the two branches of the curve
+will _intersect_, (_e.g._ pyridine and methyl iodide; Aten, _Versl. Konink.
+Akad. Wetensch. Amsterdam_, 1905, 13. 462). The smaller the degree of
+dissociation, therefore, the sharper will be the bend. (See Stortenbeker,
+_Zeitschr. physikal. Chem._, 1892, 10. 194.) From the extent of flattening
+of the curve, it is also possible, with some degree of approximation, to
+calculate the degree of dissociation of the substance in the fused state.
+(See Roozeboom and Aten, _Zeitschr. physikal. Chem._, 1905, 53. 463;
+Kremann, _Zeitschr. Elektrochem._, 1906, 12. 259.)
+
+[227] See Roozeboom, _Zeitschr. physikal. Chem._, 1889, 4. 31.
+
+[228] Tammann, _Wied. Annalen_, 1899, 68. 577.
+
+[229] Duhem, _Journ. Physical Chem._, 1898, 2. 31.
+
+[230] Gibbs, _Trans. Conn. Acad._, 3. 155; Saurel, _Journ. Phys. Chem._,
+1901, 5. 35.
+
+[231] In the case of the fusion of a compound of two components with
+formation of a liquid phase of the same composition, the temperature is a
+maximum; in the case of liquid mixtures of constant boiling-point, the
+temperature may be a minimum (p. 105).
+
+[232] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477. The formula of
+ferric chloride has been doubled, in order to avoid fractions in the
+expression of the water of crystallization.
+
+[233] Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 477.
+
+[234] A similar series of hydrates is formed by zinc chloride and water
+(Dietz and Mylius, _Zeitschr. anorg. Chem._, 1905, 44. 209).
+
+[235] Meyerhoffer, _Ber._, 1897, 30. 1810.
+
+[236] Walden, _Ber._, 1899, 32. 2863.
+
+[237] _Zeitschr. physikal. Chem._, 1903, 42. 432.
+
+[238] This composition was also confirmed by measurements of the vapour
+pressure (cf. p. 90).
+
+[239] Since all substances are no doubt volatile to a certain extent at
+some temperature, it is to be understood here that the substances are
+appreciably volatile at the temperature of the experiment.
+
+[240] For a general discussion of the partial pressures in a system of two
+components, see Bancroft, _Journ. Physical Chem._, 1899, 3. 1.
+
+[241] _Zeitschr. physikal. Chem._, 1889, 3. 11; _Rec. trav. chim.
+Pays-Bas_, 1888, 7. 152.
+
+[242] The composition of a solution is represented symbolically by placing
+a double wavy line between the symbols of the components, and indicating
+the number of atoms present in the ordinary manner: thus, I [wavy] Cl_{_x_}
+represents a solution containing _x_ atoms of chlorine to one atom of
+iodine (Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 450).
+
+[243] Since iodine monochloride in the liquid state is only very slightly
+dissociated, the bend at C is very sharp (see p. 147, footnote). See also
+the investigation of the system pyridine and methyl iodide (Aten, _Versl.
+Konink. Akad. Wetensch. Amsterdam_, 1905, 13. 462).
+
+[244] This upper branch of the curve is not shown in the figure, as the
+ordinate corresponding to 30deg would be very great.
+
+[245] Stortenbeker, _Zeitschr. physikal. Chem._, 1889, 3. 22.
+
+[246] Ramsay and Young, _Journ. Chem. Soc._, 1886, 49. 458.
+
+[247] Van't Hoff, _Lectures on Physical Chemistry_, I. p. 77 (Arnold).
+
+[248] This is different from what we found in the case of non-volatile
+solutes (p. 126). In the present case, the _partial pressure_ of the iodine
+in the vapour will be lowered by addition of chlorine, but the _total
+pressure_ is increased.
+
+[249] The diminution of volume is supposed to be carried out at constant
+temperature. The pressure and the composition of the phases must,
+therefore, remain unchanged, and only the relative amounts of these can
+undergo alteration.
+
+[250] At point _b_ the ratio of chlorine to iodine in the solution is less
+than in the monochloride, so that by the separation of this the excess of
+chlorine yielded by the condensation of the vapour is removed.
+
+[251] Roozeboom, _Rec. trav. chim. Pays-Bas_, 1884, 3. 29; 1885, 4. 65;
+_Zeitschr. physikal. Chem._, 1888, 2. 450.
+
+[252] Two curves "enclose" a field when they form with one another an angle
+less than two right angles.
+
+[253] Roozeboom, _Zeitschr. physikal. Chem._, _loc. cit._
+
+[254] Van't Hoff, _Zeitschr. physikal. Chem._, 1890, 5. 323.
+
+[255] Bancroft has proposed to restrict the term "occlusion" to the
+formation of solid solutions, and to apply "adsorption" only to effects
+which are primarily due to surface tension. Such a distinction, however,
+would probably be very difficult to carry through, for although adsorption
+may, in large measure, be due to surface tension, the behaviour of adsorbed
+substances is similar to that of substances existing in solid solutions.
+
+[256] Tammann, _Wied. Annalen_, 1897, 63. 16; _Zeitschr. physikal. Chem._,
+1898, 27. 323.
+
+[257] See, for example, Chappuis, _Wied. Annalen_, 1881, 12. 161; Joulin,
+_Annal. chim. phys._, 1881, [5], 22. 398; Kayser, _Wied. Annalen_, 1881,
+12. 526.
+
+[258] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1.
+
+[259] _Annales chim. phys._, 1874, [5], 2. 279.
+
+[260] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 1; Dewar, _Phil.
+Mag._, 1874, [4], 47, 324, 342; Mond, Ramsay and Shields, _Proc. Royal
+Soc._, 1897, 62. 290.
+
+[261] _Loc. cit._
+
+[262] It is noteworthy that the form of curve obtained for hydrogen and
+palladium bears a striking resemblance to that for the dehydration of
+colloids containing absorbed water, _e.g._ silicic acid (_vide_ van
+Bemmelen, _Zeitschr. anorg. Chem._, 1897-1900. Cf. Zacharias, _Zeitschr.
+physikal. Chem._, 1902, 39. 480).
+
+[263] _Zeitschr. physikal. Chem._, 1890, 5. 322.
+
+[264] Kuester, _Zeitschr. physikal. Chem._, 1895, 17. 367. Bodlaender,
+_Neues Jahrbuch f. Mineralogie_, 1898-99, Beilage Band, 12. 92.
+
+[265] Bruni and Padoa, _Atti Accad. Lincei_, 1902 [5], 11. 1; 565.
+
+[266] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 385; Bruni, _Rend.
+Accad. Lincei_, 1898, 2. 138, 347. For a general account of "solid
+solutions" the reader is referred to Bruni, "_Ueber feste Loesungen_"
+(Ahrens'sche Sammlung), and to Bodlaender, _loc. cit._ For the formation
+and transformation of liquid mixed crystals, see A. C. de Kock, _Zeitschr.
+physikal. Chem._, 1904, 48. 129.
+
+[267] In discussing the various systems which may be obtained here,
+Roozeboom (_loc. cit._) made use of the variation of the thermodynamic
+potential (p. 29) with the concentration. In spite of the advantages which
+such a treatment affords, the temperature-concentration diagram has been
+adopted as being more readily understood and as more suitable for an
+elementary discussion of the subject.
+
+[268] These curves are also called the "liquidus" and the "solidus" curve
+respectively.
+
+[269] Kuester, _Zeitschr. physikal. Chem._, 1895, 17. 360.
+
+[270] Kuester, _ibid._, 1891, 8. 589.
+
+[271] It should be remarked that the behaviour described here will hold
+strictly only when the solid mixed crystals undergo change sufficiently
+rapidly to be always in equilibrium with the liquid. This, however, is not
+always the case (see Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494;
+van Wyk, _Zeitschr. anorg. Chem._, 1905, 48. 25), and complete
+solidification will not in this case take place at the temperature
+corresponding with the line _dc_ in Fig. 50, but only at a lower
+temperature.
+
+[272] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 469.
+
+[273] Reinders, _Zeitschr. physikal. Chem._, 1900, 32. 494.
+
+[274] Hissink, _Zeitschr. physikal. Chem._, 1900, 32. 542.
+
+[275] Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 430.
+
+[276] Cady, _Journ. Physical. Chem._, 1899, 3. 127.
+
+[277] See Roberts-Austen and Stansfield, _Rapports du congres international
+de physique_, 1900, I. 363.
+
+[278] Heycock and Neville, _Proc. Roy. Soc._, 1903, 71. 409. For the
+partial liquefaction of mixed crystals on cooling, see also A. C. de Kock
+(_Zeitschr. physikal. Chem._, 1904, 48. 129).
+
+[279] Armstrong, _Watt's Dictionary of Chemistry_ (Morley and Muir), III.,
+p. 88. See also Lowry, _Jour. Chem. Soc._, 1899, 75. 211.
+
+[280] See Bancroft, _Journ. Physical Chem._, 1898, 2. 143; Roozeboom,
+_Zeitschr. physikal. Chem._, 1899, 28. 288.
+
+[281] Hylotropic substances are such as can undergo transformation into
+other substances of the same composition (Ostwald, _Lehrbuch_, II. 2. 298).
+
+[282] Also called Equilibrium Point (Lowry).
+
+[283] For a discussion of these systems, see Roozeboom, _Zeitschr.
+physikal. Chem._, _loc. cit_.
+
+[284] See Bancroft, _loc. cit._, p. 147; Wegscheider, _Sitzungsber. Wiener
+Akad._, 1902, 110. 908.
+
+[285] Reference may be made here to the term "stability limit," introduced
+by Knorr (_Annalen_, 1896, 293. 88) to indicate that temperature above
+which liquefaction and isomeric change takes place. As employed by Knorr
+and others, the term does not appear to have a very precise meaning, since
+it is used to denote, not the temperature at which these changes can occur,
+but the temperature at which the change is rapid (vide _Annalen_, 1896,
+293. 91; 1899, 306. 334); and the introduction of an indefinite velocity of
+change renders the temperature of the stability limit also somewhat
+indefinite. The definiteness of the term is also not a little diminished by
+the fact that the "limit" can be altered by means of catalytic agents.
+Since, as we have seen, the stable modification can always undergo isomeric
+change and liquefy at temperatures above the natural freezing point, but
+not below that point; and, further, the less stable modification can
+undergo isomeric transformation and liquefy at temperatures above the
+eutectic point, but will not liquefy at temperatures below that; it seems
+to the author that it would be more precise to identify these two
+points--the natural freezing point and the eutectic point--which are not
+altered by catalytic agents, with the "stability limits" of the stable and
+unstable modification respectively. A perfectly definite meaning would
+thereby be given to the term. In the case of those substances which do not
+undergo appreciable isomeric change at the temperature of the melting
+point, the stability limits would be the points G and H, Fig. 60.
+
+[286] Cameron, _Journ. Physical Chem._, 1898, 2. 409.
+
+[287] Carveth, _Journ. Phys. Chem._, 1898, 2. 159. See also Dutoit and
+Fath, _Journ. chim. phys_., 1903, 1. 358; Findlay, _Trans. Chem. Soc._,
+1904, 85. 403.
+
+[288] Hollmann, _Zeitschr. physikal. Chem._, 1903, 43. 129.
+
+[289] For other examples of the application of the Phase Rule to isomeric
+substances, see _Journ. Physical Chem._, vols. 2. _et seq._; Findlay,
+_Trans. Chem. Soc._, 1904, 85. 403.
+
+[290] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 30. 410.
+
+[291] See also Saposchnikoff, _Zeitschr. physikal. Chem._, 49. 688;
+Kremann, _Monatshefte_, 1904, 25. 1215, 1271, 1311.
+
+[292] J. C. Philip, _Journ. Chem. Soc._, 1903, 83. 821.
+
+[293] _Cf._ also Paterno and Ampolla, _Gazzetta chim. ital._, 1897, 27.
+481.
+
+[294] Philip, _loc. cit._, p. 826.
+
+[295] Philip, _loc. cit._, p. 829. Compare curves for iodine monochloride,
+Fig. 42, p. 162.
+
+[296] Kuriloff, _Zeitschr. physikal. Chem._, 1897, 23. 676.
+
+[297] Ladenburg, _Ber._, 1895, 28. 163; 1991.
+
+[298] Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 494; Adriani,
+_ibid._, 1900, 33. 453.
+
+[299] Adriani, _Zeitschr. physikal. Chem._, 1900, 33. 453.
+
+[300] A. Findlay and Miss E. Hickmans.
+
+[301] Kipping and Pope, _Journ. Chem. Soc._, 1897, 71. 993.
+
+[302] See Roozeboom, _Zeitschr. physikal. Chem._, 1899, 28. 512; Adriani,
+_ibid._, 1900, 33. 473; 1901, 36. 168.
+
+[303] In this connection reference should be made more especially to the
+paper by Roberts-Austen and Stansfield, "Sur la constitution des alliages
+metalliques," in the _Rapports du congres international de physique_, 1900,
+I. 363; J. A. Mathews, _Journ. of the Franklin Inst._, 1902; Gautier,
+_Compt. rend._, 1896, 123. 109; Roberts-Austen, "Reports of the Alloys
+Research Committee," in _Journ. Inst. Mechan. Engineers_, from 1891 to
+1904; and the papers by Heycock and Neville, published in the _Journ. Chem.
+Soc._, and the _Trans. Roy. Soc._ since 1897; also Neville, _Reports of the
+British Association_, 1900, p. 131. Reference must also be made to the
+important metallographic investigations by Tammann and his pupils, and of
+Kurnakoff (_Zeitschr. anorgan. Chem._, vol. 40 and onwards), and also to
+those of Shepherd, _Journ. Physical Chem._, 8. A bibliography of the alloys
+is given in _Zeitschr. anorgan. Chem._, 1903, 35. 249.
+
+[304] Kurnakoff and Puschin, _Zeitschr. anorgan. Chem._, 1902, 30. 104.
+
+[305] Gautier, _Bull. Soc. d'Encouragement_, 1896 [5], 1. 1312.
+
+[306] Heycock and Neville, _Phil. Trans._, 1900, 194. 201.
+
+[307] Gautier, _loc. cit._ See also Roberts-Austen and Rose, _Proc. Roy.
+Soc._, 1903, 71. 161.
+
+[308] Heycock and Neville, _Journ. Chem. Soc._, 1897, 71. 414.
+
+[309] See Roberts-Austen, _Introduction to Metallurgy_, 5th edit., p. 102;
+Bakhuis Roozeboom, _Journ. Iron and Steel Inst._, 1900, II. 311; _Zeitschr.
+physikal. Chem._, 1900, 34. 437; von Jueptner, _Siderology_, p. 223
+(translation by C. Salter); van't Hoff, _Zinn, Gips, und Stahl_, p. 24, or
+_Acht Vortraege ueber physikalische Chemie_, p. 37. Further, Roozeboom,
+_Zeitschr. Elektrochem._, 1904, 10. 489; E. Heyn, _ibid._, p. 491;
+Carpenter and Keeling, _Journ. Iron and Steel Inst._, 1904, 65. 224.
+
+[310] The melting point of pure iron is given by Carpenter and Keeling
+(_Journ. Iron and Steel Inst._, 1904, 65. 224) as 1505deg.
+
+[311] _Zeitschr. fuer Elektrochem._, 1904, 10. 491.
+
+[312] See also Hiorns, _Journ. Soc. Chem. Ind._, 1906, 25. 50.
+
+[313] Bancroft, _Jour. Physical Chem._, 1902, 6. 178; Bell and Taber,
+_ibid._, 1906, 10. 120.
+
+[314] The method to be followed when the third component enters into the
+solid phase will be explained later.
+
+[315] Tammann, _Zeitschr. anorg. Chem._, 1903, 37. 303; 1905, 45. 24.
+Reference may be made here to the registering pyrometer of Kurnakoff,
+_Zeitschr. anorg. Chem._, 1904, 42. 184.
+
+[316] In this connection, see Doelter, _Physikalisch-chemisch Mineralogie_
+(Barth, 1901); Meyerhoffer, _Zeitschr. f. Kristallographie_, 1902, 36. 593;
+Guthrie, _Phil. Mag._, 1884 [5], 17. 479; Le Chatelier, _Compt. rend._,
+1900, 130. 85; and especially E. Baur, _Zeitschr. physikal. Chem._, 1903,
+42. 567; J. H. L. Vogt, _Zeitschr. Elektrochem._, 1903, 9. 852, and _Die
+Silikatschmelzloesungen_, Parts I. and II. (Christiania, 1903, 1904). See
+also N. V. Kultascheff, _Zeitschr. anorg. Chem._, 1903, 35. 187.
+
+[317] G. G. Stokes, _Proc. Roy. Soc._, 1891, 49. 174; Gibbs, _Trans. Conn.
+Acad._, 1876, 3. 176; Roozeboom, _Zeitschr. physikal. Chem._, 1894, 15.
+147.
+
+[318] This figure has been taken from Ostwald's _Lehrbuch_, II. 2. 984.
+
+[319] Roozeboom, _Zeitschr. physikal. Chem._, 1893, 12. 369.
+
+[320] C. R. A. Wright, _Proc. Roy. Soc._, 1891, 49. 174; 1892, 50. 375.
+
+[321] The distribution coefficient will not remain constant because, apart
+from other reasons, the mutual solubility of chloroform and water is
+altered by the addition of the acid.
+
+[322] Bancroft, _Physical Review_, 1895, 3. 21; Schreinemakers, _Zeitschr.
+physikal. Chem._, 1897, 23. 652, and subsequent volumes.
+
+[323] C. R. A. Wright, _Proc. Roy. Soc._, 1889-1893.
+
+[324] C. R. A. Wright, _Proc. Roy. Soc._, 1892, 50. 390.
+
+[325] Bodlaender, _Berg- und Huettenmaenn. Ztg._, 1897, 56. 331.
+
+[326] C. R. A. Wright, _Proc. Roy. Soc._, _loc. cit._
+
+[327] Schreinemakers, _Zeitschr. physikal. Chem._, 1900, 33. 78.
+
+[328] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 27. 95.
+
+[329] Schreinemakers, _Zeitschr. physikal. Chem._, 1899, 29. 577.
+
+[330] Schreinemakers, _Zeitschr. physikal. Chem._, 1898, 25. 543.
+
+[331] Charpy, _Compt. rend._, 1898, 126. 1569. Compare the curves for the
+system KNO_{3}--NaNO_{3}--LiNO_{3} (H. R. Carveth, _Journ. Physical Chem._,
+1898, 2. 209). Also alloys of Pb--Sn--Bi (E. S. Shepherd, _Journ. Physical
+Chem._, 1902, 6. 527).
+
+[332] It should be remembered that in the triangular diagram a _line_
+parallel to one of the sides indicates, at a given temperature, a constant
+amount of the component represented by the opposite corner of the triangle;
+and, hence, points in a _plane_, parallel to one face of a right prism,
+will indicate for different temperatures, variation in the amounts of two
+components, but constancy in the amount of the third.
+
+[333] _Gazzetta chim. ital._, 1898, 28. II. 520.
+
+[334] Bruni, _Gazzetta chim. ital._, 1898, 28. II. 508; 1900, 30. I. 35.
+
+[335] _Zeitschr. physikal. Chem._, 1900, 36. 168.
+
+[336] For a discussion of these systems, see van't Hoff, _Bildung und
+Spaltung von Doppelsalzen_ (Leipzig, 1897).
+
+[337] Van Leeuwen, _Zeitschr. physikal. Chem._, 1897, 23. 35.
+
+[338] Meyerhoffer, _Zeitschr. physikal. Chem._, 1889, 3. 336; 1890, 5. 97.
+
+[339] Reicher, _Zeitschr. physikal. Chem._, 1887, 1. 220.
+
+[340] For other examples of the formation and decomposition of double salts
+at a transition point, the reader is referred to the work by van't Hoff,
+already cited, on the _Bildung und Spaltung von Doppelsalzen_; or to
+Bancroft, _Phase Rule_, p. 180.
+
+[341] Bancroft, _Phase Rule_, p. 183.
+
+[342] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 514.
+
+[343] The influence of pressure on the transition point in the case of
+tachydrite has been determined by van't Hoff, Kenrick, and Dawson
+(_Zeitschr. physikal. Chem._, 1901, 39. 27, 34; van't Hoff, _Zur Bildung
+der ozeanischen Salzablagerungen_, I. p. 66--Brunswick, 1905). This salt is
+formed from magnesium chloride and calcium chloride at 22deg, in accordance
+with the equation--
+
+  2MgCl_{2}.6H_{2}O + CaCl_{2}.6H_{2}O = Mg_{2}CaCl_{6}.12H_{2}O + 6H_{2}O
+
+Increase of pressure raises the transition point, because the formation of
+tachydrite is accompanied by increase of volume; the elevation being
+0.016deg for an increase of pressure of 1 atm. The number calculated from
+the theoretical formula (p. 57) is 0.013deg for 1 atm.
+
+If one calculates the influence of the pressure of sea-water on the
+temperature of formation of tachydrite (which is of interest on account of
+the natural occurrence of this salt), it is found that a depth of water of
+1500 metres, exerting a pressure of 180 atm., would alter the temperature
+of formation of tachydrite by only 3deg. The effect is, therefore,
+comparatively unimportant.
+
+[344] Roozeboom, _Zeitschr. physical. Chem._, 1887, 1. 227.
+
+[345] _Zeitschr. physical. Chem._, 1887, 1. 227.
+
+[346] Van't Hoff and Mueller, _Ber._, 1898, 31. 2206.
+
+[347] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+165.
+
+[348] For a full discussion of the solubility relations of sodium ammonium
+racemate, see van't Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 81.
+
+[349] _Annales chim. phys._, 1848 [3], 24. 442.
+
+[350] See Van't Hoff and van Deventer, _Zeitschr. phys. Chem._, 1887, 1.
+165.
+
+[351] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 121.
+
+[352] Roozeboom, _Zeitschr. physikal. Chem._, 1888, 2. 518.
+
+[353] Meyerhoffer, _Zeitschr. physikal. Chem._, 1890, 5. 109. On the
+importance of the transition interval in the case of optically active
+substances, see Meyerhoffer, _Ber._, 1904, 37. 2604.
+
+[354] In connection with this chapter, see, more especially, van't Hoff,
+_Bildung und Spaltung von Doppelsalzen_, p. 3, _ff._; Roozeboom, _Zeitschr.
+physikal Chem._, 1892, 10. 158; Bancroft, _Phase Rule_, p. 201; 209.
+
+[355] The same restriction must be made here as was imposed in the
+preceding chapter, namely, that the two salts in solution give a common
+ion.
+
+[356] For example, addition of ammonium chloride to solutions of ferric
+chloride (Roozeboom, _Zeitschr. physikal. Chem._, 1892, 10. 149).
+
+[357] It must, of course, be understood that the temperature is on that
+side of the transition point on which the double salt is stable.
+
+[358] Excess of the double salt must be taken, because otherwise an
+unsaturated solution might be formed, and this would, of course, not
+deposit any salt.
+
+[359] Meyerhoffer, _Ber._, 1904, 37. 2605.
+
+[360] Meyerhoffer, _Ber._, 1897, 30. 1809.
+
+[361] Meyerhoffer, _Ber._, 1904, 37. 2604.
+
+[362] Bancroft, _Phase Rule_, p. 203; Roozeboom, _Zeitschr. physikal.
+Chem._, 1891, 8. 504, 531; Stortenbeker, _ibid._, 1895, 17. 643; 1897, 22.
+60; 1900, 34. 108.
+
+[363] Roozeboom, _Zeitschr. phys. Chem._, 1899, 28. 494; _Ber._, 1899, 32.
+537.
+
+[364] As, for instance, strychnine racemate, a compound of racemic acid
+with the _optically active_ strychnine. This would be resolved into
+strychnine _d_-tartrate and strychnine _l_-tartrate, which are not
+enantiomorphous forms.
+
+[365] Van't Hoff and Meyerhoffer, _Zeitschr. physikal Chem._, 1898, 27. 75;
+1899, 30. 86. Fig. 113 is taken from the latter paper.
+
+[366] Solid models constructed of plaster of Paris can be obtained from Max
+Kaehler and Martini, Berlin.
+
+[367] Instead of the present method of obtaining potassium chloride by
+decomposing carnallite with water, advantage might be taken of the fact
+that carnallite when heated to 168deg undergoes decomposition with
+separation of three-fourths of the potassium chloride (van't Hoff, _Acht
+Vortraege ueber physikalische Chemie_, 1902, p. 32).
+
+[368] Roozeboom and Schreinemakers, _Zeitschr. physikal. Chem._, 1894, 15.
+588.
+
+[369] These curves represent only portions of the isotherms, since the
+systems in which a ternary solution is in equilibrium with solid hydrogen
+chloride or a hydrate, have not been investigated.
+
+[370] The numbers printed beside the points on the curves refer to the
+number of the experiment in the original paper.
+
+[371] Lash, Miller and Kenrick, _Journ. Physical. Chem._, 1903, 7. 259;
+Allan, _Amer. Chem. Journ._, 1901, 25. 307.
+
+[372] Allan, _Amer. Chem. Journ._, 1901, 25. 307.
+
+[373] Hoitsema, _Zeitschr. physikal. Chem._, 1895, 17. 651; Allan, _loc.
+cit._
+
+[374] Rutten, _Zeitschr. anorgan. Chem._, 1902, 30. 342. Compare the system
+BeO--SO_{3}--H_{2}O; Parsons, _Zeitschr. anorgan. Chem._, 1904, 42. 250.
+
+[375] _Zeitschr. anorgan. Chem._, 1904, 40. 146.
+
+[376] Schreinemakers, _Zeitschr. physikal. Chem._, 1893, 11. 76; Bancroft,
+_Journ. Physical Chem._, 1902, 6. 179.
+
+[377] _Zeitschr. anorgan. Chem._, 1904, 40. 148.
+
+[378] _Zeitschr. physikal. Chem._, 1903, 43. 354.
+
+[379] These equilibria were obtained by Boudouard, _Annales chim. phys._,
+1901 [7], 24. 5. See also Hahn, _Zeitschr. physikal. Chem._, 1903, 42.
+705; 44. 513.
+
+[380] G. Preuner, _Zeitschr. physikal. Chem._, 1903, 47. 385.
+
+[381] See Hahn, _Zeitschr. physikal. Chem._, 1903, 42. 705; 44. 513;
+Boudouard, _Bull. Soc. chim._, [3], 25. 484; Bodlaender, _Zeitschr. f.
+Elektrochem._, 1902, 8. 833; R. Schenck and Zimmermann, _Ber._, 1903, 36.
+1231, 3663; Schenck and Heller, _ibid._, 1905, 38. 2132; _Zeitschr. f.
+Elektrochem._, 1903, 9. 691; Haber, _Thermodynamik technischer
+Gasreaktionen_, p. 293 (Munich, 1903).
+
+[382] A very useful summary of the investigations carried out by van't Hoff
+and his pupils on the formation of the Stassfurt salt-beds is given by E.
+F. Armstrong, in the _Reports of the British Association for 1901_, p. 262.
+See also van't Hoff, _Zur Bildung der ozeanischen Salzablagerungen_
+(Brunswick, 1905).
+
+[383] See especially Meyerhoffer, _Silzungsber. Wien. Akad._, 1895, 104.
+II. _b_, 840; Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899,
+28. 453; 31. 370. The investigation of the equilibria between reciprocal
+salt-pairs alone (three-component systems) is of great importance for the
+artificial preparations of minerals, as also in analytical chemistry for
+the proper understanding of the methods of conversion of insoluble systems
+into soluble by fusion (see Meyerhoffer, _Zeitschr. physikal. Chem._, 1901,
+38. 307).
+
+[384] See Meyerhoffer, _Zeitschr. physikal. Chem._, 1899, 28. 459.
+
+[385] Compare the reciprocal salt-pair NaCl--NH_{4}HCO_{3} (p. 321). In
+this case the upper limit of the transition interval was found by
+extrapolation of the solubility curve for NaHCO_{3} + NH_{4}Cl +
+NH_{4}HCO_{3} and NaHCO_{3} + NH_{4}Cl + NaCl to be 32deg (Fedotieff,
+_Zeitschr. phys. Chem._, 1904, 49. 179).
+
+[386] Loewenherz, _Zeitschr. physikal. Chem._, 1894, 13. 464.
+
+[387] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.
+
+[388] As the quantities of the salts are expressed in _equivalent_
+gram-molecules, the molecule of sodium and potassium chloride must be
+doubled in order to be equivalent to sodium sulphate and potassium
+sulphate.
+
+[389] _Sitz-Ber. der kgl. preuss. Akad. der Wiss._, 1903, p. 359. Van't
+Hoff, _Zur Bildung der ozeanischen Salzablagerungen_, I. p. 34 (Brunswick,
+1905).
+
+[390] _Zeitschr. fuer Kristallographie_, 1904, 39. 155.
+
+[391] Meyerhoffer and Saunders, _Zeitschr. physikal. Chem._, 1899, 28. 479.
+
+[392] _Zeitschr. physikal. Chem._, 1904, 49. 162.
+
+[393] Another commercial process, in the study of which good service is
+done by the Phase Rule, is the caustification of the alkali salts (G.
+Bodlaender, _Zeitschr. fuer Elektrochem._, 1905, 11. 186; J. Herold,
+_ibid._, 418).
+
+[394] _Zeitschr. physikal. Chem._, 1900, 35. 32.
+
+[395] Mention may also be made here of the equilibria between magnesium
+carbonate and potassium carbonate, although these do not form a reciprocal
+salt-pair (Auerbach, _Zeitschr. fuer Elektrochem._, 1904, 10. 161).
+
+[396] O. N. Witt and K. Ludwig, _Ber._, 1903, 36. 4384; Meyerhoffer,
+_ibid._, 1904, 37. 261, 1116.
+
+[397] _Zeitschr. physikal. Chem._, 1905, 53. 513. Compare also, _ibid._,
+1903, 38. 307.
+
+[398] See Schwarz, _Beitraege zur Kenntnis der umkehrbaren Umwandlungen
+polymorpher Korper_ (Goettingen, 1892); or, Roozeboom, _Heterogen.
+Gleichgewicht_, I. p. 125. Also Barnes and Cooke, _Journ. Physical Chem._,
+1902, 6. 172.
+
+[399] Van't Hoff and van Deventer, _Zeitschr. physikal. Chem._, 1887, 1.
+173.
+
+[400] Reicher, _Zeitschr. fuer Krystallographie_, 1884, 8. 593.
+
+[401] _Zeitschr. physikal. Chem._, 1895, 17. 153.
+
+[402] _Zeitschr. physikal. Chem._, 1899, 28. 464.
+
+[403] Meyerhoffer and Saunders, _ibid._, p. 466.
+
+[404] See Van Eyk, _Zeitschr. physikal. Chem._, 1899, 30. 446.
+
+[405] See in this connection the volume in this series on
+_Electro-chemistry_, by Dr. R. A. Lehfeldt.
+
+[406] Barnes and Cooke, _Journ. Physical Chem._, 1902, 6. 172.
+
+[407] For a description and explanation of these, the reader should consult
+the volume in this series by Dr. Lehfeldt on _Electro-chemistry_; and van't
+Hoff, _Bildung und Spaltung von Doppelsalzen_, p. 48 _ff._
+
+
+
+
+       *       *       *       *       *
+
+
+
+
+Changes made to the printed original.
+
+Pages 30-31. "Fig. 3, p. 27.": 'p. 25." in original. So also page 33, "Fig.
+2, p. 27".
+
+Page 57. "pp. 29 and 35": 'pp. 25 and 38" in original.
+
+Page 65. "p. 57.": 'p. 60" in original (twice).
+
+Page 166. "there is the point C_{1}": C' in original.
+
+Page 225. "C is an eutectic point": 'eutetic' in original.
+
+Page 228. "Although this view put forward by Heyn": 'Athough' in original.
+
+Page 232. "the period of constant temperature for the eutectic point c":
+'the eutectic point e' in original.
+
+Page 249. "two liquid layers between 13deg and 31deg": 'betwen' in
+original.
+
+Page 257. Tables entries 4 and 7. "naphthol": 'napthol' in original.
+
+Page 287. "from which the model is constructed": 'he model' in original.
+
+
+
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