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The phase rule and its applications

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TEXT-BOOKS OF PHYSICAL CHEMISTRY

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Text-Books of Physical Chemistry.

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THE PHASE RULE

AND ITS APPLICATIONS

BY

ALEX. FIXDLAY, MA., Ph.D., D.Sc.

LECTURER ON PHYSICAL CHEMISTRY, UNIVERSITY OK BIRMINGHAM

WITH ONE HUNDRED AND THIRTY-FOUR FIGURES IN THE TEXT

THIRD EDITION

brary

College of Liberal Arts

Boston Univi rsity

LONGMANS, GREEN, AND CO

39 PATERNOSTER ROW, LONDON- NEW YORK, BOMBAY, AND CALCUTTA IQII All rights reserved

SO)

DEDICATED

TO

FRANCIS ROBERT JAPP, LL.D., F.R.S.

PROFESSOR OF CHEMISTRY, UNIVERSITY OF ABERDEEN,

IN GRATITUDE FOR EARLY TRAINING

AND ADVICE

PREFACE TO THE THIRD EDITION

In the present edition no changes of a fundamental nature have been made, although here and there a paragraph or section has been added dealing with subjects of recent study. Such are the sections on Metastability in Metals, and Influence of Foreign Substances on the Critical Solution Temperature. The whole book; however, has been subjected to a revision, and such corrections or additions made as were necessary to bring the points discussed into harmony with the most recent work.

A. F.

February ; 19 1 1

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 HorT, 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

PREFACE 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-mathe- matical, 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 investiga- tion 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 (" Lehr- buch "), 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 PREFACE

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.

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 depart- ments 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.

CONTENTS

CHAPTER I

PAGE

Introduction i

General, I. Homogeneous and heterogeneous equilibrium, 5. Real and apparent equilibrium, 5.

CHAPTER II The Phase Rule

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

A. Water. Equilibrium between liquid and vapour. Vapo rization 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. Meta stable triple point, 38. Fusion curve of rhombic sulphur, 39 Bivariant systems, 39. C. Tin, 41. Transition point, 41 Enantiotropy and monotropy, 44. D. Phosphorus, 46. Enan tiotropy combined with monotropy, 51. E. Liquid Crystals, 51 Phenomena observed, 51. Nature of liquid crystals, 52. Equi librium relations in the case of liquid crystals, 53.

21

77

CONTENTS xiii

CHAPTER IV General Summary 55

Triple point, 55. Theorems of van't HofTand of Le Chate- lier, 57. Changes at the triple point, 58. Triple point solid — solid — vapour, 62. Sublimation and vaporization curves, 63. Fusion curve — Transition curve, 66. Suspended trans- formation. Metastable equilibria, 69. Velocity of transforma- tion, 70. Metastability in metals, 73. Law of successive reactions, 75.

CHAPTER V

Systems of Two Components — Phenomena of Disso- ciation

Different systems of two components, 78. Phenomena of Dissociation. Bivariant systems, 80. Univariant systems, 8l. Ammonia compounds of metal chlorides, 83. Salts with water of crystallization, 86. Efflorescence, 87. Indefiniteness of the vapour pressure of a hydrate, 88. Suspended transfor- mation, 90. Range of existence of hydrates, 91. Constancy of vapour pressure and the formation of compounds, 91. Measurement of the vapour pressure of hydrates, 92.

CHAPTER VI

Solutions

Definition, 94. Solutions of Gases in Liquids, 95. Solutions of Liquids in Liquids, 96. Partial or limited miscibility, 97. Phenol and water, 99. Methylethylketone and water, 102. Triethylamine and water, 103. General form of concentration-temperature curve, 103. Influence of foreign substances on the critical solution temperature, 104. Pressure- concentration diagram, 105. Complete miscibility, 107. Pressure-concentration diagram, 107.

CHAPTER VII

Solutions of Solids in Liquids, only One of the

Components being Volatile 109

General, 109. The saturated solution, in. Form of the solubility curve, in. A. Anhydrous Salt and Water. The solubility curve, 114. Suspended transformation and supersaturation, 116. Solubility curve at higher temperatures, 117. (1) Complete miscibility of the fused components. Ice as solid phase, 119. Cryohydrates, 120. Changes at the quadruple

94

XIV CONTENTS

PAGE

point, 122. Freezing mixtures, 123. (2) Partial miscibility of the fused components. Supersaturation, 127. Pressure-tempera- ture diagram, 129. Vapour pressure of solid — solution — vapour, 129. Other univa riant systems, 1 30. Bivariant systems, 132. Deliquescence, 133. Separation of salt on evaporation, 133. General summary, 134.

CHAPTER VIII

Solutions of Solids in Liquids, only One of the

Components being Volatile 136

B. Hydrated Salt and Water, (i) The compounds formed do not have a definite melting point. Concentration- temperature diagram, 136. Sodium sulphate and water, 137. Suspended transformation, 140. Dehydration by means of anhydrous sodium sulphate, 141. Pressure-temperature diagram, 141. (2) The compounds formed have a definite melting point. Solubility curve of calcium chloride hexahydrate, 148. Pressure-temperature diagram, 153. The indifferent point, 153. The hydrates of ferric chloride, 1 54. Suspended transformation, 158. Evaporation of solutions at constant temperature, 158- Inevaporable solutions, 160. Illustration, 161.

CHAPTER IX Equilibria between Two Volatile Components . 164

General, 164. Iodine and chlorine, 164. Concentration- temperature diagram, 165. Pressure-temperature diagram, 168. Bivariant systems, 170. Sulphur dioxide and water, 172. Pressure-temperature diagram, 173. Bivariant systems, 176.

CHAPTER X Solid Solutions. Mixed Crystals 178

General, 178. Solution of gases in solids, 179. Palladium and hydrogen, 181. Solutions of solids in solids. Mixed crystals, 183. Formation of mixed crystals of isomorphous substances, 185. 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, 186. Melting-point curve, 187. [b) The freezing-point curve passes through a maximum. Example, 189. (c) 'J he freezing-point curve passes through a minimum. Example, 191. Fractional crystallization of mixed crystals, 191. II. The two components do not form a continuous series of mixed crystals, (a) The freezing-point curve exhibits a transition point, 193. Example, .

CONTENTS XV

PACK

'93> (b) The freezing-point curve exhibits a eu tec tic point, 194. Examples, 195. Changes in mixed crystals with the tempera- ture, 195.

CHAPTER XI Equilibrium between Dynamic Isomerides. . . . 198

Temperature-concentration diagram, 199. Transformation of the unstable into the stable form, 204. Examples, 206. Benzaldoximes, 206. Acetaldehyde and paraldehyde, 207.

CHAPTER XII

Summary. — Application of the Phase Rule to the

Study of Systems of two Components . . .210

Summary of the different systems of two components, 210. (1) Organic compounds, 215. (2) Optically active substances, 219. Examples, 220. Transformations, 223. (3) Alloys, 223. Iron — carbon alloys, 226. Determination of the composition of compounds without analysis, 232. Formation of minerals, 235.

CHAPTER XIII Systems of Three Components 237

General, 237. Graphic representation, 238.

CHAPTER XIV Solutions of Liquids in Liquids 243

1 . The three components form only one pair of partially tniscible liquids, 243. Retrograde solubility, 248. The influence of temperature, 250. 2. The three components can form two pairs of partially miscible liquids, 252. 3. The three components form three pairs of partially miscible liquids, 254.

CHAPTER XV Presence of Solid Phases 256

A. The ternary eutectic point, 256. Formation of compounds, 258. B. Equilibria at higher temperatures. Formation of double salts, 261. Transition point, 261. Vapour pressure. Quintuple point, 264. Solubility curves at the transition point, 267. Decomposition of the double salt by water, 270. Transition interval, 273. Summary, 274.

TACK

-75

xvi CONTENTS

CHAPTER XVI

Isothermal Curves and the Space Model . .

Non-formation of double salts, 275. Formation of double salt, 276. Transition interval, 280. Isothermal evaporation, 281. Crystallization of double salt from solutions containing excess of one component, 283. Formation of mixed crystals, 284. Application to the characterization of racemates, 285. Representation in space. Space model for carnallite, 287. Summary and numerical data, 290. Ferric chloride— hydrogen chloride — water, 293. Ternary systems, 294. The isothermal curves, 297. Basic salts, 299. Bi203— N2Os— H20, 301. Basic mercury salts, 304. Indirect determination of the com- position of the solid phase, 305.

CHAPTER XVII Absence of Liquid Phase 308

Iron, carbon monoxide, carbon dioxide, 308*

CHAPTER XVIII Systems of Four Components 315

Reciprocal salt-pairs. Choice of components, 316. Transi- tion point, 317. Formation of double salts, 318. Transition interval, 318. Graphic representation, 319. Example, 320. Ammonia-soda process, 323. Preparation of barium nitrite, 330. Barium carbonate and potassium sulphate, 331.

APPENDIX

Experimental Determination of the Transition

Point 335

I. The dilatometric method, 335. II. Measurement of the vapour pressure, 338. III. Solubility measurements, 339. IV. Thermometric method, 341. V- Optical method, 342. VI. Electrical methods, 342.

Name Index 345

Subject Index 35 1

THE PHASE RULE

CHAPTER I

INTRODUCTION

General. — Before proceeding to the more systematic treat- ment 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 tem- perature is raised. But, although the pressure of the vapour increases as the temperature rises, its value at any given tem- perature 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 ulti- mately 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

T. P. C. B

2 THE PHASE RULE

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, 2H20), 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

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.

INTRODUCTION 3

into the relations which exist between pure liquids and solu- tions? 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 " equi- libria 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 pro- ducts 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 tem- perature), decomposition ceases, and calcium carbonate exists side by side with calcium oxide and carbon dioxide. More- over, 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,1 and is independent of the amounts ot solid and gas present. So far, therefore, ammonium chloride behaves like calcium carbonate. If, however, one of the 1 Ramsay and Young, Phil Trans., iS86, 177. 8;.

4 THE PHASE RULE

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 dis- sociation alters the equilibrium, while in other cases it does not. With the help of the Phase Rule, however, a general interpre- tation of this difference of behaviour can be given — an inter- pretation 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 deter- mine 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 equi- libria 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

INTRODUCTION x

separation of different minerals from a fused rock-mass.5 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 equi- librium 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 equi- librium 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 con- centration).

Real and Apparent Equilibrium. — In discussing equi- libria, also, a distinction must be drawn between real and

1 See, more especially, Vogt, Die Silikatsckmehlosungen. (Christiania, 1903, 1904.)

6 THE PHASE RULE

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 equi- librium, 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 instantan- eously 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.

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 ; 1 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 2 and Berthollet 3 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,4 to inaugurate the period of quantitative study of equilibria. The law which these investigators enunciated

1 Trans. Connecticut Acad., 1 8 74-1 878.

2 Lehre von der chemischen Verwandtschaft der Korper, 1777.

3 See Ostwald's Klassikcr, No. 74.

4 Etudes sur les affinites chimiques, 1S67 ; Ostwald's Klassiker, No. 104.

8 THE PHASE RULE

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 equi- librium. 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 com- plexity was known.

Ten years after the law of mass action was propounded by Guldberg and Waage, Willard Gibbs,1 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 equili- brium 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,2 a method of treatment which had first' been employed by Horstmann.3 In deducing the law of equilibrium, Gibbs regarded a system as possessing only three independently variable factors4 — 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

1 Died April, 1903.

2 For a mathematical treatment of the Phase Rule the reader is referred to the volume in this series on Thermodynamics, by F. G. Donnan.

3 Liebig's Anna/en, 1873, 170, 192; Ostwald, Lehrbuch, II. 2. III.

4 The action of gravity and other forces being excluded (see p. 5).

THE PHASE RULE 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 mechani- cally 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 consti- tutes 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 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.1 Thus it is a familiar fact that the pressure of a vapour in contact with a

1 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.

io THE PHASE RULE

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,1 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 syno- nymous with the chemical elements or compounds present, i.e. with the co?istitnents 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.2

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

1 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, Zeiischr. psysikal. C/ieni., 1901, 37. 385). The vapour pressure of small crystals is also greater than that of large ones (Pawlow, Zeifschr. physikal. Chem., 1909, 68. 316).

2 See Ostwald, Lehrbnch, II. 2. 476, 934 ; Roozeboom, Zeitschr. physikal. Chem., 1894, 15. 150 ; Heterogene Gleichgrtoichte^ I. p. 16 ; Weg- scheider, Zeitschr. physikal. Chem., 1903, 43. 89.

THE PHASE RULE n

regarded as components, because, in the first place, they are 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 neces- sitates 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 tempe- rature, we are not dealing with a condition of true equi- librium. 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 j 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 ex- periment (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 equi- librium 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

CaCOa= CaO + C02

12 THE PHASE RULE

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 tJiere are to be chosen the smallest number of independently variable constituents by means of which the composition of each phase par- ticipating in the state of equilibrium can be expressed i?i 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 inde- pendently 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 compo?ie?its, 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 CaC03 and CaO, the com- position of each phase can be expressed by the following equations :■—

CaC03 = CaCOa + oCaO CaO = CaO + oCaC03 C02 = CaC03 - CaO

As we see, then, both zero and negative quantities of the components have been introduced ; and similar expressions would be obtained if CaC03 and C02 were chosen as com- ponents. The matter can, however, be simplified and the use of negative quantities avoided if CaO and C02 are chosen ; and it is, therefore, customary to select these as the com- ponents.

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 quan- tities of these, such a choice is not always possible when deal- ing with systems of a higher order (containing three or four components).

From the example which has just been discussed, it might

THE PHASE RULE 13

appear as if the choice of the components was rather arbitrary. On examining the point, however, it will be seen that the ar- bitrariness affects only the nature, not the number, of the com- ponents ; 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 com- ponents or of the second order ; and if three phases are neces- sary to give the composition of a fourth coexisting phase, the system is one of three components, or of the third order.1

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 appli- cation of the method just given for determining the number of components.

Consider the system consisting of Glauber's salt in equi- librium with solution and vapour. If these three phases are analyzed, the composition of the solid will be expressed by Na2S04, 10H0O ; that of the solution by Na,S04 + .*H20, while the vapour phase will be H20. 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.2S04, ioH20 and H20), 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 Na2S04, 7H20. In these cases there may 1 Ostwald, Lehrbuch, II. 2. 47S.

i4 THE PHASE RULE

be chosen as components Na^d and H20, and Na2S04, 7H20 and H20 respectively. In both cases, therefore, there are two components. But the two systems (Na2S04, ioH20— H20, and Na2S04, 7H20— H20) can be regarded as special cases of the system Na2S04— H20, 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 com- ponents is the same ; but by choosing Na2S04 and H20 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 —

(i) The components are to be chosen from among the con- stituents 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 sma/tcst 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 compo- nents, the choice being influenced by considerations of sim- plicity, suitability, or generality of application.1

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 occu- pying the same volume (the concentration, therefore, remaining 1 See also Hoitsema, Zeitschr. fhysikal. Chem. 1895, 17. 651.

THE PHASE RULE 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 tempera- ture 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 1 of a system as the number of the variable factors, temperature, pressure, and concen- tration of the components, which must be arbitrarily fxed 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

1 The term " decree 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.

16 THE PHASE RULE

variability or variance of a system, and describe a system as being invariant, univariant, bi variant, multivariant,1 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 import- ance 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 relatio?i 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.2

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 fol- lows : 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 ?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 —

P + F = C + 2, orF = 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

1 Trevor, Jour. Physical Chew., 1902, 6. 136.

2 Ostwald, Principles of Inorganic Chemist 'rj; translated by A. Findlav. 3rd edit., p. 7. (Macmillan, 1908.)

77//-; PHASE RULE ]?

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 asso- ciated 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 = i, 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,, and the gaseous phase, CO, 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 svstems no matter how dissimilar the systems may outwardly appear' bimilarly, all bivariant systems will exhibit analogous behaviour • and generally, systems possessing the same degree of freedom' wil 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, roultivariant

c

18 THE PHASE RULE

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.1

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 equili- brium ; 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

1 See the volume in this series on Thcrmodynarnics by F. G. Donnan.

THE PHASE RULE 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 there- fore 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 tempera- ture 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) -f 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, equili- brium occurs when the potential of each component is the same in the different phases in which it is present. If, there- fore, we choose as standard one of the phases in which all the components occur, then in any other phase in equilibrium with

20 THE PHASE RULE

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 — i) equations ; and for C components, therefore, we obtain C(P — i) equations.

But we have seen above that there are P(C — i) + 2 variables, and as wTe 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

CHAPTER III

TYPICAL SYSTEMS OF ONE COMPONENT

A, Water,

For the sake of rendering the Phase Rule more readily intel- ligible, 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 be- haviour 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 com- ponent, because all three phases have the same chemical com- position, represented by the formula H20. 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 abscissas and the pressures as ordinates. In such a diagram invariant systems will be repre- sented 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 co- existence of liquid and vapour. According to the Phase Rule (p. 1 6), a system consisting of one component in two phases has one degree of freedom, or is univariant. We should there- fore expect that it will be possible for liquid water to coexist with water vapour at different values of temperature and

22

THE PHASE RULE

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 pres- sure; and if in our diagram we join by a continuous line all the points indicating the values of the pressure corre- sponding to the different temperatures, we shall obtain a curve (Fig. i) representing the variation of the pressure with the temperature. This is the curve of vapour pres- sure, or the vaporization curve of water.

Now, the results of ex- periment are quite in agree- ment 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,1 Regnault,- Ramsay and Young," Juhlin,4 Thiesen and Scheel,5 and others. In the following table the values of the vapour pressure from — io° to -f-ioo° are those calculated from the measurements o( Regnault, corrected by the measurements of Wiebe and Thiesen and Scheel;6 those from 120" to 270° were determined

1 Pogg. Annalen, 1844, 61. 225. - Memoires de PAca-J., 26. 751.

. 3 Phil. Trans., 1884, 175. 461 ; 1892, A, 183. 107.

4 Bihang Svcnska Akad. HandL, 1 89 1, 17. I. 1.

5 Ahhandl. physikal.-tcch. Reichsansta.lt, 1 900,. 3. 71.

6 Ostwald-Luther, PJiysiko-chemische Messungen, 2nd edit.

56.

der Physik, 1907 [4], 22. 609.

TYPICAL SYSTEMS OF ONE COMPONENT 23

by Ramsay and Young, while the values of the critical pressure and temperature are those determined by Battelli.1

Vapour Pressure of Water.

Temperature.

Picssure in cm. mercury.

Temperature.

Pressure in cm. mercury.

-IO°

0*213

120°

148*4

o°

0-458*

1 30°

201'9

+ 20°

1-752

1500

356-S

40°

5'5l6

200°

1162-5

6o°

14-932

2500

2973"4

8o°

35'54

27O0

4110-1

IOO°

76 OD

364-3° (critical

14790-4 (194-6 atm.)

temperature)

(critical pressure).

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

1 Annates chim. et phys., 1S92 [6], 26. 425.

* The vapour pressure of water at o° has recently been very accurately determined by Thiesen and Scheel {toe. cit.\ and found to be 4-579 + o'ooi mm. of mercury (at o°), or equal to 0-006025 atm.

24 THE PHASE RULE

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 i94'6 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 dis- appears in the form of vapour. Even at temperatures con- siderably lower than o°, 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, repre- senting the conditions of equilibrium between a solid and its vapour, is called a sublimation curve. At temperatures repre- sented by any point on this curve, the solid (ice) will sublime or pass into vapour without previously fusing. Since ice melts at o° (vide infra), the sublimation curve must end at that tem- perature.

The following are the values of the vapour pressure of ice between o° and — 500.1

1 Juhlin, Bihang Svenska Akad. Hand/., 1891, 17. I. 58. See also Ramsay and Young, loc. at.

TYPICAL SYSTEMS OF ONE COMPONENT 25 Vapour Pressure of [ce.

Temperature.

Pressure in nun. mercury.

Temperature.

Pressure in mm. mercury.

-50° -4o° -3o°

-20° -IO°

0-050 0'I2I 0-3I2

o-fc'o6 1-279 1-999

-s°

-6° -4°

-2°

o°

2*379 2-821

3 '334 3-925 4-602

Equilibrium between Ice and Water. Curve of Fusion. —There is still another univariant system of the one com- ponent 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 tempera- ture 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,1 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 2 (Lord Kelvin), who found that under a pressure

1 Tratis. Roy. Soc. EJin., 1S49, 16. 575. ■ Proc. Roy. Soc. F.Jiu., 1850, 2, 267.

26

THE PHASE RULE

of 8*i 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,1 and the results obtained by him are given in the following table and represented graphically in Fig. 2.

Fusion Pressure of Ice.

Temperature.

Pressure in kilogms. per sq. cm.2

Change of melting point for an increase of pressure of 1 kilogm. per sq. cm.

-o°

-2-5°

-0

-7-5°

- io-o°

-12-5°

-15-0°

-17-5°

— 20 -o°

-22-1°

I

336

890

"55

1410 1625

1835 2042 2200

0*0074°

0-OC90° O-O09I0 0-0094°

0-0100° 0-0116° 0-0119°

0-0121° 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*007 6°,3 or an increase of pressure of 135 atm. is required to produce a lowering of the melting point of i°. We see further that the fusion curve bends slightly as the pressure is increased, which signifies that the variation of

1 Annalen der Physik, 1899 [3], 68. 564 ; 1900 [4], 2. 1, 424. See also Dewar, Proc. Roy. Sac, 1880, 30. 533.

2 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.

3 Tammann, loc. cit., 1900, 2. I, 424 ; cf. Goossens, Arch. neerland> 1886, 20. 449.

TYPICAL SYSTEMS OF ONE COM PON EXT

the melting point with the pressure changes; at — 150, when the pressure is 1625 kilogm. per sq. cm., increase of pressure

Pressure

in atm

£u-

lrr»

1 *-«

/ 1>

, / 0

Ice 11 & in i L%

1 / *-

' / <L>

Jcel&n/p

Water

1 /

JceT&TiiV

\*s

\*

\SP

Ice I

v£

\Ck

\o

V

■22

Fig. 2

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.

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 o° 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 equi- librium with vapour at the same pressure. Since this is so, they must, of course, be in equilibrium

28 THE PHASE RULE

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}

The triple point, however, does not lie exactly at o° C, for this temperature is denned 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*007 6°; 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 o° (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 com- ponent, 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 pres- sure 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 1 J. Thomson, Proc. Roy. Soc, 1874. 22. 28.

TYPICAL SYSTEMS OF ONE COMPONENT 29

vapour of the same pressure, they must, of course, also be in equilibrium with one another.

Bivariant Systems of Water.— If we examine Fig. 4, we see that the curves OA, OB, OC, which represent diagram- matically 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

I

Water /

Ice

II /

a'-''

■f° m

Vapour

B

0°

F

IG. 4.

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.1 Thus, the phase common to the two systems

1 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 Urn; and a bivariant system by an area.

3o THE PHASE RULE

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 o°, and it has so far not been found pos- sible 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 solidifi- cation occurring. This was first discovered in 1724 by Fahren- heit,1 who found that water could be exposed to a temperature 0f —9-4° without solidifying; so soon, however, as a small particle of ice was brought in contact with the water, crystalli- zation commenced. Superfused or supercooled water — i.e. water cooled below o° — 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 solidi- fying, 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 throuo-hout which this condition exists is called the metastable reo-ion. Supercooled water, therefore, is in a metastable con- dition. If the supercooling be carried below a certain tem- perature, 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 consider- ably 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,2 and is represented in Fig. 3,

1 Phil. Trans., 1724, 39. 78.

2 Juhlin, loc. cit., p. 61 ; cf. Ramsay and Young, loc. cit. : Thiesen and Scheelj loc. cit.

TYPICAL SYSTEMS OF ONE COMPONENT 31

p. 25, 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.

o°

4-618

4-602

0-016*

-2°

3 '995

3'925

0-070

-4°

3-450

3'334

0'Il6

-8°

2-558

2'379

0-179

-100

2-197

1-999

0*198

-15°

1-492

1-279

O 213

-20°

1-005

0-806

0-199

At all temperatures below o° (more correctly +0*007 6°), 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 o°, the (meta- stable) sublimation curve of ice, if it could be obtained, would be higher than the vaporization curve of water. This shows, therefore, that at o° 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

* This small difference is due to experimental errors in the determina- tion of the vapour pressures ; a differential method betrayed no difference between the vapour pressure of ice and of water at o°.

32 THE PHASE RULE

the vaporization curve. The existence of a break was, how- ever, shown by James Thomson1 and by Kirchhoff2 to be demanded by thermo-dynamical considerations, and the pre- diction 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.3

From what has just been said, we can readily understand why ice and water cannot exist in equilibrium below o°. 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.4

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, how- ever, 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 5 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 IL, ice III., forming invariant systems and existing in equilibrium only at a definite

1 Phil. Mag., 1874 [4], 47. 447 ; Proc. Roy. Soc, 1873, 22. 27.

2 Pogg. Annalen, 1858, 103, 206.

3 See Phil. Trans., 1884, 175, 461.

* This phenomenon of distillation from the supercooled liquid to the solid has been very clearly observed in the case of furfuraldoxime (V. Goldschmidt, Zcitschr. f. Krystallographie, 1897, 28. 169).

5 Annalen der Pliysik, 1900 [4J, 2. I, 424. Quite recently {Zcitschr. physikal. Che///., 1910, 72. 609) Tammann has studied more fully the con- ditions under which the different crystalline forms of ice can be realized ; and he has also shown that there exists another crystalline form, ice IV., related to, but less stable than, ice I.

TYPICAL SYSTEMS OF ONE COMPONENT 33

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 temperature and pressure; and lastly, the bivariant systems, ice II. and ice III. Several of these systems have been inves- tigated 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. 25.1 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 exist- ence of a substance in two or more different crystalline forms, is called polymorphism. Polymorphism was first observed by Mitscherlich 2 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.3 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 in- creased 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 co- exist, either alone or in presence of the liquid and vapour phase. For the purpose of illustrating the general behaviour of

1 A similar triple point has been determined by Tammann in the case of phenol (Anna/en der Physiky 1902 [4J, 9. 249).

2 Antiales chim. et phys., 1821, 19. 414.

* Lehmann, Molekularphysik, I. 153; Arzruni, PhysikaUscht Chemie der Ktystalle. (Graham-Otto, Lehrbuch der Chemie, I. 3.)

T. P. C D

34 THE PHASE RULE

such systems, we shall study the systems formed by the different crystalline forms of sulphur, tin, and benzophenone.

Sulphur exists in two well-known crystalline forms — rhombic, or octahedral, and monoclinic, or prismatic sulphur. Of these, the former melts at 114*5° j the latter at 1200.1 Further, at the ordinary temperature, rhombic sulphur can exist un- changed, whereas, on being heated to temperatures somewhat below the melting point, it passes into the prismatic variety. On the other hand, at temperatures above 9 6°, prismatic sulphur can remain unchanged, whereas at the ordinary tem- perature 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 theoreti- cally possible : —

I. Bivariant Systems : One co7tipo?wit in one phase.

(a) Rhombic sulphur.

(b) Monoclinic sulphur.

(c) Sulphur vapour.

(d) Liquid sulphur.

II. Univariant Systems : One co7nponent 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. (/) 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. 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 o° the vapour pressure of the stable system is lower than that of the 1 Brodie, Proc. Roy. Soc, 1855, 7. 24.

TYPICAL SYSTEMS OF ONE COMPONENT 35

C (757°, 1288 atm.)

metastable, and therefore that at the triple point there is a break in the vapour pressure curve of such a kind that above 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 con- sider that it does possess a certain, even if very small, vapcur pressure,1 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 pass- ing through the triple point, there- fore, there will be a change of the one form into the other. This point is represented in our diagram (Fig.

Fig. 5.

5) by the point O, the two curves AO and OB representing diagrammatically the vapour pressures of rhombic and mono- clinic sulphur respectively. If the vapour phase is absent and the system maintained under a constant pressure, e.g. atmo- spheric pressure, there will also be a definite temperature at which ihe two solid forms are in equilibrium, and on passing

1 That solid sulphur docs possess a certain vapour pressure has beenshown by Hallock, who observed the formation at the ordinary temperature of copper sulphide in a tube containing copper and sulphur {Amer. Jour. Scf.\ l&&9 [3L 37. 405). See also Zenghelis, Zeitschr. physikaL Chan., 190.4. 50. 219 : 1906, 57. 90.

36 THE PHASE RULE

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 Reicher1 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? 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. The determination of the transition point is thereby greatly simplified.

1 Zeitschr. fur JCrystallographie, 1884, 8.593.

8 Van't Hoff, Studies on Chemical Dynamics, p. 163.

TYPICAL SYSTEMS OF ONE COMPONENT 37

Suspended Transformation. — In many respects the tran- sition 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.1 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. 69).

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°.- 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. ;).

1 See, however, footnote 2, on p. 69.

2 Reicher, loc. cit. See also Tammann, Annalen der Physii, 1899 [3], 68. 663 ; Bronsted, Zeitschr. phvsikal. Chc/n., 1906, 55. 380.

38 THE PHASE RULE

Triple Point — Monoclinic Sulphur, Liquid, and Vapour. Melting Point of Monoclinic Sulphur. — Above 95*5°, mono- clinic sulphur is, as we have seen, the stable form. On being heated to 1200, 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 15 1°, and a pressure of 1320 kilogm. per sq. cm., or about 1288 atm.1 It, therefore, forms another triple point, the existence of which had been predicted by Roozeboom,2 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, pris- matic sulphur separates out on cooling fused sulphur, but at temperatures above 1510 and under pressures greater than 1288 atm., the rhombic form would be produced.3

Triple Point — Rhombic Sulphur, Liquid, and Vapour. Metastable Triple Point. — On account of the slowness with

1 Tammann, Annalen der Physik, 1899 [3], 68. 633.

■ Rec. Trav. Chim. Pays-Pas, 1887, 6. 314.

1 Cf. van't Hoft, Lectures on Physical Chemistry, I., p. 27 (Arnold).

TYPICAL SYSTEMS OF ONE COMPONENT 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 con- dition, 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 melt- ing point ; for we have already learned that at lower tempera- tures 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 — mono- clinic 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. 6,1 a being the curve for monoclinic sulphur and liquid ; £, that for rhombic sulphur and liquid; and <r, 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,

1 Annalen der Physik, 1S99 [3], 68. 663.

4o

THE PHASE RULE

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 meta- stable condition.

ISO"

160

1000

Fig. 6.

Other crystalline forms of sulphur have been obtained,1 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).

1 Brauns, Jahrbuchfiir Mineralogie, 1899-1901, 13. Beilage, p. 39.

TYPICAL SYSTEMS OF ONE COMPONENT 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-T868), 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.1 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." 2 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 analo- gous 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, how- ever, the confusion which obtained has been cleared away, and the "mysterious" behaviour of tin brought into accord with other phenomena of transformation.3

Transition Point.— Just as in the case of sulphur, so also in the case of tin, there is a transition point above which the

1 Fritsche, £er., 1869, 2. 112, 540.

2 De viirabilibus Atiscultatiombus, Cap. 51 (v. Cohen, Zcitschr. physikal. C/iem., 1 901, 36. 513).

3 E. Cohen and C. van Eyk, Zeitschr. phy:ikal. Chem., 1899, 30. 601 ; Cohen, ibid., 1900, 33. 59; 35. 588; 1901, 36. 513; Cohen and E. Goldschmidt, ibid., 1904, 50. 225 ; Cohen, ibid, 190S, 63. 625.

42 THE PHASE RULE

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 has been found by Cohen, who employed the dilatometric method (Appendix) to be i8°. 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 short period in the year, the mean tempe- rature of our climate is below 180, and white tin is, therefore, at the ordinary temperature, in a metastable condition. The change, however, into the stable form at the ordinary tempe- rature, 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.1

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.

Trie 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 o°. 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.2 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

1 Zdtschr. physikal. Chem., 1908, 63. 625.

2 Zeitschr. physikal. Chem., 1900, 33. 58.

TYPICAL SYSTEMS OF ONE COMPONENT 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."

Fig. 7.

44 THE PHASE RULE

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 2 7-20,1 and those of ben- zophenone at 2 6° and 480.2 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 mo?iotropy; enantiotropic substances being such that the change of one form into the other is a reversible pro- cess (e.g. rhombic sulphur into monoclinic, and monoclinic sulphur into rhombic), and monotropic substances, those in which the transformation of the crystalline forms is irrever- sible.

These differences in the behaviour can be explained very well in many cases by supposing that in the case of enan- tiotropic substances the transition point lies below the melting point, while in the case of monotropic substances, it lies above the melting point.3 These conditions would be represented by the Figs. 8 and 9.

In these two figures, 03 is the transition point, Ox and 02 the melting points of the metastable and stable forms

1 Stortenbeker, Zeitschr. physikal. Chew., 1889, 3. n ; Rec. Trav. Chim. Pays-Bas, 1888, 7. 152.

2 Zincke, Ber., 187 1, 4. 576.

3 Ostwald, Zeitschr. physikal. Chem., 1897, 22. 313.

TYPICAL SYSTEMS OF ONE COMPONENT 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 ; l in this case we have what is known as pseudomonotropy. It is possible that graphite and diamond,'2

A

/(£

°y~;

<v

/Ml

"'

/03

y\

Fig. 8.

Fig. 9.

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 deter- mination of the melting point is carried out rapidly, the point Oi, 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

1 Roozeboom, Das Hcterogaie Gleichgewidit, I. p. 177. 3 Roozeboom, ibid-, p. 179.

46

THE PHASE RULE

not allowed to rise above the melting point of the latter, the liquid may solidify. The stable solid modification thus ob- tained 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 de- terminations of the vapour pressures of liquid white phosphorus, and of solid red phosphorus,1 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 in cm.

Temperature.

Pressure in atm.

Temperature.

Pressure in atm.

1650 1800

20O° 219° 23C° 29O0

12

20*4

26-6 35'9 51*4 76*0

3600 440° 494° 503° 5110

3-2

rs

iS-o

2 I '9

20 '2

3600 440° 487° 5lO°

53i° 55o°

577°

O'l 175

6-8 io-S 16-0 310 56-0

These values are also represented graphically in 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

1 Schiotler, Fogg. Annalen, 1850, 81. 276; Trocst and Hautefcuille, Annales de Chim. et Phys. 1874 [5], 2. 153 ; Ann. Scient. Ecole Norm. 1S68 [2], II. 266.

TYPICAL SYSTEMS OF OXE COMPONENT 47

360° 400

440° 480° 520° 560c

Fig. io.

of light,1 oi by the presence of certain substances, e.g. iodine,2 just as the velocity of transformation of white tin into the grey modification was in-

Pressurf

creased by the presence fnatm. of a solution of tin am- monium chloride (p. 40). At the ordinary tem- perature, therefore, white phosphorus must be con- sidered as the less stable (metastable) form, for although it can exist in contact with red phos- phorus 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 3 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 phos- phorus can be best represented by the diagram, Fig. it.4

In this figure, BOx represents the conditions of equilibrium of the univariant system red phosphorus and vapour, which ends at Ol5 the melting point of red phosphorus. By heating in capillary tubes of hard glass, Chapman5 found that red phosphorus melts at the melting point of potassium iodide, i.e. about 6300,6 but the pressure at this temperature is unknown.

At 0J5 then, we have the triple point, red phosphorus, liquid, and vapour, and starting from it, we should have the

1 Pedler, Trans. Chem. Soc, 1890, 57. 599.

2 Brodie, Trans. Chem. Soc, 1853, 5, 2S9.

3 This is a familiar fact in the case of the solubility in carbon disulphide.

4 Roozeboom, Das Heterogene Gleichgeivicht, I. p. 170.

5 Trans. Chem. Soc, 1899, 57. 754.

■ Carnelley, Trans. Chem. Soc, 1876, 29. 489; 1S7S, 33. 275. V. Meyer and Riddle, Ber., 1893, 26. 2443.

43

THE PHASE RULE

vaporization curve of liquid phosphorus, OiA, and the fusion curve of red phosphorus, OxF. Although these have not been determined, the latter curve must, from theoretical considera- tions (v. p. 58), slope slightly to the right; i.e. increase of pressure raises the melting point of red phosphorus.

When white phosphorus is heated to 44° it melts.1 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.2 This A point is the intersec- tion of three curves, viz. sublimation curve, vaporization curve, and the fusion curve of white phosphorus. The fusion curve, 02E, has been determined by Tammann 3 and by Fig. 11. G. A. Hulett,4 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 5000; 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 at higher temperatures impossible. Since, however, the differ- ence between white phosphorus and red phosphorus disappears

1 Boeseken has found the melting point under a pressure of 8 mm. to be 4477° {Rec- trav- chiiti., 1907, 26. 289).

- Riecke, Zeitschr. physikal. Chem., 1890, 6. 411.

3 Annalen do- Physik., 1898 [3], 66. 492.

4 Zeitschr. physikal. Chem., 1899,28. 666.

TYPICAL SYSTEMS OF OXE COMPONENT 49

in the liquid state, the vapour pressure curve of white phosphorus must pass through the point 01} the melting point of red phos- phorus, and must be continuous with the curve Ovk, 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 BOj and dA 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).1

This is borne out by the fact that measurements of the vapour density of phosphorus vapour at temperatures of 5000 and more, show it to have the molecular weight represented by P4,2 and the same molecular weight has been found for phosphorus in solution.3 On the other hand, it has recently been shown by R. Schenck,4 that the molecular weight of red phosphorus is at least P8, and very possibly higher.

In the case of phosphorus, therefore, it is more than pos- sible that we are dealing, not simply with two polymorphic forms of the same substance, but with polymeric forms, and that there is no transition point at temperatures above the

1 See Naumann, /So:, 1872, 4. 646; Troost and Hautefeuille, Compt. raid., j.868, 66. 795 ; 186S, 67. 1345; Roozeboom, Das Heterogenic Gleich- gewicht, I. pp. 62, 171.

2 Mitscherlich, lied. Anna/en, 1834, 12. 137 ; Deville and Troost, Compt. rend.t 1863, 56. 891.

3 Beekmann, Zeitschr. Physikal. C/iem., 1S90, 5. 79 ; Hertz, ibid., 6. 358.

4 Ber.t 1902, 35. 351. Cf. also, K. Schaum, Anna! en der C/iem., iS98, 300. 221 ; R. Wegscheider and Kaufler, Sitzungsber. kaiser I. Akad. Wissensch. in U'ie/i, 1901, 110, II. 606. Sec also Boeseken, Proc. A'. Akad. Wetensch. Amsterdam, 1907,9. 613.

T. P. C E

50 THE PHASE RULE

absolute zero, unless we assume the molecular complexity of the two forms to become the same.1 The curve for red phos- phorus 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 6300 under high pressure also exists in a polymeric form, greater than P4, then the metastable vaporization curve of white phos- phorus would not pass through the melting point of red phos- phorus, as was assumed above.2

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 phos- phorus, 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 3500 is less than the vapour pressure of white phosphorus at 2000, it is possible to carry out the distillation from a colder part of the tube to a 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 3240 and

o -j

35° •

Relationships similar to those found in the case of phos- phorus are also met with in the case of cyanogen and

1 ^With regard to the nature of red phosphorus, see Cohen and Olie, jun., Zeitschr. physikdl. C/iem., 1909, 71. 1 ; Jolibois, Compt. rend., 1910, 151. 382.

2 See also Roozeboom, Das Heterogene Gteichgewicht, I. p. 177.

3 Annates de Chim. et P/iys., 1874 [5], 2. 154.

TYPICAL SYSTEMS OF ONE COMPONENT 51

paracyanogen, which have been studied by Chappuis,1 Troost

and Hautefeuille,2 and Dewar,3 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 mono- clinic and rhombic sulphur exhibit the relationship of enan- tiotropy, 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.4

E. Liquid Crystals.

Phenomena observed. — In 1888 it was discovered by Reinitzer5 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 /-azoxyanisole and /-azoxyphenetole, were, later, found to possess the same property of having apparently a double melting point.6 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

1 Compt. rend., 1 887, 104. 1505. - Compt. rend., 1868, 66. 795.

3 Phil. Mag., 1884 [5], 18. 210. See also Roozeboom, Das Heterogene Gleic/igewicht, I. p. 1 77.

4 Brauns, Neues Jahrbuch filr Mineralogie, 1900, 13. Beilage Hand, p. 39 ; Roozeboom, Das Heterogene GleichgcioicJit, I. p. 181.

5 Monatshefte, 1888, 9. 435.

* Gattermann, Ber., 1890, 53. 1738.

52 THE PHASE RULE

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, 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 hetero- geneous and are to be classed along with emulsions.2 This view was no doubt largely suggested by the fact that the aniso- tropic 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,3 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.4

Further, no proof of the heterogeneity of liquid crystals has yet been obtained, but rather all chemical and physical inves- tigations indicate that they are homogeneous.5 No separation

1 Zeitschr. physikal. Chem., 1889, 4. 468; Annalen der Physik, 1900 [4], 2. 649.

2 Quincke, A nnalen der Physik, 1894 [3], 53.613; Tammann, A nnalen der Physik, 1901 [4], 4. 524 ; 1902, 8. 103 ; Rotarski, ibid., 4. 528.

3 Anna/en der Physik, 1900 [4], 2. 649.

4 Annalen der Physik, 1 902 [4], 8. 911.

5 See, more especially, O. Lehmann, Annalen der Physik, 1900 [4], 2. 649; Reinitzer, Sitzungsber. kaiserl. Akad. zu IVien., 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 mil account of the subject, the reader is referred to the work by Lehmann, frliissige Knstalle (Engelmann,

TYPICAL SYSTEMS OF ONE COMPONENT 53

Solid Crystals

Vapour

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.1

Equilibrium Relations in the Case of Liquid Crystals.— Since, now, we have seen that we are dealing here with sub- stances in two crystal- p line forms (which we may call the solid and liquid J crystalline form), which possess a definite transition point, at which trans- formation of the one form into the other occurs in both direc- tions, we can repre- sent 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).

1904), or the smaller monograph by Schenck, gristallinische Flussigkeiten und Flussige Kristallc (Engelmann, 1905); Lehmann, Zcitschr. phvsikal. Chan., 1910, 71. 355 ; 73. 598 ; Bose, Phvsikal. Zcitschr., 1908, 9. 70S ; 1909, 10. 32, 230; Nernst, Zcitschcr. Elektrochem^ 1910, 16. 702; Yorlander, Per., 1908, 41. 2033.

1 A. C. de Kock, Zcitschr. phvsikal. Chan., 1904,43. 129.

- 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 crys- talline 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 dis- continuous ; crystalline bodies possess a certain regular orientation of their molecules and a directive force, while in amorphous bodies these are wanting (see Lehmann, Annalhn der Physik, 1900 [4], 2. 696. See, how- ever, von Weimarn, Koltoid Zcitschr., 1908, 3. 282 ; 1909, 4. 27, 123, 198, 252, 315 ; 5. 62, 117, 150, 212 ; Doelter, ibU., 19 10, 7. 29.

Fig. 12.

54

THE PHASE RULE

In Fig. 12 there is given a diagrammatic representation of the relationships found in the case of /-azoxyanisole.1

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 transi- tion point with the pressure is + 0*03 20 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 1 35*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.

Some of the substances hitherto found to form liquid crystals are 2 : —

Substance.

Transition

Melting

point.

point.

Cholesteryl benzoate .

145-5°

178-5°

Azoxyanisole ......

118-3°

135-9°

Azoxyphenetole .....

134-5°

1 68'i°

Condensation product from benzaldehyde

and benzidine .....

234°

2600

Azine of /-oxyethylbenzaldehyde

1720

1960

Condensation product from /-tolylaldehyde

and benzidine .....

2310

—

/-Methoxycinnamic acid ....

1690

1850

1 Hulelt, loc. cit.

2 Roozeboom, Das Hctcrogetie Glcichgcwicht, I. p. 144. See also Schenck, Kristalliiiische Fills sigkeiten und flussige JOistalle, p. 8 (Engel- man, 1905). A number of other substances exhibiting similar properties have been prepared by Vorl'ander, Ber.t 1906, 39. 803 ; 1907, 40. 1415.

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 sub- stance, 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 l are also possible when, as in the case of

1 The possible number of triple points in a one-component system is

■l ..i- • w(w — i)(u — 2) , . ,, , , ,

given by the expression — - - '> 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.

56 THE PHASE RULE

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, (i) 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.1 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 prolonga- tion) 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,2 only two of these have been experimentally obtained in the case of the triple point solid — liquid — vapour. At present, there- fore, we shall consider only these two cases (Figs. 13 and 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

1 Duhem, Zeitschr. physikal. Chem., 1891, 8. 371. Cf. Roozeboom, Das Iletero°ene Gleichgeivicht, p. 94 ff.

2 Roozeboom, Das Heterogene Gleichgavicht, I. p. 99.

GENERAL SUMMARY

57

greatest between the solid and vapour. In this case the cuive 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. 25 and $8)- We see further from the two figures, that O in Fig. 13 gives the highest temperature at which the

Fig. 13. Fig. 14.

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.1 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.2 The first of these, usually

1 Roozeboom, Zcitschr. physikal. Chem., 1S88, 2. 474.

2 These changes can be predicted quantitatively by means of the thermo- dynamic equation, £ = ^- • provided the specific volumes of the

ai i\v2 — vx)

phases are known, and the heat effect which accompanies the transformation of one phase into the other.

58 THE PHASE RULE

known as van't Hoff s law of movable equilibrium? states : When the temperature of a system in equilibrium is raised, that reaction takes place which is accompanied by absorp- tion 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 : 2 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 accom- panied 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 : 3 Lf 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

1 Studies on Chemical Dynamics, translate! by Ewan, p. 218. * Le Chatelier, Compt. rend., 1884, 99. 7S6.

3 See Principles of Inorganic Chemistry, translated by Findlay, 3rd edit., p. 134. (Macmillan, 1908.)

GENERAL SUMMARY 59

change in the temperature or pressure of the system can occur, but o?iIy changes in the relative amounts of tJie pJiases ; that is to say, the effect on the system of change in the external con- ditions 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 tem- perature at constant volume and constant pressure, and then the effect of alteration of the pressure both when the tempera- ture remains constant and when it varies.

When the volume is kept constant, the effect of the addi- tion of heat to a system at the triple point S-L-V differs some- what 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 -f- V->S. Above the temperature of the triple point the

6o

THE PHASE RULE

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 ulti- mately 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.

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 (iso- thermal 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 com- munication 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, with- out 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

Vapour

Fig. 15.

GENERA L S UMMA R V

Ci

obtained either S-L or L-V according as the vapour 01 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 tempera- ture in the case of systems of the first type (Fig. 13), and by a

Solid

Solid

T , Vapour

Fig. 16.

Fig. 17.

rise in temperature in the case of systems of the second type (Fig. 14).

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 THE PHASE RULE

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?- 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.2

Triple Point Solid— Solid — Vapour. — The triple point solid — solid — vapour is one which is of considerable import- ance. 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 i 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.

1 Roozeboom, Zeitschr. physikal. C/n-w., 188S, 2. 474. 8 Roozeboom, Das Heterogene Gleichgezvicht, I. p. 189.

GENERAL SUMMARY

63

In the following table is given a list of the most import- ant polymorphous substances, and the temperatures of the transition point.1

Substance.

Ammonium nitrate —

/3-rhombic — > a-rhombic . o-rhombic — > rhombohedral Rhombohedral — > regular .

Mercuric iodide ....

Potassium nitrate ....

Silver iodide

Silver nitrate .....

Sulphur

Tetrabrommethane

Thallium nitrate —

Rhombic — > rhombohedral Rhombohedral — > regular .

Thallium picrate ....

Tin

Transition temperature.

35°

830 125° 1260 1 290

145° 1600

95'5° 46 S°

8o°

142-5°

46°

18°

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 sublima- tion 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 //-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,2 and in some cases, e.g. cam- phor,3 the vapour pressure is considerable.

1 Roozeboom, Das Heterogene Gleichgewichi, I. p. 125. See also Zawid- ski, Zeitsehr. physikal. Chcm., 1904, 47. 727 ; van Eyk, ibid., 1905, 51. 720.

2 Roberts- Austen, Proc. Roy. Soc, 63. 454 ; Spring, Zeitschr. physikal. Chem., 1894, 15. 65. See also p. 35.

3 Ramsay and Young, Phil. Trans., 1884, 175. 461 ; Allen, Tram. Chcm. Soc, 1900, 77. 413.

64 THE PHASE RULE

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 par- ticipating in an equilibrium. A dissociating substance, there- fore, in contact with its vaporous products of dissociation (e.g. ammonium chloride in contact with ammonia and hydrogen chloride), will likewise constitute a uni variant 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.1 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. Con- versely, 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 tem- peratures and pressures, so long as the system remains univariant. 1 Ramsay and Young, Phil. Trans. 18S6, 177. 87.

GENERAL SUMMARY 65

U 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.1 At O solid may appear, and the temperature of the system will then remain constant until the liquid phase has disappeared (p. 60) ; the system will then follow the curve OA until the solid phase disappears, and we are ulti- mately 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 0 the solid will melt, and the system will ultimately pass to the curve OB or to OC (p. 60).

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 tem- perature 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 sub- stance is greater than the atmospheric pressure at any tempera- ture 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;'2 the

1 This is exemplified in the well-known experiment with the cryophorus.

2 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, veer., 1901 [2], 6. 244).

T. P. C. F

66 THE PHASE RULE

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 Young1 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 ab- solute temperatures corresponding to equal vapour pressures

T T

is constant, i.e. ^r =-^7. When the two substances are not 1 1 1 2

closely related, it was found that the relationship could be

T To

expressed by the equation ~= ~- -\- c{t} — t) where ns a

1 1 A 2

constant having a small positive or negative value, and /' and / 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 repre- sents 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 Chat- elier (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 tem- perature will be required in order to counteract the effect of increase of pressure ; or, in other words, the melting point will

1 Phii.. Mao;., 1S86, 21. 33. See also S. A. Moss, Physical Revieiu, J903, 16. 356.

GENERAL SUMMARY 67

be lowered by pressure.1 In the second case, the passage of the liquid to the solid state is accompanied by a diminution of volume j 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.-'

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, Bunsen3 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 ob- tained. 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 i°.

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.4 Tammann 5 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

1 This is found also in the case of bismuth. See Tammann, Zcitschr. anorgan. C/ie/n., 1904, 40. 54.

2 See p. 57, footnote.

3 Pgg. Annalen, 1850, 81. 562.

1 Barus, Amer. Jour. Set'., 1892, 42. 125 ; Mack, Compt. rend., 1898, 127. 361 ; Hulett, Zdtschr. physikaL C/iem., 1899, 38. 629.

5 Annalen der Physik, 1899 [3], 68. 553, 629 ; 1900 [4], 1. 275 ; 2. 1 ; 3. 161. See also Tammann, Kristallisieren und Schme!zen (Leipzig, 1903).

68 THE PHASE RULE

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 * 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. Experi- mental evidence, however, does not appear to favour this view.2

The transition point ', like the melting point, is also influ- enced 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.3 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 u-rhombic form of ammonium nitrate, however, the transition point is lowered by pressure, as shown by the following table.4

Temperature.

Pressure.

85-85° 84-38° 83-03° 82-29°

I atm. 100 ,, 200 ,, 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, there-

1 Ostwald, LeJirbuch, II. 2. 373 ; Poynting, Phil. Mag., 1881 [5], 12. 2 ; Planck, Wied. Annallen, 1882, 15. 446.

2 Bakhuis Roozeboom, Das Heterogene Glcichgcwicht, I. p. 91.

z See, for example, the calculation by Bronsted in the case of sulphur (Zeitschr. physikal. Chem., 1906, 55. 374).

* Lussana, // nuovo Cimento, 1895 W' *• I05-

GENERAL SUMMARY 69

fore, a form similar to that assumed by Tammann for the fusion curve.1

Suspended Transformation. Metastable Equilibria.— Hitherto we have considered only systems in stable equili- brium. We have, however, already seen, in the case of water, that on cooling the liquid .down to the triple point, solidifica- tion 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 tempera- tures 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 " hang- ing " 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 "sus- pended " 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. How- ever, even in the presence of the more stable phase, transforma- tion 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.

1 Tammann, Zeitschr. physikal. C/iem., 1903, 46. 818.

2 In this connection sec Pawlow, Zeitschr. physikal. Chem., 190S, 65. I, 545 ; Tammann, ibid., 1909, 68. 257 ; Berthoud, J. Chim. P/iys., 1910, 8, 337.

7o THE PHASE RULE

Thus, for example, although calcite is the most stable form of calcium carbonate at the ordinary temperature,1 the less stable 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.1 As the result of these investigations, it was found that, in the case of supervised salol, the very small amount of 1 X io"? gm. of the solid phase was sufficient to induce crystallization. Crystalliza- tion 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 slug- gishness 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 oases than in others, and appears to decrease with increase of the valency of the element.2 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

1 Foote, Zeitschr. physikal. Chew., 1900, 33. 740.

2 Ostwald, Zeitschr. physikal. Che/n., 1897, 22. 289.

3 Van't Hoff, Arch. neer.t 1901, 6. 471.

GENERAL SUMMARY 71

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 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.1

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 super- saturated 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/

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;3 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 in- creasing indefinitely the higher the temperature is raised. Below the transition point, however, the two factors act in

1 See, for example, the determinations of the solubility of rhombic and monoclinic sulphur, by J. N. Bronsted, Zeitschr. pkysikai. Chan., 1906, 55. 378.

2 Zeitschr. pkysikal Ckem., 1899, 32. 506.

3 Kastle and Reed, Amer. Chan. Jour., 1902, 27. 209.

72 THE PHASE RULE

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 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.1

Lastly, the presence of small quantities of different sub- stances— 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 Gernez2 on the velocity of crystallization of phosphorus and sulphur. Since that time, the velocity of crystallization of other super- cooled liquids has been investigated ; such as acetic acid and phenol by Moore ; 3 supercooled water by Tumlirz ; 4 and a number of organic substances by Tammann,5 Friedlander and Tammann,6and by Bogojawlenski.7

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 crystalli- zation is constant. As the degree of supercooling increases,

1 Zeitschr. physikal. Chem., 1 900, 35. 581.

2 Compt. rend., 1 882, 95. 1278 ; 1884, 97. 1298, 1366, 1433.

3 Zeitschr. physikal. Chem., 1893, 12. 545.

4 Sitzungsber. Wiener Akad., 1894, 103. \\a. 226.

5 Zeitschr. physikal. Chem., 23-29. See also Kiister, ibid., 25-28. • Zeitschr. physikal. Chem., 1897, 24. 152.

Ibid., 1898, 27. 585.

GENERAL SUMMARY 73

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 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.1

When the velocity of crystallization is studied at tempera- tures 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.2 The rule is, however, by no means a universal one. Thus it has been found by F. Dreyer,3 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.4

Metastability in Metals. — That a substance may exist, sometimes for a very long period, in a metastable condition, and that the velocity of transformation from the metastable to the stable condition is increased not only by " inoculation " or " infection " with the stable form but also by rise of temperature,

1 See W. Gueitler, Zeitschr. anorgan. Chem.t 1904, 40. 20S ; Tam- mann, Zeitschr. Elektrochem., 1904, 10. 532.

2 E. von Pickardt, Zeitschr. physikal. Chem., 1902, 42. 17.

3 Zeitschr. physikal. C/u-m., 1904. 48. 407.

4 M. Tad. a, .Lead. Lined, Atti, 1904, 13. 329.

74 THE PHASE RULE

are facts of the greatest importance in the industrial appli- cation of many metals. It is well known that when copper, for example, is subjected to considerable mechanical stress, as when it is rolled, hammered, or drawn into wire, its physical properties undergo change. It becomes harder and less pliable, and its tensile strength increases. When, however, this hardened copper is heated to a temperature of about 3000 it speedily becomes soft again and returns to its former state. A similar behaviour is found also in the case of other metals, such as gold, silver, platinum and lead.

That a metal which has been subjected to considerable mechanical stress passes into a less stable form, is shown by the fact that the metal after having been strained possesses a higher solution pressure than the normal metal. That is to say, it acts as an anode to a piece of the normal metal when the two are placed in a solution of a salt of the metal.1

The change which occurs in the nature o£ the metal when mechanically worked, has been studied by a" number of investi- gators, and it has been found 2 that in the process of working, the metal crystals become more or less completely destroyed, and pass into an amorphous or quasi-amorphous state. A metal in such a state will therefore be related to the normal crystalline metal, in much the same way as supervised solid or glass is re- lated to the stable crystalline form. Just as a metastable glass on being heated passes with greater or less velocity into the stable crystalline state (" devitrifies "), so the metastable, amorphous metal on being heated passes into the stable crystalline form ; and the velocity of transformation will increase with elevation of temperature. At no point can there be a true equilibrium, and, consequently, contrary to the view expressed by some, there cannot be a definite transition point at which reciprocal trans- formation of one form into the other can occur.

In most cases the velocity of transformation of the meta-

1 Hambuechen, Electrometallurgist and Elecirochemist, 1902, 1. 37 ; Spring. Bull. Acad. Roy. Belg., Classe des Sciences, 1903, p. 1066 (see Cohen and Inouye, Zeitschr. physikal. C/icm., 19 10, 71. 301).

2 See, for example, Beilby, Proc. Roy. Soc, 1902, 72. 218 ; Ewing and Rosenhain, Phil. Trans.t 1900, A., 193. 353.

GENERAL SUMMARY 7$

stable to the stable form takes place with very great slowness at the ordinary temperature, so that articles made of tin, brass, and other metals by rolling or moulding under pressure, may possess a considerable degree of permanence. It has, however, been shown by Cohen ] and by Cohen and Inouye,- who have recently made a close investigation of the behaviour of worked tin, that the change may sometimes occur in a comparatively short space of time even at the ordinary temperature, especially if the metal is " inoculated" with some of the stable crystalline form. In the case of other metals or alloys, e.g. brass, this spontaneous transformation is also well-known, and may be the cause of much loss.:!

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 solvents, in the prismatic form, the less stable prismatic crystals then undergoing transformation into the rhombic form ; 4 a similar behaviour has also been observed in the transformation of the monotropic crystalline forms of sulphur."'

Many other examples might be given. In organic chemistry, for instance, it is often found that when a sub- stance 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

1 Zeitschr. physikal. Chem.t 1909,68. 214.

2 Ibid., 19 10, 71. 301.

s See Cohen unci Inouye, loc. cit.

4 Deville, Compt. reiid., 1852, 34. 561 ; Payen, ibid,, 1852, 34. 508 : Debray, ibid., 1858, 46. 576. It has also been found by Jaffe {Zeitschr. physikal. Chan., 1903, 43. 465) that when spontaneous crystallization from solution occurs, the less stable form always separates first when purification has been carried sufficiently far.

'° Brauns, Neues JaJirbuch fur Mineralogiet 1899, 13. (Beilage Band) 84.

76 THE PHASE RULE

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 x has put forward the laiu 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 obtain- able. 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 the less stable modification is produced, appear to be rather various.2

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. :;

1 Lchrbuch, II. 2. 445. See also Principles of Inorganic Chemistry, 3rd edit., p. 211 ff.

■ Schaum and Schbnbeck, Annalcn der P/iysik. 1902 [4], 8. 652. See also Chr. Fuchtbauer, Zeitschr. physikal. Che/n., 1904, 48. 549.

3 See Skrabal, Zeitschr. E/ektrochem., 1908, 14. 529, Zeitschr. phvsikal. Chew., 1910, 73. 171.

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. t 2 and 1 3 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 j 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 equi- librium 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.

73

THE PHASE RULE

Different Systems of Two Components. Phase Rule

P + F = C + 2

•Applying the

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 con- stitute 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 pos- sible in the case of two-compo- nent systems. Two compo- nents existing in only one phase consti- tute a tervari- ant 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.

Since a two-component system may undergo three possible 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

SYSTEMS OF TWO COMPONENTS 79

variables — pressure, temperature, and concentration of com- ponents (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 (//-diagram); one in the plane containing the pressure and concentration axes (e.g. AFor DF), the change of pressure with the concentration, the temperature remaining constant (^-diagram), while in the plane containing the con- centration and the temperature axes, the simultaneous change of these two factors at constant pressure would be represented (^--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, tempera- ture, 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 phen- omena 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 ; but has been selected from considerations of simplicity and clearness.

8o THE PHASE RULE

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. NH3 + HC1 = NH4 CI, 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).1

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 : NH4C1( = NH3+HC1) ; vapour : wNH3 4- #HC1. 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 NH3 + HC1, NH4C1 + NH3, and NH4C1 + HC1 ; 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 NH3 and HC1, 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 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).

I^ therefore, we have a solid substance, such as ammonium 1 Ramsay and Young, PJul. Trans., 1 886, 177. 87.

PHENOMENA OF DISSOCIATION 81

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.1 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 ammon- ium hydrosulphide, ammonium cyanide, phosphonium bromide, and other substances.2

Univariant Systems. — In order that a system of two com- ponents 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.

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^CaO

1 See volume in this series on Chemical Dynamics, by Dr. J. \V. Mellor.

2 Isambert, Compt. rend., 1S81, 92. 919 ; 1882, 94. 958 ; 1SS3, 96. 643. Walker and Lumsden, Jour. Chem. Soc, 1S97, 71. 42S.

T. P. C. G

8 2

THE PHASE RULE

4- COo. 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 corre- sponding 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 studied first by Debray,1 and later by Le Chatelier.2 Recently more exact determinations have been carried out by Pott:i and by ZavrierT,4 who obtained the following corresponding values of temperature and pressure : —

Temperature.

Pressure

in cm. mercury.

725°

71

750°

icro

8is°

23-0

84b0

34'2

86o°

42*0

8900

61 0

9100

75'S

912°

79-15

926-

I02 "2

From this table we see that it is only at a temperature of about 910° that the pressure of the carbon dioxide becomes equal to atmospheric pressure. In a vessel open to

1 Compt. rend., 1867. 64. 603. " Compt. rend., 1883, 102. 1243.

3 Inaugural Dissertation. See Riessenfeld, Joum. de Chim. phys., 1909,7. 561.

4 Journ. de Chim. phys., 1 909, 7. 3 1.

PHENOMENA OF DISSOCIATION 33

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, 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 temperatures 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,3NH3 and 2AgCl,3NH3, 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 Y that at temperatures below 15', silver chloride combined with ammonia to form the compound AgCl,3NH3, while at temperatures above 200 the compound 2AgCl,3NH3 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. 1 Compt. rend., 1868, 66. 1 259.

84 THE PHASE RULE

Regarded from the point of view of the Phase Rule, we see that we are here dealing with two components, AgCl and NH3. On being heated, the compounds decompose according to the equations : —

2(AgCll3NH3)^2AgClJ3NH3 + 3NH3. 2AgCl,3NH3$2AgCl + 3NH3.

There are, therefore, three phases, viz. AgCl,3NH3; 2AgCl, 3NH3, and NH3, in the one case; and 2AgCl,3NH, ; AgCl, and NH3 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,3NH3 and 2AgCl,3NH3, are present, i.e. until the compound richer in ammonia is com- pletely decomposed, when there will be a sudden fall in the pressure to the value corresponding to the system 2AgCl, 3NH3 — AgCl — NH3. The pressure will again remain con- stant 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,3NH3 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, 3NH0 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.1

The dissociation pressures of the two compounds of silver

1 Horstmann, Ber., 1876, 9. 749.

PHENOMENA OF DISSOCIATH W

85

chloride and ammonia, as determined by Isambert,1 are given in the following table: —

AgCI,3NH3.

2AgC!,3NH3.

Temperature.

Pressure.

Temperature.

Pressure.

o°

29 '3 cm.

20 -o°

9'3 cm.

io-6°

50-5 »

31-0°

12-5 „

.17-5°

65-5 »

47-0°

26-8 ,,

24-0°

937 „

s8-5°

52-8 „

28-0°

135-5 .,

69-0°

78-6 „

34-2°

i7i-3 »

71-5°

94'6 ,,

48-5°

241-4 >,

77-5°

H9-8 „

5i-5°

4i3'2 ,,

83-5°

I59-3 >,

54'o°

464-1 n

86-i°

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 pres- sure 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 neces- sary to know, not merely what is the si/bstance widergoing disso- ciation, but also what is the solid product of dissociation formed. For the definition of the equilibrium, the latter is as important us the former. We shall presently find proof of this in the case

1 Uc. cit.

86

THE PHASE RULE

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 crystalliza- tion, we meet with phenomena which are in all respects similar to those just studied. A salt hydrate on being heated dis- sociates into a lower hydrate (or anhydrous salt) and water vapour. Since we are dealing with two components — salt and water l — in three phases, viz. hydrate a, hydrate b (or anhy- drous salt), and vapour, the system is univariant, and to each temperature there will correspond a certain, definite vapour

pressure (the dissociation pres- sure), which will be indepen- dent of the relative or absolute amounts of the phases, i.e. of the amount of hydrate which has already undergone disso- ciation or dehydration.

The constancy of the dis- sociation pressure had been proved experimentally by several investigators2 a num- ber of years before the theo- retical basis for its necessity nad been given. In the case of salts capable of forming more than one hydrate, we should obtain a series of dissociation curves (//-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 : CuS04,5H30^> CuS04,3HaO + 2H0O. Curve OB: CuS04,3H2O^CuS04)H20 + 2H0O. Curve OC: CuS04,H20 ^CuS04 + H20.

Let us now follow the changes which take place on

D

Fig. 19.

1 For the reasons for choosing anhydrous salt and water instead of salt hydrate and water as components, see p. 14.

2 See Ostwald, Lehrbuch, II. 2. 527,

PHENOMENA OF DISSOCIATION 87

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 CuS04,H20 will be formed, and as there are now three phases present, viz. CuS04, CuS04,H20, and vapour, the system becomes univariant ; and since the tempera- ture is constant, the pressure must also be constant. Con- tinued 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 bi- variant, and passes along the line EF ; the pressure gradually increases, therefore, until at F the hydrate 3H0O is formed, and the system again becomes univariant ; the three phases present are CuS04,H20, CuS04,3H30, vapour. The pressure will remain constant, therefore, until the hydrate iH20 has dis- appeared, when it will again . increase till G is reached; here the hydrate 5H0O is formed, and the pressure once more remains constant until the complete disappearance of the hydrate 3HX) has taken place.

Conversely, on dehydrating CuS04,5H20 at constant temperature, we should find that the pressure would maintain the value corresponding to the dissociation pressure of the system CuS04,5H20— CuS04,3H20— vapour, until all the hydrate 5H0O had disappeared; further removal of water would then cause the pressure to fall abruptly to the pressure of the system CuS04,3H20—CuS04,H20— 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 500.

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

8S

THE PHASE RULE

A

R

n

n

E

Fig. 20.

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 evi- dent that a crystalline salt hydrate will effloresce when exposed to the air, if the partial pressure of the water P vapour in the air is lower than the dissociation pres- sure 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.

Indefiniteness of the Vapour Pressure of a Hydrate- Reference has already been made (p. 85), 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 CuS04,H20 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

PHENOMENA OF DfsSOCIATloX

89

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 o/ily when the second solid phase produced by the dissociation is give?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 7iext 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.1

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,6H20 can dissociate into water vapour and either of two lower hydrates, each containing four molecules of water of crystallization, and designated respectively as CaCL,4H20a, and CaCL,4.H.,0/?. Roozeboom 2 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.

CaCl2,6H20 ; CaCl2)

CaC]2,6H20 ; CaCl2>

4H20a ; vapour.

4H20/3 ; vapour.

-15°

C027 cm.

0'022 cm.

0

0-092 ,,

0*076 ,,

4- 10

OT92 „

0-I62 „

20

0-37S „

0*315 »

25

0-508 ,,

o-432 ».

29'2

—

0-567 »•

29*8

o-6So ,,

1 Ostvvald, Lehrbuch, II. 2. 538.

' Zsitschr. physikal. C/icm., 1889, 4. 43.

9o THE PHASE RULE

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 wre 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 Na2S04,ioH20 could be kept unchanged in the open air, although the vapour pressure of the system Na2S04,ioH20 — Na2S04 — 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 Na2S04 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. 83). Horstmann,1 for example, found that the pressure of ammonia in contact with 2 AgCl,3NH3 could be raised to a value higher than the dissociation pressure of AgCl,3NH3 without this com- pound 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.

1 Bey., 1876, 9. 749.

PHENOMENA OF DISSOCIATION 91

Range of Existence of Hydrates In Fig. 19 the vapour 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.1

Constancy of Vapour Pressure and the Formation of Com- pounds.— 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 dis- sociating compound was present, independently of the degree of the decomposition (p. 87). 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 2 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,H20 and 3PbO,H20.

The method has also been applied to the investigation of the so-called palladium hydride/ and the results obtained

1 See, for example, van't Hoff, Lectures on Theoretical and Physical Chemistry, I. p. 62 (Arnold). - Jour. Chern. Soc, 1877, 32. 395.

3 Hoitsema, Zeitschr. physikal. Chern., 1S95, 17. 1.

92

THE PHASE RULE

appear to show that no compound is formed. Reference will, however, be made to this case later (Chap. X.).

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 tensimetcr}

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, ^r. bromonapthalene. If it is desired to measure the dissociation pres- sure of, say, a salt hydrate, concentrated sul- phuric acid is placed in the flask e, and a quantity of the hydrate,- well dried and pow- dered," 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 in- dependent of the degree of dissociation of the hydrate (p. 87). The small bulbs d and e Fig. 2i. 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 ex- hausted through f by means of a mercury pump, and sealed off. The apparatus is now placed in a perpendicular position

1 Zeitschr. physikal. CAem,t 1887, 1.

:895> 17- 52- See also>

.Schottky Zeitschr. physikal. Chew., 1908, 64. 415. For the determination of vapour pressures by a dew-point method, see Cumming, Trans. Chem. Soc, 1909, 95. 1772.

- It is important to powder the salt, since otherwise the dehydration of the hydrate and the production of equilibrium occurs with comparatively great tardiness.

PHENOMENA OF DISSOCIATION 93

in a thermostat, and kept at constant temperature until equi- librium 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.

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.1 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.2

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 treatment does not come within the scope of the Phase Rule, which deals with heterogeneous equilibria.

1 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).

2 Van't Hoff, Zeitschr. physikal. Chan., 1890, 5. 323 ; Ostwald, Uhrbuch, I. 606.

SOLUTh

95

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 per- manently 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 com- ponents 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 pre- dominates, 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 con- dition 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 with the gas. In the light of the Phase Rule, now, such a system is bivariant (two components in two phases); and two

96 THE PHASE RULE

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 con- centration 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 diredioti 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 con- centration 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.

Solutions of Liquids in Liquids.

When mercury and water are brought together, the two liquids remain side by side without mixing. Strictly speaking,

SOLUTIONS

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.1 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 immis- cibility 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, 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

1 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 tem- perature of 280° the mercury dissolves in appreciable quantity ; for on heat- ing a tube containing bromonaphthalene over mercury the latter sublimes through the liquid bromonaphthalene and condenses on the upper surface of the tube.

T. P. C. H

93 THE PHASE RULE

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 com- ponents 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 pres- sure of the vapour can undergo change ; or, if the composition of the solution remains unchanged, the pressure and the tem- perature 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 defi- nite 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 tem- perature, 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 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

SOLUTIONS

been studied by Guthrie,1 and more especially by Alexejeff2

and by Rothmund.3 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 no\V the tem- perature 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 tempera- tures 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 homogene- ous solution is again obtained. The phenomena are exactly analogous to those already described. Since, now, in the second case the concentration of the phenol in the solution gradually decreases, while in the former case it gradually increases, a

1 Phil. Mag., 1884 [5], 18. 22 ; 495. ■ Wicd. Aniialeii, 1886, 28. 305. 3 Zeiischr. physikal. Chem., 1898,26.433.

A For a list of partially miscible liquids, see Ti miner mans, Bull. Soc mint. Bclg., 1906, 20.

IOO

THE PHASE RULE

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.

From what has been said above, it will be seen that at any temperature below the critical solution temperature, two conjugate solutions con- taining water and phenol in different concentration can exist together, one contain- ing excess of water, the other excess of phenol. The following table gives the com- position of the two layers, and the values are represented graphically in Fig. 22.1

Phenol and Water.

Cs is the percentage amount of phenol in the first layer. C2 ,, » » second layer.

6S-4 t

Fig. 22.

Temperature.

Gi.

c2.

20°

8-5

72*2

3O0

87

69-9

40°

97

66-8

50°

I2*0

627

55°

I4-2

60 0

6o°

17-5

56*2

65°

227

497

68-4°

36-1

36-1

The critical solution temperature for phenol and water is 1 Rothmund, loc. cit.

SOLUTIONS 101

6S'4 , the critical concentration 36*1 percent, 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 there- fore 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.1

From Fig. 22 it is easy to predict the effect of bringing together water and phenol in any given quantities at any tem- perature. 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 repre- sented by y, the other that represented by y'. The system is now univariant, and on further addition of phenol, the compo- sition 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 tem- perature at which two liquid phases consisting of phenol and 1 See Timmermans, Bull. Soc. chim. Bcl^., 1909, 23. .\^j.

102

THE PHASE RULE

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 ap- pearing. Starting with x, the temperature is first raised above the critical solution temperature; phenol is then added until the concentration reaches the point x2. On allowing the tem- perature to fall, the system will then pass into the con- dition represented by x\

Methylethylketone and Water. — In the case just described, the solu- bility 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 : x —

Methylethylketone and Water.

-10' O + IO 20 40 60 80 100120140160

Fig. 23.

Temperature.

Ci per cent.

C2 per cent.

- 10°

34'5

897

+ IO°

26-1

900

300

21-9

89-9

50°

17-5

89-0

70°

l6'2

857

900

161

84-8

no°

177

80 -o

130°

21-8

71-9

140°

26*0

64*0

151-8°

44 -2

44'2

Rothmund, loc. cit.

SOLUTIONS

:©3

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 io°) in the solubility of water in mcthyl- ethylketone. Minima of solubility have also been found in other cases.

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 180, triethylamine and water mix together in all propor- tions; but, on raising the temperature, the homo- geneous solution becomes turbid and separates into two layers. In this case, therefore, the critical solu- tion temperature is found in the direction of lower temperature, not in the direction of higher.1 This behaviour is clearly shown by the graphic representation in Fig. 24, and also by the numbers in the following table : —

Triethylamine and Water.

10- 20" 30 40 50J 60 70J 30 90

Fig. 24.

Temperature.

Ci per cent.

C2 per cent.

700

1-6

50°

2-9

—

30°

5'6

96

250

7'3

95-5

20°

i5'5

73

+ 18-5°

± 30

±30

General Form of Concentration-Temperature Curve.— From the preceding figures it will be seen that the general

1 A similar behaviour is found in the case of diethylamine and water (R. T. Lattey, Phil. Mag., 1905 [6], 10, 397). See also, Rothmund, Zcitschr. pliysikal. Chcm.t 1898, 26. 433 ; Flaschner, ibid., 190S, 62, 493.

104 THE PHASE RULE

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 6o° and above 2io°, nicotine and water mix in all proportions.1 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 it2

With regard to the closed end of the curve it may be said that it is continuous ; the critical solution point is not the inter- section 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 com- ponents, 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. Q5).3

Influence of Foreign Substances on the Critical Solution Temperature. — For a given pressure, the critical solution temperature is, as we have seen, a perfectly defined point. It is, however, altered to a very marked extent by the addition of a foreign substance (impurity), which dissolves either in one

1 C. S. Hudson, Zeitschr. physikal. Chew., 1904, 47. 11; examples of closed solubility curves have been obtained by Flaschner {Trans. Chem. Soc, 1909, 95. 668) and by Flaschner and McEwen (ibid., 1908, 93. 1000).

2 See also, Dolgolenko, Zeitschr. physikal. Chem., 1908, 62. 499.

3 For a general discussion of the equilibria in the case of partially miscible liquids, see Biichner, Zeitschr. physikal. Chem., 1906,56. 257.

soLrrwxs

to«

or both of the partially miscible liquids.1 When the third substance (impurity; dissolves in only one of the two liquids, the mutual solubility of the latter is diminished ; consequently, the critical solution temperature is raised in the case of mixtures having a superior critical solution point, and lowered in the case of mixtures having an inferior critical solution temperature. Thus, in the case of phenol and water, addition of about o'6 per cent, of sodium chloride raises the critical solution point by about 10 . On the other hand, if the added substance dissolves in both of the partially miscible liquids, the mutual solubility of the latter is increased, and consequently, a superior critical solution temperature is lowered, while an inferior critical solution temperature is raised.

The sensitiveness of the critical solution temperature to impurities is of great importance, and has found very valuable application not only in analytical practice for distinguishing between related substances, such as different oils and fats, but also in the laboratory as a delicate test for the purity of a liquid.2

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 com- ponents. On studying the total pressure of a system, it is found that two cases can be obtained."

So long as there is only one liquid phase, the system is bi variant. The pressure therefore can change with the con- centration and the temperature. If the temperature is maintained constant, the pressure will vary only with the concentration, and this variation can therefore be represented by a curve.

1 See more especially, Timmermans, Zeitschr. physikal. Chew., 1907, 58. 129; Schiikarcff, ibid., 1910, 71. 90.

-See Crismer, Bull. Je V Association Jcs Chimislcs, 1S95, 9- U5 i 1S96, 10. 312 ; 1904, 18. 1 ; 1906, 20. 294 ; Zeitschr. physikal. Chew., 1895, 20, 390; Flaschner, Trans. Chew. Soc., 1909, 95. 677; Dolgolenko, Zeitschr. physikal. Chew., 1908, 62. 499.

3 Konowaloff. Wied. Annalcn, 1SS1, 14. 219. Ostwald, Lehrbuch, II. 2. 6S7. Bancroft, Phase Rule, p. 96.

io6

THE PHASE RULE

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 phe- nomenon which is very generally found in the case of partially miscible liquids, e.g. ether and water.1 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.

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.2 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 liquid phases is less than that of each of the components is not possible.

Concentration

Fig. 25.

1 Konowaloff, toe. cit.

2 Roozeboom, Zeitschr. physikal. Chem.

I, 8. 526 ; Rec. Trav.

Chim. Pays. -Bos, 1884, 3. 38.

SOLUTIONS 107

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.1

Complete Miscibility. — Although the phenomena of com- plete 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 mis- cibility and those in which only one homogeneous solution is formed. As has been already pointed out, the solubility re- lations 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 tempera- tures below -i4-4°.2 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 before the system becomes denned. For this reason the Phase Rule affords only a slight guidance in the study of such

1 Konowalni't", loc. cit. Cf. Bancroft, Phase Rukt p. 100.

2 Phil. Mag., 1S84 [5], 18. 503.

io8 THE PHASE RULE

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 composi- tion 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.1

1 See, for example, Walker, Introduction to Physical Chemistry, 3rd edit., p. 86 (Macmillan, 1903). Consult also Young, Fractional Distil- lation (Macmillan, 1903), or Kuenen, Verdampfung und Verfliissigungvon Gemischen (Barth, 1906), where the subject is fully treated.

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, tempera- ture, or concentration of the components (in the solution 2), 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 surround- ings, 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, 1 Since this is the only phase of variable composition present.

HO

THE PHASE RULE

vapour will condense to liquid, in order that the pressure may remain constant, and so much of the solid will pass into solu- tion 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 con- centration 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.1

Change of volume by- dissolving i gm. of salt in the saturated solution.

Solubility (at 180) (grams salt in 1 gram of solution).

Salt.

Pressure = 1 atm.

Pressure = 500 atm.

Sodium chloride . . . Ammonium chloride . . Alum

— 0-07

+o-io

-0*067

0-264 O272 OTIS '

0-270

0-258

0-142

(/ = 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 prac- tical purposes, therefore, the solubility as determined under atmospheric pressure may be taken as equal to the true

1 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).

SOLUTIONS OF SOLIDS IN LIQUIDS in

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 modifica- tions, 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.1 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 1 Van't Hoff, Arch, nierland. 1901 [2], 6. 471.

113

THE PHASE RULE

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.

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.1

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 theo- . rem (p. 57) increase of solubility with the tem- perature must occur in those cases where the pro- cess of solution is accom- panied by an absorption of heat ; and a decrease in the solubility with rise of temperature will be found in cases where solution occurs with evolutio?i of heat. Where there is no heat effect accompanying solution,

1 Tilden and Shenstone, Phil. Trans. 1884, 175. 23 ; Hulett and Allen, Jour. Amer. Chem. Soc. 1902, 24. 667 ; Andrea, Jour. prak. Chem. 137. 474 ; Lumsden, Jour. Chem. Soc, 1902, 81. 350 ; Mylius and v. Wrochem, Ber. 1900, 33. 3689.

Temperature

Fig. 26.

SOLUTIONS OF SOLIDS IN LIQUIDS 113

change of temperature will be without influence on the solu- bility; 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.1

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 : 2 —

Number of gram-molecules of

CuCl2, 2H2O dissolved in 198

Heat effect.

gram-molecules of water.

j

+37 K

2*02

+ 66 „

4-15

+ 105 „

7-07

+ 117 „

9'95

+ 117 „

II

+ 91 ,,

18-8

-10 ,,

19-6

-31 ,.

2475

-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. Judgino- 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

1 E. von Stackelberg, Zeitschr. physikal. Chetn. 1896,20. 159; 1898,26. 533; Lumsden, Jour. Chem. Soc, 1902, 81. 350; Holsboer, Zeitschr. physikal. Chem., 1902, 39. 691.

2 Reicher and van Ueventer, Zeitschr. physikal. Chem. 1890, 5. 559 ; cf. Ostvvald, Lehrbuch, II. 2. 803.

T. P. C. I

ii4 THE PHASE RULE

the last heat of solution is negative (as expressed above), i.e. solution of the salt in the almost saturated solution is accom- panied 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 solu- bility curve can be predicted. We shall presently meet with examples of this.1

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 o° and ioo°, the solid phase in contact with the solution being the anhydrous salt. For the representation of these equilibria, the concentration-temperature

1 It has been shown that the formula of Ramsay and Young (p. 66) can be applied (with certain restrictions) to the interpolation and extra- polation of the solubility curve of a substance provided two (or three) points on the curve are known. In this case T, T„ etc., refer to the tem- peratures 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. Chew., 1903, 42. no.)

SOLUTIONS OF SOL IDS IX LIQUIDS

1 1

diagram is employed, the concentration being expressed as the number of grams of the salt dissolved in ioo grams of water, or as the number of gram-molecules of salt in ioo 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.

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 enantio- 66 tropic crystalline forms, the range of stability of these co forms being limited by defi- nite temperatures (transition temperature^. Since the tran- sition point is not altered by 50 a solvent (provided the latter is not absorbed by the solid phase), we should find on 40 studying the solubility of these substances in water that 1 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 ca* of ammonium nitrate l

1 W. Miiller and P. Kaufmann, ZeiUchr. physikal. Cliem. 1903, 42. 497.

Fig. 2:

u6

THE PHASE RULE

and thallium picrate.1 The following table contains the values of the solubility of ammonium nitrate obtained by Miiller and Kaufmann, the solubility being expressed in gram-molecules NH4N03 in ioo 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 super- saturation, 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

25-050

48-19

35 -o°

59-80

28-0°

51-86

36-0°

61 -oo

30-00

54-40

37-5°

62-90

30-2°

54-6i

38-0°

63-60

3i-9°

57-20

39-o°

65-09

32-1°

57-6o

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 /3-rhombic into the a rhombic form of ammonium nitrate (p. 6^).

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, 1 W. O. Rabe, Zeitschr. physikal. Ch^m^ 1901, 38. 175.

SOLUTIONS OF SOLIDS IN LIQUIDS

117

therefore, a small quantity of the more stable form is intro- duced 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 im- portance 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.1

Solubility Curve at Hig'her Temperatures. — On passing to the consideration of the solubility curves at higher tempera- tures, 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 Mis ability of the luised Components. The best example of this which has been studied, so far as anhydrous salts and water are concerned, is that of silver

~ -550

__

.

J^.

/

/

/

Fig. 28.

nitrate and water. The solubility of this salt at temperatures

1 \\ Lth regard to the limits of supersaturation and the spontaneous crystallization of the solute from supersaturated solutions, see Jafl'r, Zeitschr. physikal. CInm., 1903, 43. 565 ; Miers and Isaac, Trans. Chent. Soc, 1906, 89.413; 190S, 93. 3S4 ; Hartley, Jones and Hutchinson, ibid., 825 : Jones, //W., 1739 ; Fouquet, Compt. rend.t 1910, 150. 2S0.

ii:

THE PHASE RULE

above ioo° has been studied chiefly by Etard 1 and by Tilden and Shenstone.2 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 ioo parts of solution.

_"0

/

46*2

jO

52-1

+ 5°

56-3

10°

6l*2

20°

678

40'5°

76-8

73°

84-0

135°

92-8

1 82°

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 1600 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 tempera- tures, therefore, the concentration of silver nitrate in the solution gradually increases, until at last, at a temperature of 2080,3 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 tempera- ture at which the solid salt can exist in contact with the liquid;

1 Annates chim. phys., 1894 [7], 2. 524.

2 Phil. Trans., 1884, 175. 23.

3 Hissink, Zeitschr. physikal. Chem., 1900, 32. 54^

SOLUTIONS OF SOLIDS IN LIQUIDS 119

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 o°. The melting point, will, how- ever, 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 equilib- rium. 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 Middelberg1 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 com- ponents 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 1 Zdtschr. physikal. C/icm., 1903, 13. 313.

120 THE PHASE RULE

point curve of ice. 0 is the quadruple point at which the invariant system exists, and may be regarded as the point of

intersection of the solu- bility curve with the freezing - point curve. Since this point is fixed; the condition of the system as regards tem- perature, vapour pres- sure, and concentration of the components (or composition of the so- lution), is perfectly de- finite. 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}

Cryohydrates.2 — On cooling a solution of common salt in water to a temperature of -30, Guthrie observed that the hydrate NaCl,2H20 separated out. This salt continued to be de- posited until at a temperature of -220 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 com- position of different samples of the solid phase deposited from the solution at constant temperature.3

1 Guthrie, Phil. Mag., 1875, [4], 49. 1 ; 1884, [5], 17. 462.

2 See Roloff, Zeitschr. physikal. Chcm., 1895, 17. 325 ; Guthrie, loc. cit.

3 Guthrie, Phil Mag., loc. cit. Cf. Ostwald, Lehouch, II. 2. 843.

SOLUTIONS OF SOLIDS IN LIQUIDS

121

No.

Temperature of solidification.

NaCl. Per cent.

2

3 4 5 6

- 21° to -22°

— 22°

- 22°

-23° -23° -23°

23-72 23-66

2373 23-82

23'34

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 : l —

Cryohydrates.

Salt.

Cryohydric point.

Percentage of anhydrous salt in the cryohydrate.

Sodium bromide ....

-24°

4**33

Sodium chloride . .

-22°

23-00

Potassium iodide . .

- 22°

52-07 40'So

Sodium nitrate . .

- 17-5°

Ammonium sulphate

41-70

Ammonium chloride

-15°

19-27

Sodium iodide . .

-15°

59-45

Potassium bromide .

-13°

32-15

20*03

Potassium chloride .

-n-4°

Magnesium sulphate

" 5°

21-86

Potassium nitrate

-2-6°

1 1 20

Sodium sulphate . .

-0-7°

4-55

1 Guthrie, Phil. Mag., 1875 [4], 49. 269.

122 THE PHASE RULE

The chemical individuality of these cryohydrates was, however, called in question by Pfaundler,1 and disproved by Offer,2 who showed that in spite of the constancy of the melt- ing 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,3 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 compen- satory changes will take place within the system whereby the

1 Bet., 1877, 20. 2223.

s Sitz.-Ber. Wien. Akad., 1880, 81. II. 1058.

» Guthrie, Phil. Mag., 1875 [4], 49. 206.

SOLUTIOXS OF SOLIDS IN LIQUIDS 123

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 evapora- tion. 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 tsystem 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 tempera- ture that the four phases ice — salt — solution — vapour can co- exist. 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

124 THE PHASE RULE

composition of the solution; in the former case, salt will be deposited l 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 pro- duced. 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 Mis ability 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 propor- tions, we find that the relationships are not quite so simple as

1 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.

SOLUTIOXS OF SOI. IPS IN LIQUIDS 125

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 dis- continuity in the solubility curve on passing from the cryohydric point to the melting point of the second (non-volatile) com- ponent.

Paratoluidine dissolves in water, and the solubility increases as the temperature rises.1 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 phe- nomenon of melting under the solvent. This melting point will, of course, be lower than the melting point of the pure substance.

0 •5 -2

c

5 *

-1-2°«

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, C2H4(CN)2 and water, which has been fully studied by Schreinemakers.2

If to the system ice — water at o° 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 tem- perature 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.

1 Walker, Zeitsehr. physikal. Chem., 1S90, 5. 193.

2 Zeitsehr. physikal. Chem., 1897, 23. 418.

126 THE PHASE RULE

If the temperature be now raised so as to cause the disappear- ance 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 be. 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 concentratioiis 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 </, 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 540 the melting point of the pure nitrile is reached.

Return now to the point e. 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.1 Such a system will exhibit relationships similar to those already studied in the previous chapter. As the tem- perature rises, the mutual solubility of the two fused com- ponents becomes greater, until at d (5 5 '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 1 Provided the solid nitrile is not present in too great excess.

SOLUTIOXS OF SOLIDS IX LIQUIDS 127

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 bcy corresponding to the given temperature. On adding the nitrile to water at temperatures between c and dy 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 ce 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 bed is heated to a temperature above dy 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 j the system can then be cooled down to a temperature represented by any point between the curves de'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,

128

THE PHASE RULE

these two liquid layers, as a matter of fact, make their appear- ance. 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 solidifi- cation occurring. The question, therefore, arises as to the relative solubilities of the solid and the supercooled liquid at the same temperature.

The answer to this question can at once be given from what we have already learned (p. 115), 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,

oj0r

rU I , ill 1 W

1 cz^

TOO** 90 SO 70

30 UO 50 60

80 SO 100 no 120

b, b, and c,c = solid \ FIG. 31.

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 solu- bility, it follows that the supercooled liquid, being the less stable form, must have the greater solubility. This was first proved experimentally by Alexejeff1 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 » Wied. Anna/en, 1886, 28. 328. Cf. Ostwald, Lehrbuch, II. 2. 872.

SOLUTIONS OF SO/JDS IX LIQUIDS 129

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.1

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 correspond- ing 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. in). 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 co?icc}Uratio?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 con- sidered, viz. where the concentration increases with rise of

1 Walker, Zeitschr. physikal. Cherti., 1S90, 5. 193. Schieinemnkers, ibid., 1897, 23. 417. Roozeboom, Ree. tray. chim. Pays-Bays, 1SS9, 8. 257. Bruner, Zeitschr. physikal. Chew. 1S97, 23. 542.

T. P. C. tr

i3o

THE PHASE RULE

Fig. 22.

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. 32.1 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 in- creases with rise of tem- perature, 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.

Other Univariant Systems.— Besides the univanant system

1 Van't Hoff, Lectures on Theoretical Chemistry, I. p. 4* Lehrbuch, II. 2. 824.

Ostwald.

SOLUTIONS OF SO T.I PS IN LIQUIDS 131

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 equi- librium 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 con- centration 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 compo?ients 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

1 32 THE PHASE RULE

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 jce — 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.

soirriOAs of solids in hoc ids

133

Deliquescence. — As is evident from Fig. 32, salt can exist in contact with water vapour at pressures under those repre- sented by OAM F. If, however, the pressure of the vapour is increased until it reaches a value lying on this curve at tempera- tures 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.1

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 re- presents atmospheric pressure. If, now, an unsaturated solu- tion, 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 oft', solid salt will be deposited, and the solution will evaporate to dryness.

If, however, the atmospheric pressure is represented not by aa but by bbt then, as Fig. 32 shows, the maximum vapour

1 Ostwald, Principles of Inorganic Chemistry^ translated by A. Findlay, 3rd edit., p. 455 (Macmillan, 1908) ; Skirrow and Calvert, Zeitsckr. physikal. C/io/i., 1901, 37. 217.

i34 THE PHASE RULE

pressure of the system salt — solution — vapour never reaches the pressure of i 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 atmo- spheric 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 i atm., and on evaporating these solutions, therefore, in an open vessel, there is no separa- tion 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 dis- cussed 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 sub- stances ; 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,

SOLUTIONS OF SOLIDS IN LIQUIDS 135

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.

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.

i. 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

SOLUTIONS OF SOLIDS IX LIQUIDS

1j7

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 ex- ample of the relationships met with in such cases, we shall first of all consider the systems formed of sodium sulphate and water.

Sodium Sulphate and Water. — At the ordinary tempera- tures, sodium sulphate crystallizes from water with ten molecules of water of crystallization, 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, re- presented graphically by the curve AC (Fig. 33), being given in the following table.1 The numbers denote grams of sodium sulphate, calculated as anhydrous salt, dissolved by 100 grams of water.

-60

C

£50

£40

yi //b~~-

D

S 30

-

5

£20

-''F

10

-~~K

20" Fig.

Solubility of Na2S04,ioH.,0.

Temperature.

Solubility.

o°

5 02

IO°

15°

9 00 13-20

1 8°

16-80

20°

19-40

25o

28-00

30°

40-00

33o

50-76

34°

5500

1 Vide Loewel, Annates chini. phys., 1857 [3], 49. 32. Cf. Lowenherz, Zeitschr. physikal. C/iem., 1895, ^. 82.

33

THE PHASE RULE

On continuing the investigation at higher temperatures, it was found that the solubility no longer increased, but decreased with rise of temperaftire. 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 : — l

Solubility of Anhydrous Sodium Sulphate.

Temperature.

Solubility.

1 8°

53"25

20°

5276

25o

5i"53

30°

5Q"37

33o

4971

3*°

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 330.

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 Na2S04,7H20, 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) : —

1 Loewel, loc. cit. Gay-Lussac, Annates chim. phys., 1819, 11. 296. For the solubility at higher temperatures, see Tilden and Shenstone, Phil. Trans., 1884, 175. 23. Etard, Annates chim. phys., 1894 [7], 2. 548.

SOLUTIONS OF SOLIDS IN LIQUIDS 139

Solubility of Na2S04,7H20.

Temperature.

Solubility.

o°

19*62

IO°

3""49

15°

37 43

1 8°

41-63

20°

4473

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 inter- section 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 solu- tion has a certain definite composition. The temperature of this point is 32'482° on a mercury thermometer, or 3 2 "379° on the hydrogen thermometer.1

1 Richards, Zcitschr. physikal. CJu/11., 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. no), the position of the two points will not be greatly different. The quadruple point was found by Cohen

i4o THE PHASE RULE

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 bring- ing 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 decahy- drate, and will deposit this salt if a " nucleus " is added to the solution. From this we see that at temperatures above 3 2 "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 poi7it 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

(Zeitschr. physikal. C/iem., 1894, 14. 90) to be 32"6° and 308 mm. of mercury.

SOLUTIONS OF SOLIDS IN LIQUIDS \\\

represents only metastable equilibria. Sodium sulphate, there- fore, 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 tempera- ture of about 1 8°, it is readily seen, from Fig. 33, that at a temperature of, say, 200 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 fur- nishes 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 deca- hydrate 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 330 the latter is unstable, and cannot be formed in pre- sence 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.

1 42 THE PHASE RULE

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 cir- cumstances, equal to or higher than that of the latter. An example of this is found in sodium sulphate decahydrate.

On heating Na2S04,ioH20, 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 measure- ments;1 the differences in pressure being expressed in milli- metres of a particular oil.

Temperature: 29-0° 30-83° 3179° 32'09° 32'35° 32'6°

Difference of pressure : 23-8 io-8 5'6 3*6 r6 o

At 32-6°. therefore, the vapour pressures of the two systems

Na,S04, 1 oH20—Na.S04— vapour Na2S04 — solution — vapour

are equal; at this temperature the four phases, Na2S04,ioH20 ; Na2S04 ; 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.2 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 tran- sition point of the decahydrate and anhydrous salt, i.e. with the reaction Na2S04, ioH2O^Na2S04 + ioH20.

Since at the point of partial fusion of the decahydrate four

1 Van't Hoffand van Deventer, Zeitschr. physikal. C/iem., 1887, 1. 185. Cf. Cohen, ibid., 1894, 14. 88.

8 Debray, Compt. rend., 1868, 66. 194.

SOLUTIONS OF SOLIDS IN I. lor IDS

143

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.1 The temperature is, of course, practically the same as that at which the two solu- bility curves intersect (pp. 115, 137). If, however, the vapour phase disappears, the system becomes univariant, and the equi- librium temperature undergoes change with change of pressure. The transition curve has been determined by Tammann,- and shown to pass through a point of maximum temperature.

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. :; The curve ABCD represents the vapour-pressure curve of the saturated solution of anhydrous sodium sul- phate. GC is the pressure curve of decahydrate + anhydrous salt, which, as we have seen, cuts the curve ABCD at the tran- sition 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.4 As at temperatures below this point the solubility of the deca- hydrate is less than that of the anhydrous salt, the vapour pres- sure 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).

1 Richards, Zeitschr. physikal. Chew., 1 898, 26. 690. A number of other salt hydrates, having transition-points ranging from 200 to 7S0, which might be used for the same purpose, have been given by Richards and Churchill, ibid., 1899, 28. 313 ; Richard and Wells, ibid., 1906, 56. 34S.

- Zeitschr. physikal. Che///., 1903, 46. SlS.

1 Van't Hoff, Lectures on Physical Chemistry) I. p. 67.

4 Cohen, Zeitschr physikal. Chem., 1 894, 14. 90.

Fig. 34.

144 THE PHASE RULE

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 tem- peratures below 32*5° the system decahydrate — solution — vapour is more stable than the system anhydrous salt — solu- tion— 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 240 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 dissocia- tion 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.1 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.

1 Ziz, Schweiggei's Journal, 18 15, 15. 166. See Ostwald, Lehrbuch- II. 2. 717.

SOLUTIONS OF SO /JDS IX LIQUIDS

145

In the case of sodium sulphate there is only one stable hydrate. Other salts are known which exhibit a similar be- haviour; 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,1 and in some cases there is more than one metastable hydrate. This is found, for example, in the case of nickel iodate,2 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

Fig.

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.3 In the preceding cases, the dissociation-pressure curve of the hyd rated salt cuts the vapour-pressure curve of the saturated

1 See, for example, the solubility determinations published in Wissen* VhaftHche Abhandl. der physikaliseh-technischen Reich sanst alt, Vol. III., or

in the Berichte, for the years 1897-1 901. - Meusser, Ber., 1901, 34. 2440. 3 Mylius and von Wrochem, i>Vr., 1900, 33. 3693. T. P. C. L

146

THE PHASE RULE

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 hydrates1 (Fig. 36).

At temperatures above o°, barium acetate can form two stable hydrates, a trihydrate and a monohydrate. The solu-

64

8 7t

7=

T

• 1

/

^>

**- "A

c C

h 1 1 y\

^v.

i^Roi

» 5 SALT

>%

9

/ /+*

A**

JA

6o°

8o°

bility of the trihydrate increases very rapidly with rise of tem- perature, and has been determined up to 26'i°. At temperatures above 2470, 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 suc- ceeds that of the trihydrate, and exhibits a conspicuous point of minimum solubility at about 30°. Below 24*7° the

1 Walker and Fyffe, Jour. Chem. Soc, 1903, 83. 180.

SOLUTIONS OF SOLIDS IX LIQUIDS 147

monohydrate is the less stable hydrate, but its solubility has been determined to a temperature of 220. At 410 the solubility curve of the monohydrate intersects that of the anhydrous salt, and this is therefore the transition temperature for the mono- hydrate 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 solu- bility of barium acetate as determined by Krasnicki.1 Since, however, three different solid phases can exist under the conditions of experiment, it is evident, from what has already been stated (p. 114), 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 eq?ial 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 accom- plished by actual weighing of the dissolved salt or by deter- minations of the density of the solution, as well as by other methods.

1 Monatshef'ty 18S7, 8. Coi.

[48 THE PHASE RULE

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 thio- sulphate, NaoSA^H.O, sodium acetate, NaC2H302,3HoO) 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 5 6°; in the case of sodium acetate trihydrate, 5 8°. 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.1— Although calcium chloride forms several hydrates, each of which possesses its own solubility, it is nevertheless the solu- bility curve of the hexahydrate which will chiefly interest us

at present, and we shall therefore first discuss that curve by itself.

The solubility of this salt has been determined from the cryo- hydric point, which lies at about — 550, up to the melting point of the salt.2 The solubility in- creases with rise of temperature, as is shown by the figures in the

Temperature following table, and by the (dia-

grammatic) curve AB in Fig. 37. In the table, the numbers under the heading "solubility" denote the number of grams of CaCL dissolved in 100 grams

1 The equilibria between calcium chloride and water have been most completely studied by Roozeboom {Zeitschr. physikat. Chem., 1S89, 4. 31).

- Hammed, Sitzungsber. Wien. Akad, 2t0 Abteil, 1878, 78. 59. Rooze- boom, Zeitschr. physikal. C/iem., 1889, 4. 31.

SOLUTIOXS OF SOLIDS IX LIQUIDS

of water; those under the heading " composition," the number of gram-molecules of water in the solution to one -ram molecule of CaCL

Solubility of Calcium Chloride Hexahydrate.

Temperature.

Solubility.

Composition.

-.0

42-5

145

~25o

500

123

-IO°

55"o

II'2

o°

595

10-37

10°

65 0

9 '49

20°

745

8-28

250

820

7-52

2S'5°

Q0'5

681

29-5°

95'5

6-46

30-2°

1027

6-oo

29-6°

109-0

57o

29-2°

II2-8

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' 20 ; and since the composition of the solution is the same as that of the solid salt, viz. 1 mol. of CaCL 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 tempera- ture 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 con- sidered in that it possesses a refrqflex 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 lhat

i5o THE PHASE RULE

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 sub- stance. 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 tempera- ture 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 equi- librium 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.1

1 Lidbury, Zcitschr. 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 un- dissociated, 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 dissocia- tion, 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 Rooze- boom and Aten, Zcitschr. physikal. Chan., 1 905, 53. 463 ; Kremann, Zcitschr. Elehlrcchcm., 1906, 12. 259. Findlay, Trans. Faraday Sac, 1907, 3.

SOLUTIONS OF SO/. IDS IX LIQUIDS

mi

Although in taking up the discussion of the equilibria

between calcium chloride and water, it was desired to call attention to the form of the solubility curve in the case of salt hydrates possessing a definite melting point, ueverthe- Temperature

-50° 0° 50° 100° 150° 20cf

0

/

A

J*'

E

y

5-

kLft

\

^

6>

XO

bN

H

-6H20

^NK

b

Tj\\'

"^F>^

■4*1

H20

\

\

\

«

•c

\

S

s^

\

5 70

\

<S

\ft

5

;<

\

•£

\

&

e

^O

L

o

GT

\

\

-

2H

,c

)

^

^

ft<

o

\

H

5

E^

H:

o

c

S

A

M

_,

\S^

oS

rvvr1, \j^

°120

D X^, ^

CO

5

r —

K

C;

i n,

^130

"^^tn

25 30 35 40 45

Fig. 38.

less, for the sake of completeness, brief mention may be made of the other systems which these two components can form.

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 moh cules

152 THE PHASE RULE

of water of crystallization ; these, are distinguished as a-tetra- hydrate, and y3-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 tempera- ture of 29'8° represented by the point H, it cuts the solubility curve of the a-tetrahydrate. This point is therefore a quad- ruple point at which the four phases hexahydrate, a-tetrahydrate, solution, and vapour can coexist. It is also the transition point for these two hydrates. Since, at temperatures above 29-8°, the a-tetrahydrate is the stable form, it is evident from the data given before (p. 149), as also from Fig. 38, that the portion of the solubility curve of the hexahydrate lying above this temperature represents metastable equilibria. The realiza- tion 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 a-tetrahy- drate is ioo*6 parts of CaCl2 in 100 parts of water.

The retroflex portion of the solubility curve of the hexa- hydrate extends to only i° below the melting point of the hydrate. At 29*2° crystals of a new hydrate, /3-tetrahydrate, separate out, and the solution, which now contains 112*8 parts of CaCL to 100 parts of water, is saturated with respect to the two hydrates. Throughout its whole extent the solubility curve EDF of the /3-tetrahydrate represents metastable equilibria. The upper limit of the solubility curve of /?-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 a-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 2600. The whole diagram, therefore, shows a succession of stable hydrates, a metastable hydrate, a metastable melting point and retroflex solubility curve.

SOLUTIONS OF SOLIDS IX LIQUIDS

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. 141), except that the rounded portion of the solu- bility curve of the hexahydrate would be represented by a similar rounded portion in the pressure curve.1 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 a- or the /?-tetrahydrate (p. 89).

The Indifferent Point. — We have already seen that at 3o'2° calcium chloride hexahydrate melts congruently, and that, pro- vided 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 com- position 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 Invariant. 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 otie temperature, change of temperature producing complete transformation in ' See Roozeboom, Zeitschr. physikal. C/icni., 18S9, 4. 31.

154 THE PHASE RULE

one or other direction. The variability of the system has therefore bee?i 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 systeui NH4C1 ^ NH3 + HC1 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 uni- variant 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;1 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,2 and it has been shown3 that at a given pressure the temperature in the indifferent point is the maximwn or minimum temperature possible at the particular pressure 4 (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.5

1 Tammann, Wied. Annalen, 1899, 68. 577.

2 Duhem,/0«r«. Physical Chem., 1898, 2. 31.

3 Gibbs, Trans. Conn. Acad., 3. 155 ; Saurel, Joarn. Phys. Chem.,

i9OI> 5- 35-

4 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).

5 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.

SOLUTIONS OF SOLIDS IX LIQUIDS 155

Ferric chloride can form no less than four stable hy drates, viz. Fe2Cl6,i2H20, Fe.,Clc,7H,0, Fe,Cl(i,5lLO, and FeoClG,4H20, and each of these hydrates possesses a definite, stable melting point. On analogy with the behaviour of cal- cium 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

-60 -40" -20

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 tlie different saturated solutions which can be formed is given in the following tables, the composition being expressed in mole- cules of Fe2Cl6 to 100 molecules of water. The figures printed in thick type refer to transition and melting points.

i56

THE PHASE RULE

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. Fe,Cl0,i2H2O. Fe2Cl6,7H20.

Tempera-

Composi-

ture. '

tion.

+ -55°

+ 2 75

-40°

2-37

-27-5°

1-90

-20-5°

1*64

-IO°

I'OO

0°

0

Fe2Cl9

5H20.

Tempera-

Composi-

ture.

tion.

12°

12-87

20°

1395

27°

14-85

30°

1512

35o

I5'64

5°o

17-50

55°

I9"i5

56°

20-00

55°

2032

Tempera- ture.

55°

■4i°

•270 o°

Composi- tion.

±2 75 281

10°

4 54

20°

510

30°

5 "93

35°

678

36-5°

7 '93

37°

833

36°

929

33°

io'45

30°

1 I'20

27-4°

1215

20°

12-83

10°

13-20

8°

1370

Tempera- ture.

Composi- tion.

66°

29 20

70°

75° 8o°

29-42 2892 29-20

IOO°

2975

Tempera- ture.

Composi- tion.

20°

27-4°

"'35

1215

320

32-5°

30°

13-55 14-29 1512

25°

I5-54

Fe2Cl6,4H20

Fe2Cla (ANHYDROUS).

Tempera-

Composi-

ture.

tion.

50°

19-96

55°

2032

6o°

2070

69°

21-53

72-5°

23-35

735°

25 00

72-5°

26-15

70°

27-90

66°

29 20

The lowest portion of the curve, AB, represents the equilibria between ice and solutions containing ferric chloride. It re- presents, in other words, the lowering of the fusion point of ice by addition of ferric chloride. At the point B (-55°), tne cryohydric point (p. 120) is reached, at which the solution is in equilibrium with ice and ferric chloride dodeoahydrate. As

SOLUTIONS OF SOLIDS IN LIQl , !57

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 dodecahy- drate. 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 tempera- ture 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 (5 6°) 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 550, 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 lias an intermediate value. Thus, if a solution represented by x1 is cooled down, the composition will remain unchanged as indicated by the horizontal dotted line, until the point 1) 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

,53 THE PHASE RULE

the system remaining constant until complete solidification has taken place. If, on the other hand, a solution of the composi- tion Xj 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 altera- tion 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 x2 or x4. 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 back- wards to a temperature of 8°, or about 190 below the tem- perature 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 150 ; 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

SOLUTIONS OF SOLIDS IN LIQUIDS

r: ,

15 20c

Fig. 40.

evaporating dilute solutions of fcrri.- chloride at constant tem perature, a remarkable series of changes is observed, which, however, will be understood with the help of Fig. 40. Suppose an unsaturated so- lution, the composition of which is represented by the point xlt is evaporated at a temperature of about 1 7 - iS°. As water passes off, the composition of the solution will follow the dotted line of constant tem- perature, until at the point where it cuts the curve BC the solid hydrate Fe,Cl,;,i2lI I I 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.

But the heptahydrate is not always formed by the dehydra- tion 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 rela- tively more ferric chloride. If, therefore, evaporation is con- tinued, 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 repre- sented 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 — 320

160 THE PHASE RULE

would be still more bewildering. The composition of the different solutions formed will be represented by the perpen- dicular line x2i 2345. Evaporation will first cause the separa- tion 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 dehy- dration will again cause liquefaction, when the concentration of the solution will be represented by the point 4 ; the hepta- hydrate will ultimately disappear, and then will ensue the deposition of the pentahydrate, and complete solidification will result. On evaporating a solution, therefore, of the composi- tion x2, the following series of phenomena will be observed : solidification to dodecahydrate ; liquefaction • solidification to heptahydrate ; liquefaction ; solidification to pentahydrate.1

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.2 In all these cases the solubility curve will show a point of maximum tempera- ture, 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 un- changed, 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

» Roozeboom, Zeitschr. physikal. Chem., 1892, 10. 477. 2 A similar series of hydrates is formed by zinc chloride and water (Dietz and Mylius, Zeitschr. anorg. Chem., 1905, 44. 209).

SOLUTIONS OF SOLIDS l.\ LIQUIDS

161

by fusion of the ice, and the quantity of solution will remain constant. Such solutions are called inevaporable.1

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 2 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 3 made a complete investi-

Tvniperature

Fig. 41.

gation 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 — 727°- Addition of potas- sium iodide lowered the freezing point, but the maximum de- pression 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 1 MeyerhofTer, Ber., 1897, 10. 1810. ' Walden, Ber.t 1S99, 32. 2S63. 1 Zeitschr. physikal Chan., 1903, 42. 432. T. P. C. **

!62 the phase rule

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 concentra- tion of the potassium iodide was continuously increased, the temperature of equilibrium first fell and then slowly rose, until at 4-0*2 6° (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 inter- pretation 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. 148), which was found to be that represented by the formula KI,i4S02. B is therefore the eutectic point, at which solid sulphur dioxide and the compound KI,i4S02 can exist together in equilibrium with solution and vapour. The curve DE is the solubility curve of the red crystalline solid, and the

SOLUTIONS OF so I. ins i\ LIQUIDS

163

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^SO.,.1 D is, therefore, the eutectic point at which the compounds KI,i4S03 and KI, , can coexist in equilibrium with solution and vapour. Hie curve DE does not exhibit a retroflex portion ; on the contrary, on attempting to obtain more concentrated solutions in equi- librium with the compound KI,4S02> a new solid phase (probably potassium iodide) was formed. Since at this point there are four phases in equilibrium, viz. the compound KI,4S02, potassium iodide, solution, and vapour, the system is invariant. E is, therefore, the transitio?i point for KI,4S02 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. 125). 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 ob- tained with triethylamine and water (p. 103). 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,i4S02 and KI,4SO0; 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 : —

Point.

Temperature.

A (m.p. of S02) .... B (eutectic point) . . . C (m.p. of KT,I4S02) . . E (m.p. of KI,4S02) . . G (KI + two liquid phases} H (critical solution point) K (KI + two liquid phases)

-727-

-23-4° +0-26° (about) 8S°

77-3° (about) 88°

Composition of the solution per cent. KI.

0S6 17-63

39*33 24'0 12 27

1 This composition was also confirmed by measurements of the vapour pressure (cf. p. 92).

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 1 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 intro- duced in the present discussion.2 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 Stor- tenbeker,3 form a very complete example of equilibria in a two-component system. We shall first of all consider the

1 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.

2 For a general discussion of the partial pressures in a system of two components, see Bancroft, Journ. Physical Chem., 1899, 3. I.

3 Zeitschr. physikal. Chem.t 1889, 3. 11 j Pec. trav. chitn, Pavs-Bas, 1888, 7. 152.

TWO VOLATILE COMPONENTS

165

relations between concentration and temperature, with the help of the accompanying diagram, Fig. 42.

x-3,or

1 — ! — 1 — 1 — 1 — 1 — 1 — r

Composition of th solutions I ( <U n equilibrium with th tolid phased

TiTT

20° 30

SO" CO" 10" SO" 90° 100° 110" 120'

Fro. 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,1 is represented by the ordinates. In the diagram, A represents the melting point of pure iodine, 1140. 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

1 The composition of a solution is represented symbolically by placing a double wavy line between the symbols of the components, and indi- cating the number of atoms present in the ordinary manner : thus, I zzzz C\x represents a solution containing x atoms of chlorine to one atom of iodine (Roozeboom, Zeitschr. physikal. C/iern., 18S8, 2. 450).

1 66 THE PHASE RULE

with which solid iodine is in equilibrium at different tempera- tures. This curve can be followed down to o°, but at tempera- tures 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 mono- chloride. The composition of the latter solution is represented by the cur