s — "^ LJ ^ CD — a ^s ^E rn ^ a ^^ ^ 0 COMMENTARY ON THE SCIENTIFIC WRITINGS OF J. WILLARD GIBBS PH.D., LL.D. FORMERLY PROFESSOR OF MATHEMATICAL PHYSICS IN YALE UNIVERSITY m TWO VOLUMES I. THERMODYNAMICS Dealing with the Contents of Volume One OF THE Collected Works EDITED BY F. G. DONNAN Professor of Chemistry in University College University of London ARTHUR HAAS Professor of Physics in the University of Vienna NEW HAVEN • YALE UNIVERSITY PRESS LONDON • HUMPHREY MI LFORD • OXFORD UNIVERSITY PRESS 1936 Copyright, 1936, by Yale University Press Printed in the United States of America All rights reserved. This book may not be reproduced, in whole or in part, in any form (except by reviewers for the public press), without written permission from the publishers. AUTHORS OF VOLUME I DONALD H. ANDREWS PROFESSOR OF CHEMISTRY, JOHNS HOPKINS UNIVERSITY, BALTIMORE, MARYLAND J. A. V. BUTLER LECTURER, CHEMICAL DEPARTMENT, UNIVERSITY OF EDINBURGH E. A. GUGGENHEIM LECTURER IN CHEMISTRY, UNIVERSITY COLLEGE, LONDON H. S. HARNED PROFESSOR OF CHEMISTRY, YALE UNIVERSITY, NEW HAVEN, CONNECTICUT F. G. KEYES PROFESSOR OF PHYSICAL CHEMISTRY, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS E. A. MILNE BOUSE-BALL PROFESSOR OF MATHEMATICS, UNIVERSITY OF OXFORD GEORGE W. MOREY GEOPHYSICAL LABORATORY OF THE CARNEGIE INSTITUTION, WASHINGTON, D. C. JAMES RICE LATE ASSOCIATE PROFESSOR OF PHYSICS, UNIVERSITY OF LIVERPOOL F. A. H. SCHREINEMAKERS FORMERLY PROFESSOR OF INORGANIC CHEMISTRY, UNIVERSITY OF LEIDEN EDWIN B. WILSON PROFESSOR OP VITAL STATISTICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS iii FOREWORD These volumes have been prepared with a two-fold purpose, — to honor the memory of J. Willard Gibbs, and to meet what is believed to be a real need. They are designed to aid and sup- plement a careful study of the original text of Gibbs' writings and not, in any sense, to make such a study unnecessary. The writing and printing of this commentary have been carried out under the auspices of Yale University, and have been financed in part from University funds and in part by generous contributions received from Professor Irving Fisher of Yale, to whom credit is also due for having conceived and initiated the movement for a memorial to Willard Gibbs of which this commentary is the direct and, thus far, the principal result. In January, 1927, an informal meeting was held of members of the Yale faculty interested in the creation of such a memorial. The proposal to publish a commentary on Gibbs' writings met with favor, and a committee was appointed to study the matter. After an extended investigation, in the course of which per- sonal opinions were obtained from a large number of authori- ties, both in this country and abroad, on the desirability of such a commentary and on various questions of policy, the committee reported favorably, and was thereupon instructed to carry the plan into effect. Definite arrangements were completed in February, 1929, and work began during that year, but it was not until four years later that the manuscript of both volumes was ready for the press. Each of the two volumes deals with the portion of Gibbs' writings contained in the like-numbered volume of The Col- lected Works of J. Willard Gibbs. Volume I, "Thermody- namics" is essentially interpretative and explanatory, but in- cludes a discussion of recent developments concerning Gibbs' thermodynamic principles and many examples, drawn from the modem literature, of their application to concrete problems. vi FOREWORD Volume II, "Theoretical Physics", contains an analysis, appre- ciation and interpretation of Gibbs' work in this field, espe- cially his statistical mechanics, and, in addition, a discussion of the relation of Gibbs' statistics to the modern quantum statis- tics. The volumes are separately indexed and except for a few cross-references are entirely independent of each other. May this commentary, the product of devoted and conscien- tious labors on the part of its authors and editors, prove truly helpful to those who wish to follow the paths opened up by Willard Gibbs, and promote a better and more widespread appreciation of the value of his services to science. The Committee on the Gihhs Commentary John Johnston Herbert S. Harned Leigh Page William F. G. Swann Ralph G. Van Name, Chairman Yale University May, 1936 PREFACE TO VOLUME I ''^;.. The present Volume of the Commentary deals with Gibbs' thermodynamical papers, and principally with the famous paper on The Equilibrium of Heterogeneous Substances. In this immortal work, Gibbs, building on the sure foundations laid by Carnot, Mayer, Joule, Clausius and Kelvin, brought the science of generalised thermodynamics to the same degree of perfect and comprehensive generality that Lagrange and Hamilton had in an earHer era brought the science of generaUsed dynamics. The originality, power and beauty of Gibbs' work in the do- main of thermodynamics have never been surpassed. The gen- erahty and abstract nature of the reasoning have, however, made the understanding of his methods and results a difficult task for many students of science. This has been particularly true of students of chemistry, who in general are deficient in mathematical training and are not as a rule familiar with the methods and results of generafised classical dynamics — a very necessary mathematical precursor to the study of generafised thermodynamics. This state of affairs has been very unfor- tunate in the past, since the work of Gibbs contained a complete and perfect system of chemical thermodynamics, i.e., a system of thermodynamics peculiarly well adapted to the most general and complete application to chemical problems. What, for ex- ample, could exceed, in simplicity and generality, Gibbs' expres- sions, in terms of his chemical potentials, for chemical equilibrium in a homogeneous phase or the distribution equilibrium of inde- pendent components throughout a system of coexistent phases? Although the physicist will undoubtedly find much of the greatest interest and value in the present volume, this Com- mentary is intended for the use of students of physical chemistry as well as physics. The Articles contained in it are not there- fore merely running comments on and illustrations of Gibbs' equations, but constitute in each case a thoroughgoing discus- sion of the corresponding part of Gibbs' work, the object of which is so to smooth the path for the reader of the original viii PREFACE papers that the methods and results of Gibbs will be intelligible to and available for the use of all serious students of both chem- istry and physics. The only exception to this mode of treat- ment will be found in the interesting Article C of the present volume, where our distinguished collaborator, Professor E. B, Wilson, considered it more advantageous to give an outline of Gibbs' own lectures on thermodynamics than a detailed discus- sion of Papers I and II of Volume I of The Collected Works of J. Willard Gibbs. Readers who have followed the reasoning given by Gibbs in his lectures will find no difficulty in under- standing the graphical developments of Papers I and II. In order further to lighten the work of the mathematically inexpert reader, the present volume contains a short Article (B) deahng with certain mathematical methods. In this connec- tion reference may be also made to Chapter II of the Special Com- mentary on Gibbs' Statistical Mechanics by A. Haas, dealing with the algebra of determinants and contained in Volume II of the Commentary. One of the objects of Article F of the pres- ent Volume is to famifiarise students with certain mathematical difficulties, e.g. the difference between Gibbs' use of the opera- tors 8 and A. Some points of detail may now be considered. In the Table of Contents and in the titles of the Articles of the present Volume, the expression "Gibbs, I, pp." refers to the relevant part of Volume I of The Collected Works of J. Willard Gibbs (two volumes), Longmans, Green, and Co., 1928, or to the like- numbered volume and page of The Scientific Papers of J. Willard Gibbs, Longmans, Green, and Co., 1906.* This ap- plies also to occasional references in the text. In each Article the current numbers referring to the particular author's equations are given between curved parentheses, whereas the numbers referring to the equations as given by Gibbs in the original paper are enclosed between rectangular brackets. When * The Collected Works is a reprint of the Scientific Papers, with iden- tical pagination and contents except that it includes (in Volume II) Gibbs' Elementary Principles in Statistical Mechanics, which was not printed in the Scientific Papers. References to this particular portion, however, occur in this Commentary only in Volume II and in Article J of Volume I. PREFACE ix coincidence occurs, as is very frequently the case, the necessary double numbering is given, e.g. Equation (a) [g]. Here a is the author's number, g is Gibbs' number. The same method is followed in the reference numbers of equations given in the text. The notation employed by Gibbs for the principal thermo- dynamic quantities has been retained in general, the few devia- tions from this procedure being indicated at the appropriate places in the text. In order to facilitate comparison with the usage of a number of other writers on thermodynamics, a comparison Table of Symbols is given (Article A). This Article also contains a comparison Table of the names as- signed to the principal thermodynamic quantities by Gibbs and a number of other writers. Of the Articles contained in this Volume, all, with the excep- tion of A and C, refer to Paper III of Volume I of the Collected Works, i.e., the paper on The Equilibrium of Heterogeneous Sub- stances, and Papers (Sections) V, VII, VIII, and IX. Article D deals with the general thermodynamic system of Gibbs, as expounded in Gibbs, I, pp. 55-144; 419^24. Special parts of this section of Paper III are further discussed and illustrated in Articles E, F, G, and H, whilst Articles I, J, K, L and M deal with the remaining portions of Paper III (and Sections V, VII, VIII and IX) of Volume I of the Collected Works. Readers of this Volume will find in Volume II of the Com- mentary a general survey of Gibbs' thermodynamical methods and results (by A. Haas), as well as an account of certain sub- sequent work (by P. S. Epstein). In the present Volume we have not dealt with such later developments as the Nernst Heat Theorem and related topics, since a proper understanding of the present state of this subject requires a considerable knowledge of Statistical Mechanics. These matters are dealt with by P. S. Epstein in Volume II of the Commentary. Besides the condensed survey of Gibbs' thermodynamical methods and results contained in Volume II of the Commentary, students will find an excellent account in the book of E. A. Guggenheim, entitled Modern Thermodynamics by the Methods of Willard Gibbs (Methuen & Co., London, 1933). X PREFACE The thermodynamical writings of Gibbs have proved a golden source of knowledge and inspiration to later workers. This mine is by no means exhausted. It is the confident belief of the Editors that those who are led by the present book to a study of the relevant parts of Gibbs' work will find therein much that is as yet imperfectly understood and experimentally undeveloped. Gibbs was no mere generaliser of the work of others, but a pro- found and original investigator who opened new domains of knowledge to the mind of man. As is well known, Gibbs himself endeavored to obtain a rational foundation for thermodynamics in his splendid develop- ment of the science of Statistical Mechanics, founded by Clerk Maxwell and Boltzmann (see Volume II of the Commentary). Nowadays, by means of the quantum concept and the newer methods of theoretical physics, the older Statistical Mechanics has been transformed into a new science of Quantum Statistics and Quantum Mechanics. Although without doubt this won- derful new development penetrates much more deeply into the analysis of the phenomenal world than the older science of thermodynamics, there is no reason to deny the term rational to the earher method. It deals with the phenomenal world in a different manner, but it remains, within its rightful domain, an enduring and powerful weapon of the human mind. More- over, the modern development of physical theory tends more and more to revert to the essential method of thermodynamics, which abstains from "mechanical" pictures of individuahsed entities interacting in space and time, and describes phenomena by means of a generafised functional analysis. Thermo- dynamics was indeed the essential precursor of the modern method. It will ever be the imperishable achievement of Gibbs to have developed this earlier scientific method to the fullest extent of its power. Modern physical chemistry utihses in constantly increasing measure the newer developments of theoretical physics. Never- theless, thermodynamics is one of the principal foundations on which the structure of "classical" physical chemistry rests. Every well-trained student of pure or applied chemistry must therefore possess a thorough working knowledge of its principles PREFACE xi and methods. In this essential task he will j5nd no surer or better guide than the original papers of J. Willard Gibbs. In the work of producing this Commentary we have been fortunate in enlisting the cooperation of a number of very able collaborators, to each of whom has been entrusted a special section of the Volume. To all these collaborators we desire to express our very high appreciation of the work which they have accomphshed. Our work as Editors has been greatly lightened by the extreme care which the members of the Gibbs Committee have bestowed on the correction of the proofs and on many other matters of importance. For this valuable help we are extremely grateful. Last, but not least, we wish to express, on behalf of ourselves and our collaborators, our deep sense of the honor which the Gibbs Committee has conferred upon us all. Should our joint labors succeed in liberating the beautiful work of Gibbs from the abstract tour d'ivoire in which it has been for so long con- cealed from many students of science, then great will be our reward. London and Vienna, F. G. DoNNAN January, 1936 Arthur Haas CONTENTS Foreword v Preface vii A. Note on Symbols and Nomenclature, F. G. Donnan . 1 B. Mathematical Note, J. Rice 5 1 . The Method of Variations Used for Determining the Conditions under Which a Function of Several Variables Has a Maximum or Minimum Value. . 5 2. Curvature of Surfaces 10 3. Quadric Surface 15 C. Papers I and II as Illustrated by Gibbs' Lectures on Thermodynamics (Gibbs I, pp. 1-54), E. B. Wilson 19 I. Introduction 19 II. OutUne of Gibbs' Lectures on Thermodynamics 19 III. Further Notes on Gibbs' Lectures. Photographs of Models of the Thermodynamic Surface 50 D. The General Thermodynamic System of Gibbs (Gibbs I, pp. 55-144 ; 4 19-424) ,J.A.V. Butler 61 I. Introduction 61 1. General Thermodynamic Considerations 61 II. The Criteria of EquiHbrium and Stabihty 70 2. The Criteria 70 3. Equivalence of the Two Criteria 71 4. Interpretation of the Conditions 72 5. Sufficiency of the Criteria of Equilibrium 74 6. Necessity of the Criteria of EquiUbrium 78 III. Definition and Properties of Fundamental Equations ... 79 7. The Quantities ^, ^, x 79 8. Differentials of e, \p and f 86 IV. The Conditions of Equilibrium between Initially Existent Parts of a Heterogeneous System 92 9. General Remarks 92 10. Conditions of Equilibrium When the Component Substances Are Independent of Each Other .... 92 11. Conditions of Equilibrium When Some Compon- ents Can Be Formed out of Others 96 12. Effect of a Diaphragm (Equilibrium of Osmotic Forces) 102 ^l:7S73 xiv CONTENTS V. Coexistent Phases 105 13. The Phase Rule 105 14. The Relation between Variations of Temperature and Pressure in a Univariant System 108 15. Cases in Which the Number of Degrees of Freedom is Greater Than One. (a) Systems of Two or More Components in Two Phases Ill (6) Systems of Three Components in Three Coexistent Phases 115 \T. Values of the Potentials in Very Dilute Solutions 116 16. A Priori Considerations 116 (a) m2 Is Capable of Negative As Well As Posi- tive Values 117 (6) niils Capable Only of Positive Values 117 17. Derivation of the Potentials of a Solution from Their Values in a Coexistent Vapor Phase 120 18. Equilibria Involving Dilute Solutions 124 (a) Osmotic Pressure 124 (6) Lowering of the Freezing Point 125 (c) Lowering of the Vapor Pressure of a Solvent by an Involatile Solute 127 VII. The Values of Potentials in Solutions Which Are Not Very Dilute 128 19. Partial Energies, Entropies and Volumes 128 20. The Activity 131 21. Determination of Activities from the Vapor Pres- sure 132 22. The Lowering of the Freezing Point 135 23. Osmotic Pressure of Solutions 138 VIII. Conditions Relating to the Possible Formation of Masses Unlike Any Previously Existing 141 24. Conditions under Which New Bodies May Be Formed 141 25. Generalized Statement of the Conditions of Equi- librium 145 IX. The Internal Stability of Homogeneous Fluids 146 26. General Tests of Stability 146 27. Condition of Stability at Constant Temperature and Pressure 148 28. Condition of Stability Referred to the Pressure of Phases for Which the Temperature and Poten- tials Are the Same as Those of the Phase in Question 150 X. Stability in Respect to Continuous Changes of Phase. . . 152 29. General Remarks 152 30. Condition with Respect to the Variation of the Energy 153 CONTENTS XV 31. Condition with Respect to the Variation of the Pressure 156 32. Conditions of Stability in Terms of the Functions x// and f 156 33. Conditions with Respect to Temperature and the Potentials 159 34. Limits of Stabihty 161 XL Critical Phases 163 35. Number of Degrees of Freedom of a Critical Phase . 163 36. Conditions in Regard to Stability of Critical Phases 164 XII. Generalized Conditions of Stability 166 37. The Conditions 166 38. Critical Phases 172 XIIL Equilibrium of Two Components in Two Phases 175 39. The Equilibrium 175 40. Konowalow's Laws 177 XIV. Phases of Dissipated Energy. Catalysis 178 41. Dissipated Energy 178 E. Osmotic and Membrane Equilibria, including Electrochemical Systems (Gibbs I, pp. 83-85; 413-417), E. A. Guggenheim 181 1. Introduction 181 2. Proof of General Conditions of Membrane Equi- librium 183 3. Choice of Independent Components 185 4. Choice of Independent Variables 186 5. Mols and Mol Fractions 187 6. Ideal Solutions 188 7. Non-ideal Solutions 190 8. Osmotic Equilibrium 192 9. Incompressible Solutions 193 10. Relation between Activity Coefficients 193 11. Osmotic Coefficients 194 12. Osmotic Equilibrium in Terms of Osmotic Coeffi- cient 196 13. Extremely Dilute Solutions 197 14. Electric Potential Difference between Two Identi- cal Phases 198 15. Electric Potential Difference between Two Phases of Different Composition 199 16. Combinations of Ions with Zero Net Electric Charge 200 17. Ideal Solutions of Ions 201 18. Non-ideal Solutions of Ions 201 19. Mean Activity Coefficient of Electrolyte 202 xvi CONTENTS 20. Membrane Equilibrium of Ideal Ionic Solutions . . 203 21. Membrane Equilibrium of Non-ideal Ionic Solu- tions 205 22. Contact Equilibrium 206 23. Purely Chemical Cell 206 24. Electrochemical Cells 208 F. The Quantities i^, x and f, and the Criteria of Equilibrium (Gibbs I, pp. 89-92), E. A, Milne 213 1. Stability Tests 213 2. The Work Function 214 3. The Free Energy Function 216 4. The Heat Function 220 5. Physical Properties of the Thermodynamic Func- tions \i', r, X 223 6. The Heat Function at Constant Pressure 223 7. The Heat Function in General 224 8. The Work Function i/- at Constant Temperature . . 226 9. The Free Energj^ Function f at Constant Tem- perature and Constant Pressure 227 10. Further Illustration 229 G. The Phase Rule and Heterogeneous Equilibrium (Gibbs I, pp. 96-100), G. W. Morey 233 I. Introduction 233 11. Equation [97] and the Phase Rule 233 1. Equation [97] 233 2. Derivation of the Phase Rule 234 III. Application of Equation [97] to Systems of One Com- ponent 236 3. The Pressure-Temperature Curve of Water 236 IV. Application of Equation [97] to Systems of Two Com- ponents 241 4. Application of the Phase Rule to a System in Which No Compounds Are Formed. H2O- KNO3 241 5. AppUcation of Equation [97] to a System in Which No Compounds Are Formed. H2O-KNO3 242 6. The EquiUbrium, KNO3 + Solution + Vapor. ... 243 7. The Maximum Pressure of the Equilibrium, KNO3 + Solution + Vapor 246 8. The Maximum Temperature of the Equilibrium, KNO3 + Solution + Vapor 247 9. The Second BoiUng Point 248 10. The Equilibrium, Ice + Solution + Vapor 249 11. The Equilibria, Ice + KNO3 -h Vapor, and Ice + KNO3 + Solution 250 CONTENTS xvii 12. Derivation of an Equation in Which the Argument Is Pressure, Temperature, and Composition. ... 251 13. Derivation of an Equation Applying to the Solu- bility (t-x) Curve 252 14. Correlation of the t-x and p-t Curves 253 15. EquiUbrium Involving SoUd Solutions 254 16. AppHcation of Equation [97] to a System in Which Compounds Are Formed. H20-CaCl2.. 256 17. The Minimum Melting Point of a Dissociating Compound 257 18. Correlation of the t-x and p-t Curves 258 19. The Equilibrium between a Dissociating Hydrate and Its Products of Dissociation 259 20. The Equilibrium, Two SoUds + Liquid 261 21. The Equilibrium, SoUd + Solution + Vapor 261 22. Types of Invariant Points and Univariant Systems. 262 23. Equilibrium Involving Two Immiscible Liquids. Water-phenol 263 V. Application of Equation [97] to Systems of Three Com- ponents 267 24. Transformation and Interpretation of Equations. . 267 25. Equilibrium, K20Si02-^H20 + Solution + Vapor 269 26. Coincidence Theorem 274 27. Equilibrium, K20-2Si02-H20 + K2O -28102 + Solution + Vapor 276 28. Equilibrium, K20-Si02-*H20 + KjO-SiOa -h Solution + Vapor 278 29. Equilibrium, K2O -28102 + K20-4Si02-H20 + Solution + Vapor 279 30. The Order of p-t Curves around an Invariant Point 280 31. Generalized Theorem Concerning the Order of p-t Curves around an Invariant Point 283 32. Generalizations Concerning p-t Curves 286 33. Order of the p-t Curves in the Ternary System, H2O-K2O -8102-8102 288 H. The Graphical Representation op Equilibria in Binary Systems by IVIeans of the Zeta (Free Energy) Function (Gibbs I, pp. 115-129), F. A. H. Schreinemakers 295 I. Introduction 295 II. The ^-x Diagram and the f-Curve (Free Energy Curve) 295 III. Binary Systems in Which Besides Liquids Only the Solid Components W and X Can Occur 304 xviii CONTENTS IV. Binary Systems in Which Besides Liquids Only the Solid Components W and X and a SoUd Compound May Occur 315 V. Note by F. G. Donnan. (Analytical Addendum to the Geometry) 322 I. The Conditions of Equilibrium for Heterogeneous Masses under the Influence of Gravity (and Centrifugal Force) (Gibbs I, pp. 144-150), D. H. Andrews 327 J. The Fundamental Equations of Ideal Gases and Gas Mixtures (Gibbs I, pp. 150-184; 372-403), F. G. Keyes 337 I. General Considerations 337 1. Pure Ideal Gases 337 2. Mixtures of Ideal Gases 339 3. Ideal Gas Concept as Related to the Behavior of Actual Gases under Diminishing Pressure 339 4. Constancy of Specific Heat 341 5. Concluding Statement 341 6. Comment on Gas Law for Real Gases 341 7. Choice of Units of Mass and Energy 343 8. Definition of Temperature 343 9. Constants of Energy and Entropy 344 10. \p Function for an Ideal Gas 345 11. f Function for an Ideal Gas 347 12. X Function for an Ideal Gas 348 13. Vapor Pressures of Liquids and Solids 349 14. Effect of Presence of a Neutral Gas on Vapor Pressure 353 15. Defect in the Sum Rule for Vapor Pressures 355 16. Gibbs' Generahzed Dalton's Law 356 17. Entropy of an Ideal Gas Mixture 357 18. Implications of Gibbs' GeneraHzed Dalton's Law Apart from Ideal Gas Behavior 358 19. Ideal Gas Mixture in a Potential Field 363 20. Vapor Pressure of a Liquid under Pressure from a Neutral Gas 363 21. AppUcation to "Gas-Streaming" Method of Meas- uring Vapor Pressures 365 22. Heat of Evaporation of a Liquid under Constant Pressure 367 23. Fundamental Equations from Gibbs-Dalton Law. 369 24. Case of Gas Mixtures Whose Components Are Chemically Reactive 369 CONTENTS xix II. Inferences in Regard to the Potentials in Liquids and Solids 370 25. Henry's Law 371 26. Raoult's Law of Vapor Pressure and the Thermo- dynamic Theory of Dilute Solutions 372 III. Considerations Relating to the Increase of Entropy Due to the Mixture of Gases by Diffusion 375 IV. The Phases of Dissipated Energy of an Ideal Gas-Mix- ture with Components Which Are Chemically Related 377 27. Restatement of the Above in Different Notation. . 379 V. Gas Mixtures with Convertible Components 382 28. A More General Apphcation of the Gibbs-Dalton Rule 387 29. General Conclusions and the Equation of State of an Ideal Gas Mixture Having Convertible Com- ponents 388 VI. On the Vapor-densities of Peroxide of Nitrogen, Formic Acid, Acetic Acid, and Perchloride of Phosphorus 391 K. The Thermodynamics of Strained Elastic Solids (Gibbs I, pp. 184-218), J. Rice 395 I. Exposition of Elastic SoUd Theory So Far As Needed for Following Gibbs' Treatment of the Contact of Fluids and Solids 395 1. Analysis of Strain 395 2. Homogeneous Strain 402 3. Heterogeneous Strain 417 4. Analysis of Stress 417 5. Stress-Strain Relations and Strain-Energy 429 6. Thermodynamics of a Strained Homogeneous Solid 444 II. Commentary 455 7. Commentary on Pages 184-190. Derivation of the Four Equations Which Are Necessary and Sufficient for the Complete Equilibrium of the System 455 8. Commentary on Pages 191-197. Discussion of the Four Equations of Equilibrium 470 9. Commentary on Pages 197-201. The Variations of the Temperature of Equilibrium with Respect to the Pressure and the Strains. The Variations of the Composition of the Fluid 477 10. Commentary on Pages 201-211. Expression of the Energy of a SoUd in Terms of the Entropy and Six Strain-Coefficients. Isotropy 481 11. Commentary on Pages 211-214. Approximative Formulae for the Energy and Free Energy in the Case of an Isotropic SoUd 492 XX CONTENTS 12. Commentary on Pages 215-219. Solids Which Absorb Fluids. Elucidation of Some Mathe- matical Operations 502 L. The Influence of Surfaces of Discontinuity upon THE Equilibrium of Heterogeneous Masses. Theory of Capillarity (Gibbs I, pp. 219-331; 331- 337), J. Rice 505 I. Introductory Remarks 505 1. The Surface of Discontinuity and the Dividing Surface 505 2. The Mechanical Significance of the Quantity Denoted by iXly X2, Xn) = 0, where (f> is another given functional form. Considering a definite set of values for the variables, say Xi = qi, X2 = q2, . . . . Xn = Qn wc compare the value of the function for this set with the value for any neighbouring set, such as Xi = qi -\- Sqi, a-2 = ?2 + Sq^, x„ = g„ + 5g„, where 5gi, 5^2, .... 5g„ are infinitesimal quantities. These infinitesimal quan- tities are not completely arbitrary in their ratios to one another; for we have to choose them to satisfy the conditions (qi, qz, qn) = 0, {qi + 8qi, q2 + 5^2, qn + 8qn) = 0. It is convenient to write for 8qi, 8q2, . . . . 5g„ the symbols ^^i, ^^2, . . . . d^n where 6 is an infinitesimal positive magnitude whose value can be reduced without limit and ^i, ^2, .... ^n are finite quantities. The difference between the value of the function / for the set of values (xr = qr) and the value for the set (Xr = ?r + 8qr) IS * fiqi + bqi, qi + bq2, ?„ + 8q^ - f(qi, q2, qn). * The enclosing bracket in (xr = (/r) or (g,) indicates that we mean Xi = qi, Xi = q2, . . . . Xn = Qn, orqi, qz, . . . qn. 6 RICE By Taylor's theorem this is equal to ART. B ^X'"^^'^^ r = 1 s = 1 d^rbg. hqrhq^ + etc. where we write f{q) briefly for fiqi, q^, .... g„). This difference we now write in the form + 2! ss mq) 1 :, . . ^.^a[4-etc. (2) As 0 is reduced in value, the numerical magnitude of the term in 0 preponderates more and more over the terms in 6'^, 9^, .... (apart from discontinuities arising in the differential coefficients, a state of affairs which we cannot discuss here). The sign of this term will therefore determine whether f(q + dq) is greater or less than/(5). If /(g + 5g) is greater than /(g) for any values of {qr + 8qr) consistent with the condition imposed, it is neces- sary that ^ 5/(g) ^qr r = 1 ^r 0 (3) for apy possible sets of values of (Ir)^ since if the expression on the left-hand side of (3) were positive for a set of values of (|r), it would be negative for the set with opposite signs, and so f(q + 8q) would not be greater than f(q) for all possible sets of (qr + 8qr). If the quantities (^r) were perfectly arbitrary this would necessitate the n conditions, c>f(q)/dqr = 0. However, they are not arbitrary; for by (1) they satisfy the condition S c>4>(q), , e ss c)V(g) .=1 ^Qr "" ■ 2!(^^jfr(g) ^1 ^^r ^r = 0. (4) MATHEMATICAL NOTE Suppose we multiply (3) by d(q) bf(q) Idcj^iq) bqi J bqi bq2 / ^^2 bf(q) jdckiq) bqn I bqn (6) since they make all the coefficients of ^2, ^3, . • • • ^n in (5) indi- vidually zero. Exactly the same argument shows that if the function f{xi,X2, .... a;„) has a minimum value for the set of values (xr = qr) the same conditions (6) hold. It follows therefore that in order to determine the sets of values of the variables for which the function f(x) is maximum or minimum in value, subject to the condition, (f>{x) = 0, we have to solve the n equations bXi bxi (x) = 0, bf(x) jb4>{x) ^ bX2 I bX2 bfix) \bct>{x) dXn dXn (7) Any solution of these equations yields a set of values for "max- min" conditions. A special case of this result, which is the one actually required for the considerations arising in Gibbs' Equilibrium of Hetero- geneous Substances * concerns the situation in which the condi- tion imposed on the variables is that their sum should be a constant, i.e. Xl + X2 + Xn — C = 0. See Gibbs, I, pp. 65 and 223. 8 RICE ART. B In this case all the d(l)(x)/dxr are unity and equations (7) take the form {x) = 0, dxi dxo ' ' ' ' dxn (8) n n r = 1 s = 1 In order to distinguish between the sets of values which yield a maximum and those which yield a minimum, we must con- sider the terms in the expansion of f{q +- bq) — f(q) which in- volve 6 and higher powers of d. Thus we now write f(q + dq) - f{q) = |^ + higher powers of d, (9) where ttrs is the value of the second differential coefficient b^f(x)/dxT dxs when a set of values (qr) obtained from the equa- tions (7) are substituted for the variables (xr). Now if this set of values yields a minimum, then the right-hand side of (9) must be positive for any possible values of ^r- If we now assume that the term in 6"^ preponderates in value over the remaining terms in 6^, 6*, etc. (which will be the case if the differential coefficients satisfy the usual conditions) then the condition for a minimum value is that the quadratic expression in {^r) an ^1^ + ^22 ^2^ + 2 ai2 ^1 ^2 + should be positive in value for any set of values of (^r) which satisfy the condition imposed. Actually the conditions which make the quadratic expression positive for any values of (^r) unrestricted by any condition have been worked out by the mathe- matician; so these conditions will be sufficient for the criterion of minimum in our problem, though they may not be absolutely necessary for our restricted values. The conditions can be stated as follows. Consider the determinant of the n*^ order MATHEMATICAL NOTE ail C^12 .... CLln Oil a^l .... CLin 9 flfil dni .... Ctr Now consider: (1) All the leading constituents an, 022, 033, .... a„„; (2) All the minor determinants obtained by selecting any two rows and the two corresponding columns, for instance CItt dra (3) All the minor determinants obtained by selecting any three rows and the three corresponding columns, for example Or a. a„ Or a. Or ttrn a. ttr, and so on; (r) All the minor determinants obtained by selecting any r rows and the r corresponding columns ; and so on; (n) The determinant itself. If the quadratic expression is a "positive definite form," i.e. positive in value for all values of (^r), then all the determinants in (1), (2), (3), .... (n) must be positive in value. If on the other hand the set of values qi, q2, . ■ ■ . qn for the variables xi, X2, . . . . Xn yield a maximum, then the quadratic expression in (^r) must be a "negative definite form," i.e. nega- tive in value for all values of (^r). The conditions are that the determinants in (1), (3), (5), (7) etc. are all negative in value, while those in (2), (4), (6), (8), etc., are all positive. If neither of these conditions holds, then the set of values a;i = Qi, X2 = q2, . . . . Xn = qn does not yield a true maximum or 10 RICE ART. B minimum condition and the consideration of the problem goes beyond the Hmits of possible discussion here. For the proof of these results see any text of modern algebra, for example Bocher's Introduction to Higher Algebra, Chapters IX-XII. For reference to these conditions in the Collected Works, see Gibbs, I, pp. 111,112,242. 2. Curvature of Surfaces. The average curvature of a plane curve between two points A and B is defined as the quotient of the external angle between the tangents at A and B by the length of the arc AB. From a kinematic point of view it is the average rate of rotation of the tangent per unit length travelled by the point of contact. If the point B approaches indefinitely near to A, the limiting value of the average curvature is defined to be the curvature at the point A. In the case of a circle this is obviously the reciprocal of the radius at all points. For any curve at any point the curvature has the dimension of a recipro- cal length, and so, on dividing the value of the curvature at a point on a curve into unity, we obtain a definite length which is then referred to as the "radius of curvature" at that point. Clearly where the curvature is relatively large the radius of curvature is relatively small; thus the extremities of the major axis of an ellipse are the points on it at which curvature is great- est but radius of curvature least ; at the extremities of the minor axis, curvature is least, radius of curvature greatest. The measurement of curvature at a point on a surface is based on this simple idea for a curve. Thus we conceive the tangent plane and the normal line to be drawn at a point P on the sur- face, and we then consider any line through P lying in this plane. An infinite number of planes can be drawn cutting the tangent plane in this hue. These planes will cut the surface in an in- finite number of curves, and we w'ill readily recognise that suffi- cient information concerning the curvature of these curves at the point P will give us all the vital information concerning the curvature of the surface at P. Two obvious details in the con- struction of one such curve can be varied at will; we can alter the angle between the tangent plane at P and the plane drawn through the line in the tangent plane (the tangent line as we MATHEMATICAL NOTE 11 may call it) and we can alter the direction in the tangent plane of the tangent line. In the first place a well-known theorem, known as Meunier's theorem, connects the radii of curvature of different sections through the same tangent line: the radius of curvature of an oblique section through a tangent line at P is equal to R cos (/> where R is the radius of curvature at P of the normal section, (i.e. the section containing the normal line at P as well as the tangent line) and (j> is the angle between the normal section and the oblique section. Thus if we know the radius of curvature of the normal section through the chosen tangent line at P we im- plicitly know the radius of curvature of any given oblique section through it. In the second place if we now vary the direction of the tangent line the radius of curvature of the normal section varies in a manner which is well known and quite simply described. Call- ing the curvature of the normal section c (where c is of course equal to i2~0 it is known that c varies continuously in value be- tween a maximum limit and a minimum as the tangent line is rotated. It attains its maximum value twice in a complete rotation of the line, the two directions corresponding to this maximum being directly opposite to one another. The mini- mum is attained for the two opposite directions at right angles to the former. Taking the two lines thus marked out on the tan- gent plane as axial lines PXi, PX^ in the plane, we can indicate the direction of any other line in the tangent plane by the angle 6 which it makes with PXi, say. It is known that c, the curva- ture at P of the normal section through this line, is given by c = Ci cos^ 6 -{- C2 sin^ d, where C: and c^ are the curvatures at P of the normal sections through PXi and PX2. The values Ci and C2 are known as the "principal curvatures" of the surface at the point P. In this way we see that our complete knowledge concerning the curva- ture of a surface at a point P is summarized in a knowledge of the two principal curvatures at that point. One simple result of some importance follows very easily from the equation just written: if c and c' are the curvatures of two normal sections at 12 RICE ART. B a point which are at right angles to one another then c + c' is a constant quantity at the point and is equal to Ci + C2. On page 229 of Vol. I Gibbs uses an important theorem concerning the increase in size of a small portion of a surface produced by an elementary displacement of each element of the Fig. 1 surface by an amount BN in the direction of its normal. Let the element of surface he ABE F (Fig. 1) bounded by normal sections which are at right angles to one another. Let C be the "center of curvature" of the element AB of one of the sections, i.e., the position in the limit where the normals in the plane to the curve at the points A and B meet.* Let C be the center of curvature * The reader unacquainted with the geometry of surfaces is warned that for the sake of simplicity we have neglected a detail which is of no MATHEMATICAL NOTE 13 of the arc ^F in the other plane at A which is at right angles to the plane at ABC. Let the element of area be displaced to the position XYZW where AX = BY = EZ = FW = 8N. If the elementary angles Z ACB and Z AC'F are denoted by a and /3 then the area of the element of surface ABEF is equal to the product oi AB and AF, i.e., it is Ra X R'0. If we denote this by s and the area of XYZW by s + 5s we see that s = RR'a^, s + 8s = (R-\- 8N) (R' + 8N) a^. Therefore, neglecting products of the variations, we obtain the result 8s = (R -\- R') 8N a|3 = s{c + c') 8N. But since c + c' = Ci + C2 it follows that 8s = (ci + cz) s 8N, a result used by Gibbs in obtaining equation [500]. It is used again on page 280 in the lines immediately succeeding equation [609] (where J'a 8Ds is replaced by y*o-(ci + C2)8NDs) and also on page 316. If the equation of a surface in Cartesian coordinates is given in the form 2 = fix, y) importance for our purpose. But in order to avoid producing a wrong impression the writer must point out that if a plane section is drawn con- taining the normal to the surface at A, it is in general not true that the normal in this plane to the curve AB at B is also the normal to the surface at B. In our example where we are considering elementary arcs and areas of small size, this feature may be ignored without detriment to the argument. 14 RICE ART. B the sum of the principal curvatures at a point x',y' z' on the surface can be calculated as follows: Let p and q represent the values of the differential coefficients bf/dx and df/dy when the values x', y' are substituted for x, y, and let r, s, t be the values of the second differential coefficients d^f/dx^, d^f/dxby, d^f/dy"^ with the same substitutions; then , _ (1 + 9^) r + (1 + p^) ^ - 2 pqs "'^''~ (1 + P^ + 3^)i This formula is used in obtaining equation [620] on page 283. Its proof will be found in any text of analytical solid geometry. On page 293 of Gibbs, Vol. I, there is a reference to the total curvatures of the sides of a plane curvilinear triangle. The total curvature of an arc of a plane curve is equal to the external angle between the tangents at its extremities and must be care- fully distinguished from the average curvature of the arc which is the quotient of its total curvature by its length. The angles of the curviHnear triangle abc (Fig. 2) are YaZ, ZhX, XcY. Their sum exceeds the sum of the angles of the plane triangle ahc by Z Xbc-\- Z Xch -]- ZYca-\- Z Yac+ Z Zah + Z Z6a which is equal to the sum of the external angles at X,Y, Z between the tangents. This result is cited on page 293 of Gibbs, I. In conclusion it should be realised that Ci and C2 for a surface may have different signs so that the expression Ci + d may sometimes actually denote the numerical difference of the prin- cipal curvatures of a surface at a point. This occurs when the two principal sections produce curves which are convex to dif- ferent parts. For example if one considers a mountain pass at its top lying between hills on each side, a vertical section of the surface of the mountain at the top of the pass made right across the traveller's path is concave upwards, while one made at right angles to this following the direction of traveller's path is con- cave downwards. The principal centres of curvature are on opposite sides of the surface in such a case and the principal radii of curvature are directed to opposite parts. The radii have opposite signs and the principal curvatures likewise. A surface is said to be "anticlastic" at such a point (as opposed to "synclastic," when the centres of curvatures are on one side and MATHEMATICAL NOTE 15 Ci and C2 have the same sign) . The surface of a saddle is another example. This will show the reader that a reference, as on page 318, to a surface for which ci + C2 = 0 does not of necessity imply that the surface is plane. Quite a number of interesting investigations have been made by geometers on the family of surfaces which have the general property Ci + C2 = 0. An interesting example of a surface of "zero curvature" may be visualised thus. Imagine a string hanging from two points of support, in the curve known as a "catenary," and a horizontal line so far below it that the weight of a similar string stretching from the lowest point of the catenary to this line would be equal to the tension of the string at its lowest point. If one conceives Fig. 2 the catenary curve to be rotated around this horizontal line, the resulting surface of revolution is an anticlastic surface such that its principal radii of curvature at each point are equal in magnitude but oppositely directed. 8. Quadric Surface* The equation of a quadric surface, that is ellipsoid or hyperboloid, is ax2 + by^ + cz^ + 2 fyz -{- 2 gzx -\- 2 hxy = k * To be read in conjunction with pp. 404, 410 of Article K of this Volume. 16 RICE ART. B when the origin of the axes is at the centre of the surface. It can be proved that the equation of the plane which is tangent to the surface at the point Xi, y\, Zi on the surface is (axi + %i -\-gZi) X + (/ixi + hyi + fzi) y + (gxi 4- fyi + czi) z = k. Hence the direction-cosines of the normal to the surface at the point Xi, yi, Zi are proportional to the three expressions aXi + hyi + gzi, hxi + byi + fzi, gxi -\- fyi + czu (10) Another result which is required concerns the changes in the coefficients in the equation of the surface if the axes of reference are transformed to another set of three orthogonal lines meeting at the centre. If the coordinates of a point are x, y, z referred to the old axes and x', y', z' referred to the new, the values of x, y, z can be worked out in terms of x', y', z' and the nine direc- tion cosines of the new axes with reference to the old. On put- ting these values for x, y, z in the above expression, we obtain the equation of the quadric surface referred to the new axes as a'x'^ + by^ + c'z'^ + 2f'y'z' + 2 g'z'x' + 2 h'x'y' = k, where the values of a', h', c',f', g', h' can be obtained in terms of a, h, c, f, g, h and the nine direction cosines. The following three results can then be proved : a' -\- b' + c' = a -}- b + c, b'c' + cW + aV - P - g'^ - h"" = be + ca -\- ab — p — q^ — h^, \iM) a' h' g' a h g h' b' r = h b f g' f c 9 f c The interested reader will find the proof in any standard text of analytical geometry. MATHEMATICAL NOTE 17 A special case of considerable importance arises when the second set of axes of reference are the principal axes of the quad- ric surface. In that case it is known that/', g', h' are each zero and the equation of the surface has the form a'x'^ + by^ + c'z'^ = k. The results written above then become a' -\-b' + c' = a-^b -\- c, b'c' + c'a' + a'b' = be -\- ca + ab - f - g' - h\ a'b'c' = a h 9 h b f 9 f c } (12) c PAPERS I AND II AS ILLUSTRATED BY GIBBS' LECTURES ON THERMODYNAMICS [Gibbs, I, pp. 1-54] EDWIN B. WILSON I. Introduction As Papers I (pp. 1-32) and II (pp. 33-54) are properly charac- terised by H. A. Bumstead in his introductory biography (Gibbs, I, pp. xiv-xvi) as of importance not so much for any place they made for themselves in the literature as for the prep- aration and viewpoint they afforded the author as groundwork for his great memoir on the Equilibrium of Heterogeneous Substances, it will perhaps be most appropriate to illustrate them by an outline of Gibbs' course on thermodynamics as he gave it towards the end of his life. From such a sketch one may possibly infer what Gibbs himself considered important in the papers and what illustrations he himself thought it worth while to lay before his auditors. In this outline the notes of Mr. L. I. Hewes (now of the U. S. Bureau of Public Roads, San Francisco) who took the course in the academic year 1899-1900 will be followed.* II. Outline of Gibbs' Lectures on Thermodynamics Lecture I {October 3, 1899). The measurements in our subject fall into two sets, thermometry and calorimetry. Ordinary units of heat and scales of temperature. Constant pressure and constant volume thermometers. Gas thermometers with con- * I took the course two years later in 1901-1902; my notes were lost, but unless my recollection is mistaken the course did not differ except by the inclusion, toward the end, of a few lectures on statistical mechanics and a more rapid advance in the earlier parts (see Note on p. 50). 20 WILSON ART. C stant volume, pressure varying with the temperature, give best results. Clausius in his 1850 memoir brought order into the sub- ject of thermodynamics — with references to Clausius in the original and in translations, and to Maxwell's Theory of Heat. Lecture II. Heat capacity (specific heat) at constant pres- sure and at constant volume. Work, dW = pdv. Relation between heat and work — first and second laws of thermody- namics. We take the second law first (Carnot's law). Carnot was a French army officer, son of a minister of war. He pub- lished his results at about 28 years of age. His father was also a mathematician and wrote on geometry and mechanics. (He was uncle of the late President Carnot. ) Carnot's father named him Sadi after the Persian poet. Carnot's results meant an im- portant question solved and interpreted.* The Carnot cycle or Carnot engine, a reversible cyclic process: Given a cyHnder im- pervious to heat, except for the bottom which is a perfect con- ductor, filled with some medium (as air). Given a large hot and a large cold reservoir at assigned temperatures. Place the cylinder on the cold reservoir until the medium has taken the temperature of that. Carry out the following process. (1) Insulate the cylinder and compress the medium until the tem- perature has risen to that of the hot reservoir and then place the cylinder in contact with this reservoir. (2) Decompress the medium while the cylinder remains in contact with the reservoir thus absorbing heat and doing work at constant tem- perature. (3) Insulate and further decompress the medium until the temperature is lowered to that of the cold reservoir. (4) Place the cylinder in contact with the cold reservoir and compress to original volume. The result of the process is that some heat has been removed from the hot reservoir, som» has been given to the cold reservoir, and some external work has been done. Lecture III. Carnot's law: The same results are obtained with any medium when working between the same temperatures, or all reversible engines are exactly equivalent between the same * The class notes of Mr. Hewes, carefully written up, show that Gibbs did not think it infra dig. to go into interesting bits of scientific history. GIBBS' PAPERS I AND II 21 temperatures. If you have two engines both using the same amount of heat, they must do the same amount of work. For if they do not, running one direct and the other reversed will do a net amount of work without the use of heat or any other change in the system from cycle to cycle, which would consti- tute a perpetual motion machine — a reductio ad absurdum. There is no perfectly reversible engine, but one can be approxi- mated and for the purposes of reasoning one may be postulated. We assume that heat has to do with motion of the particles of a body. We have little doubt that matter consists of very small discontinuous particles and there is no reason they should not move. In regard to molecular motion forces are conservative; there are no frictional losses. Lecture IV. Continuation of discussion of evidence of fric- tionless character of molecular motion. Count Rumford thought heat not a substance. Joule determined the mechan- ical equivalent of heat; J = 772 ft. pds. W = JQ. We may as well measure Q directly in mechanical units as Q = W. Carnot failed to estabhsh the law Q" = Q' + W, namely, that the difference between the heat received and the heat given up was (proportional to) the work done. Joule seems not to have been entirely clear about the conversion of heat into work. Clausius was the first to set these matters straight. Lecture V. Discussion of meaning of first and second laws, and of various ways of stating them, by Tait, Clausius and Kelvin, illustrating each from considerations of the Carnot cycle. If Q" be the heat taken in at one temperature and Q' that given out at the other and W the work done; and if q", q', w be the similar quantities for another engine working between the same temperatures the quantities Q" , Q', W must be pro- portional to q", q', w. For we could by multiphcation (engines in parallel) make Q' = mq'. Now reversing one of the engines (or the set in parallel) the net heat taken or given to the cold reservoir would be nil and if the work were not also nil we should be obtaining work from heat at the single temperature of the hot reservoir which is contrary to Kelvin's statement of the second law. Hence W = imo and since by the first law Q" — Q' = W 22 WILSON AKT. c and q" — q' = w we must have Q" = mq", which proves the theorem.* Lecture VI. The first and second laws may be used to define a thermometric scale. For any two engines working between the same temperatures tx and h the heats received and given up satisfy the proportion Qi qx and hence these ratios may be taken as ti/ti. Thus ti Qi U Qs 1 .1 <• Q^ ^3 - = — , ~ ~ TT' ^^^ therefore 7^" = ""• ti Qi ti Qi Qi ti This shows that t may be taken as proportional to Q or Q._Qy This is called the absolute thermodynamic scale and the only remaining freedom is to define the unit. The first law is not confined to reversible cycles but the second law is. If we have two engines with Q" — Q' = W (reversible or not) and q" — q' = W (reversible) and run the second backward so that no work is done, the net heat Q" — q" leaves the higher temperature and the equal amount Q' — q' is received at the lower temperature. As heat cannot go without work from lower to higher temperature, Q" - q" = Q' - q' ^0. Hence Q" - q" ^Q' -q' t" - t' ' the equahty sign holding only when the numerators vanish, i.e., for the reversible case. But as q"/q' = t"/t' we have Q" Q' -7;- ^ -7 for any cycle. * The slow development of the analytical part of the subject was note- worthy. It was Gibbs' intention that the student should thoroughly grasp the physical, historical, and logical background through ample discussion. GIBBS' PAPERS I AND II 23 If in place of Q', the heat given up at t', we use —Q' as the heat absorbed at I', the relation becomes •^ + — < 0 With the understanding that Qi represents the heat absorbed at the temperature f » summation shows that 2yi:S0 or j j&O is a statement of the second law, the equality sign holding for the reversible engine. The corresponding statement of the first law is 2 Qi = W or fdQ = W. Lecture VII (Oct. 23). The characteristic equation /(p, y, t) = 0. The -pv diagram; isothermals and adiabatics. The work done in a circuit is the area of the circuit. fdQ=fdW, f!^SO. Jo Jo Jo t If we define the energy as ei - €0 = / (dQ - dW), Jo e is independent of the path since the circuit integral of dQ — dW is zero. In like manner for reversible engines the quantity Jo dQ ^71 — Tjo — ; — is independent of the path. It is called the entropy and like the energy is known except for an additive constant determinable when the arbitrary common origin of the paths is known. Then dW = pdv, de = dQ - dW, drj = dQ/t, dQ = tdt], dc = tdf] — pdv. Of the seven quantities, five, namely, t, p, v, e, r; have particular values at any point of the diagram; the other two, Q, W have no certain values, being dependent on the path to that point. 24 WILSON ART. C Lecture VIII. Discussion of pv diagram. To get the heat Qab absorbed along a path from AtoB draw the adiabatic from B and the isothermal from A intersecting in C and forming a curvilinear triangle ABC. Then Qab = area ABC + (rjc - t?^)^^. The ^Tj-diagram. Isometric and isopiestic Hues. Carnot's cycle a simple rectangular figure. We may draw diagrams other than the py-diagram or the ^Tj-diagram for other purposes but they do not have the advantage of simple areal interpretations.* The energy surface e = /(rj, v) as a function of entropy and volume. de de dri dv Lecture IX. Review of fundamental concepts. Lecture X. Mathematical transformations. 'dQ\ .dt/,' Specific heats C'p = ( — ) , C„ = ( - \dt/ p \ ( Elasticities E^ = - v(y\ Et = - v(-f) • Proof of Cp/Cv = Erj/Et given first by calculus as usual and second geometrically by means of anharmonic ratios in the in- finitesimal figure OV, OH, OT, OP formed by the intersection of a fine VHTP with the isometric, adiabatic, isothermal and iso- piestic issuing from a point of the py-diagram. The second proof is as follows: f}p — Vo Vp — Vh PH Cp Cy \dtJ 1 p r tp r]v to rio tp r\Y __ tr Vh PT VH \dtj K tv — to tv — tr VT * To this stage very little of the elaborate discussion of Paper I has been given. And no illustrative material. The lecture jumps right to Paper II. It may be particularly noted that the scale factor y was not treated, nor the fij-diagram discussed at this stage in the course, though they were treated in Paper I. CABBS' PAPERS I AND 11 25 The first and last steps depend merely on the infinitesimal char- acteristic of the figure and the intervening step on the definition of the iso-Hnes. Next, similarly, /dA \dv/ Vh — Vo Vb — Vp HP Er, \dv/^ _ Vh — Vo _ Vh — Vr _ HY_ Et ~ fdp\ Pt — po Pt — Pp TP Vt — Vo Vt — Vr TV Lecture XI. About anharmonic ratios and in particular their independence of the choice of the secant fine VHTP inferable from the physical interpretation above. Gases, pv = f(t). Laws of Boyle and Charles, Mariotte and Gay-Lussac. f(t) = at. Practical measurement of Cp. Theoretical measurement of Cv Measurements of E^ and Et. Lecture XII. Velocity of sound and its relation to the thermodynamic constants. Experiment with standing waves and lycopodium powder (Kundt's tube). It is found that for a gas C„ and Cp/Cv are constant within close limits over a wide range of the pv diagram. The equation de = dQ - dW = dQ - pdv reduces to de = dQ = Cvdt for constant volume and integrates into e = Cvt + V(v) where the constant of integration is a func- tion of the volume. Similarly for constant pressure we have 6 = Cpt — pv -{- P(p). Comparing, and using pv = at, V(v) - Pip) = (Cp - C„ - a)pv/a. This indicates Cp — C„ — a = 0 and F — P = 0, so that if the zero of energy is taken at ^ = 0 we have V = P = 0 and the equations of the gases are v € = C,t = Cpt — pv, a = Cp — Cv Lecture XIII. Review of fundamental equations. Discus- sion of differences between gas thermometer scale and absolute temperature defined by Carnot cycles. Further integration of 26 WILSON ART. C fundamental equations. For adiabatic changes de = — pdv may be put in form Cv— = — a — , or C„ log e = — a log y + H{-n), € V or for any change, de dv dH Cv — = — a 1 — r- dr], e V dt] which, by the equations e = Cvtfpv = at, becomes dH de = - pdv + t—- dr] = dQ - dW = td-q - dW . drj Hence dH/dr] = 1 and H = r] -\- const; with the constant taken as Cj, log Cv this makes* Cv log — = 77 — a log y , the equation between e, rj, v. Lecture XIV. The differential de = tdr] — pdv gives (de\ _ /de\ _ _ ^ _ /dt\ _ _ /dp\ \dr]/^ ' \dv/^ ' d'r]dv \dv/ ^ \dr]/^ Consider the functionf \p = e — trj and d\p = —rjdt — pdv. Then \dt/,~ '^' \dv/ ~ ^' dtdv ~ \dv)t ~ \dt/,' * On comparison with the development, Gibbs, I, 12-13, formulas A to D, it will be seen that there are slight differences, but the method here given was followed by Gibbs in his course on thermodynamics in differ- ent years. t I do not recall, and there is no evidence in the notes, that Gibbs gave names to the functions ^p, x, f such as free energy, heat function, or thermodynamic potential. He appears not to have referred to the function * = 77 — (c + pv)/t = — f/< which is widely used as a potential. GIBBS' PAPERS I AND II 27 Consider the function x = e + py and dx = tdr] + vdp. Then \dr]/p ' \dp/,, ' dr]dp \dp/^ \dr]/p Consider ^ = e — trj -{- pv and d^ = — rjdt + vdp. Then (^\ = - (^\ = -^ = - ('h\ = (^\ Kdt/p ''' \dpJt ^' dtdp \dp)t KdtJp The four Maxwell relations. For perfect gases 7] = Cvlogp -{- {Cv + a) log y — C„ log a = (7„ log i + a log y, \p = Cvt — Cvt log t — at log V, with similar expressions in f and x- The fundamental forms imply that e is a function of t?, y; that ;^ is a function oi t, v; that X is a function of 77, p; and that f is a function of t, p. Lecture XV. Avogadro's law. This differs from the laws thus far considered in that it relates to the invisible, molecular, properties of a gas instead of to the observable properties. The equation of a gas becomes pv = A{m/M)t where m is the mass of the gas and M is the molecular weight. Lecture XVI. A gas mixture has the equation \Mi Mi Mj The translational kinetic energy of the molecules is proportional to the pressure and therefore to the temperature. Lecture XVII. The geometric interpretation of p and t on the thermodynamic surface €(17, v). The use of the surface is to aid in thermodynamic investigations. The equation of the sur- face is known for a perfect gas, but the idea of it is equally applicable to any substance which need not be in a homogeneous state. Discussion of a substance in a liquid and vapor phase; ruhngs on the surface; the py-diagram. 28 WILSON ART. c Lecture XVI IL The solid-liquid and solid- vapor lines; the "triple-point" and the triply tangent plane. The relation dp Q dt {vv — VL)t for the invariant system consisting of liquid and vapor. Lecture XIX. Integrate de = td-q — pdv from liquid to vapor phase, t and p being constant. €r — iL = t{T}v — -til) — p(vr — Vl) or ^Y = tv — triv + PVV = €;. — tr}L + PVL = fi. The function f has the same value. The interpretation of f as the intercept of the tangent plane on the e-axis. The equation ,. ,. . dp rjv - riL Q dtv = d^L gives — = = -• dt Vv — Vl [Vv — Vijt The discontinuity of dp/dt at the freezing point. Discussion of the physical meaning of the Maxwell relations. Lecture XX.* In the py-diagram the isothermals in the vapor state start from large values of v approximately like the hyper- bolas pv = at; SiS V decreases their form is modified somewhat because when the vapor becomes dense the relation pv = at is somewhat inexact If the vapor starts to condense for values p = p',v = v' the isothermal becomes a straight line p = p' and so remains until condensation is completed aX p = p' = p" and V = v" < v'. From this point as v decreases the iso- thermal rises rapidly because a Hquid is compressed only with rapidly increasing pressure. The locus of the points {p\ v') and * To this point the lecturer had been following his two Papers I and II (Vol. I, pp. 1-54) with numerous omissions, with very few modifica- tions, and with considerable elaboration of the physical principles and facts underlying the subject. From here on he goes into a very consider- able development, which though perfectly natural and now found in other books, is not found in his writings. It seems that these applica- tions of his own may have so great an interest as to justify following them in considerable detail in the order of his thought. GIBBS' PAPERS I AND II 29 (p", v") forms a curve which we call the critical locus. If the temperature is high enough there will be no condensation. It has been seen that f is constant for the rectilinear portion of the isothermal including its extremities which lie upon the critical locus. For any path connecting these two limiting points (p', v') and {p", v") with p' = p" upon the isothermal t the total change of f must be nil. Now 6" - e' = fdt = fdQ - fpdv, n" - V = fdQ/t, p"v" - p'v' = fipdv + vdp). If the second equation be multiplied by —t' = —t" and the three be added (c" - t"y)" + p"v") - W - t'v' + p'v') = fdQ - t' fdQ/t + fvdp = 0. Hence for any path joining the two points / ^-^^ dQ -\- vdp = 0. In particular if the path be taken as a line v = v' rising above the critical point to p = p" ', a line p = p'" to the value V = v", and finally the Hne v = v" to p = p" (the three lines forming three sides of a rectangle of which the straight por- tion of the isothermal is the base), the value of fvdp is {v" — v') {p' — p" ') and thus for this path / ^—-^dQ + {v"-v'){p' -p'") =0. 6 We have seen that pv = aMs a law satisfied within wide limits. The law a at V = -,+ 1.2 V — b proposed by van der Waals, reduces essentially to pv = at when V is large and is found to be an improvement on that equation 30 WILSON ART. C for smaller values of v. For large values of t the isothermals in the py-diagram are concave upwards throughout their course from V = CO to y = 6 where they become infinite ; for small values of t the concavity changes and indeed the curves have a maximum and minimum. An isothermal of this type may have some degree of realization; for the phenomena of the super-cooled vapor in which condensation does not start and of super-heated liquid in which vaporization does not start are known, and indicate that under suitable conditions the isothermals of the vapor state may cross the critical line as the volume is reduced and the isothermal of the liquid state may also cross that line when the volume in- creases. The part of the isothermal of van der Waals which lies between the minimum and maximum and for which dp/dv is positive cannot be expected to be realized, as a positive value of dp/dv represents a mechanically unstable condition. If none- theless one writes d^ = — rjdt -\- vdp and integrates along an iso- thermal one has f " — f' = J'vdp and as for coexistent states f " — f ' = 0, one must have for such states J'vdp = 0. This means that from any van der Waals isothermal the line p = p' = p", which is the physical isothermal corresponding to coexistent states for the same temperature, must cut off equal areas, one below the line and the other above it. If the series of isothermals be drawn there are three interest- ing loci, the critical locus which gives the limiting conditions of coexistence of vapor and liquid phases, the locus of maxima and minima, and the locus of the point at which the rising (unrealiz- able) part of the isothermal cuts the hne p = p' = p". Lecture XXI. The word "unstable" is used in thermo- dynamics in not quite the same sense as in mechanics. If we have a supersaturated solution crystalhzation may not start; the substance may be stable within limits to certain variations, but will start to crystallize rapidly if a minute crystal be intro- duced, i.e., the solution may be unstable to the introduction of the crystal phase. So in superheated water, there may be stability with respect to various processes, but not with respect to the introduction of a bubble of steam. Entropy has been defined for a body considered homogeneous ; the restriction may be removed. There would be no difficulty GIBBS' PAPERS I AND II 31 with respect to coexistent homogeneous phases such as a sub- stance part liquid and part vapor which has been under discus- sion; we should add the entropies as well as the volumes and energies of the two parts. It is, however, necessary to proceed with some caution because entropy and energy have arbitrary origins and it is essential that the entropy and energy in one phase should be consistent with those in any other phase into which the substance may go or from which it may come. Sup- pose we have a substance in various phases, and not necessarily all in one working unit. Suppose the substance receives amounts Qi, Q2, • • • • of heat at temperature ^1, iz, . . . . , negative values of Q meaning that heat is returned to the reservoir. Also a certain amount of work is done by the substance or on it. The number of temperatures ti, ^2, . • • • of the reservoirs from which the substance receives heat may be infinite. Let the substance work on a cyclic process or on cyclic processes which may or may not be reversible. With this entire system we combine a per- fect (reversible) thermodynamic engine or a number of such engines to take the quantities of heat Q2, .... all to a reservoir of the given temperature ti. The quantities may be sche- matized as follows : Reservoir tempera- tures tl, tzi tzf ti, .... Heat absorbed by system Qi, Q2, Qs, Qi, Heat used by engines — Q2, —Qs, —Qi, Heat yielded by en- gmes - Q2, 7 Qh -Qi, ti tz ti Work done by engines — - — Q2, — - — Qs, — - — Qi, t2 t3 ti Work done by system Qi, +Q2, +Q3, +Q4, 32 WILSON ART. C As the whole complex consisting of the system and the engines is cyclic, the total work done, which is Q1 + 7Q2 + 7Q3 + 7Q4+...., t2 t3 h must be negative or zero as we cannot obtain work by a cyclic process without creating a perpetual motion machine. Hence dividing by ti, which is positive, we have «! + e^ + Q' + «' + ....=s«so, or /-so, tl ti ts 14 t r-f the equality sign holding only when the system is reversible. Now let s be any state of reference of the body for which we take 1? = 0; then any states 1 and 2 which can be reached from s by a reversible process will have the entropies - r dQ t' and the difference between the entropies will be where there is obviously one reversible way to go from 1 to 2, namely, that via s reversing the path from 1 to s above and following the path from s to 2. For example, if we have a satu- rated solution in equilibrium with some crystals, the application of heat will dissolve the crystals maintaining a saturated solu- tion until such point as the crystals are all dissolved and the further application of heat will render the solution unsaturated. Next, if heat be withdrawn the solution will become saturated and then possibly somewhat supersaturated rather than crystal- lizing. This process is reversible ; if the solution were supersatu- rated appUcation of heat would render it unsaturated. The transition from the state of saturation in the presence of crystals to an unsaturated state through the application of heat is how- ever not necessarily reversible because of the phenomenon of supersaturation; but there is generally some way to induce GIBBS' PAPERS I AND II 33 crystallization so that we can consider that the state of satura- tion in the presence of crystals may be reached reversibly. If this is the case it is easy enough to define the difference in entropy between a state of supersaturation and the state of saturation in the presence of crystals. Consider next a process which goes on within a wholly iso- lated system doing no work and receiving no heat. If that system can exist in two states 1 and 2 such that the path from 1 to 2 is irreversible but the path from 2 to 1 is reversible we can represent the difference in entropy at 2 and at 1 as 772 — 171. Then r^A+r^A^O and 7l t J2 t ~ irrev. rev. ^ T72 — 771. But if the irreversible process goes on entirely within the system there will be no heat dQ absorbed by the system, dQ = 0, and hence 0 ^ T72 — Tji or 172 = 171- Hence if an isolated system changes from state 1 to state 2, the entropy in state 2 must exceed that in state 1 (except when the change is reversible, when 772 = 171). It is assumed that there is some way to reach both states 1 and 2 reversibly from a third state. Take the case of the supersaturated solution. This may go over of itself into the state of a saturated solution with crys- tals. We have seen that we can reach the supersaturated states reversibly (i.e., we can reach any attainable degree of supersaturation reversibly). We can reach the state of satu- ration in the presence of crystals by merely placing the saturated solution and the crystals in juxtaposition. We have thus the possibility of defining the entropy 772 of the mixture of saturated solution and crystals and the entropy 771 of the supersaturated solution. The difference 772 — 771 will be positive. It is assumed that the mixture of saturated solution and of crystals in all its characteristics is that which would result from the spontaneous crystalhzation of the supersaturated solution in complete iso- lation. 34 WILSON ART. C The thermodynamic surface e(r}, v) represents the various states of a substance. There is a plane tangent to the surface at three points representing the three phase possibihties, sohd, Hquid, vapor. If the energy, entropy and volume of unit masses of the substance in contact with each other in solid, liquid and vapor state are es, vs, Vs] cl, vl, Vl', tv, -qv, Vy, respec- tively, then the energy, entropy and volume of a unit mass of which ms is solid, rtiL is liquid, mv is vapor are e = mses + rriLf-L + rrivtv, V — msrjs + mLr]L + nivVv, V = msVs + MlVl + mvVv, with 7ns + niL + mv = 1. There are developable surfaces "cor- responding to the equihbrium between liquid and vapor, be- tween solid and liquid, and between sohd and vapor. There are curved surfaces to represent the pure phases vapor or liquid or sohd. The thermodynamic surface is constituted of all these parts. In addition to this there may be parts of the surface which may be actually realized to some extent corresponding to supersaturation when the liquid fails to crystallize and super- heating when the liquid fails to vaporize. Such parts of the surface must lie inside the surface as viewed from the positive end of the entropy axis because they must represent states in which the entropy is less than it is in states into which the substance may spontaneously go. Let A and B be any two points of the thermodynamic surface which represents the entirely stable states. The segment AB must lie within (or on) the surface as viewed from the positive entropy axis. For consider any point P on AB and instead of the unit of substance for which the surface is given consider a mixture of AP/AB units of the substance in the state represented by A with PB/AB units of substance in the states represented by B. The energy and volume and entropy of the mixture are _ AP PB GIBBS' PAPERS I AND II 35 AP PB '^ = ab'^^ab'^^ _ AP PB Shut up in the volume v and isolated, changes will go on in the mixture which while unable to change e or y will increase 77. Thus the unit of the substance will come to equilibrium at a point on the thermodynamic surface e = tp, v = Vp, ri "^ tjp. As the proof holds for any point P no point between A and B can lie in the surface unless they all do. It follows that if a tangent plane is drawn to the surface at any point which repre- sents an entirely stable state of the body no point of the surface can lie on that side of the plane for which entropy is greater. Physically, in any change that would increase rj but involves the formation of a state widely different (such as a new phase) there is a certain reluctance* to take the step and this phenomenon * Lewis and Randall in their Thermodynamics, and the Free Energy of Chemical Substances, McGraw-Hill (1923), say, on p. 17: "In the work of Gibbs and some other writers upon thermodynamics, some proc- esses are supposed to be of infinite slowness, but this view of the exist- ence of a so-called "passive resistance" is apparently not supported by experimental evidence . . . . " The term "passive resistance" is appar- ently not used by Gibbs in Papers I and II; but that he would have re- garded the reluctance to change exhibited in the phenomena of super- cooling, superheating and supersaturating as due to such resistances is rendered likely by his definitions and illustrations when he first intro- duces the term, namely, in Paper III (Gibbs, I, p. 58) where he writes: "In order to apply to any system the criteria of equilibrium which have been given, a knowledge is requisite of its passive forces or resistances to change, in so far, at least, as they are capable of preventing change. (Those passive forces which only retard change, like viscosity, need not be considered.) ... As examples, we may instance the passive force of friction which prevents sliding when two surfaces of solids are pressed together, . . . , that resistance to change which sometimes pre- vents either of two forms of the same substance (simple or compound), which are capable of existing, from passing into the other. ..." It cer- tainly does not appear from this phraseology that Gibbs was supposing the processes which he associated with the term passive resistance to be of infinite slowness; indeed his underlining of the word preventing and his 36 WILSON ART. C gives rise to states which for some variations behave as stable states but for others give indications of not being entirely stable.* excepting those passive forces which only retarded change seem clearly to indicate that there was a state of no process whatsoever associated with the passive resistances rather than one of very slow process. And again in the discussion of Certain Points Relating to the Molecular Con- stitution of Bodies (Gibbs, I, pp. 138- 144) he seems to be drawing a pos- sible logical distinction between passive resistances which prevent change and those which only slow it down, though they may slow it down very greatly. He certainly does seem to postulate that there may be real states of equilibrium which are not states of dissipated energy and which do not even with infinite slowness go over into such states. Lewis and Randall would appear to postulate that there are in reality no such states, that only states of dissipated energy are states of equilibrium. They may be entirely right without Gibbs being in any way wrong. It is important to have the solutions for both ideal cases — that in which the change is absolutely prevented and that in which it is completely con- sumated. A case in practice may well be intermediate between the two so that both solutions might be inapplicable. Gibbs speaks as though hydrogen and oxygen placed together at room temperature would never unite to form water vapor; while Lewis and Randall expect them to unite (almost completely, though slowly) according to their equation (22), p. 485, viz., H2 + 5O2 = H20(^) ; A F°2is = —54507, and so, too, we may pre- sume that if hydrogen were shut up by itself they would expect it to go over into helium. There is, of course, no practical difference between the two postulates when the reaction is slow enough, but it would seem that Gibbs' form would be at least as convenient practically as that of Lewis and Randall. * The logical difference between stability and slowness in attaining the stable state must be kept in mind. Thus a liquid in the presence of its vapor may be very slow in evaporating to the point where the vapor is saturated and the equilibrium is established. Things do not dry im- mediately simply because there is not equilibrium between their state of wetness or dryness and the humidity in the atmosphere. In thermo- dynamics time is disregarded, the processes are permitted to take place infinitely slowly. Indeed finite velocities may introduce irreversibility. For example in the simple Carnot cycle in the decompression stage 2 (Lecture II) it is specified that the decompression is isothermal, which means that it is slow enough so that the medium remains at the tempera- ture of the reservoir. If the medium were a perfect gas pv = at, the work would he W = Spdv = at log (?;2/fi). But if the decompression be fast enough the medium would expand practically adiabatically (and GIBBS' PAPERS I AND II 37 Lecture XXII (December 18, 1899). A detailed discussion of the characteristics of the thermodynamic surface with respect to increasing entropy.* Lecture XXIII {January 11, 1900). The surface hes on the negative entropy side of any tangent plane. If the surface in the immediate vicinity of the point of tangency lies on the nega- tive entropy side of the plane, the substance is in a stable state for infinitesimal variations from the state represented by the point of tangency. In like manner as an isolated system tends to a state of minimum energy it follows that if the surface lies upon that side of the tangent plane upon which energy increases the state represented by the point of tangency will be one of stable equilibrium ; if at a considerable distance from this point the plane again cuts the surface we have a kind of instability (the state is not entirely stable) but there is still stability for small variations. then heat up from the reservoir). The work would be less, say w. By the time the medium had absorbed the heat from the reservoir its energy would however be the same. For the two processes we have therefore Q — W = q — w or Q — q = W — w>0 or Q>q. When the heat Q is transferred from the reservoir to the medium isothermally at tempera- ture t, the medium gains entropy to the amount Q/t and the reservoir loses the same amount of entropy. In the adiabatic decompression and subsequent heating the medium gains the same amount of entropy Q/t but the reservoir loses only q/t so that the system consisting of reservoir and medium gains the amount {Q — q)/t of entropy. To put this in another light suppose there are two like cylinders one in condition vi, t which expands adiabatically to state V2, t and then heats up as above and the other in state V2, t which is compressed isothermally in contact with the reservoir to (^i, t) as in stage (4) of the Carnot cycle. The operation of the two will result in work W — w being done on the media. In the final condition the two cylinders have only interchanged states. The reservoir has gained the heat Q — q equivalent to the work done and the system consisting of the two cylinders and medium will have gained the entropy (Q — q)/t representing the irreversibility in the process. * This was essentially a review and illustration of the close of the pre- vious lecture, consideration being also given to the kind of isothermals encountered in van der Waals' equation. It does not seem worth while to follow this detail here, though it was helpful to the class in gaining a better appreciation of the subject matter. The long Christmas vacation intervened at this point in the course. 38 WILSON Conditions for stability. Let z = /(.r, y). dz dz z = 2o + -^^x + —Ay ax dy ART. C + H^. ^^' + 2 ^ AxLy + ^, Ay^ + d^z ^ v^ ' dxdy dy"^ Tangent plane dz dz Zp = Zo + -- Ax -\- -- Ay, dx dy ^- ^P = Ht^, ^^' + 2£^Aa:-A2/ + ^,A2/2J + .. dH d^ dxdy dy^ Neglecting higher powers, the condition that z > Zp, except for Ax = Ay = 0, is first dh , d'z ^,>0 and ->0. and then by completing the square also dx"^ dy^ \dxdy/ > 0. For the limit of stability this last condition is zero. Re- place 2 by e and x, yhy r],v and remembering de = idr] — pdv the conditions are dh fdp\ dh (dt\ dv^ \dv/r, dti^ \dr] dh d^e dv^ df] 2 / d^e Y _ _ (dp) (dt\ _ /dpV \dvdr]' \dv/„ \dri/^ xd-q/^ V The first condition means that when the change is adiabatic p must decrease as v increases, and the second means that at con- stant volume the temperature must rise if heat is supplied. The third condition may be transformed. Note first that i!i - _ (^\ _ (^\ dr\dv \dr]/^ \dv/^ GIBBS' PAPERS I AND II 39 Now for constant volume p generally increases if heat is sup- plied, and under adiabatic conditions the temperature generally rises under compression; hence generally this second derivative is negative. But for water under the temperature of maximum density the results are reversed and the derivative is positive. Next , fde\ dh ^ dh ^ dp = — d\—- } = — — — drj — dv = — Bdt] — Adv, \dV/r, dvdrj dv^ dh dh dt = d[^^] = -—dr) -j- -— dv = Cdr, + Bdv. drf dvdt] (-) = Solve for dt] and dv; then /^\ _ _ AC - B-" \dv)t ~ C Xdri/r, A AC - 52 /dp\ _ AC - B'^ /dt\ \dii] Jt B \dv/p B Now as C > 0, AC — B^ >0, this means that on an isothermal p must decrease with increasing v. So, too, at constant pressure the temperature must increase with a supply of heat. In the general case where B <0, supplying heat and maintaining a con- stant temperature must decrease the pressure, or at constant pressure the temperature must increase with the volume. Note that equating the last two expressions and inverting the deriva- tives yields the Maxwell relation obtained from the function f . Lecture XXIV. Discussion of van der Waals' equation.* * The development may not seem logical and was probably adopted for pedagogic reasons. As early as Lecture XVII the py-diagram for vapor, liquid, and vapor-liquid phases was introduced, leading from physical reasoning to the definition of critical locus and the conception of that sort of stability or instability which is represented by the super- cooled vapor or superheated liquid. On this basis in Lectures XVIII- XIX properties of the thermodynamic surface were discussed. In Lec- ture XX the equation of van der Waals was cited as affording possible conceptual though largely unrealizable isothermals through the critical region, and this type of isothermal was kept to the fore, in parallel with 40 WILSON ART c Here a Rt P = - -, + 7' (1) V^ V — 0 /dp\ _ 2a _ Rt \dv/t ~ V' ~ (v -by~^ ^^^ at the limit of stability. Eliminating t, the locus in the pv plane is* a 2ab . . p = -,--r' (3) v^ v^ We have also the equation \dvyt Qa 2Rt = - ~T + 7 ^3 = 0 (4) to represent the inflections of the isothermals. Equations (1), (2), (4) have a common solution, which must be also a solution of (3), and this is the critical point. If (1) be regarded as a cubic in v the critical point is that for which the cubic has three equal roots. For this point the actual physical isothermal representing complete equilibrium, in the detailed discussion of the thermodynamic surface including the questions of stability (whether entire or limited) in Lectures XX-XXIII. This general discussion completed, the lecturer returns to a considerable development and illustration with the aid of the equation of van der Waals. * The limit of stability is defined by {dp/dv)t = 0, i.e., when AC — B' = 0. It may be observed that by this definition there may lie within the limit of stability states with negative values of p, i.e., with tensions instead of pressures. From (3) we have v = 2b when p = 0. Then Rbt/a = 1/4. In terms of the critical values v/vc = 2/3, t/tc = 27/32. Thus for temperatures below 27ic/32 = .Siitc the van der Waals' iso- thermal dips down to negative values of p. Indeed as v decreases toward b, p in (3) decreases toward —a/b^ = —27pc, and t toward zero. Al- though all negative values of p represent instability in vapor phases, we do know that under careful experimental conditions liquids can be made to support very considerable tensions without going over into the vapor phase, thus parts of these isothermals for negative p can be realized qualitatively even if the quantitative relations are quite inadequately represented by (1). GIBBS' PAPERS I AND II 41 1 a 8a 2762' ^^ " 27 Rb' la 8a Vc = 3o, P' ~ 7^77' ^c ~ > (5) and 6=^^ a = 3po^;c^ 7^ = ^^^ (6) 3' " ' 3 f, c There is no great difficulty in determining pc, tc from observa- tion. Sketch of possible methods. The determination of Vc is more difficult because infinitesimal changes in v near Vc produce changes of p, t from pc and tc which are infinitesimals of higher order and hence slight changes in p and t from pc and tc produce large variations in v from Vc, — as may be seen geometrically from the shape of the isothermals in the vicinity of the critical point. However, we may determine Vc by the known value oiR. Lecture XXV. Discussion of the accuracy with which van der Waals' equation represents the physical facts. The critical locus may be obtained from the condition that Sv^v along the isothermal from one of its intersections (p, v^ with the critical locus to the other {p, v^ must be equal to p{v2 — Vi) by the areal of property previously proved. Hence p{v2 -V,) - ~ -^ - + nt log -^— - = 0. (7) V2 vi V2 — 0 Equation (1) holds for p, Vi, t and for p, V2, t. Eliminate p, t. Then V2 + vi , yi - & , Vi , V2 log -I + — = 0. Let ^^2 — i^i 1^2 — 6 Vi — b V2 — b Vr-b _ V2-b ^'~ b ' ^'- b Then with P = F1/F2 we have V2 21ogP _ L _ 1' P - 1 P 7i = PV2. 42 WILSON AET. C At the critical point Vi = V2, log P = 0. We may take P ^ 1. Furthermore a (V, + 1)2 (72 + 1)2 b'p FiFs - 1 and" F3 = a iVi + 1)2 {V2 + 1)2 /^i 7i + F2 + 2 bpViV2 V,V2 - 1 The critical locus may therefore be plotted from the following computation form p V2 Vi bRt/a b^p/a V, 1.0 2.0 2.0 .296 .0370 2.00 .9 2.11 1.90 .296 .0370 2.00 .8 2.24 1.79 .296 .0368 2.00 .7 2.40 1.68 .295 .0365 2.01 .6 2.60 1.56 .294 .0360 2.02 .5 2.86 1.43 .292 .0351 2.03 .4 3.23 1.29 .290 .0338 2.06 .3 3.79 1.14 .285 .0316 2.10 .2 4.77 .95 .277 .0279 2.17 .1 7.23 .72 .259 .0210 2.36 .05 11.21 .56 .238 .0146 2.61 .02 20.76 .42 .211 .0080 3.04 .01 33.98 .34 .191 .0048 3.44 .005 56.79 .28 .173 .0027 3.91 .002 115.24 .23 .153 .0012 4.60 .001 200.58 .20 .139 .0007 5.17 * The intermediate value V3 where the ascending branch of the iso- thermal cuts the horizontal is obtainable from bWiViVz = ViV2Vi — b{viVi + i;ij;3 + r2f3) + b^ivi + fj + Vz) — b^ which may be evaluated at once from van der Waals' equation. GIBBS' PAPERS I AND II 43 One may plot in the same diagram the isothermals from b^ _ Rht/a _ 1 a ~ V ~ (7 + 1)2' and the locus of the limit of stability from ¥p 2V 1 a (7 + 1)3 (7 + 1)2 The table is good for any substance satisfying van der Waals' equation. Lecture XXVI. li \}/ = e —tr], d\p = —'i]dt —pdv, and _ _ (^\ - ("^ ?L \dv / 1 \v'^ V — h may be integrated to find ), ^ = -^ - ntAog(,v -h)+^ (t), (8) V v = - (^)^ = R log (v-h)- $' (t), (9) e = _ ^ + $(^) -t^'{t), (10) V ^•^ (I). = -'*"«• (!') If the volume is very great the specific heat for constant volume is ordinarily constant, say c. Then —^'{t) = c log t + const., and the constant may be taken as zero without loss of gen- erality. Hence *(0 = d - d log t, (12) and for a substance satisfying van der Waals' equation we have \p = -- - Rt log {v - h) + d - d log t, (13) V 7] = R\og(v - h) -{- c log t, (14) e = - - + d, (15) V 44 WILSON ART. c The last two equations consist of sums of a function of v and a function of t. The thermodynamic surface is r] = R log (y — 6) + c log (16) c or ^=-- + ^(^73^0- (17) This surface is that which corresponds to following the sub- stance through its partly stable and its unstable states which correspond to the parts of the isothermals within the critical locus; it is, therefore, not precisely the thermodynamic surface discussed in Lecture XXI. We may obtain ^ = e — trj + pv a,s f = -- - Rt log {v - b) -\- ct - d log t + pv. (18) V This is not the desired form, which should involve p and t, but the elimination of v would require the solution of a cubic equa- tion. The condition for corresponding states is ^2 = Ti and this reduces to (7) which was obtained above. Corresponding states. By introducing the values of a, 6, J? in terms of pc, Vc, tc into the equation and using P = p/pc V = v/vc, T = t/tc, van der Waals' equation takes the form which is of the same form for all substances, but with pressure, volume and temperature expressed as multiples of the (different) critical values for the (different) substances. Lecture XXVII. The tangent plane to the thermodynamic surface is e — eo = t{-n — Vo) — p(v — Vo). GIBBS' PAPERS I AND II 45 The slopes of the plane are t in the erj plane or planes parallel thereto and —pin the ev plane or any parallel plane. Further — dp = Adv + Bdr], dt = ■ Bdv + Cd-n, with dv^' dvdri^ '-%' and then (dp\ B /dp\ A \dtJ, C \dt/. ~ B These two quantities are in general different but at the limit of stability they are equal and in particular at the critical point. Both these quantities are easy to measure. If we have coexist- ent phases the tangent plane is rolling on the surface with con- tact at two points and the successive positions intersect in the line giving the two points of contact and representing the diifer- ent states in which the two phases can exist in different propor- tions at the same pressure and temperature. At the critical point according to van der Waals' equation. R R ^ tc Sb = = — and — = — • V - b 2b pc R Hence \dt/^ t d log V , , - = -—^ = 4. (20) p d log I Now we may experimentally determine the values of p and t for states of coexisting phases and make a graph in which we plot log p against log t. If then van der Waals' equation were satis- fied we should find that as we approached the critical point the slope of the curve approached 4. This value does not, as a matter of fact agree with that found by experiment, which points rather to 5 or 6 or 7. Various modifications of the equation have been proposed by Clausius and others. We could treat any of these proposals by similar methods. No entirely satis- factory equation of state has been proposed. The usefulness of 46 WILSON ART. c the various forms depends on the particular inquiry to which they are applied. Lecture XXVIII. Returning to van der Waals' law, ( dp\ R dt/v V — This is not quite true, of course, but it is surprisingly correct in many cases over a very wide range. For very great densities it cannot be expected to hold, and we have to exclude dissocia- tion at very high temperatures, and those states in which the substance is congealed. Now in the -pt plane a line of constant volume becomes straight. It is easy to determine correspond- ing values of p and t under conditions of constant volume and observe how straight the curves in p against t are. At the limit of stability we had {dp/dv)t = 0, i.e., maxima or minima of the isothermals in the yv plane. Keeping t constant in the p^-dia- gram corresponds to a vertical displacem.ent. If {dip/dv) « > 0 it is seen that the lines of increasing volume on the p^-diagram lie one above the other in the direction of increasing pressure; in the limit when {dp/dv)t = 0 the successive lines of constant volume intersect. These lines will therefore envelop a locus which consists of points pv for which (dp/dv)t = 0, i.e., for states at the limit of stability. This locus has a cusp which is the crit- ical point. In the region within the cusp and near to it there are three tangent lines of the envelope through each point, i.e., for a given pair of values p, t there are three lines of constant volume along which one may proceed. Taking van der Waals' equation in the form (19), the equations 8 8^ - T -T V V - 1/3' UfA ~ Y^ {V - 1/3)2 will give the cuspidal locus on elimination of V from 9(V-l/3)''^ 3 7-1/3 3 2 4 73 ' 72 "■ " 73 72 73 The plot of P against T is more readily made from this para- metric form than from the equation obtained by eliminating V. GIBBS' PAPERS I AND II 47 The point P = 1, T = 1 corresponding to F = 1 is the critical point. As (-) \dT/v 8/3 V - 1/3' = 4. The values of V, T, P and (dP/dT)v are entered in the table which clearly shows the cusp at (1, 1, 1) and from which the envelope may be plotted easily. V T P {dP/dT)y 2/3 27/32 = .84 0 8 3/4 25/27 = .93 16/27 = .59 32/5 5/6 243/250 = .97 108/125 = .86 16/3 1 1 1 4 7/6 675/686 = .98 324/343 = .94 16/5 4/3 243/256 = .95 27/32 = .84 8/3 3/2 49/54 = .91 20/27 = .74 16/7 2 25/32 = .78 1/2 = .50 8/5 3 16/27 = .59 7/27 = .26 1 00 0 0 0 Lecture XXIX. We return to the consideration of coexistent phases, basing the development upon the condition ^2 = Ti or €2 - Cl - t(V2 - Vl) + V(V2 - Vi) = 0. For 62 — €i we use (10) ; for 772 — Vi we use dp _ 7/2 — rji _ Q 1 dt 1^2—1^1 t Vi — Vi previously derived. Thus the condition may be given the form a --^ + 1 = 0. pviVi p dt But the three roots of van der Waals' equation for p = const, satisfy /Rt \ v^ — I \- 0 j v a ah ^ + - V =0, P V 48 WILSON ART. c and hence yij;2i^3 = ah/p and ^3 d log p d log t - 1. The value Vs is that at which the rising (unstable) part of the iso- thermal cuts the horizontal line and is not attainable by experi- ment. But on substituting this in the equation we have by -f- /d log p _ \2 dlogp _ 2 Vdlogi / dlogt which is sometimes useful in working with coexistent phases when we are willing to put conjfidence in the equation of van der Waals. The general equation of state p = F'{v) + tf'iv), of which van der Waals' is a special case, maybe discussed. For this (dp/dt)v is again a function /'(t;) of v and at constant vol- ume is constant, so that the isometric lines in the p^diagram are straight. We have ,/, = -F(v) - tf(v) + $(^), € = -F(v) -f$(0 - t^'{t). If we use for $ (t) the expression ct — ct logt, thene = —F{v) + d. At any rate both e and 77 consist of a function of the volume plus a function of the temperature. It is to these equations that we naturally look for some improvement upon van der Waals'. Lecture XXX. Let us make the hypothesis that there is an equation of state which is independent of the substance, pro- vided only we measure p, v, t in the appropriate units. What results could be obtained? There is one state of the substance which is physically defined, namely, the critical state. It is therefore P = p/pc, V = v/vc, T = t/tc which are the variables GIBBS' PAPERS I AND II 49 which must be used and the equation must be between P, V, T. Such an expression as pv — must be the same for all substances. PcVc tc If m denote the mass and M the molecular weight we have p t V pci'cM p V M Pc tc Vc Um ' tm equal for all substances. (The last two expressions must be measured in the same units for the different substances, but the first three may be measured in any units.) So, too, t_ /dp\ ^ t_ /dri\ ^ 1 /dQ\ p\dt/v p \dv/t p \dv/t would be alike. Also V \dt/p V \dp/t V 'dQ' \dp/ For coexistent phases there would be certain expressions in- variant of the substance. p\dt/v p\dv/t pv2 — Vi As f 1 = ^2 we may state that the ratios (€2 - ei) : ^(772 - Vi)-Piv2 - Vi) are the same for all substances when 2 and 1 stand for the vapor and the liquid phase, each in the presence of an infinitesimal quantity of the other. By examining data for different sub- stances one may see how far the departure from constancy is and thus gain some idea of in how far it might be hopeful to seek for equations of state which would satisfy the requirement that in proper units the equation should be the same for the different substances. 50 WILSON ART. 0 III. Further Notes on Gibbs' Lectures. Photographs of Models of the Thermodynamic Surface These thirty lectures as given in the academic year 1899-1900 represent the development, discussion, and application of the matter in Papers I and II so far as Gibbs covered it. In the year 1901-1902 he covered the same ground in just fifteen lec- tures. He continued with a lecture on dynamical similarity and the theory of models which he applied to the consideration of intermolecular forces and the problem of corresponding states, and then launched into the topic of heterogeneous substances (Paper III). It will be seen that although he laid great stress on the physical and on the logical aspects of thermodynamics, and spent a good deal of time on van der Waals' equation as a type of equation of state, he did not indulge in many numerical appli- cations, nor discuss practical engineering consequences of the theory. He used chiefly the pt-diagram, giving scant mention to the temperature entropy diagram. An interesting and helpful episode in the course was the illus- tration of the discussion of the thermodynamic surface by a model of the surface for water, which had been sent him by Maxwell. Four photographs of this model taken from different points of view are reproduced here. The legends indicate the direction of the axes. Maxwell's highly favorable comments on the work of Gibbs and the concrete evidence which he gave of his opinion through the construction of the model of the thermodynamic surface prob- ably did more at the time to convince physicists of the impor- tance of Gibbs's contributions than the reading of so long, so novel, so closely reasoned and withal so difficult a memoir as that on Heterogeneous Equilibrium. It is of interest in this connection to give the record of the award by the American Academy of Arts and Sciences of its Rumford Medal to Gibbs. At the meeting of May 25, 1880, Professor Lovering presented the following report from the Rumford Committee.* "The mechanical theory of heat, which treats of heat as being, not a pecular kind of matter called caloric, but as being some form or forms * The Committee consisted of Wolcott Gibbs, E. C. Pickering, J, M. Ordway, John Trowbridge, J. P. Cooke, Joseph Lovering, G. B. Clark. GIBBS' PAPERS I AND II 51 of molecular motion, has made necessar}' and possible a new branch of mechanics, under the name of thermo-dj'namics. This theory has not only introduced new ideas into science, but has demanded the applica- FlG. 1 Fig. 2 Fig. 3 Fig. 4 The Thermodynamic Surface (Maxwell's Model) Fig. 1. Vertical axis; energy (e). Axis of volume (?0 toward the front and left. Axis of entropy (tj) toward the right. Fig. 2. Vertical axis; energy (e). Axis of volume (r) toward the front and right. Axis of entropy (77) toward the right and back. Fig. 3. Vertical axis; energy (e). Axis of volume (v) toward rear and left. Axis of entropy (r/) toward front and left. Fig. 4. Vertical axis; volume {i'). Axis of entropy (rj) toward front and left. Axis of energy (e) toward the right. tion, if not the invention, of special mathematical equations. Clausius has devoted thirty j^ears to the develoi)ment of thermo-dynamics, and at the end of his ninth memoir he expresses, in two brief sentences, the 52 WILSON ART. c fundamental laws of the universe which correspond to the two funda- mental theorems of the mechanical theory of heat : 1 . The energy of the universe is constant; 2. The entropy of the universe tends towards a maximum. "Professor J. Willard Gibbs, in his discussion of the 'Equilibrium of Heterogeneous Sul)stances/ derives his criteria of efiuilibrium and sta- bility from these two theorems of Clausius, and places the two generali- zations of Clausius in regard to energy and entropj' at the head of his first publication. Having derived from his criteria some leading equa- tions, and having defined his sense of 'homogeneous' and its opposite, he applies these equations: — "1. To the internal stabilitj^ of homogeneous fluids. "2. To heterogeneous masses, under the influence of gravity or other- wise; such as gas-mixtures, solids in contact with fluids, osmotic forces, capillarity, and liquid films. "3. Finally, he considers the modifications introduced into the con- ditions of equilibrium by electromotive forces. "His treatment of the subject is severely mathematical, and incap- able of being translated into common language. The formulas, how- ever, are not barren abstractions, l)ut have a physical meaning. "The laws of thermo-dynamics reach down to the heart of physics and extend tlieir roots in all directions. It is now understood that the energy of a system of bodies depends on the temperature and physical state, as well as on the forms, motions, and relative positions of these bodies. The Rumford Committee congratulate the Academy on the opportunity they now enjoy of awarding the Rumford Premium for a contribution to physical science of far-reaching importance; not antici- pating, but already realizing, the approval which this award must receive from all who are conversant with the subject. "For the Committee, "Joseph Lovering, ChairmanJ' The medal was awarded at the meeting of January 12, 1881, Professor Lovering having in the interim been elected president of the Academy. His address as Chairman of the Committee was in part* as follows. "On the mechanical theory of heat, as a foundation, has been erected * The material here quoted is from Proc. Amer. Acad. Arts Sci., 16, pp. 407-408 and 417-421. The introductory portion which deals with the history of the award is omitted. GIBBS' PAPERS I AND II 53 the grandest generalization of physical science, the Conservation of Energy. The results of observation and calculation agree, whenever a comparison is practicable, if the calculation is made upon the assump- tion that the totality of energy in a system, potential as well as dynam- ical, is as unchangeable as the totality of matter. This sweeping gen- eralization includes and interprets Grove's experimental demonstration of the correlation and convertibility of the different forms of energy, known under the familiar names of gravity, elasticity, light, heat, elec- tricity, magnetism, and chemical affinities. The conversion of heat (which is supplied to an indefinite amount by the consumption of the forests and the coal-beds) into ordinary mechanical energy or work, is of the highest significance to the advancing civilization of the race; but heat cannot be transformed into work without the transformation of a larger amount of heat of high temperature into heat of low temperature. This passage of heat from hot to cold bodies, without doing work, rein- forced by the conduction and radiation of heat, creates the tendency to what is now called the dissipation of heat. This is what the writer in the London Spectator meant when he called hSat the communist of the universe, the final consummation of this dissipation being a second chaos. Sir William Thomson has computed that the sun has lost through its radiations hundreds of times as much mechanical energy as is represented by the motions of all the planets. The energy thus dispensed to the solar system, and from it to remoter space, 'is dissi- pated, always more and more widely, through endless space, and never has been, and probably never can be, restored to the sun without acts as much beyond the scope of human intelligence as a creation or anni- hilation of energy, or of matter itself, would be.' Therefore, unless the sun has foreign supplies, in the fall of meteors or otherwise, where its drafts will be honored, its days are numbered. "What I have attempted to state in language as little technical as possible is tersely expressed by Clausius in two short sentences: 'The energy of the world is constant.' 'The entropy of the world (that is the energy not available for work) tends constantly towards a maximum.' "Professor J. Willard Gibbs takes his departure from these two propositions when he enters upon his investigation on the 'Equilibrium of Heterogeneous Substances.' Any adequate theoretical treatment of this complex subject must be, necessarily, highly mathematical, and intelligible only to those familiar with the analytical theory of heat. To assist the imagination, Professor Gibbs has devised various geomet- rical constructions; especially one, of a curved surface, in which each point represents, through its three rectangular coordinates, the volume, energy, and entropy of a body in one of its momentary conditions. 54 WILSON ART, C The late Professor J. C. Maxwell (whose early death is ever a fresh grief to science) devoted thirteen pages of the fourth edition of his 'Treatise on Heat' to the elucidation and application of these construc- tions; and it is understood that he embodied in a visible model the equations in which Professor Gibbs expressed his strange surface. In a lecture delivered before the Chemical Society of London, Professor Maxwell gave publicly the endorsement of his great name to the merits of these researches which we are now met to honor. He says: 'I must not, however, omit to mention a most important American contribu- tion to this part of thermo-dynamics by Professor Willard Gibbs, of Yale College, U. S., who has given us a remarkably simple and thor- oughly satisfactory method of representing the relations of the different states of matter, by means of a model. By means of this model, prob- lems which had long resisted the efforts of myself and others may be solved at once.' "It is now my pleasant duty to present, in the name of the Academy and with their approving voice, the gold and silver medals to the Re- cording Secretary, Professor Trowbridge, who has been commissioned by Professor Gibbs to represent him on this occasion. I cannot but think that if Count Rumford were living, he would regard with peculiar pleasure this award. For the researches of Professor Gibbs are the consummate flower and fruit of seeds planted by Rumford himself, though in an unpromising soil, almost a century ago. In transmitting these medals to Professor Gibbs, by which the Academy desires to honor and to crown his profound scientific work, be pleased to assure him of my warm congratulations and the felicitations of all the Fellows of the Academy, here assembled to administer Count Rumford's Trust." In reply to the President's address, the Recording Secretary then read the following letter from Professor Gibbs :— "To THE American Academy of Arts and Sciences: — "Gentlemen, — Regretting that I am unable to be present at the meet- ing to which I have been invited by your President, I desire to express my appreciation of the very distinguished honor which you have thought fit to confer upon me. This mark of approbation of my treat- ment of questions in thermo-dynamics is the more gratifying, as the value of theoretical investigation is more difficult to estimate than the results obtained in other fields of labor. One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity. The success of the investigations in this respect is a matter on which GIBBS' PAPERS I AND II 55 he who makes them may be least able to form a correct judgment. It is, therefore, an especial satisfaction to find one's methods ap- proved by competent judges. "The leading idea which I followed in my paper on the Equilibrium of Heterogeneous Substances was to develop the roles of energy and en- tropy in the theory of thermo-dynamic equilibrium. By means of these quantities the general condition of equilibrium is easily expressed, and by applying this to various cases we are led at once to the special conditions which characterize them. We thus obtain the consequences resulting from the fundamental principles of thermo-djTiamics (which are implied in the definitions of energy and entropy) by a process which seems more simple, and which lends itself more readily to the solution of problems, than the usual method, in which the several parts of a cyclic operation are explicitly and separately considered. Although my results were in a large measure such as had previously been demon- strated by other methods, yet, as I readily obtained those which were to me before unknown, or but vaguely known, I was confirmed in my belief in the suitableness of the method adopted. "A distinguished German physicist has said, — if my memory serves me aright, — that it is the office of theoretical investigation to give the form in which the results of experiment may be expressed. In the present case we are led to certain functions which play the principal part in determining the behavior of matter in respect to chemical equi- librium. The forms of these functions, however, remain to be deter- mined by experiment, and here we meet the greatest difficulties, and find an inexhaustible field of labor. In most cases, probably, we must content ourselves at first with finding out what we can about these functions without expecting to arrive immediately at complete expres- sions of them. Only in the simplest case, that of gases, have I been able to write the equation expressing such a function for a body of vari- able composition, and here the equation only holds with a degree of approximation corresponding to the approach of the gas to the state which we call perfect. "Gratefully acknowledging the very favorable view which you have taken of my efforts, I remain, gentlemen, very truly yours, "J. WiLLARD GiBBS. "New Haven, Jan. 10, 1881." It is noticeable that with the exception of mere mention of the chief divisions of the great memoir in the report recommend- ing the award there is neither in the report nor in the address of the chairman any reference to the content of that memoir, let 56 WILSON ART. C alone any critique of its importance to science; the references are to the previous state of thermodynamics and to the thermo- dynamic surface and Maxwell's model of it, i.e., to material by Gibbs contained in his Paper II, which we have been discussing. It may be recalled that in December 1878, more than two years prior to President Lovering's address, Gibbs had published in the American Journal of Science an Abstract of his memoir (Gibbs, I, Paper IV) from which certain important descriptive material might have been culled more readily than from the original. That the Rumford Committee realized that a great contribution had been made by Gibbs and that they promptly recognized it by their recommendation of the award of the medal is clear, but in how far they appreciated the nature and signifi- cance of the contribution is not indicated.* Particularly interesting in the reply by Gibbs is his reference to the fact that it is only for gases that he has been able to write the equation expressing the thermodynamic functions for a body of variable composition. Perhaps his great attention in his course to van der Waals' equation was because, although its accuracy for liquid and vapor phases is not so great as that of the gas equation for gases, it offered some fair approximation to the representation of a decidedly less restricted state of matter and led to equations expressing the thermodynamic functions for more general bodies of variable composition. It is custom- ary for the recipient of the medal to make a considerable address expounding as well as he can to a general academic audience the significance of some of his contributions. What would Gibbs have said about the memoir on Heterogeneous Equilibrium had he been able to be present? Would he have alluded to some of the important possible applications of his work on osmotic equi- librium or to the significance of his phase rule (obviously a matter easy to make graphic to the kind of audience he would * In the first footnote of the Abstract (Gibbs, I, p. 358) Gibbs points out that Massieu "appears to have been the first to solve the prob- lem of representing all properties of a body of invariable composition which are concerned in reversible processes by means of a single func- tion"— a fact that was probably unknown to him at the time of printing Paper II. GIBBS' PAPERS I AND II 57 have had) or would he have gone into the matter of the electroly- tic cell, or the theory of dilute solutions, or the mass law? Per- haps he would have followed the lead of the address of the Chairman and confined himself chiefly to contributions of others. It is not without interest that in the period from 1872 to 1891 he is not recorded as offering any course on thermodynamics which could be presumed to include any of the matters in his thermo- dynamic papers, although from 1886 on he announced a course on the a priori deduction of thermodynamic principles from the theory of probabilities, which in view of his paper of 1884 (Gibbs, II, Pt. II, p. 16) may safely be assumed to have dealt with statistical mechanics. Was he concentrating his attention, as Clausius and Maxwell had done and as Boltzmann and Kelvin were doing, on the attempt to deduce thermodynamic behavior from dynamical properties of matter and possibly to find some equation expressing the thermodynamic functions of a body of variable composition other than perfect gases? It is not often that we find a great scientist neglecting in his lectures his own most important contributions at a time when they are of as great interest to others as Gibbs' contributions were to the ris- ing physical chemists of the decade from the early eighties to the early nineties of the past century. Certainly the subject matter of his Papers I and II to which he gave half his time during the year 1899-1900 in the course above summarized was no more difficult, no less suitable for instruction than the courses he did offer on mathematical physics to students who could not have been expected to have much if any physics beyond the first general course, or much if any mathematics beyond the differ- ential and integral calculus.* It has been seen that Gibbs, as he taught thermodynamics, late in his life, made much use of the pressure-volume diagram, discussed briefly the entropy-temperature and pressure-temper- ature diagrams, but ignored the volume-entropy diagram (except as its properties may be considered to be implied in those of the thermodynamic surface). He made no use of the concept of * The list of courses offered by Gibbs from 1872 to the time of his death is given in my "Reminiscences of Gibbs by a Student and Col- league" in the Scientific Monthly, 32, 210-227, (1931). 58 WILSON ART. C efficiency, so dear to the engineer, nor of that of availabihty of energy, upon which some authors base their discussion of en- tropy; as the equivalents of these ideas must be imphed in any development of the subject, it is only the terminology and view- point, not the essentials, which were omitted. He dealt at length with the properties of the thermodynamic surface, but did not cover all the detail which was included in his second paper; there was no particular reason why all of it should be covered. As for what we find in the current literature with respect to the subject matter of these two initial papers one may state that the temperature-entropy diagram is now treated at length in engineering treatises on the steam engine* in which many detailed illustrations, both graphical and numerical, may be found. Physicists and chemists do not seem to use the temper- ature-entropy diagram to any great extent. The thermo- dynamic surface was perhaps given more attention by Maxwell in his little book on Heat (4th edition) to which reference has been made than is now customary with writers of texts on the physics or chemistry of heat.f This neglect is certainly not due to any failure to appreciate the contributions of Gibbs any more * See for example the article on the Steam Engine in the Encyclopedia Britannica or the treatise An Introduction to Thermodynamics for En- gineering Students hy John Mills (Ginn and Co.) or Thermodynamics of the Steam Engine and Other Heat Engines by C. H. Peabody (John Wiley and Sons) . It is far from clear that the use of the temperature-entropy diagram in such works derives directly from the presentation in Gibbs' Paper II. t For example, in the excellent Einfuhrung in die theoretische Physik, Berlin, 1921, Bd. II, Th. 1, by C. Schaefer, the theory of heat is presented in 562 pages. Yet the temperature-entropy diagram seems not to appear, nor the thermodynamic surface to be mentioned. There are fourteen references to Gibbs in the index, mentioning the following topics: The Gibbs paradox of increase of entropy on mixing gases, the total energy e, the phase rule, definition of components, the electro- motive force of a cell, and statistical mechanics. None of these refer- ences is to Paper I or II. In the Thermodynamics of G. N. Lewis and M. Randall, McGraw-Hill, 1923, there is equal citation of Gibbs for much the same topics but again no mention of the i77-diagram or thermo- dynamic surface. GIBBS' PAPERS I AND II 59 than the failure to include in some modern treatise on mechanics many of the geometrical proofs of the Principia is an indication of the author's lack of appreciation of Newton. Science goes on its way, picking and choosing and modifying. The trend of the last fifty years is not toward Papers I and II. Interesting as they are historically, and important because of the preparation they afforded Willard Gibbs for writing his great memoir III, there is no present indication that they are in themselves signifi- cant for present or future science ; for better or for worse we have adopted other ways of preparing for the exposition of the theory and for the use of the results of that memoir which in so many of its parts is indispensable today and in still others as yet inadequately explored may become indispensable in the future. D THE GENERAL THERMODYNAMICAL SYSTEM OF GIBBS [Gibbs, I, pp. 55-lU; U9-m] J. A. V. BUTLER I. Introduction 1. General Thermodynamic Considerations. At the head of his memoir, "On the EquiHbrium of Heterogeneous Sub- stances," Gibbs quotes the first and second laws of thermo- dynamics, as stated by Clausius: * "Die Energie der Welt ist constant. Die Entropie der Welt strebt einem Maximum zu." From these two principles he proceeds to deduce, with rigor and in great detail, the conditions of equilibrium in heterogene- ous systems containing any number of substances. As an introduction to his method, we shall first outline the earlier development of the laws of thermodynamics and discuss their bearing on the question of equilibrium in material systems. The first law of thermodynamics, or the Principle of the Conservation of Energy, was first stated in a general form by Helmholtz in his memoir "On the Conservation of Force" (1847). Starting with a denial of the possibility of perpetual motion, and making use of the experimental results of Davy, Joule and Mayer on the production of heat by the expenditure of mechanical work and in the passage of electric currents through conductors, Helmholtz arrived at the generalisation that the sum of the energies of the universe is constant and when energy of one kind disappears, an equivalent amount of other kinds of energy takes its place. Lord Kelvin, in 1851, introduced the concept of the intrinsic energy of a body as the sum of the total quantities of heat and 61 62 BUTLER ART. D work which can be obtained from it. Since it is not possible to remove the whole of the heat from a body, or to change it into a state in which we may be sure that no further work may be obtained from it, for practical purposes we may define a stand- ard state in which the energy is taken as zero. Then the energy of a body in any given state is taken as the sum of the quantities of heat and work which must be supplied to bring the body from the standard state into the given state. The energy of a body or system of bodies in a given state is a definite quantity and is independent of the way in which it is brought into that state. For if it were possible for a system of bodies to have different amounts of energy in the same state, it would be possible to obtain energy without the system or any other bodies undergoing change, which is contrary to the Principle of Conservation of Energy. Consider two states of a system in which its energy is e' and e". The change of the energy of the system, i.e., the energy which must be supplied from outside, when it passes from the first to the second state, is Ae = e" — e'. Since e" and t' depend only on the initial and final states of the system, Ae is independent of the way in which the change of state occurs. In general, the energy of a system may change (1) by receiving or giving heat to other bodies, and (2) by performing work against ex- ternal forces. If, in a change of state, the system absorbs a quantity of heat Q from outside bodies and performs work W against external forces,* its energy change is Ae = Q - PF. (1) Now, although the energy change of a system in passing from a given initial state to a given final state is constant and inde- pendent of the way in which the change occurs, the same is not true of Q or W. But of the possible ways of conducting the change, there will usually be one for which PF is a maximum and, therefore, Q also a maximum. As a simple illustration, consider the fall of a body to the * Heat evolved by the system and work done on the system by ex- ternal forces are counted as negative. THERMODYNAMIC AL SYSTEM OF GIBBS 63 earth under the influence of gravity. "V^Hien the body falls unimpeded no work is obtained and the whole of its energy is converted into heat when it collides with the earth. If we arrange a pulley so that, in its descent, the falling body raises another mass we shall obtain work corresponding to the weight of the mass raised. There is a limit to the amount of work which can be obtained in this way, for the first body will only continue to fall as long as its weight is greater than that of the body which is raised. The maximum work is obtained when the weight raised is only infinitesimally less than that of the faUing body. In other words, we obtain the maximum work when the force tending to cause the change (in this case, the gravitational force on the falling body) is opposed by a force which is only smaller by an infinitesimal amount. Similar considerations apply to changes of other kinds. For example, in the expansion of a gas into an evacuated space, there is no opposing force and no work is obtained; but if the expansion of the gas is opposed by a mechanical force acting on a piston, work is obtained which has a maximum value when the force on the piston is only infinitesimally less than that required to balance the pressure of the gas. When the force on the piston exactly balances the gas pressure, no change occurs; but when the former is reduced by an infinitesimal amount the gas will expand and will continue to do so as long as the applied force is slightly less than that required to balance the gas pressure. Under these conditions we obtain the maximum work from the gas expansion. A change carried out in such a way is called a reversible change, since an infinitesimal increase in the forces opposing the change will be sufficient to make them greater than the forces of the system and will cause the change to proceed in the reverse direction. If we take the system of bodies through a complete cycle of operations, so that its final state is identical with its original state, the total energy change is zero, so that by (1), 2Q - ZTF = 0 ; i.e., the algebraic sum of all the quantities of heat absorbed by the system is equal to the algebraic sum of the amounts of work done against external forces. 64 BUTLER art. d In 1824 S.Carnot made use of such a process to determine the amount of work obtainable by an ideal heat engine, drawing heat from a heat reservoir at a temperature t' and giving it out at a lower temperature t". In this process, the body or "work- ing substance" is put through a cyclic series of operations, consisting of two isothermal and two adiabatic stages : (1) The working substance is put in contact with the heat reservoir at the temperature t' and is allowed to expand, thereby performing work against the opposing forces and, since its temperature remains constant, absorbing a quantity of heat Q' from the heat reservoir. (2) The working substance is thermally insulated so that it cannot receive or give up heat to its surroundings, and allowed to expand further, whereby work is obtained and the tempera- ture falls to t". (3) The working substance is put in contact with a heat reservoir at t", and is compressed until it reaches a state from which it can be brought into its original state without communi- cation of heat. In this stage work is expended on the substance and a quantity of heat —Q" passes from it to the heat reservoir. (4) The working substance is thermally insulated, and brought into its original state by the expenditure of work. In this process a quantity of heat Q' has been taken from the heat reservoir at t' and a quantity of heat — Q" given to the heat reservoir at t". Since the working substance has been returned into its original state the total work obtained is equal to the sum of the quantities of heat absorbed, i.e. W = Q' + Q". The ratio of the work obtained to the heat absorbed at the Q' + Q" higher temperature, i.e. ^ is termed the efficiency of the process. Carnot postulated, (1) that a cyclic process, in which every stage is carried out reversibly, must be more efficient than any irreversible cycle working between the same temperature limits can be, and (2) that all reversible cycles working between the same temperature limits must be equally efficient, whatever THERMODYNAMICAL SYSTEM OF GIBBS 65 may be the nature of the working substance or of the change it undergoes. The proof of these propositions given by Carnot was unsatisfactory, for he adhered to the caloric theory of heat and did not admit that, when work is obtained, an equivalent amount of heat must disappear. Clausius, in 1850, showed that their proof, in fact, involves another principle which he stated as follows: "It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature." Suppose that it were possible to have two such reversible cyclic processes, working between the same temperature limits, one of which was more efficient than the other. Then in the operation of the first process a quantity of heat Qi may be absorbed at the higher temperature and a quantity of work W obtained. This work may be used to operate the second process in the reverse direction so that it absorbs heat at the lower temperature and gives it out at the higher temperature. Let the amount of heat given out at the higher temperature for the expenditure of work W, in this cycle be Q2. Then by hypothesis, W/Qi > W/Q2, or, Q2 > Qi. Therefore the second cycle returns more heat to the heat res- ervoir at the higher temperature than is absorbed in the first cycle, and it would be possible by the use of the two cyclic processes, without the action of any outside agency, to cause heat to pass from the lower to the higher temperature, which is contrary to the principle stated above. This principle is one of several alternative ways of stating the second law of thermodynamics. We may observe that the passage of heat from a hotter to a colder body is a spontaneous process by which a system, which is not in a state of equilibrium, proceeds towards equihbrium. Applied generally to all kinds of changes, the principle may be stated in the following way: Mechanical work can always be obtained when a system changes from a state, which is not a state of equilibrium, into a state of 66 BUTLER ART. D equilibrium. Conversely, it is impossible to obtain mechanical work, over and above the work expended from other sources, by the change of a system, which is in equilibrium, into another state. We have seen that the maximum work is obtained from a spontaneous change when it is carried out by a reversible process. But a reversible process proceeds infinitely slowly, since at every stage the forces of the system are nearly balanced by opposing forces. When changes occur in Nature at a finite rate, the forces of the system must be appreciably greater than the opposing forces. Such changes are essentially irreversible and the maximum work of which they are capable, which Kelvin called the available energy, is not obtained. In an irreversible process only part of the available energy is obtained as work, the remainder is dissipated. Kelvin (1852) therefore stated the second law of thermodynamics as the Principle of the Dissipation of Energy : "1. There is at present in the material world a universal tendency to the dissipation of mechanical energy. "2. Any restoration of mechanical energy, without more than an equivalent of dissipation, is impossible in inanimate material processes, and is probably never effected by means of organised matter, either endowed with vegetable life or subjected to the will of an animated creature." To return to Carnot's cycle, Kelvin had pointed out in 1848 that Carnot's theorem may be employed to define an absolute scale of temperature. Since the ratio of the work obtained in a reversible Carnot cycle to the heat absorbed at the higher tem- perature depends solely on the temperatures of the two bodies between which the transfer of heat is effected, we may write Qt = ii', i")i where ^{t' , t") is a function of t' and t" alone. Kelvin defined absolute temperature so that t' — t" {t', t") = —^- THERMODYNAMIC AL SYSTEM OF GIBBS Then, W/Q' Q' + Q" t' - t" Q' ~ t' ' so that, Q" t" Q' t' and therefore, Q' Q" t' + r - «' 67 i.e. the sum of the quantities of heat absorbed by the working substance in a reversible Garnot cycle, each divided by the absolute temperature at which it takes place, is zero. In 1854, Kelvin and Clausius independently showed that this result may be extended to any reversible cyclic process whatever, since any reversible cyclic process whatever may be resolved into a number of simple Carnot cycles. Thus, we may write: where dQ is the element of heat absorbed at the temperature t in any reversible cycle, and the integration is extended round the cycle. Let us now designate by A and B two reversible paths by which a body or system of bodies may be brought from a state (/) to a state (//) . We may take the system through a reversible cycle by changing it from state (7) to state (77) by path A and returning it to its original state (7) by path B. Therefore, = 0 B or, by changing the direction of the second term, /•(") dQ Jin t 1 =^T^• J A Jm t Jb 68 BUTLER ART. D The integral, / dQ/t has therefore the same value for all re- versible paths by which the system may be changed from state (7) to state (//). Its value for a reversible path is thus a definite quantity, depending only on the initial and final states of the system, and it may be regarded as the difference between the values of a function of the state of the system in the two states considered. This function was termed the entropy of the system by Clausius in 1855. We may therefore write: •(") dQ = V' — 1 t (2) where 77^ and rj'^ are the values of the entropy in states (/) and For an infinitesimal change of state, (1) may be written in the form: de = dQ - dW. Now if the change of state is reversible, according to (2), dQ = tdrj ; also if the work is done by an increase of volume dv against a pressure p, dW = pdv, so that de = tdr] — pdv. (3) We may observe that all infinitesimal changes of state of a system, which is in equilibrium, fulfil the condition of reversi- bility, for equilibrium is a state in which the forces of the system are balanced by the opposing forces, and in an infinites- imal change the system can only be removed to an infinites- imal extent from a state of equilibrium. Equation (3) there- fore applies generally to infinitesimal changes of a system which is in a state of equilibrium. We will now consider the changes of a system of bodies in relation to the changes which necessarily occur in surrounding bodies. When the sytem undergoes a reversible change from a state (7) to a state (77), the entropy change, as we have seen, is: r^u -n^ ^ \ dQ/t, THERMODYNAMICAL SYSTEM OF GIBBS 69 where dQ is the element of heat absorbed at temperature t. This heat must come from surrounding bodies, and the process can only be perfectly reversible when each element of heat is absorbed from a body which has the same temperature as the system itself. Therefore — / dQ/t represents the entropy Jin change of the surrounding bodies, so that when a reversible change takes place the sum of the changes of entropy of the system and its surroundings is zero. On the other hand, if the change of the system is irreversible, its entropy change is still 77" — rj^, since this quantity depends solely on the initial and final states and not on the way in which the change occurs, but it is no longer equal to / dQ/t. JU) Since less work is obtained from the system in an irreversible change than in a reversible change, the heat absorbed is also less, and therefore: dQ/t {system) < 7?" " 1?'^ in or Jc nil) 77" — TJ^ — / dQ/t (system) > 0. J {n The decrease in entropy of the surroundings cannot be greater than/ dQ/t (,y,tem), since an element of heat c?Q can only be Jin absorbed from a body having a temperature equal to or greater than the momentary temperature t of the system. The total entropy change of the system and its surroundings is therefore positive, i.e. when an irreversible change takes place, the entropy of the universe is increased. We have seen that irreversible changes may take place spontaneously in the universe or in any isolated system which is not in a state of equilibrium, so that we arrive at Clausius' statement of the second law of thermodynamics; "The entropy of the universe tends to a maximum." It is evident that the second law of thermodynamics affords a 70 BUTLER ART. D criterion of equilibrium, which may be stated in several different ways. The statement of Clausius, that the entropy of an isolated system tends to a maximum, implies that equilibrium is reached when the entropy has the maximum value which is consistent with its energy, and when there is no possible change, the energy remaining constant, which can cause a further increase of entropy. Also, the entropy of a system remains constant if the latter does not undergo any irreversible changes and if it does not receive any heat from its surroundings. Any change of its energy under these conditions must be the result of work done on or by the system against external forces. We have seen that if a system is not in equilibrium, it may undergo changes from which work can be obtained and which therefore result in a decrease of energy. A system is therefore in equilibrium, if there is no possible change, which does not involve a change of entropy, whereby its energy can be decreased. In making use of these criteria of equilibrium we need only consider infinitesimal changes, for every finite change must begin by being an infinitesimal one and if no infinitesimal change is possible it is evident that no finite change can occur. If (Srj),, (5e), represent the change of entropy and energy in any infinitesimal change of the system in which the energy and entropy respectively remain constant, the two criteria of equilib- rium stated above may be expressed by the statement that {b-n), ^ Oand (5e), ^ 0, for all possible changes. Gibbs first discusses in detail the equivalence and validity of these criteria, and the conditions to be observed in using them. An analysis of his discussion is given in the following chapter, but the reader who does not wish, at this stage, to consider these elaborate arguments need only read Section 4 on the Interpretation of the Conditions and may then proceed to the discussion of their application which begins with Chapter III. 11. The Criteria of Equilibrium and Stability 2. The Criteria. Gibbs begins his discussion of the equifib- rium of heterogeneous substances by stating in the following THERMODYNAMICAL SYSTEM OF GIBBS 71 propositions the criterion of equilibrium for a material sys- tem which is isolated from all external influences: I. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system which do not alter its energy, the variation of its entropy shall either vanish or be negative. This condition of equilibrium may be written (5v). ^ 0, (4) [1] where {8r})( denotes a variation of entropy, the energy remaining constant. II. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system which do not alter its entropy, the variation of its energy shall either vanish or be positive. This condition may be written (5e), ^ 0, (5) [2] where (8e) „ denotes a variation of energy, the entropy remaining constant. He proceeds to prove, that these two propositions are equiva- lent to each other, that they are sufficient for equilibrium, and that they are necessary for equilibrium. We shall quote largely from Gibbs' own exposition, interpolating explanatory remarks where they seem to be helpful. 3. Equivalence of the Two Criteria.* "It is always possible to increase both the energy and the entropy of the system, or to decrease both together, viz., by imparting heat to any part of the system or by taking it away. For, if condition I is not satisfied, there must be some variation in the state of the system for which 5t7 > 0 and 8e = 0." Therefore, by taking heat from the system in its varied state we may decrease the entropy to its original value and at the same time diminish the energy, so that we reach a state for which 3?7 = 0 and 8e < 0. Gibba, I, p. 56, lines 20-37. 72 BUTLER ART. D Thus, if there are possible variations which do not satisfy I, there must also be possible variations which do not satisfy II. Thus if condition I is not satisfied, condition II is not satisfied. Conversely, it is shown that if condition II is not satisfied, condition I is not satisfied, so that the two conditions are equivalent to each other. 4. I nteryr elation of the Conditions* Before proceeding to the proof of the sufficiency and necessity of the criteria of equilib- rium, Gibbs discusses the interpretation of the terms in which the criteria are expressed. In the first place, "equations which express the condition of equilibrium, as also its statement in words, are to be inter- preted in accordance with the general usage in respect to differ- ential equations, that is, infinitesimals of higher orders than the first relatively to those which express the amount of change of the system are to be neglected." That is, if be is change in the energy produced by a change bS in the state of the system, and if dt/dS is the limiting value of bt/bS when bS becomes infinitely small, the value of 5e is taken as (de/dS) • bS, infinitesimals of higher orders, such as dh/dS'^, being neglected. Biit different kinds of equilibrium may be distinguished by noting the actual values of the variations. The sign A is used to indicate the value of a variation, when infinitesimals of the higher orders are not neglected. Thus, Ae is the actual energy change pro- duced by a small, but finite variation in the state of the system. The conditions of the different kinds of equilibrium may then be expressed as follows; for stable equilibrium (A7?)e < 0, i.e., (Ae), > 0, (6) [3] (i.e. the entropy is a maximum at constant energy and the energy a minimum at constant entropy for all possible varia- tions); for neutral equilibrium there must be some variations in the state of the system for which (At,), = 0, i.e., (Ae), = 0; (7) [4] = Gibbs, I, p. 56, line 38; p. 58, line 40. THERMODYNAMICAL SYSTEM OF GIBBS 73 (i.e. which do not change the entropy at constant energy, or the energy at constant entropy), while in general (At?), ^0, i.e. (Ae), ^0; (8) [5] and for unstable equilibrium there must be some variations for which (At?), > 0, (9) [6] i.e. there must be some for which (Ae), < 0," (10) [7] (i.e. in respect to some variations the entropy has the properties of a minimum, and the energy of a maximum), while the general criteria of equilibrium: (577), ^ 0, i.e. (8e), ^0; (11) [8] are still satisfied. Secondly, in these criteria of equilibrium only possible varia- tions are taken into account. Changes of state involving the transport of matter through a finite distance are excluded from consideration, so that an increase in the quantity of matter in one body at the expense of that in another, is regarded as possible only when the two bodies are in contact. If the system consists of parts between which there is supposed to be no thermal communication, the entropy of each part is regarded as constant, since no diminution of entropy of any of these parts is possible without the passage of heat. In this case the condition of equilibrium becomes (56)v, ," , etc. ^0, (12) [9] where 77', r]", etc. denote the entropies of the various parts between which there is no communication of heat. Otherwise, "only those variations are to be rejected as impossible, which involve changes which are prevented by passive forces or analogous resistances to change." It is neces- sary to consider what is meant by this limitation. 74 BUTLER ART. D Systems are frequently met with which are not in equilib- rium, yet which appear to remain unchanged for an unlimited time. Thus, a mixture of hydrogen and oxygen appears to remain unchanged, although it Ls not in a true state of equilib- rium, for a small cause such as an electric spark may cause a change out of all proportion to its magnitude. In such a case the change of the system into a state of equilibrium is supposed to be prevented by "passive forces or resistance to change," the nature of which is not well understood. It is evident that only those forces or resistances which are capable of preventing change need be considered. Those like viscosity, which only retard change, are not sufficient to make impossible a variation which they influence. The existence of such passive resistances to change can easily be recognised. Thus, it is possible that a system composed of water, oxygen and hydrogen which is not in equilibrium with regard to changes involving the formation of water, will remain unchanged for an indefinite period. This equilibrium can be distinguished from that caused by "the balance of the active tendencies of the system," i.e., when the tendency of hydrogen and oxygen to combine is balanced by the tendency of water to dissociate, for whereas in the former case we may vary the quantities of any of the substances, or the temperature or pres- sure without producing any change in the quantity of water present in the system ; in the latter case an infinitesimal change in the state of the system will produce a change in the amount combined. Thus if we regard variations involving the combination of hydrogen and oxygen as prevented by the passive forces or resistances, and therefore impossible, we may still apply the conditions of equilibrium to discover the equilibrium state of a system containing given amounts of hydrogen, oxygen and water under these conditions. 5. Sufficiency of the Criteria of Equilibrium* Three cases are considered, corresponding to the three kinds of equilibrium. (a) "If the system is in a state in which its entropy is greater * Gibbs, I, p. 58, line 41-p. 61, line 11. THERMODYNAMICAL SYSTEM OF GIBBS 75 than in any other state of the same energy, it is evidently in equinbrium, as any change of state must involve either a de- crease of entropy or an increase of energy, which are alike impossible for an isolated system. We may add that this is a case of stable equilibrium, as no infinitely small cause (whether relating to a variation of the initial state or to the action of external bodies) can produce a finite change of state, as this would involve a finite decrease of entropy or increase of energy." (b) "The system has the greatest entropy consistent with its energy, and therefore the least energy consistent with its entropy but there are other states of the same energy and entropy as its actual state." Gibbs first shows by special arguments that in this case the criteria are sufficient for equilibrium in two respects. In the first place, "it is impossible that any motion of masses should take place; for if any of the energy of the system should come to consist of vis viva (of sensible motions), a state of the system identical in other respects but without the motion would have less energy and not less entropy, which would be contrary to the supposition." It is evident that if this last state is im- possible, a similar state in which the parts of the system are in motion is equally impossible, since the motion of appreciable parts of the system does not change their nature. Secondly, the passage of heat from one part of the system to another, either by conduction or radiation, cannot take place, as heat only passes from bodies of higher to those of lower temperature, and this involves an increase of entropy. The criteria are therefore sufficient for equilibrium, so far as the motion of the masses and the transfer of heat are concerned. In order to justify the belief that the condition is sufficient for equilibrium in every respect, Gibbs makes use of the following considerations. "Let us suppose, in order to test the tenability of such a hypothesis, that a system may have the greatest entropy con- sistent with its energy without being in equihbrium. In such a case, changes in the state of the system must take place, but these will necessarily be such that the energy and entropy remain unchanged and the system will continue to satisfy the 76 BUTLER ART. D same condition, as initially, of having the greatest entropy consistent with its energy." Now the change we suppose to take place cannot be infinitely slow, except at particular mo- ments, so that we may choose a time at which it is proceeding at a finite rate. We will consider the change which occurs in a short interval of time after the chosen time. No change what- ever in the state of the system, which does not alter the value of the energy, and which commences in the same state which the system has at the chosen time, will cause an increase of entropy. "Hence, it will generally be possible by some slight variation in the circumstances of the case" (e.g., by a slight change of pres- sure or temperature or of the quantities of the substances) to make all changes in the state of the system like or nearly like that which is supposed actually to occur, and not involving a change of energy, to involve a necessary decrease of entropy, which would render any such change impossible." "If, then, there is any tendency toward change in the system as first supposed, it is a tendency which can be entirely checked by an infinitesimal variation in the circumstances of the case. As this supposition cannot be allowed, we must believe that a system is always in equilibrium when it has the greatest en- tropy consistent with its energy, or, in other words, when it has the least energy consistent with its entropy." The essential steps of this argument may be recapitulated as follows. A system having the greatest entropy consistent with its energy must be in equilibrium, because (a) if it were not in equilibrium a change must take place, and except at particular moments must take place at a finite rate; (/3), but it is shown that in such a case, the change can be entirely checked by an infinitely small modification of the circumstances of the case; (7), therefore, an infinitely small modification makes a finite difference in the rate of change, which cannot be allowed. We may observe that the statement that the hypothetical change cannot be infinitely slow is an essential part of the argument. For, if the change which is supposed to occur were THERMODYNAMICAL SYSTEM OF GIBBS 77 infinitely slow, there would be no rea8on to disallow it because it can be entirely checked by an infinitely small modification of the case. The argument depends finally on the consideration that an infinitely small modification of the circumstances cannot cause a finite change in the rate of change of the system, for as is explicitly stated in a succeeding paragraph, this is "contrary to that continuity we have reason to expect." "The same considerations will evidently apply to any case in which a system is in such a state that A17 ^ 0 for any possible infinitesimal variation of the state for which Ae = 0, even if the entropy is not the greatest of which the system is capable with the same energy." Thus a system of hydrogen, oxygen and water is in equilibrium when (Atj), ^ 0, for all possible varia- tions, even if the entropy is not the greatest for the same amount of energy. The conditions may be such that the combination of hydrogen and oxygen to water would cause an increase of entropy in the isolated system, but if this change is prevented by passive forces or resistances to change, variations involving it are not possible, and the system is in equilibrium if (At?)^ ^ 0, for all variations which do not involve such changes. (c) When "677 ^ 0 for all possible variations not affecting the energy, but for some of these variations At? > 0, that is, when the entropy has in some respects the characteristic of a minimum." "In this case the considerations adduced in the last paragraph will not apply without modification, as the change of state may be infinitely slow at first, and it is only in the initial state that {dr])t ^ 0 holds true." None of the differential coefficients of all orders of the quantities which determine the state of the system, taken with respect to the time, can have any value other than 0, for the state of the system for which (5r?), ^ 0. For if some of them had finite values, "as it would generally be possible, as before, by some infinitely small modification of the case, to render impossible any change like or nearly like that which might be supposed to occur, this infinitely small modifica- tion of the case would make a finite difference in the value of differential coefficients which had before the finite values, or in some of lower orders, which is contrary to that continuity 78 BUTLER ART. D which we have reason to expect. Such considerations seem to justify us in regarding such a state as we are discussing as one of theoretical equihbrium; although as the equilibrium is evi- dently unstable, it cannot be realized." The argument of the last section is here applied to the higher differential coefficients of the quantities which represent the state of the system with respect to the time. Thus if 0, it is evidently a state of unstable equilibrium. 6. Necessity of the Criteria of Equilihrium* When "the active tendencies of the system are so balanced that changes of every kind, except those excluded in the statement of the condition of equilibrium, can take place reversibly (i.e., both in the positive and the negative direction,) in states of the system differing * Gibhs, I, p. 61, line 11 ; p. 62, line 8. THERMODYNAMICAL SYSTEM OF GIBBS 79 infinitely little from the state in question", the criteria are evi- dently necessary for equilibrium. For if there is any possible change for which (Srj)^ ^ 0 does not hold, since no passive forces or resistances to change are operative, this change will take place. Also, in this case, the inequality in the equations cannot apply, since for every change of the system there is a similar one of opposite sign, so that if for a certain change of state (577) e < 0 we should have (St/), > 0 for a similar change of opposite sign. In this case, we may therefore omit the sign of inequality and write as the condition of equihbrium (577), = 0, i.e. (de), = 0. (13) [10] "But to prove that the condition previously enunciated is in every case necessary, it must be shown that whenever an isolated system remains without change, if there is any infini- tesimal variation in its state, not involving a finite change of position of any (even an infinitesimal part) of its matter, which would diminish its energy . . . without altering its entropy, . . . this variation involves changes in the system which are prevented by its passive forces or analogous resistance to change. Now, as the described variation in the state of the system diminishes its energy without altering its entropy, it must be regarded as theoretically possible to produce that variation by some process, perhaps a very indirect one, so as to gain a certain amount of work (above all expended on the system)." We have seen that according to the second law of thermodynamics, a change which can be made to yield work may take place spon- taneously, and will do so unless prevented by passive forces. "Hence we may conclude that the active forces or tendencies of the system favor the variation in question, and that equilib- rium cannot subsist unless the variation is prevented by passive forces." III. Definition and Properties of Fundamental Equations* 7. The Quantities ^, f, x- At this point, Gibbs proceeds to apply the criterion of equilibrium to deduce the laws which determine equilibrium in heterogeneous systems. For this Gibbs, I, 85-92. 80 BUTLER ART. D purpose he uses the criterion in its second form, "both because it admits more readily the introduction of the condition that there shall be no thermal communication between the different parts of the system, and because it is more convenient, as respects the form of the general equations relating to equilib- rium, to make the entropy one of the independent variables which determine the state of the system, than to make the energy one of these variables."* In order to apply the criterion it is nec- essary to specify completely the possible variations of which the energy of the system is capable, and for this purpose differential coefficients, representing the change of energy of homogeneous parts of the system with the quantities of their component substances, must be introduced. The complete significance of these quantities does not appear until a later stage. It is thought that the discussion of the conditions of equiUbrium in heterogeneous systems will be more easily followed if we first define the auxiliary functions \p, f and x and derive the varia- tions of the energy, and of these quantities, in homogeneous masses. Let e, 7] and v be the energy, entropy and volume respectively of a homogeneous body at a temperature t and pressure p. We have seen that in any given state the energy and entropy of a body are definite, but since it is only possible to measure differences of energy and entropy, "the values of these quantities are so far arbitrary, that we may choose independently for each simple substance, the state in which its energy and entropy are both zero. The values of the energy and entropy of any compound body in any particular state will then be fixed. Its energy will be the sum of the work and heat expended in bringing its components from the states in which their energies and their entropies are zero into combination and to the state in ques- tion; and its entropy is the value of the integral J — for any reversible process by which that change is effected." The quantities \p, f and x, defined by the equations ^ = 6 - iT,, (14) [87] f = ,-trj-^pv, (15) [91] X = e + vv; (16) [89] * Gibbs, I, 62. THERMODYNAMIC AL SYSTEM OF GIBBS 81 have then definite numerical values in any state of the homo- geneous body. The definition xf^ = e - tr] (17) [105] may evidently be extended to any material system whatever which has a uniform temperature throughout. Consider two states of the system at the same temperature, in which ^ has the values \f/' and \p". The decrease in i/' in the change from the first to the second state is ^' - ^" = e' - t" - tW - ri"). (18) [106] Now if the system is brought from the first to the second state by a reversible process in which a quantity of work W is done by the system and a quantity of heat Q absorbed, the decrease of energy is: e' - e" = IF - Q, (19) [107] and since the process is reversible ; Q = tw - V), (20) [108] so that; ^> - ^" = W; (21) [109] i.e. the decrease in i/', in a change of state at constant tem- perature, is equal to the work done by the system when the change of state is carried out by a reversible process. Thus i^ can be regarded as the maximum work function of the system for changes at constant temperature. Equation (21) can be written as: - (A^), = W, (22) so that, for an infinitesimal reversible change of state, we may write : -(5^)t = dW, (23) [llD] In mechanics the potential 0 of a particle in a field of force is a quantity such that the work obtained in a small displacement of the particle is dW = -d4>. 82 BUTLER ART. D If the forces acting on the particle in the directions of the .r, ?/, and z axes are /i, fi, fs the work obtained in a small displace- ment is dW = -d(j) = fidx + f^dij + fzdz, so that /i = „^ ' /2 = —7 ' etc. The forces acting on the particle are thus differentials of — is the force function of the particle. The quantity \p has analogous properties and, according to (23), — \^ is the force function of the system for changes at constant temperature. A system is in equilibrium at constant temperature if there is no possible change of state which could yield work, that is, for which dW is positive, and therefore h\}/ negative. Thus, we may write as the condition of equilibrium for a system which has a uniform temperature throughout: mt ^ 0; (24) [111] that is, the variation of \f/ for every possible change which does not affect the temperature is either positive or zero. Gibbs gives a direct proof that the condition of equilibrium (24) is equivalent to the condition (5) when applied to a system which has a uniform temperature throughout, for which the reader may be referred to the original memoir,* The definition ^ = e - tv + pv (25) [116] may similarly be extended to any material system whatever which has a uniform temperature and pressure throughout. We will consider two states of the system, at the same tem- perature and pressure, in which f has the values f ' and f ", The decrease in f in the change of the system from the first to the second state is, r - r = e' - e" - tin' - V") + Viv' - V"). (26) * Gibbs, I, 90. See also this volume, page 214. THERMODYNAMIC AL SYSTEM OF GIBBS 83 Now, if the system is brought from the first to the second state by a reversible process in which work W is done by the system and heat Q absorbed, we have as before ^' - ," = W - Q, Q = t(v"-v'), so that ^' - ^" = W + p(v' - y") = W - p(v" - v'). (27) Now p(v" — v') is the work done by the system in increasing its vokime from v' to v" at the constant pressure p, and the quantity w - vW - v') = w, i.e., the maximum work of the change at constant temperature and pressure less the work done on account of the change of volume, is often known as the "net work" of the change. Just as the decrease in ^i' in a change at constant temperature is equal to the maximum work obtainable, the decrease in f in a change at constant temperature and pressure is equal to the "net work" obtainable. Thus f is the "net work function" of the system. From considerations similar to those cited in discussing \p, it can be seen that — f is the force function of the system for constant temperature and pressure. Equation (27) may be written in the form -Ar = W, (28) so that, for an infinitesimal reversible change of state, we may write -(80t,p = dW. (29) Now, a system is in equilibrium at constant temperature and pressure if there is no possible change of state for which the net work is positive. We may therefore write as a criterion of equilibrium ; mt,P^O, (30) [117] that is, a system is in equilibrium when the variation of f for every possible change of state, which does not affect the tem- 84 BUTLER ART. 1) perature and pressure, is zero or positive. It follows that it is necessary for the equilibrium of two masses of the same com- position, e.g., water and ice, which are in contact, that the values of f for equal quantities of the two masses must be equal. Thus, if the value of f for unit mass of ice were greater than the value of f for unit mass of water, at the temperature and pres- sure at which they are in equilibrium with one another, the value of f of the system could be decreased by the change, ice -^ water, at constant temperature and pressure. Since according to (30) this is impossible, the values of f for unit masses of ice and water in equilibrium with each other, must be equal. Similarly for the equilibrium of three masses, one of which can be formed out of the other two, it is necessary that the value of f for the first mass should be equal to the sum of the values of f for those quantities of the other masses, out of which the first mass can be formed. For example, 100 grams of calcium carbonate can be formed from 56 grams of lime and 44 grams of carbon dioxide. When the three substances are in equilib- rium with each other, the value of f for 100 grams of calcium carbonate must be equal to the sum of the values of f for 56 grams of lime and 44 grams of carbon dioxide. Also if a solu- tion composed of a parts of water and b parts of a salt is in equilibrium with crystals of the salt and with water vapor, the value of f for the quantity a + 6 of the solution is equal to the sum of the values of ^ for the quantities a of water vapor and h of the salt. The definition X = e + py (31) may likewise be extended to any material system for which the pressure is uniform throughout. If we consider two states of a system at the same pressure, in which x has the values x' and x", we see that x" - x' = 6" - e' + p{v" - v'), (32) [119] or Ax = Ae + pAv = Qp , (33) THERMODYNAMIC AL SYSTEM OF GIBBS 85 i.e., the heat absorbed in a change which occurs at constant pressure, when the only work done is that due to increase in volume, is equal to the increase of x- Similarly, when a system undergoes a change at constant volume, pAv is zero and, if no work is done against external forces other than the pressure, the increase of energy is equal to the heat absorbed: Ac = Q„, (34) so that the energy can be regarded as the heat function at constant volume. Various names have been given to the thermodynamic func- tions 4/, ^, X- Clerk Maxwell called rp the available energy, but a certain amount of confusion has arisen because Helmholtz in 1882* used the term, free energy, for the same quantity. G. N. Lewis,t in his system of thermodynamics, has made use of the functions A, F and H which are identical with Gibbs's ^, f, x and has used the names: A or \^: Available energy. F or ^'. Free energy. H OT X' Heat content. F. Massieut was the first to show that the thermodynamical properties of a fluid of invariable composition may be deduced from a single function, which he called the characteristic func- tion of the fluid. He made use of two such functions; which, in Gibbs' notation, are as follows : (1) (2) — e-\- ty _ _ ]A t ~ ~ t — €-{-tv - PV _ _ f . t ~ ~ t * Sitzungsber. preuss. Akad. Wiss, 1, 22 (1882). t Lewis and Randall, Thermodynamics and the Free Energy of Chem- ical Substances (1923). t Comptes rendus, 69, 858 and 1057, (1869). 86 BUTLER AKT. u Planck has also made use of the second function, which has the same properties in a system at constant temperature and pres- sure as the entropy at constant energy and volume. 8. Differentials of e, \p and f . The variations with temperature and pressure of the quantities i/' and f , for- a homogeneous body of fixed composition, are obtained by differentiating (14) and (15) and comparing with (3). Thus but since we have and Similarly, so that dyp = de — tdr] — -qdt, (35) de = tdrj — pdv, d4/ = —pdv — -qdt, (36) (f).=-. (a=- ^3. d^ = de — tdr] — 77c?/ + pdv + vdp = - ndt + vdp; (38) Now, if the system is heterogeneous, the quantity of matter in some of its parts may increase at the expense of that in other parts and we shall need to express the effect of such variations on the energy and on the quantities yp, f and x- Consider a single homogeneous mass containing the quantities Wi, m2, W3, . . . m„ of substances ^1, S2, Sz,... Sn- It is usually possible to express the composition of a mass in a number of different ways. It is immaterial which way is chosen, provided that the components are such that every possible independent variation in the composition of the mass can be expressed in terms of them. For example, possible variations in the com- position of a solution of sulphuric acid in water may equally THERMODYNAMIC AL SYSTEM OF GIBBS 87 well be expressed by taking sulphuric acid and water, or sulphur trioxide and water, as components, but sulphur, oxygen and hydrogen are not admissible as components as their amounts cannot be independently varied . The change in the value of f of this mass when the amounts of Si, S2,. . .Sn are increased by dmi, drui, . . . drrin, the temperature and pressure remaining constant, is given by dr = ( -, — 1 • dmi + I - — ) • dvii \(l17li/ 1, p, mj, etc. \CtWl2/ «. p, mi, m,, etc. ■ ■-+(r~) -^^"^ (^0) \(tmn/t, p, m„ . . . m„_i and we may write (^) \dmijt. \dmi/t. = Ml, p, ntj, etc. = jU2, etc., p, nil, wij, etc. (41) so that {d^)t,p =nidmi + M2^m2 . . . + findvin. (42) When the temperature and pressure also vary, by combining with (38), we have d^ = —r]dt-\- vdp + iJ.idmi + ju2C?W2 . . . + tindnin, (43) [92] whence, by (38), de = idt] — pdv + mdmi + Hidrrh . . . + UndrUn, (44) [86] and by (35) d}p = —rjdt — pdv + fjiidmi + HidTm . . . + Undrrin. (45) [88] The definition of mi, etc., given above, corresponds to the most familiar condition, viz., that of constant temperature and pres- sure. Since f is the free energy of the homogeneous mass, the quantity (—) \dnhjl, p, m.,, . . . m,. ^ 88 BUTLER ART. D represents the rate of increase of f with the quantity of the component S\, when the temperature, pressure and quantities of the other components remain constant. It is therefore the 'partial free energy of the first component. According to equa- tions (44) and (45), ^i is also given by Ml = (jt) ' (46) [104] and by Ml = f T^) , (47) [104] \afn,\/ 1, V, TOj, . . . m„ i.e. /ii is equal to the rate of change of e with mi, when the en- tropy, volume and quantities of the other components remain constant, and to the rate of change of \p with mi, when the temperature, volume and quantities of the other components remain constant. Now all the terms in (44) are of the same kind, that is mul- tiples of quantities {t, p, ni, etc.) which depend on the state of the system, by the differentials of quantities (t/, v, mi, etc.) which are directly proportional to the amount of matter in the state considered. We may therefore integrate (44) directly, obtaining: e = tr] — pv -\- mmi + n^rrii . . . + Urmn, (48) [93] whence by (14), (15) and (16) : \p = —pv-\- mrrii + H2ni2 . . . + Unnin, (49) [94] f = Mi^i + M2W2 . . . + Hnm„, (50) [96] X = tV + MlWl + /I2W2 . . . + Mn^n- (51) [95] A concrete picture of the process involved in this integration may be obtained as follows. If we take a homogeneous mass having entropy 7? and volume v, and containing quantities mi, nii, . . . m„ of the components >Si, 82,--. Sn, and add quantities of a mass of the same composition and in the same state; t, p, Mi> M2, etc., all remain unchanged and (44) may be apphed to a finite addition: THERMODYNAMIC AL SYSTEM OF GIBBS 89 Ae = tA-q — pAv + niArtii + HiArrh . . . + UnAtUn , where A77, Av, Ami, etc., are all proportional to the values of 7], V, mi, etc. in the original mass. We may thus continue these additions until we have doubled the amount of the original mass. Then, since At; = t], Av = v, Ami = mi, etc., the energy of the added substance is Ae = It] — pv + iumi + nim^ . . . + m»w„ , and this must be equal to the energy t, of the mass originally present. The general justification of this treatment depends on Euler's theorem on homogeneous functions. According to this theorem, a y = {xi, X2, . . . Xn), i.e., if each variable Xi, X2,- . .Xn is multiplied by a quantity k, the value of the function is multiplied by /b". The energy of a homogeneous mass is evidently a homogeneous function of the first degree with respect to 77, v, mi, m^,. . .m„. If we increase each of these quantities k times, i.e., by taking k times as much of the homogeneous substance, the energy is increased in the same proportion. Therefore by Euler's theorem, putting € = (j], V, mi,. . .w„) we have de dt de de i = VT-i-v— +mi- — ... +m„ t — > drj dv dmi dmn or t = r]t — vp -\- mi/xi . . . + mnUn, 90 since BUTLER ART. D (-) \dv/v xdmi/r,, V = t, m^ • • • mn = - P, 7Jf TTli ' ' ' trifi wij - • • m-n = Hi, etc. (53) Euler's theorem further states that if e = 0(t/, v, m\, nh, . . .m„) is a homogeneous function of the first degree 9e 9e a^ "^' m; " ~ ^' be drrii = )U], etc., are functions of zero degree. Therefore, applying Euler's theorem to one of these functions, e.g. to 9e/9mi, we have: 326 a^e dh + V — + mi :r~l + ^ a^e dmi • dr] dmi • dv + mn dm-^ drill • dm^ dh dmi • drrin = 0. (54) or dt dp dfii dfXi dfin , . V Z~~ - V -r^ -{- mi -— -\- m2-~ ... + mn z =0. (55) dmi dm-i dmi dmi dnii Therefore, in general, 7]dt — vdp + midfjLi + m2dp,2 . . . + m„c?jun = 0. (56) [97] Gibbs obtains this equation by differentiating (48) in the most general manner, viz., de = tdr] + rjdt — pdv — vdp + mdmi + midm . . . + Hndmn -\-mndHn, and comparing the result with (44), which is a complete differ- ential. Equation (56) provides a relation between the variations of the ?i + 2 quantities, t, p, m,. . .ju„, which define the state of THERMODYNAMIC AL SYSTEM OF GIBBS 91 a homogeneous mass. If the variations of n + 1 of these quantities are given any arbitrary values, the variation of the remaining quantity can be determined by (56). A single homogeneous mass is therefore capable of only n + 1 inde- pendent variations of state. Additional Relations It will be convenient to give here some additional relations which are easily obtained from the equations of the last section. By (37) or (45) we have, for a body of fixed composition and mass (indicated by the subscript m), or This equation, which has been found a very convenient expres- sion of the relation between \p and e, was first given explicitly by Helmholtz* and is known as the Gibbs-Helmholtz equation. An equivalent equation between f and x is obtained from (39) or (43), viz: (S).,. = Further, since M 37 = - 'J^ = r - X. (59) d{yP/t) # ^'~dr ^^jt-"^' we may write (58) as /d{m\ ^ _ 1 \ (II / V, m t (60) and similarly (59) becomes mm ^ _x y ai y p, m V * Sitzungsber preuss. Akad. Wiss., 1, 22 (1882); cf. Gibbs, I, 412 (61) 92 BUTLER ART. D IV. The Conditions of Equilibrium between Initially Existent Parts of a Heterogenous System* 9. General Remarks. Gibbs first considers the equilibrium of heterogeneous systems when uninfluenced by gravity, by external electric forces, by distortion of the solid bodies, or by the effects of surface tension. A mass of matter of various kinds, the conditions of equilibrium of which are to be deter- mined, is supposed to be "enclosed in a rigid and fixed envelop, which is impermeable to and unalterable by any of the sub- stances enclosed, and perfectly non-conducting to heat." It is supposed that there are no non-isotropic strains in the solid bodies, and that the variations of energy and entropy which depend on the surfaces separating the heterogeneous mass are so small in comparison with those which depend on the masses themselves that they may be neglected. The effects excluded here are examined in detail in later parts of the Memoir. Gibbs points out that "the supposition of a rigid and non- conducting envelop enclosing the mass under discussion involves no real loss of generality, for if any mass of matter is in equilib- rium, it would also be so, if the whole or any part of it were enclosed in an envelop as supposed; therefore the conditions of equilibrium for a mass thus enclosed are the general conditions which must always be satisfied in case of equilibrium." The use of such an envelop ensures that the volume of the system remains constant and that no heat is received from or given up to any outside bodies. Since a system which is in equilibrium cannot undergo any irreversible change, its entropy must, under these conditions, remain constant. In the first place, the conditions relating to the equilibrium between initially existing homogeneous parts of the mass are examined; the conditions for the formation of masses unlike any previously existing are discussed in a later section. 10. Conditions of Equilibrium When the Component Substances Are Independent of Each Other. ■\ Let the energies of the separate homogeneous parts of the system be e', e" etc. ♦Gibbs, I, 62-70. t Gibbs, I, 62^67. THERMODYNAMIC AL SYSTEM OF GIBBS 93 According to (44), the variation of the energy of the first homogeneous part tlirough a change of entropy, or of volume, or by a change of its mass, is de' = t'dt)' - v'dv' + ju/c^mi' + tii'dm^' . . . + y.n'dmn'- (62) We will first suppose that the components *Si, &, . . . Sn are chosen so that dnii, dm^', . . . drrir! are independent and express every possible variation in the composition of the homogeneous mass considered. With regard to this choice of components, we may note that if drrii, dnii etc. are all inde- pendent, the number of components is evidently the minimum by which every possible variation can be expressed. Further, some of the terms in (62) may refer to substances which are not present in the mass considered, but are present in other parts of the system. If a component Sa is present in the homogeneous mass considered, so that its quantity ma may be either increased or decreased, it is termed an actual component of the given mass. But if a component Sb is present in other parts of the system, but not in the homogeneous mass considered, so that it is a possi- bility that its quantity mb can be increased but not decreased, it is termed a possible component of the given mass. We will first consider the case in which each of the component substances Si, 82,- --Sn is an actual component of each part of the system. The condition of equilibrium of the matter enclosed in the envelop, since its entropy cannot vary, is that its energy cannot decrease in any possible variation. Thus if 5e', 5e", etc. represent the change of energy of different parts of the system in a variation of the state of the system, the con- dition of equilibrium is de' + 66" + 8t"' + etc. ^ 0 (63) [14] for all possible variations. Writing out the values of these variations in full, we have: t' 8r}' — p' y + ill 8mi + H2 8m2 . . . + Mn'5m„' -\-t"8r," - p"8v" + y.i"8mx" -\- ii2"8m2" . . . + iin"8mn" + etc. ^ 0 (64) [15] 94 BUTLER ART. D for all possible variations which do not conflict with the condi- tions imposed or necessitated by the nature of the case. These conditions may be expressed in the following equations, which are termed the equatio7is of co7idition. (1) The entropy of the whole system is constant; or bri' + h-n" + hri'" + etc. = 0, (65) [16] (2) The volume of the whole system is constant; or bv' + bv" + bv'" + etc. = 0, (66) [17] (3) The total mass of each component is constant; or bmi' + bnii" + 5mi'" + etc. = 0, ^ bm2' + bnii" + 5m2'" + etc. = 0, bnin' + bnin" + bnin'" + etc. = 0. ^ (67) [18] Now since all the quantities like brj', bv', bmi, . . . brtin may be either positive or negative, the left-hand side of (64) is only incap- able of having negative values when (65), (66) and (67) are sat- isfied, if t' = t" = t'" = etc. p' = p" = p'" = etc. Ml = Ml = Ml — etc. M2' = M2" = M2'" = etc. Hn = fin — IJ'Ti — etc. (68) [19] (69) [20] (70) [21] For example, consider the terms ixi'bmi + ixi'bmi" + iix"bmi" -f etc. Since 6mi' + bmi" + bmi'" + etc. = 0, it follows that Mi'6wi' + ii,"bnh" + ixx"'bmi"' + etc. = 0 (71) THERMODYNAMIC AL SYSTEM OF GIBBS 95 if iJLi = Hi" = Hi", etc. But if ni" were greater than hi, hi'", etc., there would be variations of the state of the system (if Hi" is positive, those for which 8mi" is positive) which satisfy (71), but for which Hi8mi' + Hi'^mi" + Hi"5mi"' + etc. > 0. But since the quantities Snii, 8mi", etc., may be both positive and negative, there are similar variations in which all these quantities have the opposite sign and for which Hi8mi' + Hi'^rrii" + Hi"^mi"' + etc. < 0. The same considerations apply to the other sets of terms of the types thy], p8v, H^8m2, etc., so that we may conclude that if (64) holds for all possible variations which satisfy (65), (66) and (67), the equalities (68), (69) and (70) must be satisfied. Equations (68) and (69) express the conditions of thermal and mechanical equilibrium, viz., that the temperature and pressure must be constant throughout the system. Equations (70), which state that the value of h for every component must be constant throughout the system, are "the conditions character- istic of chemical equilibrium." Gibbs calls the quantities Hi, H2, etc., the potentials of the substances Si, Si, etc., and ex- presses the conditions (70) in the following statement: "The potential for each component substance must he constant throughout the whole mass." We will now consider the case in which one or more of the substances Si, S2,-.. Sn are only possible components of some parts of the system. Let S2 be a possible component of that part of the system distinguished by ("). Then 8mi" cannot have a negative value, so that equation (64) does not require that H2" shall be equal to the value of H2 for those parts of the system of which S2 is an actual component, but only that it shall not be less than that value. For if H2" were greater than Ma'i Hi"', etc., the sum of the terms fii'Snh' + iJ,2"8nh" + iX2"'8m2"' + etc. would be positive if 8m2" were positive, but since 8m2" cannot be negative, this expression can never have a negative value. The condition of equilibrium (64) is therefore satisfied. 96 BUTLER ART. D In this case, Gibbs therefore writes the conditions of equilib- rium (70) in the following way: " Ml = Ml for all parts of which Si is an actual component, and Ml ^ Ml for all parts of which Si is a possible (but not actual) component, M2 = M2 !► (72) [22] for all parts of which S2 is an actual component, and M2 ^ M2 for all parts of which >S'2 is a possible (but not actual) component, etc.. Ml, M2, etc., denoting constants, the value of which is only determined by these equations." When a component is neither an actual nor a possible com- ponent of some part of the system, the terms /idm and 8m, which refer to this component in that part of the system of which it is neither an actual nor a possible component are absent from (64), and from the equations of condition (67). The condi- tions of equilibrium are otherwise unaffected. "Whenever, therefore, each of the different homogeneous parts of the given mass may be regarded as composed of some or of all of the same set of substances, no one of which can be formed out of the others, the condition which (with equality of temperature and pressure) is necessary and sufficient for equilibrium between the different parts of the given mass may be expressed as follows : — The potential for each of the component substances must have a constant value in all parts of the given mass of which that substance is an actual component, and have a value not less than this in all parts of which it is a possible component.'' 11. Conditions of Equilibrium When Some Components Can THERMODYNAMICAL SYSTEM OF GIBBS 97 Be Formed Out of others* If the substances Si, S2,. . -Sn are not all independent of each other, i.e., if some of them can be formed out of others, the number of components is no longer the minimum number in terms of which every possible variation of the state of the system can be expressed. For example, if the system contains a solution of sodium chloride in water in equilibrium with the sohd hydrate, NaCl-H20, it may be convenient to regard the hydrate as a component, as well as sodium chloride and water. Every independent variation of the system can be expressed in terms of the tw^o components sodium chloride and water, but these two components are not independently variable in the sohd hydrate. Their ratio is fixed. Consider a system containing, in addition to other sub- stances, water, sodium chloride and the solid hydrate NaCl-H20, and let the components Si, S2 and S3 be water, sodium chloride and the hydrate respectively. We will suppose that the other components S4,... Sn are independent of each other. The general condition of equilibrium, which may be written more briefly in the form 2^577 - Ipdv + 2mi5toi + 2M25m2 . . . + ^UrMn ^ 0 (73) [23] still holds, but the equations of condition 25mi = 0, S5m2 = 0, S5m3 = 0, (74) [24] do not necessarily hold, since the total amount of water and sodium chloride in the system may decrease and the total amount of the hydrate may increase. It is therefore necessary to replace (74) by equations representing the relation between the quantities of these substances. Thus, if b grams of sodium chloride combine with a grams of water to form (a + 6) grams of the hydrate, the quantity (Sms) of the hydrate contains 7 (dms) of water, and for the constancy of the actual total a + 6 am.ount of water in the system (i.e., the sum of the amount of * Gibbs, I, p. 67, line 24; p. 70, line 9. 98 BUTLER ART. D the component water and the amount of water contained in the component, hydrate), the equation 25wi + —7-7 S5m3 = 0 (75) [25] must hold. Similarly the equation 25w2 + — n 25m3 = 0 (76) [25] a -\- 0 expresses the constancy of the sum of the amount of the com- ponent sodium chloride and the amount of sodium chloride present in the hydrate. The other equations of condition, 2577 = 0, Xdv = 0, 257^4 = 0, etc. (77) [26] will remain unchanged. We may first consider variations of the system which satisfy (74). Such variations evidently satisfy (75) and (76) and constitute some, but not all of the variations of which the system is capable. Equation (73) must hold for such varia- tions, so that all the conditions of equilibrium, (68), (69) and (72) must apply to this case also. Therefore in (73), /xi, /X2, Ms have constant values Mi, M2, Ms in all parts of the system of which Si, S2 and S3 are actual components. In the general case, when these conditions are satisfied (73) reduces to Mi25mi + ikfaSSwa + MsSSms ^ 0*. (78) [27] * The proof of the equivalence of (78) with (73), given by Gibbs, may be stated as follows. When conditions (68), (69) and (72) are satisfied, and so long as 5m is zero for every substance in all parts of the system of which that substance is not an actual component, i.e., for all terms in (73) involving a value of m which may be greater than the corresponding value of M, we may write (73) in the form tE5v — pSSy + MiS5mi + M225m2 + MzHbrnt + Mi'L&nn . . . + M„S5to„ ^ 0, and since S67; = 0, 'Lhv = 0, S5m4 = 0, etc., THERMODYNAMIC AL SYSTEM OF GIBBS 99 We may eliminate ZSnii and 25w2 from this equation, by means of the equations of condition (75) and (76), so that it becomes -aMiXdniz - hMi^Lbrm + (a + b)M3X8mz ^ 0, (79) [28] so that, as XSms may be either positive or negative, -aMi - hMi + (a + 6)^3 = 0, or aAfi + 6M2 = (a + h)Mz. (80) [29] The relation between the values of the potentials, each of which is determined in a part of the system of which the substance concerned is an actual component, is thus: am + &M2 = (a + h)iiz. (81) In a more general case, suppose that the system may be considered as having n components Si, 82,- ■ ■ Sn, of which Sk, Si, etc. can be formed out of the components Sa, Sb, etc., according to the equation: a and this is evidently the relation between the /x's in a gaseous mass containing all four components. In this case we may- observe that if the gram were taken as the unit mass of aU four substances, the relation between the components would be (approximately) 73 @a + 16 ©6 = 18 ©ft + 71 ©,, where Si, S2,. . . Sn, r equations similar to (84) must be satisfied in addition to the general con- ditions (68), (69) and (72), provided that each of the compo- nents Si, 82,- . . Sn is an actual component of some part of the system. But it must be understood that a relation between the com- ponents such as (82) implies not merely the chemical identity of the substances represented, but also that the change of the substances represented by the left hand member into the substances represented by the right hand member can occur in the system and is not prevented by passive resistances to change. For example, in a system containing water and free hydrogen and oxygen, at ordinary temperatures, the combina- tion of hydrogen and oxygen to form water is prevented by "passive resistances to change," so that we cannot write l®H + 8©o = 9 ©^4, as a relation between the components, for under these conditions there can be no change in the amounts of water in the system in any possible variation of its state. Water must therefore be treated as an independent component and there will be no necessary relation between the potential of water and the potentials of hydrogen and oxygen. 12. Effect of a Diaphragm {Equilibrium of Osmotic Forces) * Consider the equilibrium between two homogeneous fluids, separated by a diaphragm which is permeable to some of the components and impermeable to others. Suppose that the two fluids are enclosed in a rigid, heat-insulating envelop as before, but that they are separated by a rigid, immovable diaphragm. We shall distinguish quantities which refer to the two sides of the diaphragm by single and double accents. As before, the total entropy of the system is constant, i.e., dv' + 8v" = 0, (85) [72] and the total quantities in both fluids of those components. * Gibbs, I, 83-85. THERMODYNAMIC AL SYSTEM OF GIBBS 103 Sh, Si, etc., which can pass through the diaphragm, is constant, i.e., dmh' + 87nh" = 0, dm/ + 6m /' = 0, etc., (86) [75] but the quantities of those components, Sa,Sb, etc., which cannot pass through the diaphragm must be constant in each fluid, i.e., 8ma' = 0, 8ma" = 0, dnib' = 0, 8mb" = 0, etc., (87) [74] and the volume of the fluid mass on each side of the diaphragm must be constant, i.e., 8v' = 0, bv" = 0. (88) [73] The general condition of equilibrium (64), which takes the form t'bt]' — p'bv' + Ha'dMa + Hhbrrih . . - + Hh'bmi,' + Hi'dnii . . . +t"8v" - p"8v" + ^a"8ma" + fJLb"8mb" . . . + fjiH"8mH" + tii"8mi" ... ^0, will now give the following particular conditions: (1) t' = t", (89) [76] (2) m;/ = m;.", m/ = Mi", etc., (90) [77] if Sh, Si, etc., are actual components of both fluids; but it is not necessary that V' = V", (91) or tia' = Ma", Mb' = Mb", etc. (92) Thus the values of the potentials of components which are present on both sides of the diaphragm and which can pass through it must be equal, but it is not necessary that the pres- sures, or the values of the potentials of those substances to which the diaphragm is impermeable, shall be the same in the two fluids. 104 BUTLER ART. D Gibbs points out that these conditions do not depend on the supposition that the volume of each fluid mass is kept constant. The same conditions of equiUbrium can easily be obtained, if we suppose the volumes variable. In this case the equilibrium must be preserved by external pressures P', P" acting on the external surfaces of the fluids, equal to the internal hydrostatic pressures of the liquids p', p". Suppose that external pressures P' and P" are appUed to the two fluids, which are separated by an immovable diaphragm, in some such arrangement as Figure 1. When the volume of the fluid (/) increases by 8v' work is done against the external pressure P' and the energy of the source of this pressure is increased by P'8v'. Similarly when the volume of fluid (//) is increased by 8v", the energy of the source of the P' P" > / i >K (I) (IL) v' i r 1 1 v" Fig. 1 pressure P" is increased by P"hv". These energy changes must be added to the energy change of the fluids in order to find the conditions of equilibrium. The general condition of equilibrium for constant entropy thus becomes 5e' + Se" + P'y + P"hv" ^ 0. (93) [79] From this equation we can derive the same internal conditions of equilibrium as before, and in addition, the external conditions : p' = P', p" = P". When we have a pure solvent Si and a solution of a sub- stance S2 in Si separated by a membrane which is permeable to THERMODYNAMICAL SYSTEM OF GIBBS 105 Si only, it is necessary for equilibrium that f = t" and m' = Hi", but not that ii2 = /X2", or that p' = p". The difference of hydrostatic pressure on the two sides of the membrane which is necessary to preserve equilibrium is the osmotic pressure of the solution, and is that which is required to make the value of potential of Si m the solution the same as its value in the solvent. We shall calculate its value in simple cases in a later section. V. Coexistent Phases 13. The Phase Rule* The variation of the energy of a homogeneous body, containing n independently variable com- ponents, has been expressed by the equation : dt = tdr\ — pdv + indrtii + /X2c?m2 ... + HndiUn. (95) In this equation, there are altogether 2n + 5 variables, viz., mi, rrhj . . . w„, /Xi, /i2, ... Hn, and €, t, 77, p, V. These quantities are not all independent, for the n -\- 2 quanti- ties, t, p, jjLi, M2, • • • Mn can be derived from the original equation by differentiation. Thus, the equations \t) = ^' C/l = - V, y^V/v, Tni,...mn \^^/ V, nn,...mn i = Hi, etc. nil,., .mn give us n -(- 2 independent relations between the 2n -\~ 5 vari- ables. The original equation (95) is an additional relation, so that if € is known as a function of 77, v, rrii,. . .nin, there are altogether n -f 3 known relations between the 2n -f 5 variables and the remainder, n -(- 2 in number, are independent. The homogeneous body may thus undergo n + 2 independent Gibbs, I, 96-97. 106 BUTLER ART. D variations, e.g., the quantities m,i,...m„, r?, v may be varied independently of each other. But if they are all varied in the same proportion, the result is a change in the amount of the body, while its state and composition remain unchanged. A variation of the state or composition of the body involves a change in at least one of the ratios of these quantities. There are n + 1 independent ratios of these n -\- 2 quantities (e.g., the ratios mi/v, m^/v,. . .m„/v, rj/v) so that the number of independent variations of state and composition of a homo- geneous body is n + 1. Gibbs calls a variation of the thermodynamic state or com- position of a body, as distinguished from a variation of its amount, a variation of the phase of the body. In a heterogene- ous system, such bodies as differ in composition or state are regarded as different phases of the matter of the system, and all bodies which differ only in quantity or form as different examples of the same phase. Thus we may say that the number of inde- pendent variations of the phase of a homogeneous body which contains n independent components is n + 1. Consider a system of r phases each of which has the same v. independently variable components. The total number of independent variations of the r phases, considered separately, is (n + l)r. When the r phases are coexistent these variations are subject to the conditions (68), (69) and (70), i.e., to (r — 1) (n 4- 2) conditions. The number of independent vari- ations of phase of which the system is capable is therefore % = (n + l)r - (n + 2) {r - 1) = n - r + 2. (96) The integer ^5 has been called the number of degrees of freedom of the system. This relation, which is now known as the phase rule, holds even if each phase has not the same n independently variable components. For if a component is a possible, but not an actual, component of some part of the system, the variation, bm, of its quantity in that part, can only be positive, whereas in the previous case it can be either positive or negative, and instead of the equality /x = Af , we have the condition n ^ M. The number of independent variations of the system is there- THERMODYNAMIC AL SYSTEM OF GIBBS 107 fore unaltered. When a component is neither an actual nor a possible component of some part of the system, the total number of variations of the phases, considered separately, is one less than {n -\- l)r and, since there is no condition as to the potential of this component in the part of the system of which it is not a possible component, the number of conditions is also reduced by one. Finally we may consider the case in which some of the components can be formed out of others. Let n, as before, be the number of independently variable components of the system as a whole, and let n + /i be the total number of substances which are regarded as components in various parts of the system. If all these latter components were independent, the number of degrees of freedom of the system would be n + A — r + 2. But, since they are not independent, there are h additional equations between their potentials similar to (84), corresponding to h equations representing the relations between the units of these substances. The number of independent variations of the system, therefore, is still n — r -{- 2. Gibbs deduced the phase rule more concisely by the following considerations, "A system of r coexistent phases, each of which has the same n independently variable components is capable of n + 2 — r variations of phase. For the temperature, the pressure, and the potentials for the actual components have the same values in the different phases, and the variations of these quantities are by [97] subject to as many conditions as there are different phases. Therefore, .... the number of inde- pendent variations of phase of the system, will he n -\- 2 — r. "Or, when the r bodies considered have not the same independ- ently variable components, if we still denote by n the number of independently variable components of the r bodies taken as a whole, the number of independent variations of phase of which the system is capable wUl still he n -\- 2 — r. In this case, it will be necessary to consider the potentials for more than n component substances. Let the number of these potentials be n -\- h. We shall have by [97], as before, r relations between the variations of the temperature, of the pressure, and of these n -{• h potentials, and we shall also have . . . . h relations between these potentials, of the same form as the relations 108 BUTLER AUT. D which subsist between the different component substances," (that is, the variations of the n + /i + 2 quantities, viz., n -\- h potentials, and temperature and pressure, are subject to r -^ h relations). We may illustrate the phase rule by reference to systems containing a single component (w = 1). If there is only one phase, |5 = 2, i.e., the temperature and the pressure may be varied independently. If there are two phases, e.g., liquid and vapor, only one independent variation of phase is possible, so that the temperature and the pressure cannot be varied inde- pendently of each other. A variation of the temperature involves a necessary variation of the pressure, if the two phases are to remain in equilibrium. If there are three phases of the substance, ^^ = 0, i.e., it is impossible to vary either the tem- perature or the pressure while the three phases remain. The conditions under which three phases of the same substance can coexist are thus invariant. Gibbs remarks that "it seems not improbable that in the case of sulphur and some other sub- stances there is more than one triad of coexistent phases" (a prediction which has been verified in numerous cases), "but it is entirely improbable that there are four coexistent phases of any simple substance." 14. The Relation between Variations of Temperature and Pressure in a Univariant System* According to (96), a system of r = w + 1 coexistent phases has one degree of freedom. The pressure and the temperature cannot therefore be varied inde- pendently and there must be a relation between a variation of the temperature and the consequent change of pressure. We will first consider a system of one component in two phases, e.g., liquid and vapor. The variations of each phase must be in accordance with (56), so that we may write v' dp' = rj' dt' + m' dfi' ,1 .Q_s v"dp" = v"dt" + m"diJL".j ^ ^ If the two phases are to remain in equilibrium, dp' = dp", dt' = dt", dp' = dtx". * Gibbs, I, 97-98. THERMODYNAMIC AL SYSTEM OF GIBBS 109 Therefore, eliminating djj.' from (97), we have (vW - v"m')dv = Wm" - rj"m')dt, or dp r\'m" — r]"m' dt v'm ' — V m (98) [131] If we consider unit quantity of the substance in each of the two phases, we may put m' = 1 and m" = 1, so that (98) becomes d'p dt Now, where Q is the heat absorbed when a unit of the substance passes from one state to the other, at the same temperature and pressure, and v" — v' is the corresponding change of volume. Thus, we obtain the Clapeyron-Clausius equation :* dv Q -n' --n" ■n" - ■n' ~ v' - v" v" - v' -n" ■ -v' = Q/t, dt t{v" - v'Y (99) Gibbs derives a general expression, similar to (98), for a system of n independently variable components, >Si, . . . aS„, in r = n + 1 coexistent phases. In this case there are n + 1 equations of the general form of (56), one for each of the existent phases. But the values of dp and dt must be the same for all phases and the same is true of djxi, c?^2, etc., so far as each of these occurs in the different equations. Thus, if each phase is regarded as being composed of some or all of the n independ- ent components, a variation of the system must satisfy the following equations: v' dp = T]' dt -{- nil dm + m2' c?/x2 . . . + w„' dfin, ' v" dp = ■(]" dt + rrix' dm + m^" d^ . . . + m„" dju„, v"'dp = v"'dt + mi'"dni + r)h"'dn2 . . . + mr/"diji„, etc. (100) [127] * Clapeyron, J. de I'ecole polytechnique, Paris, 14, 173, (1834). Clau- sius, Ann. Physik, 81, 168, (1850). Also obtained by W. Thomson, Phil. Mag., 37, 123, (1850). no BUTLER ART. D There are thus n + 1 Hnear equations between the w + 2 quantities dp, dt, dm, . . . dun, by means of which the n quantities, d^y dm, . . . dy.n can be eliminated. We thus obtain, in the notation of determinants: v' mi rrii . . . w/ v" my" m^" . . . m„'' v'" mi'" mi'" . . . mn'" dp = r\' mi m^' . . . w„' ■t]" mi" W2" . . . w„" 7/ mi m% . . . 7/in dt. (101) [129] As a simple example, we shall work out the application of this equation to a system containing as separate phases, calcium carbonate, lime and carbon dioxide. The two components lime and carbon dioxide are sufficient to express every possible variation of the system. Let the entropies, volume and quan- tities of the phases be specified as follows. Volume Entropy Quantity of carbon dioxide. Quantity of lime Gas phase Solid phase (lime) v' rrix 0 v" r," 0 Solid phase (calcium carbonate) nil mi' where m"' and m^" are necessarily in the proportion a : 6 in which lime and carbon dioxide unite to form calcium carbonate. Then, by (101), we have the following relation between varia- tions of the temperature and the pressure: v' mi' 0 v" 0 mi" dp = v"' mi'" W v' mi 0 ■n" 0 nn" n'" mi'" m (jT^^) = ^'- (115) [214] \d log W2/t. p. m, The integral of this equation may be put in the form Bm-2 M2 = A'log ' (116) [215] mi where B, like A', is independent of W2 and Wi. This equation holds for such small values of rrii/mi that d\L\ldmi in (111) has the same value as in the limiting case when m2 = 0. In such cases mi/y may be regarded as constant and we may write /i2 = A' log ' or M2 = C + A' log T/iaA, (117) where Cwi/y = 5, and C = A' log C. Suppose that the independently variable components of a homogeneous body are Sa,--. Sg and Sh, and that the quantity of Sk is very small compared with the quantities of Sa,- ■ . S, and is incapable of negative values. Then, by an extension of the argument, it can be shown that a M. = A,' log ^\ (118) but Ah and Ch may be fimctions not only of the temperature and pressure but also of the composition of the "solvent" (composed oi Sa,. . .Sg) in which Sh is dissolved. If another component Si is also present in very small amount, it is reason- able to assume that the value oi nh and therefore those of Ah and Ch are nearly the same as if it were absent. Thus the potentials of components Sh,. . • Sk, the quantities of which are very small 120 BUTLER art. d compared with the quantities of Sa, ■ ■ . Sg, can be expressed by equations of the form , , , Chnih Hh = Ah log Ilk = Ak log V Ckirik (119) [217] [218] where A//, Ch. ■ -Ak, Ck are functions of the temperature, the pressure and the ratios of the quantities nia, . . . mg. 17. Derivation of the Potentials of a Solution from Their Values in a Coexistent Vapor Phase* The part of the memoir which deals with the values of the potentials in gases does not come within the scope of this article, but since it is necessary for us to show how the potentials of the volatile components of a solution can be determined from the partial vapor pressures in a co- existent vapor phase we must first give a short derivation of the equation representing the variation of the potential of a gas with its pressure. According to the laws of Charles and Boyle the pressure, volume and temperature of unit weight of a perfect gas are related according to the equation pv = at, where a is a specific constant for each gas. For a weight m of the gas, we have pv = amt, and since, according to Avogadro's law, equal numbers of molecules of all perfect gases occupy the same volume at the same temperature and pressure, this equation becomes Amt , _ p. = — > (122) where A is a universal constant and M the molecular weight of the gas. *Gibba, I, 164-165. THERMODYNAMIC AL SYSTEM OF GIBBS 121 Let f ", f ' be the values of f for two states of the gas at the same temperature t. By (26) we have r - r = e" - t' - tin" - v') + P"v" - pV = - t W -7?'), (123) since the energy of a perfect gas at constant temperature is independent of its volume, and the product pv is also constant. In order to find the entropy change of the gas when its volume changes from v' to v" at constant temperature, we have by (3) idr] = pdv and, introducing the value of p/t given by (122), Am dv dv = ^-- (124) Integrating this from y' to v", we thus have ,, , Am , v" Am , v' , ^ ,"-V = ^log---^log - (125) or, inserting these values in (123), Amt^ v" , Amt , v' ^ +l^'°8,I = f +-M '°«» = ™'^. where C is a constant, which is a function of the temperature. The value of ^ for any volume v is thus given by the expression Amt m r = mC + — log-. (126) and the potential of the gas is therefore At m or, by (122), M = C + - log - (127) At M = m + - log p. (128) 122 BUTLER art. d A perfect gas mixture is one in which there is no interaction between the components, so that the energy is the sum of the energies which each component would possess if present in the same volume (and at the same temperature) by itself, and the entropy and pressure the sum of the entropies and pressures of the components separately under the same conditions.* In such a perfect gas mixture it is evident that the potential of each component is not affected by the presence of the other com- ponents and may also be represented by (127). When a liquid and a gaseous mass are coexistent, the poten- tials of those components which are common to the two phases must have the same values in each. Thus, if *S2 is an actual component of coexistent liquid and vapor phases and its concentration in the vapor is nii''^^ /v'^°\ its potential in the gas phase, provided that the latter has the properties of a perfect gas mixture, is given by the equation ^ , ^t m^ (129) M2 = ^2 + M^iO) log ^(o) , and this is also the value of its potential in the liquid. As an example of the determination of the potentials in a liquid by means of a coexistent vapor phase, we may consider a solution with two volatile components Si and Si. If the partial pressures of the components in the vapor are pi and P2, their potentials in the vapor by (128) are At /*! = /^(^) + ]^) log Vu (130) At M2 = fiit) -\- ^^^ log P2, (131) where Mi^"\ Mi^'^'' are the molecular weights in the vapor. These equations also give the values of the potentials in the coexistent liquid phase. At constant temperature and total applied pressure, applying (56) to the liquid phase, we have mi dfii + Mi djXi = 0, * A proof of this proposition is given by Gibbs (I, 155). or THERMODYNAMIC AL SYSTEM OF GIBBS 123 At At mi • -^^ dlogpi +nh • j^ d log pa = 0; I.e., d log pi _ (WMa^ d log P2~ ~ (mi/Mi(«') (132) This equation was obtained by Duhem,* and may be used to determine the partial pressures of one component of a binary solution when the partial pressures of the other component are known. In many cases, when the concentration of a component in the liquid phase is very small, the ratio of its concentrations in the liquid and gaseous phases is constant at a constant temperature (Henry's law), i.e., ^2(^)/i;(^) = D (m2(«>A(''0, (133) where Z) is a function of the temperature. In such cases, substituting this value of W2^°V«^^*'^ in (129), we have At rrh^^^ At nh^^'> = ^^' + i^;^ log -^- (134) Henry's law is not, however, a general law of nature. From a consideration of cases in which it fails it has been shown to be probable that it holds when the molecular weight of the solute is the same in the vapour and in the solution. We may therefore substitute M^*^^^ for M^'^^^ in (134). There is no reason to suppose that the equation so obtained, viz., At m2^^^ M2 = Ca' + ^17717 log -TJ- (135) Compt. rend., 102, 1449, (1886). 124 BUTLER ART. D does not hold in every case in which the amount of the component is very small, provided that the proper value of the molecular weight in the solution is employed. The difficulty arises here that there is no independent method by which the molecular weights in solution can be determined. The general validity of (135) is based on the fact that it has been found to hold in a very large number of cases in which M-/^'' is given the value to be expected for simple molecules according to the chemical formula. The cumulative effect of this evidence is so strong that in doubt- ful cases the value of the molecular weight in solution may be determined from (135) itself. In deducing the limiting law of the variation of the potential of a solute with its concentration we have considered a solute having an appreciable vapor pressure. But there is no reason to suppose that the behavior of involatile solutes is different in this respect and we may regard (135) as generally applicable to all components, the quantities of which cannot be negative and which are present in very small amounts, provided that the proper values of the molecular weights are used. IS. Equilibria Involving Dilute Solutions. In the last chapter of the first volume of the Collected Works (Gibbs I, Chap. IX) is printed a fragmentary manuscript of a proposed supplement to The Equilibrium of Heterogeneous Substances, in which Gibbs shows that the laws of dilute solutions obtained by van't Hoff from his law of osmotic pressure can be derived by making use of equation (135) for the potential of a solute. It will be of interest to give these demonstrations as examples of the application of the method of Gibbs to specific cases. We will consider a dilute solution formed by dissolving a small quantity, m2 grams, of a solute aS'2, in Wi grams of a solvent Si. The molecular weight of the solute in the solution is ilf2^^\ We will assume that the potential of S2 in the solution is given by (135), so that under these conditions, at constant temperature and pressure At v_ ^M2 = ^) • ± • d(^y (136) (a) Osmotic Pressure. Suppose that this solution is separated from a quantity of the pure solvent at the same temperature THERMODYNAMIC AL SYSTEM OF GIBBS 125 by a membrane which is permeable to the solvent, but not to the solute. The difference of pressure on the two sides of the mem- brane is the osmotic pressure of the solution. Let the potentials of S\ and >S2 in the solution at the temperature t and the pressure p' be Hi and ^2', and the potential of *Si in the solvent at the same temperature and pressure y" be /i/'. For equilibrium it is necessary that ^t/ = ni". All variations in the state of the solution must satisfy (56), so that for constant temperature dp' = y dni' + ^ dM2'. (137) So long as the solution remains in osmotic equilibrium with the solvent in its original state, din' = 0, so that Wo' rfp' = -7 ■ duL2'. (138) V By (136)= W , , At /W\ ../ • aM2 = ,r (,.) • d[ ^, I, hence, integrating (138), we obtain At TYli Since — • 777^, is the pressure, as calculated by (122), of m^ IMi^^'^ gram molecules of a perfect gas in the volume v' and at temperature t, this equation expresses van't Hoff's law of osmotic pressure.! (6) Lowering of the Freezing Point. Consider the equilibrium of the solution with a mass of the solid solvent. Applying (56) * Strictly, -7- • dix-^ = —7 • r^ — j—r - d —j -{ ;-•——• dp, but the V V dinh/v') V V dp last term vanishes at infinite dilution. t Z. physikal. Chem., 1, 481 (1887). M. Planck also gave a derivation of this law, Z. physikal. Chem., 6, 187 (1890). 126 BUTLER ART. D to the two phases, we have, for a variation of the solution, at constant pressure, 0 = n'dt + mi' dm' + nh'dni', (140) and for a variation of the soUd phase, at constant pressure, 0 = r,"dt + m/'d/xi". (141) In order to preserve equiUbrium so that if mi = mi", i.e., if we take quantities of the soUd and of the solution which contain equal amounts of *Si, W - v')dt = m'dfii'. (142) Now, by (136), Atv' /mA At , ,^ so that, integrating (142), we obtain At W - V) ^^ = ^i^) • ^2', (143) where At is the change of temperature when the value of m^' increases from zero to its value in the given solution. Thus the lowering of the freezing point is At mi' At^ rrii - ^ - 7^7' • Mix-. = -Q- • 72 128 BUTLER and since we have ART. D p t/ ' Ma^^) w/ ' Ma^^^' (149) i.e., the fractional lowering of the vapor pressure is equal to the ratio of the numbers of molecules of the solute and solvent. Rearranging (149), we easily obtain i.e., the ratio of the vapor pressure of the solution to that of the pure solvent at the same temperature is equal to the molar fraction of solvent. This is Raoult's law.* It is to be par- ticularly noticed that the molecular weight of the solvent which appears in these equations is that in the vapor, while the molecular weight of the solute is that in the solution. VII. The Values of Potentials in Solutions Which Are Not Very Dilute 19. Partial Energies, Entropies and Volumes. We shall now give an account of some extensions of the method of Gibbs which permit the quantitative treatment of equilibria involving concentrated solutions. The development of these extensions and the working out of practical methods for the evaluation of the potentials and other significant properties of solutions is largely due to G. N. Lewis and his collaborators.! Much of the work of these investigators has been concerned with solu- tions of electrolytes, which are the subject of a separate article * CorriTpt. rend., 104, 130 (1887); Z. physikal. Chem., 2, 353 (1888). t Outlines of a New System of Thermodynamic Chemistry, Proc. Amer.Acad.,43, 259 (1907); Z. physikal Chem., 61, 129 (1907). G. N. Lewis and M. Randall, Thermodynamics and the Free Energy of Chemical Substances, 1923. THERMODYNAMICAL SYSTEM OF GIBBS 129 in this volume. We shall only attempt to give in a concise form the significant extensions of Gibbs' method, with examples from solutions of non-electrolytes. The exact treatment of cases of equilibrium involving actual solutions is greatly facilitated by the use of some additional quantities, which we must first introduce. Consider a solution containing Wi, . . . 7n„ grams of the independently variable com- ponents /Si, . . . Sn, and let e, tj and v be the values of its energy, entropy and volume. Then, differentiating the equation ^ = e - tr] + pv with respect to mi, we have \dmi/t, p. m^, etc. \dmi/t. p. m., etc. \dmi/t, p, \dini/t. p. m^, etc. + P[ mj, etc. or where m = h - tm + pvi, (151) .. = (r-) . (152) \ami/t, p, mj. etc. \dmi)t, p. ' "ni - \ j^ ] » WTj* etc. and Vi = \aWi/ I, p, mj, etc. (154) which represent the ratios of the increments of the energy, entropy and volume of the solution to the increase of mi, when the temperature, pressure and quantities of Si,. . . Sn remain constant, are called the partial values of the energy, entropy and 130 BUTLER ART. D volume for a gram of the component Si. In the same way we may determine the partial energies, entropies and volumes for a gram of the other components. Similarly, since x = e + pr, we have Xi = €i + pvi. (155) At a given temperature and pressure, the quantities e, -q, v, x are all homogeneous functions of the first degree with respect to Ml, . . . lUn. Therefore, by (52), e = mill + rrhh • ■ • + Wne„, (156) and, by (54), rriidli + nhdh . • . + w„c?e„ = 0, (157) and similar equations may be obtained for rj, v and x-* The variations of the potentials with pressure and temperature are easily found in terms of these quantities. Thus, by (39), \dp/t. m ^* so that, differentiating this equation with respect to mi, we have 9 /ar\ dv d / d^\ dv /af\ ^ ^ or — (—\ \dp/ drrii °^ dp \dmi/ drrii \dp/ drrii dp \dmi/ drrii i.e., expressing the invariant quantities in full, \dp/t,m \dmi/ 1. p. m„ etc. Similarly, by (39), \(ll / p, m * The partial molar values of these quantities are obtained by multi- plying the values per gram given here by the molecular weight. Practi- cal methods of evaluating the partial molar quantities have been worked out by G. N. Lewis and collaborators (G. N. Lewis and M. Randall, Thermodynamics and the Free Energy of Chemical Substances, 1923). THERMODYN AMICAL SYSTEM OF GIBBS d drrii d ~ dt \dtni) dr, drrii 131 or Substituting the value of tj^i given by (151) and (153) we have = Ml - XI, (160) or Xi td{njt)\ n (161) (Compare equation (61).) 20. The Activity. The potential of a solute, the relative amount of which is very small, according to (128), is A« mi ^^ = ^ + i^ ^°s 7- This relation can only be regarded as expressing the limiting law of variation of the potential with the concentration at infinite dilution, and the foregoing considerations give us no guidance as to the modifications which may be necessary at greater concentrations. In order to represent the values of the potentials in actual solutions, G. N. Lewis has introduced a quantity a, called the activity, which may be defined by the equation At Ml = Ml" + ^^ log «i, (162) where /ii" is the potential in a chosen standard state, at the same temperature and pressure, in which the activity is taken as unity. The standard state may be chosen according to the circumstances of different cases. 132 BUTLER AUT. d For example, in the case of a binary solution of the compo- nents Si and S2, regarding Si as the solvent and *S2 as the solute, we may adopt the following conventions: (1) The activity of the solvent is unity in the pure solvent at the same temperature and pressure, i.e. ai = iVi, when A^i = 1, (163) where mi/Mi Ni = nil/ Ml + mil Ml is the molar fraction of the solvent. When the possible range of concentrations extends to iV2 = 1, as is the case with two liquids which are miscible in all proportions, the same convention may be adopted for *S2. (2) The activity of the solute is equal to its concentration when the latter is very small. The concentration may be expressed in any suitable way. If expressed as the molar fraction {N^, we have as -^ A^2, when ATj -> 0. (164) In the case of dilute aqueous solutions the concentration is often expressed as the number of mols {ui = nh/Mi), dissolved in a given weight, say 1000 grams, of the solvent. The activity may then be defined so that "2 —>■ ni, when n^ -^0* (165) 21. Determination of Activities from the Vapor Pressure. The potential of a volatile component of a solution is given, as in (129), by the equation * The molecular weight to be employed in determining the activity by (162) may have any appropriate value. But if the activity is deter- mined from the partial vapor pressure according to the method of Section 21 the molecular weight of the substance in the vapor state must be used. Also when the activity is defined by convention (2) its value can only be equal to the concentration in an infinitely dilute solu- tion if the molecular weight is that in the solution. THERMODYNAMIC AL SYSTEM OF GIBBS 133 where pi is its partial vapor pressure above the solution, and Ml its molecular weight in the vapor, provided that the vapor behaves as a perfect gas. If pi" be the partial vapor pressure in the standard state in which its activity is taken as unity, which we will consider to be the pure liquid at the same temperature, we have so that tl° = m + At : log pi\ Ml = Mi" + At log pi Pi" (166) and by (162), taking the molecular weight as that in the vapor, ai = Pi/pi'. (167) When the amount of the solute is very small, it has been shown that Raoult's law, PiM = Nr, (168) follows from the expression (126) for the variation of the poten- tial. It has been found by experiment that in some solutions this relation holds over the whole range of concentrations. The solutions which exhibit this behavior are usually composed of closely related substances, which might be expected to be less influenced by effects due to the interaction of the components than solutions of substances of different types or with widely differing properties. Consequently such solutions have been regarded as ideal solutions. Therefore, when the activity is defined as in (163), ai = A^i in ideal solutions. The fraction ai/Ni which has been termed by G. N. Lewis the activity coefficient, may be regarded as a measure of the deviation of a solution from the ideal behavior. In the case of dilute solutions for which we take a^ = ^2, when ri2 = 0, the activity coefficient is taken as ailni. Table I gives the activities and activity coefficients at 35.17° 134 BUTLER ART. D in solutions of chloroform (Si) and acetone (^2) calculated from the partial vapour pressures determined by Zawidski.* For both components, the activity is taken as unity in the pure liquid. TABLE I Activities and Activity Coefficients in Solutions of Chloroform AND Acetone (35.17°C.) Ni pi ai = pi/pi" ai/Ni Ni P2 at = cti/Ni 0.000 0 0.000 — 1.000 344.5 1.000 1.000 .0595 9.3 .032 0.538 0.9405 322.9 0.938 0.998 .1217 20.1 .069 .567 .8783 299.7 .871 .992 .1835 31.8 .108 .590 .8165 275.8 .801 .982 .2630 50.4 .172 .654 .7370 240.6 .699 .948 .3613 72.6 .248 .687 .6387 200.3 .582 .912 .4240 89.4 .305 .719 .5760 173.7 .504 .875 .5083 115.3 .394 .775 .4917 137.6 .400 .814 .5523 130.5 .440 .796 .4477 119.5 .347 .775 .6622 169.9 .577 .871 .3378 79.1 .230 .681 .8022 224.3 .765 .954 .1978 37.9 .110 .556 .9177 266.3 .909 .991 .0823 13.4 .039 .474 1.000 293.1 1.000 1.000 .000 0.0 0.0 — The activities of a non-volatile component of a binary solution can be determined from the activities of a volatile component by means of the Gibbs-Duhem equation : Since and we have Wid/il + W2C?/i2 = 0. At , At ^ JU2 "= Ala" + ^ log a2, — d log ai + — d log ai = 0. (169) Z. physikal. Chemie, 35, 129 (1900). THERMODYNAMIC AL SYSTEM OF GIBBS 135 If mi/ Ml = rii and mil Mi. = Ui, we have log 0:2' — log 0:2 = / — —-d log ori. (170) If Ni and A'"2 are the molar fractions of the two components ni d log Ni-\- riid log ^"2 = 0 (171) and, subtracting this from (169), (170) is obtained in the form log (a^'/N^') - log (a./N,) = rai'/Ni' Jm/Ni "^■dlogiai/Ni). (172) For example, Downes and Perman have determined the vapor pressures of water over aqueous cane sugar solutions.* From these measurements Permanf has calculated the activity coefficients of water (Si) by (167) and those of cane sugar (^2) by (172), takmg m/Ni = 1, when iV2 = 0. Table II gives the values at 50°. TABLE II Activities and Activity Coefficients in Cane Sugar Solutions AT 50°C. Nt pi (mm. mercury) ai/Ni Cli/N2 0 92.35 1.000 1.000 0.0060 91.74 0.9999 1.000 0.0174 90.51 0.9974 1.134 0.0238 89.55 0.9933 1.269 0.0335 88.81 0.9950 1.437 0.0441 87.52 0.9914 1.624 0.0561 85.88 0.9852 1.847 0.0677 83.51 0.9699 2.053 0.1089 76.92 0.9347 2.801 22. The Lowering of the Freezing Point. Consider the equilibrium of a solution of a solute >S2 in a solvent Si with a soUd phase consisting solely of Si. We will denote the poten- tials of Si in the solid, the pure solvent and in the solution at a ♦ Trans. Faraday Soc, 23, 95 (1927). t Ibid., 24, 330 (1928). 136 BUTLER ART. D temperature t by f4, lA and /jli. Let ^o be the freezing point of the pure solvent and t, the freezing point of the solution. For the equilibrium of the solid with the pure solvent at ^o it is necessary that Z!2 = :^«, (173) and similarly for the equilibrium of the solution with the solid at t, 6 = ^« t t' (174) By (161) so that din'jt) _ x; dt e ti' = ^- f'^-dt (175) t to J'o t^ . ^ ^ Similarly, for the pure solvent, we have 7 = 7^- rS'^^ (176) t to Jto r and by (166), if Pi and pi are the partial vapor pressures of Si over the liquid solvent and over the solution at t, and Af/°^ is its molecular weight in the vapor, we have i = f + ji^'°s^''-/p'°'" so that T-'i-Lj-" + w^>'''^^^/^'°^'- <^"' Comparing (177) and (175), it is evident that ^ \og{p,/p,^\= P^^--^-dt. (178) Mi^^^ THERMODYNAMIC AL SYSTEM OF GIBBS 137 Now, if we write t = to —A/, where A^ is the lowering of the freezing point, and represent xt and x< as functions of the temperature by means of the equations x: = x; - Co -At, x: = x- - c.-At, (179) where Co and C, are the specific heats of the pure solvent and of the solid at constant pressure, we have '°H^°Jrj. Tit^^' ''^'•(i»* Here Mi^'^^ix]^ — x'J is the heat absorbed in the melting of the molecular weight of the solid solvent at ^o- For ice and water in the vicinity of 0°C., G. N. Lewis and M. Randall* have used the values Mi^^"^ ixl - X') = 1438 calories, iWi^^^ (Co - C.) =9 calories, and integrating the right hand member of (180) in series have obtained the expression log (pi/pi") = - 0.009696 At - 0.0000051 Af, (181) which they consider accurate up to 20 or 30 degrees from the freezing point. This equation gives log ivi/v^) or log aj at the freezing point of the solution. Table III gives a comparison of the values of log(pi/p]°)t for aqueous mannite solutions, as calculated by (181) from the freezing point depressions, with the values determined directly from the vapor pressures by Frazer, Lovelace and Rogersf at 20°C. The small differences between the two sets of values are to be ascribed to the difference between the temperatures to which * Thermodynamics, p. 283 (1923). t J. Amer. Chem. Soc, 42, 1793, (1920). 138 BUTLER ART. D they refer. The change of logfpi/pi"), or logori, with tempera- ture can be obtained by dividing equation (166) by t and differentiating. Thus we find that d log (p^/p^') ^ M^ ( dMt) _ rfOiiVOl dt A \ dt dt j Mi(«> (^> (182) where Mi(xi — xi'*) is the heat absorbed when the molecular weight of the pure solvent is added to a large quantity of the solution at the temperature t. If xi is known as a function of the temperature, this equation may be integrated over a con- TABLE III Freezing Point Depressions and Vapor Pressure Lowerings of Aqueous Mannite Solutions m At log (pi/pi") at - At" (calc.) log (pi /pi") at 20° (obs.) 0.1013 0.1874 0.00182 0.00180 0.2061 0.3807 0.00369 0.00366 0.2709 0.505 0.00489 0.00481 0.5323 0.9835 0.00953 0.00945 0.546 1.019 0.00988 0.00974 siderable range of temperature, and the values of log(pi/pi") or logai at a given temperature can be evaluated from measure- ments at another temperature. In the data for mannite solu- tions it appears that log{pi/pi^) diminishes slightly as the temperature rises. In these solutions xi — Xi" is therefore a small positive quantity. 23. Osmotic Pressure of Solutions. We will consider the osmotic equilibrium of a solution of a solute *S2 in a solvent Si separated from the pure solvent by a membrane which is per- meable to Si only. Let the values of the potential of Si at a temperature t and pressure Po be /ii" in the solvent and /xi in the solution. For osmotic equilibrium, by (90), it is necessary that the potential of Si shall be the same on both sides of the mem- THERMODYNAMICAL SYSTEM OF GIBBS 139 brane, i.e., if the pressure on the solvent remains constant, the pressure on the solution must be such that the potential of Si in the solution is /xi". The variation of mi with pressure, accord- ing to (158), is fdfjA \dPjt. Vi. Therefore, if P is the pressure on the solution for osmotic equilibrium, .0 . _ r Ml -Mr = - h-dP. (183) By (166), we may write At — i,.o — ^'~ ^' ~ M:(«> log (pi/pi"), where pi" and pi are the partial vapor pressures of Si over the solvent and the solution at a total hydrostatic pressure Po, and Mi^^^ is the molecular weight of Si in the vapor. If we regard vi as constant, we have At P -Po = - J^^^ log (Pi/Pi«),* (184) where P — Po is the osmotic pressure. * Differentiating equation (183), we obtain dm = — vi-dP, and since midfii + m2dii2 = 0, this becomes .dn2 = dP, which is similar rriiVi to (138), rriiVi (the partial volume of Si in the solution) being substituted for the total volume of the solution. Assuming that Vi is constant, this At ?«2 becomes for dilute solutions which obey (136), P — Po = TnTi) niiVi M2 which may be regarded as a more exact form of (139). This equation was obtained by G. N. Lewis, /. Amer. Chem. Soc, 30, 668 (1908). Equation (184) was derived by Berkeley, Hartley and Frazer, and by Perman and Urry from A. W. Porter's theory, Proc. Roy. Soc, A, 79, 519 (1907). 140 BUTLER ART. D A comparison of the observed osmotic pressure of solutions of cane sugar, a-methyl glucoside and calcium ferrocyanide with values calculated from the vapor pressures by means of this equation has been made by Berkeley, Hartley and Burton,* taking for Vi the mean value between Po and P. The following table gives their data for solutions of cane sugar and a-methyl glucoside at 0°C. TABLE IV Concentration, grams sugar in 100 grams water loge(po/p) vi Calculated osmotic pressure Observed osmotic pressure Cane sugar 56.50 0.03516 0.99515 43.91 43.84 81.20 0.05380 0.99157 67.43 67.68 112.00 0.07983 0.98690 100.53 100.43 141.00 0.10669 0.98321 134.86 134.71 a-methyl glucoside 35.00 0.03878 0.99810 48.29 48.11 45.00 0.05153 0.99709 64.22 63.96 65.00 0.06451 0.99579 80.50 81.00 75.00 0.09253 0.99354 115.74 115.92 Perman and Urryf have expressed Vi as a linear function of P — Po, by the equation V, = h' (1 - s(P - Po)), and (184) then becomes At f^ — log (px/pi«) = - j^^ -V.' {1 - s{P - Po)} dP Ml = - h' (P - Po) 1 - (■ s(P - Po)' )■ (185) where the relatively small term sPo'^ is neglected. ♦ Phil. Trans., 213, 295 (1919). Osmotic pressures from Proc. Roy. Soc, A, 92, 477 (1916). t Proc. Roy. Soc, A, 126, 44, (1930). THERMODYNAMIC AL SYSTEM OF GIBBS 141 Table V gives a comparison of the osmotic pressures of a solution of cane sugar containing 1 gram molecule in 1000 grams solution, as calculated by equation (185), using the vapor pres- sure data of Perman and Downes,* with the direct determi- nations of Morse, t TABLE V Calculated and Observed Osmotic Pressures op Sucrose Solutions Temperature log (p„/p) n« Osmotic pressure (calculated) Osmotic pressure (observed) 30.00 — 1.002877 27.025 27.22 40.00 0.01940 1.006456 27.506 27.70 50.00 0.01914 1.010650 27.88 28.21 60.00 0.01839 1.016843 27.45 28.37 70.00 0.01848 1.0195 28.34 28.62 80.00 0.01809 1.0257 28.41 28.82 VIII. Conditions Relating to the Possible Formation of Masses Unlike Any Previously ExistingJ 24. Conditions under Which New Bodies May Be Formed. So far, the only variations which have been considered possible in applying the criteria of equihbrium are those involving infinitesimal variations of the composition or state of the masses originally present. The conditions of equihbrium so obtained are obviously necessary for equihbrium but they are not always sufficient, for an infinitesimal variation of the system may also result in the formation of bodies entirely different from those originally present, and in order to discover whether the original state is one of equilibrium it is necessary to ascertain if the criteria of equilibrium are also satisfied for variations of this kind. Gibbs defines a new part as one which cannot be regarded as * Trans. Faraday Soc, 23, 95 (1927). The value used in the calculation at 30° is obtained from the work of Berkeley, Hartley and Burton (loc. cit.). t Osmotic Pressure of Aqueous Solutions, Carnegie Institution, Wash- ington. Publ. No. 198 (1914). t Gibbs, I, 70-79. 142 BUTLER ART. D having been formed by an infinitesimal variation in the state or composition of a part of the original mass. The new parts form.ed in an infinitesimal variation of the original mass are necessarily infinitely small. Let De, D-q, Dv, Drrii,. . .Drrin denote the energy, entropy, volume and the quantities of the components 8\, . . .Sn contained in any one of these new parts. We have no right to assume that a very small new part is homogeneous or that it has a definite physical boundary. Under these circumstances in order that these quantities may have a definite meaning it is necessary to define unambiguously the boundaries of the new parts. Gibbs uses a convention similar to that which he employs in the theory of capillarity. A dividing surface is drawn round each new part in such a way that it includes all the matter which is affected by the vicinity of the new part, so that the original part or parts remain strictly homogeneous right up to this boundary surface. De, Dtj, Dv, etc., then refer to the whole of the energy, entropy, volume, etc., within the boundary surface. If we use, as before, the character 5 to express infinitesimal variations of the original parts of the system, the general con- dition of equilibrium may be written in the form (25e + 2Z)e), ^0 (186) [36] or, substituting the value of SSe taken from equation (62), SDe + 2^577 - 'L'pbv + ^/xiSmi . . . + SM«5wn ^ 0. (187) [37] Making use of this equation Gibbs deduces de novo and by a very general argument the conditions of equilibrium when the component substances are related by r equations of the type: ai ©1 + a2 ©2 ... + a„ ®„ = 0. (188) [38] We shall consider here the simpler case in which the components ^1, Si,. . . Sn are all independent of each other. There is no real loss of generality in this limitation for, as Gibbs points out, we may consider all the bodies originally present in the system and the new bodies which may be formed to be composed of the same ultimate components. THERMODYNAMICAL SYSTEM OF GIBBS 143 The conditions of equilibrium between the original parts of the system have already been established. They are: t = T,p = P, (189) Ml ^ Ml, li2^M2, ... Hn^ Mn, (190) i.e., the temperature and pressure have uniform values T and P throughout the system, and the potential of the component Si has the value Mi in all parts of the system of which *Si is an actual component and may have a value greater than Mi in those parts of which it is a possible, but not an actual com- ponent. In using (187) we suppose that the total entropy and the total volume are constant, and since also in the case under consideration no component can be formed out of others the total amount of each component is also constant. The equa- tions of condition are thus (191) [39] (192) [40] (193) 25m„ + ZDnin = 0. Inserting the values of t, p, fxi, etc., and of Zdrj, Z8v, XSmi, etc., as given by these equations, in (187), we obtain SDe - TSDt; + P^Dv - M{LDmi ... - Mn^Dnin ^ 0, (194) or De - T-Dri + PDv - MiDmi ... - Mn-Drrin ^ 0, (195) for each of the new parts. This is the condition which must be satisfied in addition to the conditions relating to the equilib- rium of the initially existing parts of the system. Gibbs shows that when there are r relations of the type (188) between the components the same condition holds, but there are then r relations of the type aiMi + a^Mi . . . + a„M„ = 0 (196) [43] between the potentials. 257? + ^Dt] = 0, 25v + SZ)y = 0, S5wi + ZDmi = 0, 144 BUTLER ART. D If it could be supposed that the relation between the energy, entropy, volume and mass of the infinitely small new part were the same as that of a large homogeneous body of similar com- position, the quantities De, Drj, Dv, Drrii, etc., would be pro- portional to the energy e, entropy 17, volume v, masses mi, etc., of the large body, and (195) could be written in the form e - Tri -\- Pv - MiMx ... - Mnmn ^ 0. (197) [53] In general however such an assumption is not permissible. For, apart from difficulties arising from the definition of the boundary surface enclosing the new part, we neglect in deter- mining the energy, entropy, etc., of a large homogeneous body the contributions which arise from the action of capillary forces at its surfaces, and it is obviously impossible to neglect these in the case of very small bodies. Nevertheless it is probable that when (197) is satisfied, (195) is also satisfied. This appears from a consideration of the meaning of (197) in which e is the energy of a body having entropy 17, volume v, masses mi, . . . nin, which is formed in a medium having the temperature T, pressure P and potentials Mi, . . . ilf „. Since the total entropy and vol- ume are supposed to remain constant in the formation of this body, — Trj + Pv — MiVfii ... - ilf „m„ is the change in the energy of the medium. The quantity rep- resented in (197) is thus the energy change of the whole system in the formation of the new body, and since there is no change of entropy in the process this must be equal to the work which would be expended in the formation of the body from the medium by a reversible process. Now work must usually be expended to reduce a body to a finer state of subdivision, so that if (197) is positive or zero for a finite body there does not appear to be any reason to suppose that it will become negative even when the particles are infinitely small. So that if (197) is satisfied it appears that (195) will also be satisfied. This argument would however break down if the energy of a mass of a body within a medium ever decreased as the size of the particles decreased (i.e., in cases of negative surface tension). THERMODYNAMICAL SYSTEM OF GIBBS 145 Substances which exhibit the phenomenon of peptisation, i.e., when a large mass of a substance spontaneously breaks up into small particles, may be examples of such behavior. How- ever in such a case large masses of the substance in the given medium would be inherently unstable and there would be no advantage in substituting (197) for (195). It is evident that (197) cannot be regarded as a necessary condition of equilibrium, for (195) may be satisfied and the system will therefore be in a state of equilibrium even when (197) is unsatisfied. Cases of this kind are met with in super- heated liquids, supersaturated solutions, etc. In the case of a supersaturated solution of a given substance (197) is negative, but we must suppose that on account of capillary forces etc. the separation of an infinitely small quantity would give rise to positive (or zero) value in (195). It is however difficult to distinguish between effects of this kind and "passive resist- ances" to change. Gibbs remarks that "such an equilibrium will, however, be practically unstable. By this is meant that, although, strictly speaking, an infinitely small disturbance or change may not be sufficient to destroy the equilibrium, yet a very small change in the initial state, perhaps a circumstance which entirely escapes our powers of perception, will be sufficient to do so. The presence of a small portion of the substance for which the condition [53] does not hold true, is sufficient to produce this result, when this substance forms a variable com- ponent of the original homogeneous masses. In other cases, when, if the new substances are formed at all, different kinds must be formed simultaneously, the initial presence of the different kinds, and that in immediate proximity, may be necessary." 25. Generalized Statement of the Conditions of Equilibrium. The conditions of equilibrium of the parts initially present, and with respect to the formation of new parts, may be summed up as follows. Since for any homogeneous mass, by (48), the equation € — trj -\- pv — Himi — /LI2W2 ... — MnW„ = 0, (198) holds when mi, m^, . . .mn refer to the ultimate components of the mass, the condition of equilibrium between the original parts 146 BUTLER ART. D can be expressed by the conditions that it shall be possible to give to T, P, Mi,...Mn in 6 - Tr? + Py - MiWi - MiTTh ... - MrMn (199) such values that the value of this expression shall be zero for every homogeneous part of the system. The equilibrium is practically stable if ^ ^ Tt) -\- Pv - Mimi - M^m^i ... - M„m„ ^ 0 (200) for any other body which may be formed from the same com- ponents, and this condition may be united with the former one in the statement that it shall be possible to give T, P, Mi,. .. Mn such values that the value of (200) for each homogeneous part of the system shall be as small as for any body whatever made of the same components. IX. The Internal Stability of Homogeneous Fluids* 26. General Tests of Stahility. Consider a homogeneous fluid, the ultimate components *Si, S2, . . . *S„ of which are pres- ent in the amounts mi, TO2, . . . m„. The conditions imposed in deducing the conditions of equilibrium are fulfilled if we suppose that the fluid is contained in a rigid envelop which is a non-conductor of heat and impervious to all its com- ponents. The conditions (199) and (200) might be employed to determine the stabflity of the fluid, but it is desirable to formulate them in a somewhat more general manner, since for the stability of the fluid it is necessary that it shall be in equilibrium both with respect to the formation of new parts as defined in the last section, and also with respect to the forma- tion of phases which may only differ infinitesimally from the original phase of the body. Gibbs states the condition of stability as follows: "7/ it is possible to assign such values to the constants T, P, Ml, Ml, . . .Mn that the value of the expression ^ - T-n + Pv - MiWi - M^nh ... - Mnrrin (201) [133] * Gibbs, I, 100-105. THERMODYNAMIC AL SYSTEM OF GIBBS 147 shall he zero for the given fluid, and shall he positive for every other phase of the same components, i.e., for every homogeneous hody not identical in nature and state ivith the given fluid {hut composed entirely of [some or all of the substances] Si, Sz, ■ . .»S„), the con- dition of the given fluid will he stable." The following proof may be given of this proposition. It is evident that if (201) is positive for every other phase of the components, its value for the whole mass must be positive when the latter is in any other than its given condition. The value of (201) is therefore less when the mass is in the given condition than when it is in any other condition. Since on account of the conditions imposed by the surrounding envelop neither the entropy, volume, or the quantities m^, W2, ...Wnfor the whole mass can change, it follows that the energy in the given condition is less than that in any other condition of the same entropy and volume. The given condition, by (5), is therefore stable. Since (201) is zero when applied to the given fluid (i.e., when e is the energy, rj the entropy, v the volume, mi, . . .mn the quantities of the components of the given fluid), it is evident that T is its temperature, P its pressure, and Mi, Mi, . . . Af „ the potentials of its components in the given state. If we wish to test the stability of the fluid with respect to the formation of some other phase we must insert for e, -q, v, mi, etc. the values of the energy, entropy, volume, and masses in a mass of the phase in question (not necessarily at the same temperature and pressure). If there is no other phase of the components for which the quantity so obtained has a positive value the given fluid is stable. It has already been shown that the expression (201) repre- sents the reversible work which must be expended in forming a phase of energy e, entropy t], volume v and masses mi, m^,... mn within a medium having the temperature T, pressure P, potentials Mi, Mi, . . . Mn. The condition of stability there- fore amounts to this: the fluid is stable if no other phase can be formed in it without the expenditure of work. When the value of the expression (201) is zero for the given fluid and negative for some other phase of the same components 148 BUTLER ART. D it is evident that the fluid is unstable. It may also happen that while T, P, Mi, Af 2, • • • Mn niay be given such values that (201) is zero for the given fluid there is some other phase for which (201) is also zero. This other phase must obviously have the same temperature and pressure, and the same values of the potentials, and is therefore a phase which could coexist with the given fluid. But Gibbs points out that although there may be phases which can coexist with the given mass, it is highly improbable that such phases could be formed within the given mass without a change of entropy or of volume. Thus although at the triple point water can coexist with ice and vapor, a quantity of water in this state enclosed in an envelop which has a constant volume and is impervious to heat is quite stable. 27. Condition of Stability at Constant Temperature and Pressure. In considering whether (201) is capable of a negative value for any phase, Gibbs points out that it is only necessary to consider phases which have the temperature T and the pressure P. For it may be assumed that the mass is capable of at least one state of not unstable equilibrium at this tem- perature and pressure, and in such a state the value of (201) must be as small as for any other state of the same matter. Therefore, if (201) is capable of a negative value, it wUl have a negative value at the temperature T and the pressure P. Also, if it is not capable of a negative value, any state for which it has the value zero must have the temperature T and the pressure P. For any body at the temperature T and the pressure P, (201) reduces to r - MiMi - Minh ... - M„w„, (202) [135] and in this form is capable of a very direct application, which is the basis of the geometrical methods employed by Gibbs in his use of curves and surfaces. Consider a series of homogeneous phases containing the two components Si and *S2 in different proportions. The ^-curve for a constant temperature t and pressure p is obtained by plotting THERMODYNAMICAL SYSTEM OF GIBBS 149 the values of f for the unit mass of the different phases (i.e., nil -\- nh = I) against the composition. Thus the point Z (Fig. 5) represents a phase for which XZ Mi Wi + nii XY and the value of f for this phase is represented by ZE. The curve AB represents the values of f for all homogeneous phases Fia. 5 when the composition is varied from that of the phase for which Wi = 1 (represented by point X) to that for which nh =1 (point Y). CD is the tangent to the f curve at the point E. It can be shown that intercepts made by this tangent on the axes at X and Y are equal to the values of Mi and M^ for the phase represented by E, i.e., XC = Mi and YD = Mi* The * If the potentials of ;Si and St in the phase E are ixi and m2, the tangent CD is characterized by the equation df = ixidm,]. + y^idmi, or since when 150 BUTLER ART. D value of niiMi + ^2^2 for any given values of Wi and rrh (for which mi -\- nii = 1) is therefore represented by the point on the line CD corresponding to these values. The expression f - Mimi - ilf 2W2 (203) is positive for every other phase of the components, other than the one under consideration, when there is no phase for which the value of f , at the same temperature and pressure, lies below the line CD. Thus if the two components form a solid com- pound, of which the composition and value of f are represented by the point P (under CD), the phase E will be unstable (supersaturated) with respect to this phase, for f — MiMi — M^rUi is negative for the phase P. But if the point representing this phase is above CD (say at P'), T ~ Mini], — 71^2^2 will be positive, and the phase E will be stable in respect to the forma- tion of this phase. Similarly if the curve AB is everywhere above the tangent CD, except at the single point of contact, the phase E is stable with respect to the other homogeneous phases, and cannot split into any of the phases represented by the points of this curve. 28. Condition of Stability Referred to the Pressure of Phases for Which the Temperature and Potentials Are the Same as Those of the Phase in Question. In the expression e - Tj] + Pv - Mmi - M2W2 - . . . (204) T, P, Ml, M2, etc. are the temperature, pressure and potentials in the fluid mass the stability of which is in question, and e, 17, V, mi, W2, etc. are the energy, entropy, volume, etc. of a given phase with regard to which the stability is being tested. These quantities are related by the equation e = tri — pv -\- iiimi H- /X2W2 + . • • , (205) where t is the temperature, p the pressure and in, /X2, etc., the potentials in the given phase. If we consider only phases for nil + VI2 = 1, d7ni = —dm2, the slope of the tangent is given by d^ — it^i ~ iMi)dm.2. Since ZE = nimi + M2W2, XC = mi'^i + M2W2 — (ixi — ni)mi = )ui. Similarly YD = juj. THERMODYNAMICAL SYSTEM OF GIBBS 151 which /. ^ T, jxi = Ml, H2 = Ms, etc., we may by substituting the value of e given by (205), reduce (204) to the expression (P - v)v- (206) In order to justify the use of this expression it is necessary to show that in testing the stability of a fluid it is sufficient to take into account only phases for which the temperature and poten- tials are the same as in the given fluid. This can be done by considering the least value of which (201) is capable at a constant value of V. Suppose that (201) has its smallest possible value, without any restriction, when evaluated for a phase having the energy e, entropy 77, volume v, masses Wi, . . .w„.* Then if e', rj', v', m/, rui', . . . m„' are the values referring to any other phase we have e' - Tv' + Pv' - Miiui' - M.nii' ... - Af„w„' ^ e — T-q -\- Pv - Mimi — MiiUi ... — Mnirin or, if both phases have the same volume, €' - e - T(7j' - 77) - Mi{mi' - mi) - Miim^' - roi) . . . ^0. Thus if the second phase can be considered as having been formed by an infinitesimal variation of the first phase, at constant volume, we may write this equation as de - Tdi) - Midmi - M^dn^ ... ^0. (207) But a variation of the energy of the first phase, at constant volume, is given by de = tdrj + nidirii + ^l2d'm2 + . . . , (208) and (207) and (208) can only both hold if t = T, m = Ml, M2 = Mi, etc. * It is supposed here that the components of the body are some or all of the components *Si, S2, ■ . -Sn. Gibbs considers the case in which the components of the new phase may be different from those of the given fluid. 152 BUTLER ART. D Therefore the phase for which (201) has the least value will be found among those having the temperature T and potentials Mi, Mi, etc., and in determining the stability of the given fluid we need only consider phases in which the temperature and potentials have these values. In this case the given fluid wfll be stable unless the expression (206) is capable of having a negative value. The conditions of stability are thus stated by Gibbs in the following very simple form: "// the pressure of the fluid is greater than that of any other phase of the same components which has the same temperature and the same values of the potentials for its actual components, the fluid is stable without coexistent phases; if its pressure is not as great as some other such phase, it will he unstable; if its pressure is as great as that of any other such phase, hut not greater than that of every other, the fluid will certainly not be unstable, and in all probability it will be stable {when enclosed in a rigid envelop which is impermeable to heat and to all kinds of matter), hut it will he one of a set of coexistent phases of which the others are the phases which have the same pressure." For example, consider a solution of carbon dioxide in water. If the pressure of a vapor phase at the same temperature, and in which carbon dioxide and water have the same potentials as in the solution, is greater than the pressure of the solution, the latter is unstable; but if the pressure of a vapor phase which satisfied these conditions is less than that of the solution, the latter is stable (with respect to the formation of a vapor phase). A vapor phase containing carbon dioxide and water at the same potentials as in the solution, and having the same temperature and pressure could obviously coexist with the solution, but a quantity of such a solution in a confined space is stable. X. Stability in Respect to Continuous Changes of Phase* S9. General Remarks. In order to test whether a homogene- ous fluid is stable with respect to the formation of phases which differ from it infinitely little (which are termed by Gibbs, * Gibbs, I, 105-115. THERMODYNAMICAL SYSTEM OF GIBBS 153 adjacent phases), we may apply to such changes the same general test as before. It is evidently only necessary to con- sider as the component substances of such phases the inde- pendently variable components of the given fluid. The con- stants Ml, M2, etc. in (201) have the values of the potentials for these components in the given fluid, for which the value of (201) is necessarily zero. Then, if for any infinitely small variation of the phase the value of {201) can become negative, the fluid will he unstable; but if for every infinitely small variation of the phase {201) becomes positive, the fluid will be stable. Gibbs points out that the case in which the phase can be varied without altering the value of (201) can hardly be expected to occur. For, in such a case, the phase concerned would have coexistent adjacent phases. This condition, which Gibbs calls the condition of stability, may be written in the form e" - t'r," + P'v" - ixi'm," ... - Mn'm„" > 0, (209) [142] where t', p', ni, m', etc. are the temperature, pressure and the potentials in the phase, the stability of which is in question, and t", 1]" , v", mi', rrii", etc., are the energy, entropy, volume and quantities of the components in any adjacent phase. Single accents are used to distinguish quantities referring to the first phase, and double accents those referring to the second. Particular conditions of stability can be obtained by trans- forming this equation in various ways. 30. Condition with Respect to the Variation of the Energy. If we add - e' -f t'r)' - p'v' + m'mi' + yii'nh' ... + Mn'w„' = 0, to (209), we obtain (e" - t') - t'{r}" - v') -h p'{v" - v') - uLi'{mi" - m/) -M2'(W - m') ... > 0, [143] which may be written in the form Ae > tAr) — pAv -f mAmi + HiAm2 . . . + UnAmn, (210) [145] 154 BUTLER ART. D where the character A is used to signify that the condition, although relating to infinitesimal differences, is not to be inter- preted in accordance with the usual convention in differential equations, in which infinitesimals of higher orders than the first are neglected, but is to be interpreted strictly, like an equation between finite differences. (See page 72.) When applying the condition (210), it is necessary that the quantities Ae, Arj, Ami, etc., should be such as are determined by an actual change of phase and not by a change in the total amount of the phase, for in that case the term on the left of (210) is zero. This can be accomplished by making v constant, and then divid- ing the remaining terms by the constant v. Then we have A— >iA — +^iA — +M2A^ V V V V ...-{- Hn A -. (211) [146] V But according to (44) we have a — = t a — -\- ^il a — +M2« — • V V V V ...+Mn^-, (212) [147] V so that, "the stability of any phase in regard to continuous changes depends upon the same conditions in regard to the second and higher differential coefficients of the density of energy regarded as a function of the density of entropy and the densities of the several components, which would make the density of energy a minimum, if the necessary conditions in regard to the first differential coeffi- cients were fulfilled.'' In a phase of one component, it is more convenient to make m constant instead of v, when (210) becomes Ae > tAif} — pAv. The meaning of this condition can be seen if the values of €, 17 and V are represented by rectangular coordinates. Let D THERMODYNAMIC AL SYSTEM OF GIBBS 155 represent a phase having energy e, entropy 77 and volume v (Fig. 6), The points representing adjacent phases form a surface. Let E be a point on this surface, representing a phase having the energy e + Ae, entropy rj -{- Arj and volume v + Ay. Fig. 6 If the tangent plane to the surface through the point D, cuts the vertical line through E at E', the ordinate of the point E' is de de e + — At? + — Av. ay] dv Since dr] t, dv P, the vertical distance EE' is thus equal to Ae — /A77 + pAv. Thus, (210) is positive when the e, 77, v surface for adjacent phases lies above the tangent plane, taken at the point repre- senting the phase in question. Any phase for which this holds true is stable with respect to continuous changes. 156 BUTLER ART. D 31. Condition with Respect to the Variation of the Pressure. Substituting the value e = t ri — p V +/xiOTi .,.-t-/x„ ntn in (209), we obtain - v"{t' - t") + v"{v' - V") - m,"{y.,' - Ml") - W(m2' - M2") ... > 0. (213) [144] This formula expresses the condition of stability for the phase to which t', p', etc. relate. But if all phases (within any given hmits) are stable, (213) will hold for any two infinitesimally differing phases (within the same hmits) and the phase (") may be regarded as the phase of which the stabiUty is in ques- tion, and (') as the infinitestimal variant of it. Then (213) can be written - r]At + vAp - miA/ii ... - m,Apin > 0, (214) [148] or Ap > ^ Ai + -^ Ami . . . + - AMn. (215) [149] V V V But by (56) dp= ^dt-\- '-^ dfJi,... + "^ d^n, (216) V V V so that "we see that it is necessary and sufficient for the stability in regard to continuous changes of all the phases within any given limits, that within those hmits the same conditions should be fulfilled in respect to the second and higher differential coefficients of the pressure regarded as a function of the tem- perature and the several potentials, which would make the pressure a minimum, if the necessary conditions with regard to the first differential coefficients were fulfilled." 32. Conditions oj Stability in Terms of the Functions \p and T- Writing e" = lA" + t'W. THERMODYNAMICAL SYSTEM OF GIBBS 157 and _ ^' _ p'v' + (jiMi ... + fj^n'mn' = 0, (209) becomes (rP" - ^') + it" - t')v" + {v" - v')v' - (mi" - m/W ... - (m„" - mn')nn' > 0. (217) [150] As in (213), when all phases within any given limits are stable, this condition holds for any two phases which differ infinitely little. When v' = v", mi = nil", . . . lUn = Mn", ir - ^') + it" - t'W > 0, (218) [151] or (^' - r) + {f - t")ri" < 0, (219) which may be written [^^P + -nM],, ^ < 0. (220) [153] Note that the phase, the stability of which is in question here is that to which t]" refers; hence Axp = 4/' — \p". Similarly, when t' = t", ir - ^') + V\v" - y') - m/(wi" - m/) ... - /xn'(w„" - w„') > 0, (221) [152] or [A^P + pAv - HiAmi ... - HnAmn]t > 0. (222) [154] The phase of which the stability is in question is now that distinguished by single accents. We may first observe that since, by (45), {d^/dt\rn = — »7> (220) requires that d^rp/dP < 0, i.e., d-q/dt or td-q/dt is positive, tdr}/dt being the specific heat of the phase in question at constant 158 BUTLER ART. D volume. Secondly, when the composition of the body remains unchanged, (222) becomes [A^ + vLv]t, „. > 0, (223) [160] and since, by (45), {dxp/clv)t,„, = ~p, this implies that {d^/dv^)t^rn > 0 or dp/dv must be negative. The conditions (220) and (223) thus express the conditions of thermal and mechanical stability of the body. The meaning of condition (222), as applied to the \p-v-m diagram for constant temperature, easily follows from considera- tions similar to those used in connection with (211). Again, by (15) and (50), (209) becomes (f" - n + v"{t" - n - v"ip" - p') - Hi (mi" - mi') ... - Hn'imn" - m/) > 0, (224) [161] from which we may obtain the conditions [Af + vM - vApU < 0, (225) [162] and [Ar - /xiAmi ... - M»Aw„],.p > 0. (226) [163] In order to show the meaning of this condition, we will consider the f-composition diagram, for constant temperature and pressure, of a two component system.* It is convenient in graphical representations (as in Fig. 7), to use as the variables expressing composition the fractional weights of the com- ponents. If we limit ourselves to phases for which Wi -{- W2 = 1, the quantities mi and rrh become equal to the fractional weights. Then for any change of phase. Ami = — Am2. The curve AB (Fig. 7) represents the f-values of homogeneous phases, at constant temperature and pressure, when m2 is varied from 0 to 1. Let the coordinates of the point D he i;, nh and the coordinates of an adjacent point E he ^ -\- A^, nii -{- Arrh. Let ST he the tangent to the curve AB, at the point D. The slope of this tangent is given by d^/drrh = M2 — Mi, so that if E' is its point of intersection with the vertical through E, the * Compare also Article H of this volume. THERMODYNAMIC AL SYSTEM OF GIBBS 159 ordinate of E' is i; -{■ (m — mOAws or f + /x2A?n2 + miAwi, since Atwj = —Ami. If Af > n^^m^ + miAwi, the point E is above the point E'. Therefore the condition of stability of the phase D, with respect to continuous changes, is that the f-curve for adjacent phases shall be above the tangent at D, except at the single point of contact. nig'O Trt^'l Fig. 7 33. Conditions with Respect to Temperature and the Potentials. Since (213) holds true of any two infinitesimally differing phases, within the limits of stabiHty, we may combine this condition, viz., rj"{t" - t') - v"{p" - p') + m,"W - Ml') . . . + mn"{lXn" - fin) > 0, and the condition obtained by interchanging the single and double accents, i.e., V'it' - t") - V'(p' - p") + W/W - Ml") . . . + m„'(Mn' - Hn") > 0, 160 BUTLER ABT. D in the condition (t" - n (v" - V) - (p" - V') W' - v') + (mi"-mi'; (wx"-mi') . . . + (m„" - Mn') {mj' - m„') > 0, (227) [170] which may be written in the form ^t^■n - ApAv + A/iiAmi . . . + Aju„Aw„ > 0. (228) [171] This must hold true of any two infinitesimally differing phases within the hmits of stabiHty. If we give the value zero to one of the differences in every term except one, it is evident that the values of the two differences in the remaining term must have the same sign, except in the case of Ap and Av, which have opposite signs. Thus we have, for example. (-) /A^\ \Ami/t, V, m^, /AM2\ \Am2Jt, V, >0; >0, >0, Ml. *"3' ( Afin\ Amn/t. V. >0; Ml. M2. •Mn — 1 (229) [166] [167] (230) [168] [169] and (: Av\ < 0. (231) Thus, when v, mi, ... rrin have any given constant values, within the limits of stability, t is an increasing Junction of rj; and when t, v, nh, . . .mn have any given constant values, within the limits of stability, fn is an increasing function of mi, etc. In general, "within the limits of stability, either of the two quantities occurring {after the sign A) in any term of [171] is an increasing function of the other, — except p and v, of which the opposite is true, — when we regard as constant one of the quantities THERMODYNAMICAL SYSTEM OF GIBBS 161 occiirring tn each of the other terms, but not such as to make the phases identical." It is evident that when v is taken as constant, there are a number of ways in which one of the quantities in each of n of the remaining n -\- 1 terms can be made zero. We can thus obtain different sets of n + 1 conditions, Hke (229) and (230). Gibbs points out that it is not always possible to substitute the con- dition that the pressure shall be constant for the condition that the volume shall be constant, without imposing a restriction on the variations of the phase. It may be pointed out with regard to the equations (229), (230), that if the sign A is replaced by d we obtain conditions which are sufficient for stability. It is evident that if the condition \dmn/i. V. ^„ /A/xA \AmnJt, V, w. . . > 0, (232) Mn — 1 > 0 (233) Mn— 1 must also hold true, i.e., the condition of stabihty is satisfied. But (233) may also hold true if = 0 (234) ' Mn — 1 (when one or more of the higher differential coefficients are positive). The expression (233) cannot hold true when the differential coefficient term (232) is negative, so that it is necessary for stability that ^ 0. (235) lin—i. 34. Limits of Stability. At the limits of stability (i.e., the limits which divide stable from unstable phases) with respect to continuous changes, one of the conditions (229), (230) must 162 BUTLER ART. D cease to hold true. Therefore, one of the differential coefficients like that in (234) must be zero. The differential coefficients dt dni dfXn jri ^: ■■■i^: »36) [181] may be evaluated in a number of different ways, according to whether the quantities which are to remain constant are chosen from the numerators or the denominators of the other terms. Gibbs shows that when the quantites which, together with V, are to remain constant are taken from the numerators of the others, their values will be at least as small as when one or more of the constants are taken from the denominators. At least one of the coefficients determined in this way will therefore be zero. But if one of these coefficients is zero it can be shown that all the others, having their constants chosen in the same way, will also be zero. Gibbs gives the following proof of this proposition. "For if (dfin/dmn)t, V, ^,. . . . ^„_u (237) [182] for example, has the value zero, we may change the density of the component Sn without altering (if we disregard infinitesi- mals of higher orders than the first) the temperature or the potentials, and therefore, by [98], without altering the pres- sure. That is, we may change the phase without altering any of the quantities t, p, m, ...Hr,. Now this change of phase, which changes the density of one of the components, will in general change the density of the others and the density of entropy. Therefore, all the other differential coefficients formed after the analogy of [182], i.e., formed from the fractions in [181] by taking as constants for each the quantities in the numerators of the others together with v, will in general have the value zero at the limit of stabihty. And the relation which character- izes the limit of stability may be expressed, in general, by setting any one of these differential coefficients equal to zero." We may write this condition in the form dfj.,,, 1 J( — 7-: = 0, (238) [183] THERMODYNAMIC AL SYSTEM OF GIBBS 163 or rd(mjv)l L dfJ'n J = 00. (239) [184] '• liU • ■ • ftn — l But, by (56), m„/v = {dp/dnn)t. w M„_i> so that (239) becomes d'^p dn„^ Similarly, we may obtain = 00 (240) [185] d^p d'^p d^p , , , "Any one of these equations [185], [186], may be regarded, in general, as the equation of the limit of stability. We may be certain that at every phase at that limit one at least of these equations will hold true." XI. Critical Phases* 35. Number of Degrees of Freedom of a Critical Phase. A critical phase is defined as one at which the distinction between two coexistent phases vanishes. For example, at the critical point of water, the liquid phase and the vapor phase become identical. Again, in Figure 8, the curves CA and CB represent the compositions of the two coexistent liquid phases in the system phenol-water at different temperatures at a constant pressure. As the temperature rises, the curves representing the compositions of the two coexistent phases approach each other, and at the point C the two phases become identical. Similar phenomena are met with in ternary mixtures. Let Si and S^ be two liquids which are incompletely miscible at a certain temperature and pressure, but which both form homogeneous solutions in all proportions with a third Hquid Sz. If we add * Gibbs, I, 129-131. 164 BUTLER AHT. D Ss to the two coexistent phases containing Si and S2, we shall obtain a series of two coexistent ternary phases, terminating in a phase at which the two phases become identical. Let n be the number of independently variable components. According to the phase rule, a pair of coexistent phases has n degrees of freedom, i.e., is capable of n independent variations. Thus, in the case of phenol and water, a pair of coexistent phases can be varied independently in two ways, i.e., we can vary both the temperature and the pressure without making one phase disappear. Now if we keep the pressure constant T X C r M N ^ ^ \p Q t. A \ \. Phenol % Fig. 8 WO and vary the temperature, we shall obtain a series of coexisting phases terminating in the critical phase. At a slightly different pressure there is a similar series of coexisting phases, terminating in a slightly different critical phase. It is evident that the number of independent variations of which the critical phase is capable is one less than that of the two coexistent phases, i.e., the number of independent variations of a critical phase, while remaining as such, is n — 1. 36. Conditions in Regard to Stability of Critical Phases. "The quantities, /, p, /xi, M2, • ■ Mn have the same value in two co- existent phases, but the ratios of the quantities 17, v, mi, m^, THERMODYNAMIC AL SYSTEM OF GIBBS 165 . . .nin are in general different in the two phases. Or, if for convenience we compare equal volumes of the two phases (which involves no loss of generahty), the quantities 77, mi, nh, . . . nin will in general have different values in two coexistent phases. Applying tliis to coexistent phases indefinitely near to a critical phase, ... if the values of n of the quantities t, p, /xi, mz, • • • Mn are regarded as constant (as well as v),* the variations of either of the others wUl be infinitely small compared with the variations of the quantities 77, mi, m^, . . . w„. This condition, which we may write in the form = 0, (242) [200] Mn-I characterizes . . . the limits which divide stable from unstable phases with respect to continuous changes." Critical phases are also at the limit which divides stable from unstable phases in respect to discontinuous changes. Thus, in Figure 8, phases represented by points inside the curve ACB are unstable with regard to the formation of the co- existent phases, represented by points on this curve. The co- existent phases thus He on the limit which separates stable from unstable phases in respect to discontinuous changes, and the same must be true of the critical phase. The series of phases determined by giving t and p the constant values which they have in the coexistent phases N and P (Fig. 8) consists of unstable phases in the part NP between the coexistent phases, but in the parts MN and PQ, beyond these phases, it consists of stable phases. But when t and p are given the constant values determined by the critical phase C, the whole series of phases XY (obtained by varying the com- position) is stable. Thus, in general, "if a critical phase is varied in such a manner that n of the quantities t, p, m, fj.2, . . .(Xn remain constant, it will remain stable in respect both to * Since two coexistent phases are only capable of n independent variations, this condition ensures that the variation considered cor- responds to the change from one coexistent state to the other, which is infinitely close to it. 166 BUTLER ART. D continuous and to discontinuous changes. Therefore, Hn is an increasing function of m„ when t, v, ni, H2, . . .At„_i have con- stant values* determined by any critical phase." If ((Ptj.n/dmJ)t. V. Ml- • Mn-1 had either a positive or a negative value, ^n would be a maxi- mum or a minimum with respect to m„, at the critical point, when (242) is satisfied. Thus, since Hn is an increasing function of nin, we have (j^) = 0, (243) [201] \am„ /t, v,^i, Hi, . . . ,i„_, but one of the higher differentials must be positive, i.e., ( -J — 3 ) ^ 0, etc. (244) [202] XII. Generalized Conditions of Stabilityf 37. The Conditions. A single phase of n components has n + 1 degrees of freedom. Therefore, if n of the quantities t, p, ni, . . -Hn are given constant values, the phase is only capable of one independent variation. If we take rj, wi, Wi, . . .w„ as the independent variables, we may write (when dv = 0) dt dt at = — di] -{- - — dm\ . (17) dmi dfi\ dfjLi dfii = —r- d-n + - — dmi . dr] ami dt + T"" dnin, + dm„ dm dnin dnin, > (245) [172] dUn dfJLn dfXn dun = ~r dv -f - — dm.1 . . . + "; — dm„. arj ami dm„ When dt = 0, dm = 0, . . . dun-i = 0, we have dlJ,n\ Rn + l (: dmn/t, v,^i,...fin-l Rn (246) [175] * t; is included to insure that a change in the amount of the critical phase is excluded, t Gibbs, I, 111-112. THERMODYNAMICAL SYSTEM OF GIBBS where Rn + i is the determinant, dh dh dh drf' dmidri drrindr] dh dh dh drjdmi drrii^ dmArrii , (2 dh d\ dh drjdnin dniidmn dm„^ 167 (247) [173] the constituents of which, by (44), are the same as the coeffi- cients of the equations (245), (thus dt/d-q = d'^e/dif, dyL„/dmn = dh/dnin^, etc.) and R^ is the determinant formed by erasing the last row and column of Rn-\-\. Similarly, the determi- nants Rn _ 1, /?„ _ 2, etc., are obtained by erasing successively the last row and column of Rn, and / dnn-i\ \dmn - 1/, Rn r, /il, . . .Mn-2i lir Rn etc. (248) [176] Now according to (230) and (232) the phase is stable if the differential coefficients (246), (248), etc. are all positive. These conditions are satisfied if the determinant (247) and all its minors, down to dh/dtf, are positive.* "Any phase for which this condition is satisfied will be stable, and no phase will be stable for which any of these quantities has a negative value." Since the conditions (230) remain valid if we replace any of the subscript /I's by m's, the order in which we erase the successive columns with the corresponding rows in the determinant is immaterial. For a body of invariable composition, it is only necessary to use the terms which are common to the first two rows and * The differential coefficients in (246), (248), etc. would also be posi- tive if all the determinants, Rn+\, Rn, etc. were negative. But the last term d^e/dr]^, by (229), cannot be negative, so none of the others can be negative. 168 BUTLER ART, D columns of (245) and (247). But in this case it is more con- venient to make dm = 0. Then we may write dt dt dt = -r dr] -\- — dv, dti dv dp dp dp = ~r dr) -\- — dv; dr] dv and, when dt = 0, the value of dp/dv is given by dH (249) drf dh dvdrj dr]dv dh dv"" (250) since, by (44), t = {dt/dr))^^^^ and p = — (c?e/dy);^,„. In stable phases, {dp/dv)i^^ must be negative. Thus, expanding (250), a phase of invariable composition is stable when d^e dh / dh' drf^ dv^ \drjdv J > 0, dh d;;^>'- (251) The physical meaning of these conditions can be seen from a consideration of the rj-v-e surface for homogeneous phases. Let rj, V, € be the coordinates on this surface of the point D, rep- resenting the phase in question. Let E be the neighbouring point on the surface, with coordinates rj + Arj, v -{- Av, e -\- Ae, and E' the point of intersection of the tangent plane through D with the vertical erected at E. (See Fig. 6.) Let the ordinate of E' he € -{- Ae'. Then, to the second order of small quantities, Ae = de de d^e dh and de de Ae' = J At? + , Av dr] dv THERMODYNAMIC AL SYSTEM OF GIBBS 169 (since de/drj, de/dv define the slope of the tangent plane at D). Thus EE' = Ae - At' d^e ■' dh O'e = ^^^^^ + d";s;^^^^ + ^^^^^' The expression on the right of this equation is positive when dh d^e / dh \2 dh dh (the last condition is a consequence of the other two), so that when these conditions are fulfilled E Hes above E'. Thus the conditions which were obtained above signify that a phase is stable with respect to continuous changes, when the rj-v-e surface for adjacent phases Ues above the tangential plane at the point representing the phase in question, except at the single point of contact. It is often more convenient to use other sets of quantities as the independent variables. Thus if we employ t, v, Wi, nh, . . .rrin as independent variables, we have when dt = 0 and dm„ =. 0,* dp dp dp dp = -rdv+T-dm^... + 7-— dmn-i dv drrii ' ' ' dm„-i dni dyL\ dni dui =» -7- dv + ~ — dnii . . . + J drtin-i, dv drrii am„_i dun-i = ~3 — dv + — — ami . . . + :; drrin-i; dv dmi dm„-i whence, when dt = 0, dp = 0, dfxi = 0, . . . d^n-i = 0, Pn > (252) /dHn-l\ Xdmn-i/t.v,^,, lin-2,mn t^n-\ (253) * In order that every variation considered shall represent a real change of phase, it is necessary to make one of the quantities v, nii, m-i, . . .ron constant. 170 BUTLER where, by (45), dV d?^p dV dt;2 dvdmi dvdm n-i dV d^ d^ ART. D Pn = drriidv dV dm-^ dV dmidnin-i d^ dm„-idv dmn-idrrii dml_i , (254) and the determinants F„_i, etc. are obtained by erasing suc- cessively the last row and the corresponding column in (254), By (231), dp/dv or ( — d^^p/dv"^) cannot be positive for a stable phase, therefore none of the determinants derived from (254) can be positive. If they are all negative the phase is necessarily stable. For two components, when dntz = 0, these conditions become d^ ^ dv^ dm^ \dvdmi) ' dmi^ >0, (255) the last of which is a consequence of the other two. Thus, if we construct a surface, the points of which have as coordinates the values of Vi, Wi, ^ for homogeneous phases having the same temperature and a constant value of W2, the condition of stability of any phase is that the surface shall be above the tangent plane taken at the point representing this phase, for all adjacent phases. Lastly, if t, p, mi, m-i, . . .nin are taken as the independent variables, and dt = 0, dp = 0, and dw„ = 0, we have dm = dfxz = dni drrii djxj dnii drrii + dmi + dni drrii dn2 dnii drrii drrio + + dm n-i djii dnin-i drrin- drn„-i, } (256) dfXn-\ = J... drrii + j drrii drui drrii dfln-l , + drrin-i. dmn-i THERMODYNAMIC AL SYSTEM OF GIBBS Therefore, by (43), /dnn-i\ \dmn-ijt. p. Ml. • • • Mn-i> where C/„_i is the determinant d^^ d^^ Un-l UnJ d^^ 171 (257) drrii dnh, dH dm^ dmn-\ dmi dm„_i dmz dm-^ dH dm-2 dm I dH drrii dm„_i dnii dtrin-i dml^i , (258) [206] and Un-2, etc. are the minors obtaiaed by erasing successively the last column and the corresponding row. A phase for which all these determinants have positive values is therefore stable. When there are three components and dmz = 0, these con- ditions become d^ ^ drrii^ dnii^ \dmi dw2/ >0, dn_ dmi' >0, dn_ dmi^ > 0. (259) If, instead of making wis constant, we use as the variables ex- pressing the composition x = Wi/(wi -{- m^ -{- mz) and y = m^/imi + m2 + ms), these conditions maybe obtained in the form dx^ dy' \dx dyj >0, d^ dx' >0, d^ dy' > 0. (260) Thus if a f-surface is constructed for homogeneous phases having the same temperature and pressure, with coordinates X, y, f, the condition of stability of any phase is that the f- surface for adjacent phases shall be above the tangent plane, taken at the point representing the phase in question, every- where except at the single point of contact. 172 BUTLER ART. D In general the condition of the Hmit of stabiHty is represented by substituting = for > in any of these equations. 38. Critical Phases* Since a critical phase may be varied without changing any of the quantities t, ni, n^, ... Mn, all the expressions (245) may be equated to zero. The solution of the equations so obtained is Rn+i = 0. (261) [203] (This also follows from the fact that a critical phase is at the limit of stability with respect to continuous changes.) "To obtain the second equation characteristic of critical phases, we observe that as a phase which is critical cannot become unstable when varied so that n of the quantities t, p, ni, )U2, ...Mn remain constant, the differential of Rn+\ for constant volume, viz., —j^ dv + -~- dmi ... + -J— ^ drrin (262) [204] dri ami otw,, cannot become negative" when n of the quantities t, p, ni, m, . . ./x„ remain constant. "Neither can it have a positive value, for then its value might become negative by a change of sign of dr], drrii, etc." Therefore the expression (262) has the value zero, when n of the expressions (245) are equated to zero. If *S is a determinant in which the constituents are the same as in i^n+i except that the differential coefficients dr) ' drrii ' ' * ' dm,, are substituted in a single horizontal line, this condition is expressed by the equation S = 0. (264) [205] This substitution may be made in any horizontal line of Rn + i- * Gibbs, I, 132-134. THERMODYNAMICAL SYSTEM OF GIBBS 173 These conditions may be expressed in terms of other sets of variables. Thus using the variables of (252), we have P„ = 0, and Qn = 0, (265) where Q„ is the determinant formed by substituting the coeffi- cients -—, -—,... ~ (266) dv ami dnin-i in any line of (254). For a system of one component, these equations become \dv^/t,m ' \dv^)t,m Again, using the variables in (256), we have as the equations of critical phases, Un-i = 0, and Vn-x = 0, (268) [208] where Fn_i is the determinant formed by substituting the coefficients d^ dE^ MJ^ 12071 drrii drrii dm n-i in any line of (258). For two components, these equations become m =0, if-) =0. (270) Instead of making W2 constant, we may use as the variable expressing the composition, a; = mi/(wi + W2). Then we have as the equations of a critical phase \dx^/t.p ' \dxyt,p As an illustration of these relations we will return to a con- sideration of the ^-composition diagram of a two component 174 BUTLER ART. D system. Suppose that at a pressure p and a temperature t', the f-x curve for homogeneous phases has the form AB (Fig. 9), with a double tangent PQ. Homogeneous phases between P Fig. 9 and Q are unstable with respect to discontinuous changes. Between R and S, the ^-curve is convex upwards, i.e., {d^^/dx%, t < 0, and these phases are unstable with respect to continuous changes. Between P and R, and between Q and S the f-curve is still concave upwards, i.e., and these phases, though unstable with regard to discontinuous changes are stable with regard to continuous changes. The points R and S, for which d'^/dx' = 0, THERMODYNAMIC AL SYSTEM OF GIBBS 175 thus represent the Hmits of stabihty with regard to continuous changes. K the temperature is varied in the direction of the critical point, the phases P and Q approach each other and at the critical temperature become identical. If CD is the f-curve at the critical temperature t", the point T representing the critical phase, where the points P, Q, R, S, all coalesce, is a point of undulation at which i(P^/dx-')p. t = 0 and {d'^/dx')p. t = 0. Finally, at a temperature t'" beyond the critical point, the f-curve is concave ever5nvhere. Now (d'^^/dx^) t, p is positive for all homogeneous phases, which are stable with regard to both continuous and discontinuous changes. It is evident that by a shght variation of the critical phase we may obtain either (1), a phase which is unstable with regard to both continuous or discontinuous changes, or (2), a phase which is stable with regard to continuous changes but unstable with regard to discontinuous changes, or (3), a phase which is stable with regard to both continuous and discontinuous changes. XIII. Equilibrium of Two Components in Two Phases 39. The Equilibrium. We can now consider in more detail the relation between temperature, pressure and composition in systems of two components. Si and S2, in two phases. Let the quantities referring to the first phase be distinguished by single accents, and those referring to the second phase by double accents. Then, for any change of state, while the phases remain in equihbrium, we have v' dv = v' dt -\- mi dm + m^' c?^2,] (272) v"dp = r}"dt + mi" dm + mi'dm-] If we consider quantities of the phases for which m^' = W/i' , we have (v" - v')dv = (r;" - ■t\')dt + (mi" - miO^Mi. (273) 176 BUTLER ART. D Now, we may express dfj,i as a function of p, t, mi by the equa- tion This equation may be applied to either of the two phases. Applying it to the first phase, we may write, by (158) and (159), \dp Jt.m ' ' \dt /p. m Hence, substituting in (273) the value of d^ given by these equations and rearranging, we find {{v" - v') - (mi" - miO vA dp = [(V - r?') - (mi" - m/) ^i'] dt + (mi" - miO ( ^Y ' dmi'. (275) Similarly, when the terms of (274) are determined by the second phase, we obtain [{v" - v') - (mi" - miO vi"\ dp - Kv" -v) - (wi" - miO vi"] dt + (mi" - miO (j^Y • dmi". (276) \dmi/p, I, mj In order to interpret these equations we may first observe that v' is the volume of the quantity of the first phase which contains mi' of the first component. Thus [v' + (m/' — m/) {dv'/dmi')] is approximately equal to the volume of that quantity of this phase which contains m/' of this substance. Hence we see that [v" — v' — (m/' — m/) y/] is approximately equal to the difference of the volumes of quantities of the two phases containing the same amount (viz., m/') of this substance. In the same way [v" — v' — (m/' — mi)vi"] is the approximate THERMODYNAMIC AL SYSTEM OF GIBBS 177 difference of volume of quantities of the two phases which contain the same amount (wi') of this component. The terms relating to the entropy can be interpreted in a similar way. Secondly, by (253) or (257) the differential coefficient (dfjLi/dmi)t. p, m, is positive in both phases.* 40. Konowalow's Laws. In the case in which the first phase is Hquid and the second a gaseous phase, the coefficients of dp in (275) and (276) are evidently positive. Then, when dt = 0, we see that (1) From (275), dp has the same sign as (m/' — m/) dm/, and from (276), dp has the same sign as (m/' — m/) dmi". Therefore dnii has the same sign as dmi". (2) Since dp has the same sign as (mi" — m/) dnii, dp and dmi have the same sign if 7ni" > m/, and opposite signs if mi' < mi. Thus we may draw the following conclusions : (1) When the composition of the liquid phase is changed, that of the vapor phase changes in the same sense. (2) If the proportion of Si is greater in the vapor than in the hquid phase, when the temperature remains con- stant the pressure is increased by the addition of Si. In the same way, it can easily be shown that when dp = 0, dt and dmi have opposite signs when mi" > mi. Therefore we have (3) If the proportion of ^Si is greater in the vapor than in the liquid phase, when the pressure remains constant the temperature is decreased by the addition of Si. (4) If the proportion of Si is the same in the vapor as in the liquid phase, the pressure is a maximum or a minimum at constant temperature, and the tempera- ture a maximum or minimum at constant pressure (See p. 113). These rules, which are illustrated by the examples shown in Figures 2 and 3, were first stated by D. Konowalow.f * It may be zero if the phase is at the limit of stability, t Wied. Annalen, 14, 48 (1881). 178 BUTLER ART. D XIV. Phases of Dissipated Energy. Catalysis* 41. Dissipated Energy. In considering the conditions of equihbrium of heterogeneous masses, changes which are "pre- vented by passive forces or analogous resistances to change" have been excluded. Thus it often happens that "the number of proximate components which it is necessary to recognise as independently variable in a body exceeds the number of com- ponents which would be sufficient to express its composition." Thus, at low temperatures the combination of hydrogen and oxygen may be regarded as prevented by passive forces, and in a system containing hydrogen, oxygen and water it is neces- sary to recognize all three substances as independently variable components. At higher temperatures, when the combination of hydrogen and oxygen is not prevented by passive forces, the state of the system is entirely determined by the temperature, pressure and the total quantities of hydrogen and oxygen present. The value of f can be expressed as a function of these four variables. The fact that part of the matter present exists in the form of water vapour does not affect the form of this function, but it is one of the facts which determine the nature of the relation between ^ and the above mentioned variables. In cases like those first mentioned^ of all the phases which may be formed from the given matter, there are some for which the energy is as small as that of any other state of the same matter having the same entropy and volume, or for which the value of ^ is as small as that of any other state of the same matter at the same temperature and pressure. These are called phases of dissipated energy. It is characteristic of such phases that the equilibrium can only be slightly disturbed by the action of a small body, or by the action of a single electric spark. The effect produced by any such action is in some way proportionate to its cause. But in a phase which is not a phase of dissipated energy, it may be possible to cause very great changes by contact with a very small body, or other action. Such changes may only be limited by the attainment of a phase of dissipated energy. * Gibbs, I, 138-141. THERMODYNAMIC AL SYSTEM OF GIBBS 179 Gibbs describes the effects which may cause a system to undergo changes of this kind in the following terms : "Such a result will probably be produced in a fluid mass by contact with another fluid which contains molecules of all the kinds which occur in the first fluid (or at least all those which contain the same kinds of matter which also occur in other sorts of molecules), but which differs from the first fluid in that the quantities of the various kinds of molecules are entirely deter- mined by the ultimate composition of the fluid and its tem- perature and pressure. Or, to speak without reference to the molecular state of the fluid, the result considered would doubt- less be brought about by contact with another fluid, which absorbs all the proximate components of the first, *Si, ... ^2, • • • Mn on the pressure p is, on the other hand, of funda- mental importance in the treatment of membrane equiUbria because in general the pressures of two phases in membrane equihbrium will be unequal. The required relation is obtained by making use of the mathematical identity dp dnih dvih dp where ^ is defined by ^ = e-tv + pv, (18) [91] OSMOTIC AND MEMBRANE EQUILIBRIA 187 and is the characteristic function corresponding to our choice of independent variables t, p, rrii, rih, ... Wn. The dependence of variations of f on those of the independent variables is given by d^ = —r]dt-\- vdp + tildmi + )U2C?W2 . . . + Undnin. (19) [92] From (19) [92] we see that drtih and dp = MA, (20) = V. (21) Substituting from (20) and (21) into (17) we obtain dnh dv dp drrih = vh, (22) where Vh denotes the increase in volume of a very large phase when one adds to it unit quantity of the species Sh, keeping the temperature and pressure constant. The volume Vh may be called the "partial volume" of the species Sh. 5. Mols and Mol Fractions. Up to this point we have purposely referred to nih as denoting the number of "units of quantity" of the species Sh without specifying what is this "unit of quantity." Willard Gibbs, living at a time when the molecular theory was less firmly established than at present, chose the same unit of mass for the unit of quantity of each species. In a letter to W. D. Bancroft (Gibbs, I, 434) he agrees, however, that "one might easily economise in letters in the formulae by referring densities (7) and potentials (n) to equivalent or molecular weights." We shall therefore adopt this procedure and take as unit quantity of each species the gram-molecule or mol in the highly dilute vapor state. None of the formulae so far given are affected, but the potentials fi now have the dimensions calories per mol instead of calories per gram, and the formulae expressing the dependence of the 188 GUGGENHEIM ART. E potentials ^t on the composition take a simpler form. Similarly Vh denotes the increase in volume of a very large phase when one adds to it one mol of the species Sh, keeping temperature and pressure constant. Therefore Vh will be called the "partial molar volume" of the species Sh- As already mentioned the potentials /zi, /i2, ... Mn will be functions not only of t and p but also of the number of mols mi, m2, . . . w„ of the various species in the phase. Actually it is clear that each n will depend on the composition of the phase but not on the absolute quantity of it. That is to say, m, 1X2, ... iin will be functions of the quantities A^i, N2, . . . Nn defined by ,. 'fni -/V 1 - 1 + m„ N2 = I + nin Mn Nn - , r (23) nil -\- nii . . . -\- rrin The quantities A^i, A''2, ■ • . Nn are called the mol fractions of the species Si, S2, ... Sn- They are, of course, not mutually independent but are subject to the identical relation A^i + A^2 . . . + A^. = 1, from which it follows that dNi + dNi ... + dNn = 0. (24) (25) 6. Ideal Solutions. A series of solutions in a given solvent are said to be "ideal" if throughout a range of concentrations extending continuously down to pure solvent the potential Hh of each species Sh whether solvent or solute obeys the formula IJih = Hh\t, V) + ^t log A^;,, (26) where Hh^{t, p) is independent of the composition of the solution and .4 is a universal constant known as the "gas constant." OSMOTIC AND MEMBRANE EQUILIBRIA 189 This definition of ideality is exactly equivalent to the condition that for a given temperature and external pressure on a solution the partial vapor pressure of each component shall be directly proportional to its mol fraction. Since A, t and Nh are all independent of p, it follows from (22) that P = ... (27) dp As, by definition, fXfP at given temperature and pressure is inde- pendent of the composition, it follows that the same is true of Vh. This means that the transference of any part of an ideal solution to another ideal solution in the same solvent takes place, at constant temperature and pressure, without volume contraction or expansion. For the dependence of Vk on the pressure p we may write Vh = Vh*(l - khp), (28) where Vh* is the value of Vh at vanishing pressure, and where it will always be allowable to assume that kh is independent of the pressure p. The compressibility coefficient kk may depend on the temperature but this need not concern us. Owing to the relations (27) and (28) we may replace (26) by M/. = y^h*{t) + pv,*{l - hxhP) + At log Nk, (29) where Hh*(t) is independent of the pressure as well as of the composition. If we now substitute from (29) into the general condition of membrane equilibrium (4) [77], we obtain w p' vh*{1 - hhP' ) + At log N,/ = p"vh*{l - hhP") + At log Nh", (30) or Nh" (p' - P") Vh* (l - KH ^^-^) = At log Nh'' (31) 190 GUGGENHEIM art. e Hence where [vh] is defined by (33) and is equal to the partial molar volume of the species Sh at the given temperature and at a pressure equal to the mean of the pressures p' and p" on either side of the membrane. Formula (32) is exact for membrane equilibrium as regards the species Sh between two ideal solutions in the same solvent, whether Sh denote the solvent species or one of the solute species. 7, Non-ideal Solutions. The range of concentrations over which solutions remain ideal varies very much according to the nature of the solvent, the nature of the various solute species and the temperature. It is however generally accepted that in the neighbourhood of infinite dilution all solutions become ideal. This provides a convenient thermodynamic treatment of solutions that are not ideal. In analogy with (26) we may write formally for any species Sh, whether solvent or solute, HH = tih\t, p) + At log Nhfhy (34) where in^H, p) is for a given solvent independent of the compo- sition. In general /;, is a function of temperature, pressure and composition, but has the simplifying property that for given temperature and pressure its value approaches unity as the dilution approaches infinity. It is called the activity coefficient of the species Sh and is a measure of the deviation of the solution from ideahty so far as the species Sh is concerned. Since ix}^{t, p) is by definition independent of the composition, and we are assuming that in the neighbourhood of infinite dilution the solutions become ideal, it follows that /xa''(^ v) must OSMOTIC AND MEMBRANE EQUILIBRIA 191 be of the same form as for ideal solutions. In accordance with (29) we may therefore write MA = MA*(0 + PVh*(l - hxhP) + At log NhSh, (35) where Hh*it) is independent of the pressure as well as the com- position; Vh* is the value of the partial molar volume of the species Sh at the given temperature, at zero pressure and at infinite dilution; kh is independent of the pressure and the com- position; while Vk*(l — Khp) is the value of the partial molar volume of the species Sh at the given temperature, the given pressure and at infinite dilution. The activity coefficient fk at given temperature and pressure tends to unity at infinite dilution. If we differentiate (35) with respect to p and use (22) we obtain Vfc = — = Vh* (1 - KhP) + At (36) or d log fh _ Vh - Vh* {I - Khp) dp ~ At (37) From this we see that the activity coefficient fh will or will not vary with the pressure at given temperature and composition, according as the partial molar volume Vh in the solution is un- equal or equal to its value Vh*{l — Khp) at infinite dilution at the same temperature and pressure. If we now substitute from (35) into the general condition of membrane equilibrium (4) [77] we obtain p'vh*{l - hxkP') -\- AtlogNh'U = p"vh*(l - hKhp") + At log Nh'Jh" (38) or ip' - P") Vh* (l - K. ^) = At log ^^'. (39) 192 GUGGENHEIM art. e Hence , „ At N,"U" ,^„, where [vh] is defined by M = ^A* I 1 - KA ^ 1 (41) and is the partial molar volume of the species Sh in an infinitely dilute solution at the given temperature and at a pressure equal to the mean of the pressures p' and p" on either side of the membrane. Formula (40) is exact for membrane equihb- rium as regards the species Sh between two non-ideal solutions of the most general type in the same solvent, whether Sh denote the solvent or one of the solute species. It is important to observe that the values of the activity coefficients to be inserted in the formula are those at the actual pressures at membrane equilibrium, that is fh at the pressure p' smdfh" at the pressure 8. Osmotic Equilibrium. If in particular the membrane is permeable to the solvent only, but impermeable to aU the solute species, the membrane equilibrium is called "osmotic equilib- rium." If the phase denoted by a double accent is the pure solvent the difference p' — p" is called the "osmotic pressure" of the solution represented by the single accent. In this case, using the suffix 0 to denote the solvent, we have N," = 1, (42) and so the osmotic pressure P in ideal solutions is given by At 1 P = p'-p" = j^log^,. (43) while in non-ideal solutions it is given by At 1 OSMOTIC AND MEMBRANE EQUILIBRIA 193 the value of fo being that at an external pressure p', and [vq] being the value of the partial molar volume of the pure solvent at the given temperature and at a pressure equal to the mean of those (p' and p") at either side of the membrane. 9: Iricompressible Solutions. If it is allowable to neglect the compressibility kq of the solvent, one need not distinguish between [vo] and vq*, and the formulae for P may be written At 1 P = — log — 45) Vo* No for ideal solutions, and At 1 , , P = — log 77-7 46 Vo* Nofo for non-ideal solutions, the value of /o being that corresponding to an external pressure p' somewhat exceeding the osmotic pressure P. From (45) we see that when compressibility is neglected the osmotic pressure of an ideal solution is independent of the external pressure on the pure solvent with which it is in osmotic equilibrium. 10. Relation between Activity Coefficients. The variations of the activity coefficients of the different species with variations of composition at a given temperature and pressure are not completely independent. For according to [98] (Gibbs, I, 88) we have at given temperature and pressure dt = 0, (47.1) dp = 0, (47.2) (fjii + m2dn2 . . . + nindixn = 0, (47.3) or, dividing by (mi + m2 . . . + m„), NidfX, + N2dtJi2 ... + NndlXn = 0. (48) If we substitute from (34) or (35) into (48), we obtain N4 log N,f, + N^d log N^U . . . + Nnd log .¥„/„ = 0. (49) 194 GUGGENHEIM art. b But Nxd log iVi + Nd \0gN2 ... + Nnd log iV„ = dNi + dNi . . . + rfiVn = 0 (50) according to (25). It follows from (49) and (50) that Nid log /i + N^d log /2 . . . + Nnd log /„ = 0. (51) From (51) we can conclude in particular that, if throughout a range of concentrations extending down to pure solvent the activity coefficients of all the solute species are unity, then this must also be the case for the solvent species. This is equivalent to the following theorem : If at given temperature and pressure but varying composition every solute species has a partial vapor pressure proportional to its mol fraction (Henry's law), then so has the solvent (Raoult's law). 11. Osmotic Coefficients. Owing to the relation (51), if the mol fraction of the solvent species is almost unity and the mol fractions of all the solute species are very small compared with unity, the value of log/o for the solvent species will generally be of a considerably smaller order of magnitude than that of log /, for any of the solute species Sg. Thus it is quite usual in a centimolar aqueous solution of a uni-univalent strong electrolyte for the activity coefficient of the solute to be less than unity by about 0.1, while the activity coefficient of the solvent in the same solution will be approximately 1.00006. Thus for purely numerical reasons the activity coefficient of the solvent species, in contrast to the activity coefficient of the solute species, may be an inconvenient function to work with. For this reason it is often convenient to define another function called the "osmotic coefficient" of the solvent, and denoted by g, by the relation or g log No = logNofo. (53) OSMOTIC AND MEMBRANE EQUILIBRIA 195 Using the sufl&x s to denote solute species and substituting (52) into (51) we obtain Nodil - g-log No) = - Nod log/o = ^Nsd\ogU (54) s If No is almost unity and all the A^,'s are very small compared with unity, we have approximately - log No= - log (i-1^n)\ = Yj Ns, (55) and (54) becomes approximately d(r^'^ n)\ + Yj Nsdlogf, = 0. (56) From this approximate relation we can conclude that 1 — g^ is likely to be of the same order of magnitude as log /,, or as 1 — /,. Thus in very dilute solutions not deviating greatly from ideality the osmotic coefficient g will have a more convenient numerical value than the activity coefficient /o of the solvent species. Substituting (53) into (35) we obtain for the chemical po- tential of the solvent in a non-ideal solution MO = Mo*(0 + PVo*(l - h xop) + gAt log No. (57) The osmotic coefficient g, like the activity coefficient /o of the solvent species, will at given temperature and pressure tend to unity at infinite dilution when the solutions become ideal. Differentiating (57) with respect to p and using (22) we ob- tain for the dependence of the osmotic coefficient on the pressure vo = vo* (1 - Kop) + At log No- J- (58) op or di _ yp - ro* (1 - KqP) dp ^ At log No * ^^ 196 GUGGENHEIM art. e Thus at given temperature and composition the osmotic co- efficient, hke the activity coefficient of the solvent, will or will not vary with the pressure according as the partial molar volume of the solvent Vq in the solution is unequal or equal to its value yo*(l — kqp) in the pure solvent at the same temperature and pressure. 12. Osmotic Equilibrium in Terms of Osmotic Coefficient. Substituting from (57) into (4) [77] we obtain as the general condition of membrane equilibrium for the solvent between two non-ideal solutions ip' - V") vo* (l - Ko ^^^') = At ig" log No" - g' log N^'), (60) or introducing [vo] the partial molar volume of the pure solvent at the given temperature and at a pressure equal to the mean of those p' and p" at either side of the membrane. At V' -V" = ^^^ ig" log No" - g' log No'), (61) the values of g' and g" being those at pressures p' and p" respectively. K we assume the membrane to be permeable to the solvent species only, and take the phase denoted by the double accent to be pure solvent, we have log N" = 0, (62) and so obtain for the osmotic pressure P At 1 ^ = "'-''" = ''Si 'OS iv'' («3) the value of g' being that at an external pressure p'. If it is allowable to neglect the compressibility of the solvent one need not distinguish between [vo] and vo*, in which case instead of (63) one may write At 1 P = 0'-,iogj,. (64) OSMOTIC AND MEMBRANE EQUILIBRIA 197 the value of g' being that at an external pressure p' somewhat greater than P. Comparing (64) with (45) we see that, when we neglect the compressibihty, the osmotic coefficient is the ratio of the actual osmotic pressure in a non-ideal solution to its value in an ideal solution of the same composition. This is the origin of the name "osmotic coefficient." 13. Extremely Dilute Solutions. If a solution, whether ideal or non-ideal, is so dUute that the mol fractions N, of all the solute species are extremely small compared with that of the solvent A^o, we may make the three approximations: log ^^ = - log (l - S ^•) = S ''•• ^^^'^ N. = ^^^ = ^'^ (66) mo Wo 4- 7 , ms s V = moVo -\- 2j ^« ^» = ^0 1'o*. (67) 8 Formula (45) for ideal solutions then takes the approximate form P = ~^rn, = At^ y., (68) where 7, denotes volume concentration. Similarly formula (46) for non-ideal solutions takes the approximate form P =gAt^y,. (69) s Formula (68) is contained in some fragmentary material by Willard Gibbs published after his death (Gibbs, I, 421, equation [7]). For its approximate validity it is necessary to assume not merely that the solution is ideal and incompressible, but also that it is extremely dilute. This formula was originally due to van't Hoff, who realised its limitations. It has unfortunately 198 GUGGENHEIM art. e been applied only too often under conditions where it cannot be even approximately correct. 14. Electric Potential Difference between Two Identical Phases. Up to this point we have tacitly assumed that all the species present were electrically neutral. The fundamental difference between the behavior of ions and of uncharged species is the following. The potential of an uncharged species in a phase at given temperature and pressure is completely determined by the bulk composition of the phase, and is independent of the presence of any impurity at the surface as long as its concen- tration in the bulk is negligible. This, however, is not the case for ions. Let us consider two phases identical with respect to temperature, pressure, size, shape and bulk composition. Then it may be that the first phase contains an excess of ions of one or more kinds over the second phase, this excess being so small that its effect on the size, shape and bulk concentration of the phase is entirely negligible. If however the total excess of ions in the first phase over those in the second has a net electric charge, the corresponding excess charge will be distributed over the surface of the first phase, and the potential of any ionic species within the phase will be affected thereby. The difference between the potential of a given ionic species in the first phase and in the second will be determined entirely by the difference in distribution of electric charge over the surfaces of the two phases and independent of the chemical nature of the excess ions. One might describe the situation roughly by saying that the excess ions in the first phase over those in the second are too few to show themselves in any manner except by their electrical effect. It is usual and convenient to refer to two such phases as "of identical composition but at different electric potentials." To emphasize the peculiar property of the potential of an ionic species, that it is not completely determined by the bulk com- position of the phase, a slightly modified symbol will be used. The potential of the ionic species Si will be denoted by [nil- The difference between its value in the two phases of identical composition will be of the form Wi]' -im]" = ZiF{V' -V") (70) OSMOTIC AND MEMBRANE EQUILIBRIA 199 where z » denotes the valency (positive or negative) of the ionic species Si and F denotes the faraday, so that ZiF is the charge of one mol of the ionic species. Finally V, Y" have values independent of the type of ion being considered, and V — V" is called the "electric potential difference" between the two phases. This may at first sight appear a strange method of defining electric potential difference between two phases of "identical" composition, but it does not seem possible to give a simpler definition that is not ambiguous. The usual definition of the mathematical theory of electrostatics is not applicable to thermo- dynamic systems, for the conditions of thermodynamic equihb- rium of ions are by no means the same as the conditions of equilibrium of "static electricity." 15. Electric Potential Difference between Two Phases of Different Composition. If we now consider the difference of the potential of a given ionic species between two phases of different bulk composition, this difference will be determined partly by the difference in the chemical composition in the bulk and partly by the distribution of electric charge at the surfaces. This may be expressed formally as M' - W = W - m/0 + ZiFiV - 7"), (71) where [m] denotes the potential of the ionic species, m* denotes the part of the potential due to the chemical composition of the phase and z,- FV the part due to the distribution of electric charge at its surface. The quantity [m,] may be called the "electro- chemical potential" of the species Si, m may be called the "chemical potential" of the species Si, and V may be called the "electric potential." When, however, we come to ask ourselves exactly what would be meant by the statement that the electric potential V had the same value in two phases of different composition, w^e would have to admit that the statement had in general no physical significance. All equifibria and changes towards equihbrium are completely determined by the electrochemical potentials IJLti], and any decomposition of [m] into two terms m and ZiFV is in general arbitrary. This attitude is in accordance with a 200 GUGGENHEIM art. e remark of Willard Gibbs (Collected Works, I, 429): "Again, the consideration of the difference of potential in the electrolyte, and especially the consideration of the difference of potential in electrolyte and electrode, involves the consideration of quan- tities of which we have no apparent means of physical measure- ment, while the difference of potential in 'pieces of metal of the same kind attached to the electrodes' is exactly one of the things which we can and do measure." Unfortunately not all chemists have been as careful as Willard Gibbs in avoiding the expres- sion "difference of electric potential" when referring to two phases of different composition. 16. Combinations of Ions with Zero Net Electric Charge. The potential [/xj of a given ionic species in a certain phase is the increase in the characteristic function when one mol of the given species is added to the phase, keeping all the other inde- pendent variables unaltered. In particular it is the increase in f when one mol is added at constant temperature and pressure. If we consider, not the addition of a single ionic species but the simultaneous addition or removal of several species, say the addi- tion of Xi mols of the species S„ where Xi may be positive or negative, then the corresponding increase in f will be ^ Xi [m]. i Making the substitution in (71) we have formally i » » Suppose now that the net electric charge of the ions added is zero. The condition for this is 2 ^i ^i = 0- (73) i If this condition is satisfied then (72) becomes i i Thus, although the chemical potential of an individual ionic species is indeterminate, certain linear combinations of the OSMOTIC AND MEMBRANE EQUILIBRIA 201 chemical potentials of ionic species are determinate and, in fact, equal to the corresponding linear combinations of the electrochemical potentials, the condition for this being that the linear combination corresponds to a combination of ions with zero net electric charge. The physical meaning of this is simply that the potential of a combination of ions with zero net electric charge is determined completely by the chemical composition in the bulk of the phase and is independent of its electrical state. 17. Ideal Solutions of Ions. At very high dilutions of ions aU equilibria are given correctly by assuming that the electro- chemical potential [^u,] of the ionic species i since by supposition the X/s satisfy the relation (73). It follows that, although the individual ionic activity coefficients /,• are physically indefinite, certain combinations of them of the form ^ ^i log fi, (80) or n (/')' (81) are completely determinate whenever the Xi's satisfy (73). 19. Mean Activity Coefficient of Electrolyte. Of the various possible products of activity coefficients of the type (81) which OSMOTIC AND MEMBRANE EQUILIBRIA 203 are physically determinate, the most important is the "mean activity coefficient" of an electrolyte. Thus for an electrolyte consisting of q+ positive ions of valency z+ and g_ negative ions of valency z-, the condition of electrical neutrality is q+z+ + q-z- = 0. (82) It follows that the quantity /±, defined by q+ log/+ + 9_ log/_ = (g+ + qJ) log/±, (83) where /+, /_ are the ionic activity coefficients, or by (/J ..+ ._ = (/+)^.(/_)s (84) is completely determinate although the ionic activity coefficients /+ and /_ are to some extent arbitrary. The function /^ is called the mean activity coefficient of the electrolyte. Another example of a combination of ionic activity coeffi- cients that is definite is the ratio of the activity coefficients of two cations, or of two anions, in the same solution and of the same valency. W. Membrane Equilibrium, of Ideal Ionic Solutions. We are now in a position to write down directly the conditions of membrane equilibrium for ionic solutions. We have merely to substitute the values of the potentials [m] in the general con- dition of membrane equilibrium [Mi]' = U.r'. (85) For ideal solutions we obtain according to (75) p' Vi*(l - ^Kip' ) + At log Ni' + Zi FV = p"vi*(l - iKip") + At log Ni" + ZiFV". (86) Introducing [v^, the partial molar volume at infinite dilution at the given temperature and at a pressure equal to the mean of those {p' and p ") at either side of the membrane, this becomes At log -^= ip' - p") k] + ZiF{V' - V"). (87) 204 GUGGENHEIM art. e Comparing formula (87) for two ionic species i and h of the same valency z, we obtain At log ^ 0 = iv' - V") ( N - k] ). (88) The right hand side of (88) will generally be small compared with At and may often with sufficient accuracy be regarded as zero. With this approximation (88) simplifies to N-' N-" Applying formula (87) to the two ionic species of an electro- lyte composed of g+ cations of valency 0+ and g_ anions of valency Z-, we obtain At\og(^-^j [jjj = (p' - p") iq^v^] - q-[v-]). (90) The right hand side of (90) will generally be small compared with At and may often with sufficient accuracy be regarded as zero. To this degree of accuracy we may replace the exact formula (90) by the approximate one (N+')'^. (NJ)"- = (N+")'^.{N -")"-. (91) If we compare (90) for the membrane equilibrium of a solute electrolyte with (32) for the equilibrium of the uncharged solvent, we obtain ^ /NV'Y fN-"Y g4-[M + q-[v-] . No" .^^. or {N+T (Njy- (N+'T (N-'T (No'y {No"y where r is defined by _ q+M + q-[v-] ' ~ [vo\ (93) (94) OSMOTIC AND MEMBRANE EQUILIBRIA 205 and is the ratio of the partial molar volume of the electrolyte to that of the solvent, both at the given temperature and at a pressure equal to the mean of those at either side of the mem- brane. At extreme dilutions the mol fraction No of the solvent differs very shghtly from unity, and (93) approximates to (91). 31. Membrane Equilibrium of Non-ideal Ionic Solutions. The corresponding formulae for non-ideal solutions are obtained similarly by substituting from (77) in the general condition of membrane equilibrium, [Mi]' = [m.]". (95) For two ionic species i and h of the same valency, we obtain in analogy with (88) At log ^1^-^ ^1^^ = (p' - v") ( k] - M), (96) Nn"h"Ni'fi' where [yj, [vh] are the values of the partial molar volumes at infinite dilution at the given temperature and at a pressure equal to the mean of those (p' and y") at either side of the mem- brane. It is to be observed that the combinations of activity coefficients occurring in (96) are the ratios of the activity coefficients for two ions of the same valency and are therefore physically definite. If the right hand side of (96) is neghgibly small compared to At, then (96) approximates to the simple relation N-' f' N-" f" Nh'Sh' Nk"U" For the membrane equilibrium of an electrolyte consisting of g+ cations of valency z+ and g_ anions of valency z-, the exact formula obtained from (77) and (95) is, in analogy with (90), = (p'-p")(9+M + 9-[y-]), (98) which involves only the mean activity coefficients /^ of the electrolyte in the two phases. If the right hand side of (98) 206 GUGGENHEIM art. e is negligibly small compared with At, then the exact formula (98) may be replaced by the approximate one (A^+0 «.(A^_') «-(/±') -'.+ "- = {N+") ".{N-") «-(/i") '.+ '- . (99) The corresponding formula for the membrane equilibrium of a single ionic species in non-ideal solutions takes the form At log ^ + At log^' = (p' - p") M + z, FiV - F'OdOO) //' but tells us nothing, as neither the term At log 77 on the left J* nor the term Zi F(y' — V") on the right is physically deter- minable. S2. Contact Equilibrium. A most important case of mem- brane equilibrium is that of two phases with one common com- ponent ion, the surface of separation forming a natural mem- brane permeable to the common ion but impermeable to all others. This may be referred to as "contact equiUbrium." For example, for two metals in contact, say Cu and Zn, there is equilibrium between the two phases as regards electrons El~ but not as regards the positive ions Cm"''"*" or Zn^'^. The equilibrium is completely defined by [M^z-]^« = [Uni-Y-, (101) the suffix denoting, as usual, the component, and the index the phase. Similarly for a metaUic electrode of Cu, dipping into a solution S containing ions of this metal, in this case Cw'''+, the contact equilibrium is completely defined by [Mcu-]"'" = [Mcu-]^ (102) the electrode and solution being in equilibrium as regards the metallic ions only. In neither of these cases of contact equilib- rium is any "contact electric potential difference" thermo- djoiamically definable. 28. Purely Chemical Cell. Consider the system composed of the following phases and membranes arranged in order, each phase being separated by partially permeable membranes from OSMOTIC AND MEMBRANE EQUILIBRIA 207 its neighbouring phases, and completely separated from the remaining phases. Phase a. Containing, inter alia, species A and B. Membrane 1. Permeable to B only. Phase /3. Containing, inter alia, species B and C. Membrane 2. Permeable to C only. Phase y. Containing, inter alia, species C and A. If all the species A, B, C are electrically neutral, the two membrane equilibria are determined completely by the con- ditions 4 = Mb. (103.1) nZ = 4, (103.2) *c f'c, but in general f^l^t^:, (103.3) that is, the phases y and a are not in equilibrium as regards the species A. If the phases y and a be now brought into contact through a membrane permeable to A only, there will be a flow of A from the one to the other in a direction determined by the sign of /x][ — n". This flow will, of course, upset the other membrane equilibria, which will readjust them- selves. The flow of A through the auxiliary membrane and the accompanying readjustments will not cease until either the phases y and a are again separated, or the conditions 4 = ^^s, (104.1) f^l = mJ, (104.2) y « Mx = M^ , (104.3) are satisfied simultaneously. We may call the system just described a cell," and the difference "purely chemical Ml - m! (105) the "chemico-motive force" of the cell for the component A. Bringing the phases y and a into contact through a membrane permeable only to A we may call short-circuiting the cell, and 208 GUGGENHEIM art. e separating these phases "breaking the circuit." When the conditions (104) are satisfied simultaneously we may say that the cell is "run down." More complicated "purely chemical cells" might be described, containing a larger number of phases, membranes and com- ponents, but the general nature of any such cell and the condi- tions of equilibrium will be similar to that of the above simple example. The "purely chemical cell" is not of practical importance and, possibly for this reason, is not usually described or discussed in text-books. It has been described here since a clear understand- ing of a "purely chemical cell" should facilitate a complete comprehension of the nature of an "electrochemical cell," which will be discussed next. It is especially to be emphasized that from a theoretical thermodynamic point of view the electric charges of the ions are rather incidental, the fundamental factors at the base of any cell, whether "purely chemical" or "electro- chemical," being the membrane or contact equilibria between successive phases. 24. Electrochemical Cells. The only essential difference between an "electrochemical cell" and a "purely chemical cell" is that in the former the membrane equilibria involve charged ions. Let us consider the following system, somewhat similar to the purely chemical cell discussed above, in which however the various species concerned are ions. Phase a. Containing ions E and A. Membrane 1. Permeable to ions A only. Phase /3. Containing ions A and B. Membrane 2. Permeable to ions B only. Phase 7. Containing ions B and E. Membrane 3. Permeable to ions E only. Phase a'. Chemically identical with phase a. The three membrane equihbria are defined completely by the conditions : . WV = M", (106.1) [fJiBp = M^ (106.2) M"' = My, (106.3) OSMOTIC AND MEMBRANE EQUILIBRIA 209 but in general [heY 9^ M". (106.4) As compared with the example of a purely chemical cell, we have included in the present system one extra phase and membrane in order that the two extreme phases or "terminals" a and a' should have the same chemical composition. We may therefore write WY - [heY = ZEFiv^' - y«), (107) and the difference of electric potential (7«' — "F") thus defined is called the "electromotive force" E of the cell. Putting the two phases a and a into contact is called short-circuiting the cell and separating them "breaking the circuit." On closing the circuit there will be an adjustment of membrane equilibria with net flow of electric charge round the circuit in a direction determined by the sign of E. This will cease when the con- ditions [Hj,f = [iia]", (108.1) Mb]^ = [(MbV, (108.2) [he]"' = [ms]^ = [m^]", (108.3) are satisfied simultaneously, when the cell is said to be "run down." We will now give a concrete example. We suppose the ionic species A to be Cw++, B to be Zn++, and E to be electrons El- and thus obtain the cell Cu a Solution S containing Cw++ and Zn ++ Zn Cu. a' We also imagine the boundaries between the phases to form natural membranes, each permeable to only one ionic species. In practice there would be irreversible deposition of copper on the zinc, and this cell would not function unless some means of preventing Cw*"^ ions from coming into contact with the metal Zn were provided. We have oversimplified the descrip- 210 GUGGENHEIM art. e tion of the cell in order to avoid a discussion of diffusion poten- tials. A workable cell would be the following : Cu Solution Si containing Cw++ and large excess of other ions Solution ^2 containing Zn+"'' and large excess of other ions Zn Cu. The diffusion potential between the two solutions Si and S2 could be made negligible by making the composition of the two solutions substantially the same apart from the Cm++ ions in the one and Z7i++ ions in the other, the concentration of these being in both cases small compared with the concentrations of the other cations. In the metallic phases we have the purely chemical, homoge- neous equihbrium conditions [/icu-P + 2[Ms,-P = Me:, (109.1) [y.zn^f' +2[M^,-f" = Mf:, (109.2) where ^^'^ and ^f^' are independent of the electric states of the respective phases. The contact equilibrium conditions are Ucu++]; = Ucu«-]"> (110.1) Uzn-]^" = Uzn-l^ (110.2) wr'^ = UEi-f"- (110.3) Combining (107), (109), (110) we obtain for the electromotive force E 2FE = [ncu+A" — [^J■cu^]" = I'cl - 4l + [/^^"-l^ - ^^cu*^^^^ (111) or, in terms of activity coefficients, 2f <-/i- E = E' + ^\og '^■^, (112) where E° is independent of the composition of the solution, the values of the mol fractions N^ and activity coefficients f^ being those in the solution. OSMOTIC AND MEMBRANE EQUILIBRIA 211 More detailed discussion of electrochemical cells would be outside our province, but the above example serves to show that the electromotive force of any cell may be computed by regard- ing the mechanism of the cell as a combination of several membrane equilibria. The electromotive force E is equal to the difference of potential of any univalent positive ion in the two terminals of the same metal at the two ends of the cell. This is the only electric potential difference that is measured, and is the only one to which any reference is made in this treatment. As already mentioned, this attitude towards the conception of electric potential is in accordance with views expressed by WiUard Gibbs. BIBLIOGRAPHY Laws of Ideal Solutions These were given in an exact form by G. N. Lewis, J. Am. Chem. Soc, 30. 668 (1908) and by E. W. Washburn, Z. physikal. Chem., 74, 537 (1910). Activity Coefficient. The definition of this useful function is due to G. N. Lewis. See Thermodynamics and The Free Energy of Chemical Substances, by G. N. Lewis and M. Randall (New York, 1923). Osmotic Coefficient. This was first used by N. Bjerrum at the Scandina- vian Science Congress 1916. See German translation in Z. Elek- trochem., 24, 325 (1918). Membrane Equilibrium. The theory of ionic membrane equilibrium was first developed for extremely dilute ideal solutions by F. G. Donnan, Z.Elektrochem., 17, 572 (1911). The exact thermodynamic treatment of solutions neither ideal nor dilute was given by F. G. Donnan and E. A. Guggenheim, Z. physikal. Chem., A 162, 346 (1932); F. G. Donnan, ibid., A 168, 369 (1934). Electrochemical Systems. Gibbs' method of treatment of equilibrium and stability was extended to electrochemical systems by E. A, Milne, Proc. Camb. Phil. Soc, 22, 493 (1925) and by J. A. V. Butler, Proc. Roy. Soc, 112, 129 (1926). Electrochemical Potentials. The use of these functions to replace the conception of electric potential difference between phases of differ- ent chemical composition is due to E. A. Guggenheim, /. Phya. Chem.. 33, 842 (1929), F THE QUANTITIES x, ^, T, AND THE CRITERIA OF EQUILIBRIUM [Gibbs, I, pp. 89-92] E. A. MILNE The following notes amount to an independent treatment of Gibbs' results in this section. They also iaclude an extension of some of his calculations so as to take account of second order terms where discussion of first order terms alone ("differen- tials") is insufficient. Some of the later calculations are adapted from Lewis and Randall's Thermodynamics. 1. Stability Tests. At the beginning of his memoir, The Equilibrium of Heterogeneous Substances, Gibbs establishes criteria of stability which may be stated as follows : Let A denote any increment of a quantity, not necessarily small. Let d denote a "differential" of the quantity, which may (non-rigorously) be identified approximately with a small increment. Then if e denotes the energy of a system, ?? its entropy, we have: For stable equilibrium, (At;), < Oor (Ae), > 0. For neutral equilibrium, in general, (At,), ^ Oor(Ae), ^0, but there exist variations for which (Atj), = Oor (Ac), = 0. For unstable equilibrium, (rfT,), = Oor(d€), = 0, 213 214 MILNE ART. F but there exist variations for which (At;). > Oor(A€), < 0. In the above, the subscript denotes that the corresponding variable is maintained constant in the variation. Gibbs proceeds, in the section under consideration (Gibbs, I, 89-92), to estabhsh the equivalence of the above to similar variational conditions involving (1) the work function yp, defined hy ^p = e — t-q, (2) the heat function x, defined by x = « + P^, (3) the free energy function f , defined hy ^ = e — tr] -\- pv. He gives a method of proof which is sound in principle, and which suggests the method to adopt, but which does not dis- tinguish between small variations and finite variations. The following includes the substance of Gibbs' results, and supplies proofs in certain cases where Gibbs left the proof to the reader. 2. The Work Function. The value of the criteria about to be discussed is that they render the general criteria more easily applicable to certain particular cases, by restricting the type of variation permitted. For example, in certain cases they impose a condition of constancy of volume in addition to constancy of entropy, in discussing changes of energy. We shall now prove that the condition W),.v^O (1) is equivalent to the condition (A6),.„^0. (2) For suppose that there exists a neighbouring state for which (Ae),., <0. We shall prove that there then exists a state for which (A^),,„ < 0. This will ensure that if we are given that (1) is true, no con- tradiction of (2) can exist; hence (1) implies (2). For, if the neighbouring state for which (Ae),, , < 0 is not X, ^, r, AND THE CRITERIA OF EQUILIBRIUM 215 one of uniform temperature, let its temperatures be equalized at constant volume. This can only increase its entropy. Now remove heat so as to reduce the entropy to the initial value, at the same volume. This process reduces the energy. Thus we have constructed a state of uniform temperature for which (Ae),,„ < 0. Now we have \p = ^ — tv, whence in general ArJ/ = Ae — tAr] — rjAt — AtArj. In our case At; = 0, and so A\f/ = Ae — r]At or A^p + v^t = Ae < 0, (3) by hypothesis. Now add or subtract heat at constant volume. For such a process the infinitesimal increment in energy, say rf'c is given by d'e = t d'-n, whilst similarly d'\p = d't - -nd't - t d'-n, i.e., d'^ = -r,d't. It follows that the fi7nte increment in \l/, namely A'\p, is given by /t+A't r, d't. (4) Accordingly, by (3) and (4), A\P + AV < - 7?Af + jv d't. J t + A't 216 MILNE ART. F Now choose A't = —At, thus restoring the initial temperature (a state for which \l/ is defined is of course necessarily a state of uniform temperature). We have then At/' + AV < - riM + i^d't, where now to denotes the initial temperature. This gives At/' + AV < - -^0 Ai + / ° Uo + f-^l (t-to) + .. .\d% where t/o denotes the initial entropy. Evaluating the integral we have At^ + AV < - h(jX ^^^^' + • • • Now ( — ) is positive. Hence, provided A^ is sufficiently small, \dt/a Ai/- + AV < 0. We have thus constructed a state for which the total (finite) increment in ^, namely (A + A')\l/, is negative, contradicting (1). Moreover it is a state of the same (initial) temperature and volume. This demonstrates that (1) implies (2). The proof of the converse may be left to the reader. The above estab- lishes for a finite change Gibbs' result [HI], established by him by less rigorous methods in equations [112] and [115] (Gibbs, I, 91). S. The Free Energy Function. In equation [117] Gibbs states without proof that the condition of equilibrium may be written We shall prove that and are equivalent. (A1A)^« ^0 (5) (Ar)^p^O (6) X, i^, r, AND THE CRITERIA OF EQUILIBRIUM 217 We will first show that (5) implies (6). To do this we will show that if there exists a state violating (6) then there exists a state violating (5). If then (5) is known to hold, there can be no state violating (6), and so (6) holds. Let us then suppose that a state exists for which (Ar)«. p < 0. Now f = ^ + py, and so Af = A^ + pAv + vAp + AvAp. Here Ap = 0, and hence Af = Ai/' + pAv < 0. Therefore AiA < -pAv. (7) Now change the volume and pressure reversibly at constant temperature. For these changes the infinitesimal increments are given by d'e = i d'r} — p d'v by the first and second laws of thermodynamics. Hence dV = d'(€ - tri) = -pd'v, since d't = 0. It follows that AV = - \ P d'v, whence p. At/' -\- A'^p < - pAv + / P d'y. 218 MILNE ART. F Now choose A'y = — Ay, thus restoring the initial volume. Then (Ai/^ + ^'^P)l, , < - pAy + \ Vd'v J v„ — Av < where po denotes the initial pressure. /dp\ At this point we encounter a difficulty. For I 7" ) is negative, and so we have apparently only established that the total incre- ment in \p, namely (A + A')\p, is less than a positive quantity. We have thus apparently not proved that it is negative. But if we examine the argument, we see that the original increment in f , namely A^, must be in general of the order Ay, and in fact there exists a constant c such that Af < clAy|, where c < 0. This means that (7) may be replaced by A^ < —pAv + c I Ay I, whence (A -\r A') ^ < c\Av\ - (jX'h ^^"^'• Hence in general [(A + A>]^. < 0, which contradicts (5) and so establishes our result. The difficulty here encountered demonstrates the great need for care in establishing thermodynamic inequalities. The reader may find it necessary to overcome a similar difficulty in the proof left to him in the preceding section. It is less difficult to prove the converse. Suppose now that we are given a state for which {AlP)t.r < 0. X, ^, r, AND THE CRITERIA OF EQUILIBRIUM 219 If this state is not one of uniform pressure, let the pressure equahze itself at constant temperature and constant volume. Then by general theory, since this is an irreversible process, the function \p must decrease in the process. (For if A" denotes the change in question, and A"Q is the heat absorbed A"r, ^ A"Q/t = A"e/t, or A"€ - t A"rt ^ 0, or A' V < 0.) Hence we have constructed a new state of uniform pressure for which (A.^),. „ < 0. Now Ar = A{^P + vv) and here Av = 0. Hence Ar = Avi' + vAj), or Af < vAj). Now change the pressure and volume reversibly at constant temperature. For this change, infinitesimal increments are given by d'e = t d'-r] — p d'v, d'f = d'{e - 7)t + vv) = V d'p, since d't = 0. Hence the new finite increment A'f is given by rpo + A'p A'^ = V d'p, J Pa 220 MILNE ART. F and accordingly /•po + A'p Af + AY 0, since the gain of entropy of the one portion must be equal to the loss of entropy of the other. It follows, by expansion by Taylor's theorem, that > 0. ' p But since /a!x\ dx = d{€ + pv) X,>P,^,AND THE CRITERIA OF EQUILIBRIUM 223 and tdt] = de + pdv, it follows in the usual way that dx = tdr] + vdp, whence Uy 1='- Hence Q = 1 1 /dt\ t ~ Cp It follows that Cp > 0. A similar argument involving ; the energy e establishes that Cv > 0. 5. Physical Properties of the Thermodynamic Functions \j/, f , x- Gibbs' statement about these may be paraphrased and extended as follows (Gibbs, I, 89, 92). If AQ represents the heat communicated to any system during any process in which the external work performed is ATT, we know always that AQ = Ae + ATF. Further, for any infinitesimal reversible change in which the masses of the ultimate constituents of the phase are unchanged, t dQ = tdt], 6. The Heat Function at Constant Pressure. Let the system undergo a change at constant pressure, in such a way that the only external work done is work of expansion. Then ATT = pAv, 224 MILNE art. f and so > Ax = Ae + pAv = Ae + ATF = AQ. Thus the increase in the heat function between any two states is equal to the heat communicated when the same change is effected (reversibly or irreversibly) at constant pressure and no other external work is done. This property gives rise to the term "heat function," (Gibbs, I, 92, equation [119].) The change in the heat function is the quantity measured by any constant- pressure calorimeter. If dt is the increase in temperature in an infinitesimal change conducted at constant pressure when no other external work is performed, then dx ^dQ^ dt ~ dt* whence \dt)^ 7. The Heat Function in General. In any change, we have Ax = Ac + A(pv), whence Ax = AQ - AF + A(pv). It may happen that some of the intrinsic energy e is converted into kinetic energy during the process, as in the expansion of a fluid through a nozzle. If q is the velocity of a typical element, then for unit mass the first law of thermodynamics must be written in the form AQ = A(ig2) + Ae + ATF, whence Ax = [AQ - A(ig2) _ AW] + A{pv) X, \P, f, AND THE CRITERIA OF EQUILIBRIUM 225 or A(x + k') = AQ - AW i- A{pv). In the case of the steady rectilinear (irreversible) flow of a fluid under its own pressure gradient, we can show that AW = Aipv). Hence for adiabatic flow of this character, where AQ = 0, we must have A(x + k') = 0 or X + iQ^ = constant. (The relation AW = Aijpv) is easily proved by considering the work done on the moving element of fluid by the adjacent elements at the two opposite ends.) If the fluid happens to be a perfect gas, we can obtain a simple expression for %• For, for any fluid whatever, L^p dp\t V - t smce d^ = d(e -\- pv — it]) = vdp — rjdt. Now, for a perfect gas, ^ = H "^ ) since pv cc t. Hence f — j =0 and dx - = Cpdi, 0/"+©/' 226 MILNE ART. F or / ^X = j Cpdt. It follows that in the adiabatic rectilinear flow of a perfect gas from rest at temperature ^o to motion with velocity q at tem- perature t, we have h Q^ = — Cpdt. J to The above somewhat miscellaneous calculations serve to illus- trate the properties of the heat function. 8. The Work Function \p at Constant Temperature. Let the system undergo a change at constant temperature, doing ex- ternal work in any way whatever (e.g., electrically), as well as by expansion against external pressure. Then A\P = A(e - tri) = Ae — tAr{, and as usual AQ = Ae + AW. If the change is reversible, AQ = tArj, and so in this case A;/' = -AW, or the increase in the work function is equal to the negative of the external work performed. (Gibbs, I, 89, equation [110].) Hence the name "work function." All reversible processes connecting two states of the same temperature yield the same amount of external work, and any irreversible process connecting them yields less work. Thus the decrease in the work function gives the maximum amount of external work obtainable in changing from the first to the second state. We can prove this in another way, from first principles, as follows. If A'Q is the heat absorbed in any change whatever, by Clausius' inequalities we have A'Q At; ^ t ' X, \^, r, AND THE CRITERIA OF EQUILIBRIUM 227 and so here, the temperature being constant, Ae - AiA = tAv ^ A'Q. In any change whatever, whether reversible or irreversible, A'Q = Ae + A'W, whence here Ae - ArA ^ Ae + A'W or A'W ^ -A^. Thus the actual amount of external work performed, A'W, cannot exceed — Aip. Now suppose a system enclosed in a fixed volume. If it undergoes of itself any process whatever, at constant tempera- ture, then necessarily A'W = 0, whence Axl^ ^ 0. Hence a necessary condition of equihbrium, subject to the condition of constant temperature and constant volume, is (AiP)t.v > 0. A state for which all possible changes satisfy this relation will be in stable equilibrium, for it cannot undergo any change of itself. This estabhshes Gibbs' criterion concerning A\f/ by an alternative method. 9. The Free Energy Function f at Constant Temperature and Constant Pressure. Let the system undergo a change at constant temperature and constant pressure, doing any external work whatever in the process. Then we have Af = A(e - trj -I- pv) = Ae — tArj + pAv. 228 MILNE art. f But AQ = Ac -f AW. If now the change is reversible, AQ = tAt], and so in this case Af = - (AW - pAv). Thus the decrease in f is equal to the excess of external work performed over the work of expansion against the external pres- sure. Hence the name "free energy" function. If any process occurs at constant pressure and constant temperature, and if A'Q is the heat absorbed and A'TF the ex- ternal work performed, whence also Hence or t Ae + pAv - Af ^ A'Q; A'Q = Ae + A'W. Ae + pAv - Af ^ Ae + A'W, {A'W - pAv) ^ -Ar. Thus the excess of external work performed over that of mere expansion cannot exceed — Af . Now suppose that the system is enclosed in an environment of constant pressure and constant temperature. Then if any process occurs of itself, the only external work is that of expan- sion, and so A'W = pAv. Therefore Af ^ 0. X, "A, r, AND THE CRITERIA OF EQUILIBRIUM 229 Hence a necessary condition that such a system shall be in stable equilibrium under the stated conditions is (Ar)p. t>0, for it then cannot undergo any change of itself. This estab- lishes Gibbs' criterion concerning Af by an independent method. 10. Further Illustration. The following original example illustrates further the properties of the ^-function. "A system, which can perform external work in any manner, is brought reversibly from a temperature ti to a temperature <2( < ^i) in such a way that it only gives up heat at the tempera- ture ti. Prove that the external work performed, AW, is given by ATF = A-A + mik - k) where Ai^ is the decrease in the work function \p between the temperatures ^i and ^2, and tji is the entropy at 0 it follows that iji > 772. If ^ ^ ^2 throughout the process, the integral is positive whether or not i is a single- valued function of 77 during the process (i.e., whether or not the system always has the same temperature at intermediate stages at which the entropy takes the same value). Consequently A'W ^ AW. It follows that AW is the maximum amount of external work that can be obtained by processes in which the temperature of the system does not fall below (2. That is, the maximum work is obtained when all the heat is given up reversibly at temperature ti, and the amount of this work is AiA + vi ih - k), A^ being the decrease in the work-function. This extends the physical significance of the work-function to processes of non- constant temperature. The absolute value tji, of the entropy appears to occur in this expression; but it must be remembered that the absolute value of the entropy occurs also in the definition of \p. The same constants used in fixing the entropy 77, must be employed in the entropy-values used ia tracing the changes in \p. G THE PHASE RULE AND HETEROGENEOUS EQUILIBRIUM [Gibbs, I, pp. 96-100] GEORGE W. MOREY I. Introduction Treatises on the Phase Rule usually deal with heterogeneous equilibrium from a purely geometrical point of view, making use of the familiar equation, F = n-\-2 — r, in which F is the number of degrees of freedom, n the number of components, and r the number of phases, as a qualitative guide, and depend- ing on the Theorem of Le Chatelier for determining the effect of change of conditions on the equilibrium. It is unfortunate that the subject has been developed in this manner, instead of by the direct application of the equations which were developed by Gibbs. The Phase Rule itself is but an incidental qualita- tive deduction from these equations, and the justification of the geometrical methods is their derivation as projections of the lines and surfaces "of dissipated energy," painstakingly ex- emplified* by Gibbs. While in the first portion of the "Equilib- rium of Heterogeneous Substances" the actions of gravity, electrical influences, and surface forces are excluded from con- sideration, these restrictions are later removed, thus rendering unnecessary the various "extended" Phase Rules which have been proposed to remedy this supposed defect. II. Equation [97] and the Phase Rule 1 . Equation [97] . The Phase Rule may be derived from Gibbs' fundamental conditions for equilibrium [15-21], but Gibbs' own treatment is intimately connected with his equation [97] * Equilibrivun of Heterogeneous Substances, Gibbs, I, 118 et seq. 233 234 MOREY ART. G vdj) = Tjdt + niid/jii + nhdm . . . + nindun, (1) [97] in which v and ?; refer to the volume and entropy of m.i + ma ... -^ Mn units of the phase considered, p and t to the pressure and temperature, and /x to "the potential for the substance in the homogeneous mass considered." The chemical potential, /x, is defined by the equations Ml ^/^\ Jdr\ ^/ix\ =(^\ (2) [104] \dmi/„,v.m \dmi/t.v.m \dmi/„,p,m \dmi/t.p,m' in which e, \p, x, and f refer, respectively, to the energy and the three Gibbs' thermodynamic functions defined by the equations \p = e - tr], (3) [87] X = e + pv, (4) [89] ^ = e - tr] -\- pv. (5) [91] The first of these, rp, is the quantity defined by Heknholtz* as the free energy, and commonly designated by that name in Continental writings; the second, x, the quantity variously known as heat content, enkaumy and enthalpy ;t the third, ^, the quantity called free energy by Lewis. J The definition of fx is evidently symmetrical with respect to e, ^, x and f , and it should not be considered as specially related to any one of these quantities. 2. Derivation of the Phase Rule. Equation (1) [97] expresses a necessary relationship at equilibrium between the intensive properties of any phase, and this relationship itself is a con- sequence of the fundamental condition for equilibrium, namely, that in an isolated system the entropy shall be a maximum for * Helmholtz, Sitzb. preuss. Akad. Wiss. 1, 22 (1882). t The term enthalpy, proposed by H. Kamerlingh Onnes (Leiden Comm. No. 109 (1909), p. 3) is, in the author's opinion, the best for the designation of this important quantity. X The thermodynamic quantities of Gibbs refer to a total mass of (mi + m2 + ... TO„) units of the phase or system in question, while some of the names subsequently applied to the Gibbs functions refer by defini- tion to a gram molecular weight. That, for example, is the diflference between Gibbs' f and Lewis' free energy. HETEROGENEOUS EQUILIBRIUM 235 the given energy and volume. The concept of phase, and the derivation of the Phase Rule, result from the appUcation of equation (1) [97] to the consideration of "the different homo- geneous bodies which can be formed out of any set of component substances." "It will be convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its quantity or form. We may call such bodies as differ in composition or state different phases of the matter considered, regarding all bodies which differ only in quantity and form as different examples of the same phase. Phases which can exist together, the dividing surfaces being plane, in an equilibrium which does not depend on passive resistances to change, we shall call coexistent. "If a homogeneous body has n independently variable com- ponents, the phase of the body is evidently capable of n + 1 independent variations." This follows from the fact that there are n + 2 independent variables, pressure, temperature, and the n quantities yiii, H2, ... Mn connected by an equation of the form of (1) [97]. "A system of r coexistent phases, each of which has the same n independently variable components is capable of n + 2 — r variations of phase," or degrees of freedom, F. "For the temperature, the pressure, and the potentials for the actual* components have the same values in the different phases, and the variations in these quantities are by [97] subject to as many conditions as there are different phases. Therefore, the number of independent variations in the values of these quantities, i.e., the number of independent variations of phase of the system, will be n + 2 — r." "Hence, if r = w + 2, no variation in the phases (remaining coexistent) is possible. It does not seem probable that r can ever exceed n -\- 2. An example of w = 1 and r = 3 is seen in the coexistent solid, liquid, and gaseous forms of any substance of invariable composition. It seems not improbable that in the case of sulphur and some other simple substances there is more than one triad of coexistent phases; but it is entirely * The distinction between "actual" and "possible" components need not be discussed in this place. See Gibbs, I, 66. 236 MOREY ART. G improbable that there are four coexistent phases of any simple substance.* An example of n = 2 and r = 4 is seen in a solution of a salt in water in contact with vapor of water and two differ- ent kinds of crystals of the salt." Coexistence of r = w + 2 phases gives rise to an invariant equilibrium, and such a co- existence is frequently called an invariant point. Invariant points are also referred to by the number of phases present ; for example, a triple point in a one-component system, quadruple point in a two-component system, etc. When r = 7i -\- 1, there are n -{- 1 equations of the form of (1) [97], one for each of the coexisting phases, and the system has one degree of freedom. We may eliminate n of the n -\- 2 independent variables, giving an equation between the two remaining. If the quantities dm, dti2, ■ ■ ■ djin are eliminated by the usual method of cross multiplication, we obtain a linear equation between the changes in pressure and temperature, which for the general case takes the form 7j' m/ rrii . . . rrin t\" mx" rri'i' . . . rrin" dp _ T?" mi" Tn?" . . . m dt v' m\' rrh' v" wi" m^" m„ m. yn ^n ^^n _ _ _ ^^n (6) [129] We shall develop in detail the application of this equation to several types of systems. III. Application of Equation [97] to Systems of One Component 3. The Pressure-Temperature Curve of Water. A simple case of heterogeneous equilibrium is that of a one-component * For an extended discussion of the possibility of the coexistence of more than n + 2 phases, see R. Wegscheider, Z. physik. Chem., 43, 93 (1903) et seq.; A. Byk, ibid., 45, 465 (1903) et seq. HETEROGENEOUS EQUILIBRIUM 237 system, such as water, in which the liquid coexists with its own vapor at a series of pressures and temperatures. There are two equations of the form of (1) [97], one for the vapor and one for the hquid. If we denote vapor and Hquid by the indices v and I, and use, as we shall hereafter, the capital letters V and H (capital eta) lor total volume and total entropy, respectively, these equations are 'V'dp = R^dt + m^'dn, and V^dp = Wdt + m^dfx. It will be remembered, from the derivation of these equations, that the quantities V and H refer to the total volume and total entropy of the mass considered ; in this case, where there is only one component, to the total volume and entropy of the m grams contained in each phase. If we divide each equation through by the mass w, they take the form v^dp = -q^dt + dfi, v^dp = 17'rfi + dny in which the lower-case letters are used to denote specific volume and specific entropy, as opposed to the total volume and total entropy, denoted by the capital letters. We can eliminate dn between these equations by subtraction, giving us (y" - v^)dp = (tj" - y]^)dt or dp rf — 7j' dt V — v^' Since dR = dQ/t, which on integration at constant tempera- ture yields AH = — , this reduces to the usual Clausius-Clapey- V ron equation dp _ AQ dt ~ t{v^ - vO • 238 MOREY art. g It will be of interest to consider the detailed application of the equation d'p r}^ — r/^ dt v" — y' to the pressure-temperature curve of water. * The thermodynamic properties of water are known to a considerable degree of precision, and tables giving the specific entropy and specific volume of water and steam are in common use by engineers. In such tables it is customary to take the specific entropy of liquid water at zero degrees centigrade as zero, but since we are always dealing with differences in entropy this is immaterial. Absolute values of entropy are not deter- minable; to determine absolute values of entropy we would have to know the value of the entropy at absolute zero,t and its variation with temperature from the absolute zero up, and we do not possess the necessary data for this. Herein Hes one of the reasons for the entropy concept being a difficult one to grasp; we are not able to measure entropy directly as we are able to measure the other quantity factors, volume and mass. For practical purposes, however, this is not material, since we are always dealing with entropy differences. In Fig. 1 are shown plotted the specific entropy of Uquid water and the specific entropy of saturated water vapor from zero to 200°C., the specific volume of water vapor at the saturation pressure in the same temperature range, and the pressure-temperature curve of the equilibrium, liquid -(- vapor. Since the slope of the p-t curve is determined by the difference in entropy between vapor and liquid, it is immaterial whether the entropy of the * From this point to the end of section (11), p. 251, the text is taken, with some omissions, alterations and additions, from the author's article, Jour. Franklin Inst., 194, 439-450 (1922) ; sections (16) to (23) inclusive (except (18) and (22)) are taken in like manner from the same article, pp. 450-460. t Absolute values of entropy may be calculated for many substances by the use of the so-called Third Law of Thermodynamics, a principle whose validity has not been completely demonstrated. HETEROGENEO US EQ UILIBRI UM 239 liquid at 0°C. is taken as zero or some other value. The entropy of the vapor is greater than that of the liquid by the entropy of vaporization, that is, the heat of vaporization divided by the absolute temperature. In the case of the volume, only the specific volume of the vapor is plotted, as that of the liquid is so small that it cannot be shown on the scale of the dia- gram. Let us now consider some actual values. so /OO /so 200 T£Mf>e/fATUff£ /-V OeSRSES Cef^TJORADe Z50 300 Fig. 1. The specific entropy of liquid water and of saturated water vapor, the specific volume of saturated water vapor, and the vapor pressure of water, plotted against temperature. At zero degrees centigrade, if the entropy of the Hquid is zero, that of the vapor is 2. 18 calories. The specific volume of water vapor in equilibrium with liquid at zero degrees is 206 liters per gram; it is evident that the volume of the liquid, 1 cc, is negligible in comparison. In the equation dp dt v" - V 4}V «)i the terms must all be of the same kind; if the slope of the p-t curve is given in atmospheres per degree, and the volume in 240 MOREY art. g liters, the entropy must be expressed in liter-atmospheres instead of in calories. The factor for this conversion is 0.0413; inserting the above values in the equation, we get dp/dt = (2.180 X 0.0413) /206 = 0.00044 atm. per degree; the corresponding experimental value is the same. At 50° the values are dp ^ (1.928 - 0.168) (0.0413) ^ dt (12.02 - 0.001) Again the experimental value is the same, and the volume of the liquid is still negligible. At 100°, the corresponding quantities are dp _ (1.756 - 0.312) (0.0413) _ „' „__ dt (1.209 - 0.001) "•"'^^^' agreeing exactly with experiment. At this temperature the volume of the liquid amounts to less than one-tenth of one per cent of the total volume ; the value of dp/dt is increasing with increasing temperature, and the explanation is evident from an inspection of the entropy and volume curves. As the tem- perature is increased the entropy of the vapor diminishes, that of the liquid increases, hence the difference decreases as the temperature increases. The numerator, the entropy of vapori- zation, is therefore diminishing, but its decrease is more than offset by the decrease in the denominator taking place at the same time because the increasing vapor pressure increases the density of the vapor, hence decreasing its specific volume. In the interval from zero to 10° the numerator decreases to 95.6 per cent of its value at zero, while the denominator decreases to only 51.5 per cent of its value at zero. The difference does not remain so marked, but for the interval 90-100° the values are 96 per cent and 70.9 per cent, respectively, and for the interval 190-200°, 96.1 per cent and 81.4 per cent, respectively. Appli- cation of the two equations of the form of (1) [97] to the uni- variant equilibrium, liquid + vapor, in the one-component sys- tem, water, shows us that not only does the pressure increase with HETEROGENEOUS EQUILIBRIUM 241 increasing temperature, but the rate of increase also increases. The p-t curve is accordingly concave upward, and the slope continues to increase. As the critical point of water is ap- proached, the difference between the properties of liquid and vapor diminishes rapidly, and vanishes at the critical tem- perature. Hence the equation for the p-f curve becomes indeterminate, and the vapor pressure curve ends. fO 1 1 i? ^ \s - % ^ ^ ^ / f J /4/n ' ^ Bm /oo \ f I /oo 200 300 Bm 400 Fig. 2. The binary system, H2O-KNO3. Diagrams A, B, and C are the projections of the curve representing the three-phase equilibrium, vapor + saturated solution + solid KNO3, in the solid p-t-x model on the pressure-composition I (p-x), pressure-temperature (p-t), and temperature-composition (i-x) planes, respectively. IV. Application of Equation [97] to Systems of Two Components 4. Application of the Phase Rule to a System in Which No Compounds Are Formed. H2O-KNO3. We will now consider the case of a simple binary system, choosing the system, water- KNO3, as an illustration. The relationship between pressure, temperature, and composition is shown in Fig. 2, A, B, and C, 242 MOREY ART. G which may be regarded as the projections of the sohd p-t-x model on the p-x, p-t, and t-x planes, respectively. It should be noted that in referring to these projections, and to the similar ones in the following figures, their conventional designa- tion in chemical literature has been followed, instead of the convention in mathematics that the symbols shall be in the order abscissa, ordinate; a:, y. The system, H2O-KNO3,* does not show liquid immiscibility, nor are solid hydrates formed, so there are four possible phases in the system; one vapor phase, one liquid phase and two solids, ice and solid KNO3. Co- existence of four phases in a two-component system gives us four equations of the type of (1) [97] between the four un- knowns, pressure, temperature, and the two chemical poten- tials, so the system is completely determined. The four phases can only coexist at one temperature and one pressure, that is, at the invariant point, often called the cryohydrate when one component is water. The invariant point can be considered as the intersection of four curves representing univariant equilibria, each of which equilibria will contain three of the phases which coexisted at the invariant point. We can have the four combinations: ice + solution + vapor, ice + potassium nitrate -f vapor, ice + potassium nitrate -\- solution, and potas- sium nitrate + solution -{- vapor. Consider each of these curves in detail, starting with the last, the solubihty curve of potas- sium nitrate in water. 5. Application of Equation [97] to a System in Which No Com- pounds Are Formed. H2O-KNO3. In the univariant equilib- rium, potassium nitrate + solution + vapor, there is only one phase of variable composition, the solution. Since potassium nitrate is not volatile at temperatures we are considering, the vapor phase is pure water; since potassium nitrate forms neither hydrates nor solid solutions with water, the solid phase is pure potassium nitrate. Let us now apply equation (1) [97] to this univariant equilibrium. In the derivation of equation (1) [97], Vdp = Udt + midfxi + WgC^Ma * The circumstance that an inversion takes place in KNOj at 127.8° is ignored, as not being pertinent to the points under consideration. HETEROGENEOUS EQUILIBRIUM 243 for a two-component system, composition was expressed as the total mass rrii and wi of the substances present, and volume and entropy as total volume and total entropy. For some purposes this is the most convenient form, but for our present discussion it is more convenient to express composition as weight per cent potassium nitrate. Since we have Wi + Wj grams of the two components water and potassium nitrate, respectively, if we divide through hy rtii -{- rrh we shall get dp = ; dt + 1 dfjLi + ; dfn. nil -{- nh mi + m2 rui -\- rUi mi + mj The coefficient of the first term, the total volume divided by the total number of grams of material, is evidently the specific volume of the phase. Similarly, the coefficient of the second term is the specific entropy. The fractions mi rtii and mi -\- nh mi + ma are the weight fractions of the components H2O and KNO3, respectively, and if we represent the weight fraction of KNO3 by X, that of H2O will be (1 — x). The equation now is vdp = rjdt + (1 — x)dni + xdm, (7) in which v and rj are specific volume and specific entropy. We will have three such equations, one for the vapor, denoted by the superscript v, one for the liquid, denoted by the superscript /, and one for the solid, denoted by the superscript s. From these equations we may eliminate dfxi and d^a by the usual methods of cross-multiplication, giving the equation x" — a;' dt , ^ x" — x\ (y' - rO - ; {v' - v^) x' — x (8) 6. The Equilibrium, KNO3 + Solution -\- Vapor* At the * The data for the system, HjO-KNOj, are taken in part from Lan- dolt-Bornstein, Physikalisch-chemische Tahellen, 1912; in part from unpublished data by F. C. Kracek and G. W. Morey. 244 MOREY ART. G cryohydrate point the weight fraction KNO3 is 0.021; since the vapor is pure water, its weight fraction of KNO3 is zero, and that of the soHd phase is unity. Substituting these values, we get The coefficient of the second term in both numerator and denominator is a fractional coefficient. Without an actual determination of the entropy of any phase, certain definite conclusions can be drawn. In the numerator, we have the entropy differences: (vapor — liquid), a positive quantity, and (solid — liquid), a negative quantity. The former is always several times the latter; in the case of this dilute solution their ratio is probably not very different from the ratio of the entropy of vaporization of water to the entropy of fusion of KNO3, which is of the order of magnitude of 20 to 1. The first term predomi- nates, and the numerator is a positive quantity of the order of magnitude of the entropy of vaporization of water at zero degrees, or a little less than 2.18. In the denominator the term affected by the fractional coefficient, the difference in specific volume of liquid and solid, is negative and is itself very small. The first term, the volume difference (vapor-liquid), is comparatively enormous; at the cryohydrate temperature and pressure it is even larger than the volume difference in pure water at its freezing point, 206 liters per gram. The slope of the pressure- temperature curve is at the beginning close to that of pure water; that of pure water is concave upward, owing to the denominator decreasing in value more rapidly than the numer- ator, and the same is true in this case. The pressure-tempera- ture curve of all systems containing a volatile component at low pressure will show a similar initial upward concavity, owing to the rapid decrease in the specific volume of the vapor phase with increasing pressure. As the temperature is raised, the fraction of KNO3 in the liquid increases, while the composition of the other phases remains the same. The specific entropy of the vapor continually HETEROGENEOUS EQUILIBRIUM 245 decreases; that of the sohd increases, as does that of the hquid. The first term in the numerator consequently decreases, the second increases, and the coefficient of the second term also increases; since the first term is positive, while the second is negative, the numerator is a continually decreasing positive quantity. The denominator is decreasing at a progressively slower rate. As the temperature is raised these effects con- tinue, until a temperature is reached at which the rate of decrease of the numerator becomes equal to that of the denomi- nator, and the curve has a point of inflection. After this it is no longer concave upward, but is concave downward, as the vapor pressure of the saturated solution is still increasing with the temperature, but at a diminishing rate. The temperature of this point of inflection is approximately 205°, and the pres- sure is about 5.3 atmospheres. The determination of the solubility curve of KNO3 in HoO is a simple matter at temperatures below 100°. As long as the vapor pressure remains less than one atmosphere, we can shake up solid and liquid in a thermostat until equilibrium is reached, suck out a sample of the supernatant liquid through a filter, and determine the composition by analysis. After the pressure has exceeded one atmosphere, other methods must be employed. Of course, if a mixture containing an excess of KNO3 is heated in an open vessel, when the vapor pressure reaches one atmos- phere the solution will begin to boil, and will evaporate to dryness. But if the mixture be heated in a closed tube, from which the water cannot evaporate, the solubility curve will be continuous until the mixture is entirely liquid ; the temperature at which the saturated solution boils at a pressure of one atmosphere is not a significant point on the solubility curve. From this point of view there is no distinction between a solubility curve and a melting-point curve, and the curve EBm can be regarded either as the solubility curve of KNO3 in H2O or as the melting-point curve of H2O-KNO3 mixtures. The first to realize this fact was Guthrie* in 1884, and the system, H2O-KNO3, was one of those that he studied. He sealed * Guthrie, Phil. Mag., 18, 117 (1884). 246 MOREY ART. G mixtures in closed tubes and observed the temperature at which the crystals disappeared. As the temperature is raised past the point of inj9ection of the p-t curve, the KNO3 content of the liquid increases and the coefficient of the second term in the numerator increases corre- spondingly. At 115°, the boiling point of the saturated solu- tion, the ratio a; V(l — a:') is about 2.5; at the point of inflection, about 4. As this coefficient continues to increase, the numer- ator decreases more and more rapidly, and the value of dp/dt decreases; but, as it is still positive, the pressure continues to increase with temperature. With a little further increase in temperature, the ratio x^/{l — x^) becomes such that the entire second term equals the first term, and the difference is zero; the numerator is now zero, so dy/dt is zero, and the curve has a horizontal tangent. Since at this point it follows that x^ _ _ yfj-Tj^ 1 — x' v' — v The ratio of the entropy difference (vapor-liquid) to the entropy difference (solid-liquid) is equal to the ratio of KNO3 to water in the saturated solution; the saturated solution at this point contains about 95.3 per cent KNO3, so this ratio is approxi- mately 95.3/4.7, or 20. The entropy of the water vapor at this temperature and pressure can be obtained from steam tables, that of KNO3 from specific heat data, and the entropy of the liquid can accordingly be calculated. It should be remembered that we are here dealing with entropy differences, not absolute entropy, and when we take off the entropy of the steam from a steam table we must remember that the assumption is made in the steam table that the entropy of liquid water at its freez- ing point is zero. 7. The Maximum Pressure of the Equilihrium, KNOz -\- Solution + Vapor. The point of maximum pressure is found at a KNO3 content of about 95.3 per cent, a temperature of HETEROGENEOUS EQUILIBRIUM 247 about 266°, and a pressure of about 7.9 atmospheres. Our equation is 0-953 , dt , ,. , 0.953. ,. ' ^'^ ~ '^ + o:or7 ^'' ~ '^ and the numerator is zero because the negative entropy differ- ence (solid-liquid), multiplied by the ratio a: V(l — 2:0 is equal to the positive entropy difference (vapor-liquid). On further increase in temperature x continues to increase, the negative second term becomes larger than the positive first term, and the numerator becomes negative. The denominator is still positive, so the p-t curve has a negative slope; pressure de- creases with increasing temperature. On further increase in temperature, the numerator continues to become more strongly negative, until at the melting point of pure KNO3 it is the entropy difference (solid-Uquid) for KNOj. 8. The Maximum Temperature of the Equilibrium, KNO3 + Solution + Vapor. The changes which have been taking place in the denominator will now be considered. The specific volume of the vapor phase at all points is much larger than that of any other phase, its smallest value at the maximum pressure being about 100 cc. per gram. As the pressure decreases from this point, the specific volume of the vapor increases; the effect of this is merely to alter the rate of decrease of pressure which takes place from this point. But as the liquid phase approaches KNO3 in composition, the amount of water becoming very small, the second term in the denominator becomes of im- portance. The specific volume difference between fused and solid KNO3 is but a few tenths of a cubic centimeter; when the water content is only 0.1 per cent, the negative volume differ- ence (solid-liquid) is multiplied by the ratio 999/1, and at 0.01 per cent water, by 10,000. As the water content decreases, the coefficient of the second term in the denominator, (v — vO> increases rapidly, the denominator approaches zero, and the slope of the p-t curve, dp/dt, becomes infinite. At this one point the curve is vertical; on further increase in temperature the 248 MOREY ART. G curve again has a positive slope. In a system of the type, H2O-KNO3, the experimental realization of this portion of the curve would be extremely difficult and we will not consider it further at present, except to point out that at zero water content the equation becomes dp 77* — rj' dt V — v^ which is the equation of the tangent to the melting-point curve of pure KNO3. The p-t curve of the saturated solutions is therefore tangent at its end to the melting-point curve of KNO3, the curve showing the change in melting point of potassium nitrate with pressure. This type of equilibrium will be considered later. 9. The Second Boiling Point. We have seen that a melting- point or solubility curve of the system, H2O-KNO3, extends from the cryohydrate E to the melting point of pure KNO3, and have followed the change in vapor pressure with composi- tion in detail. We have therefore correlated the temperature- composition or solubility curve with the pressure-temperature curve. One curve gives the change with the temperature in the composition of the liquid in equilibrium with solid and vapor, the other gives the change with temperature in the vapor pressure of the saturated solution. One other pair of the three vari- ables, composition of the liquid, temperature, and pressure, can be considered, namely, the change in vapor pressure of the saturated solution with composition. This is the pressure- composition curve; from it we see that the vapor pressure at first increases with decreasing water content of the saturated solutions, reaches a maximum at a small H2O content, then decreases rapidly with further diminution of the water content, until at its end-point at pure KNO3 the vapor pressure is that of the triple point of KNO3. We are all familiar with the fact that as the water content of the saturated solution decreases with increasing temperature the vapor pressure increases, until at the boiling point of the solution the pressure of the atmos- phere is reached. But there are two saturated solutions whose vapor pressure is one atmosphere; one has a water content of 29 HETEROGENEOUS EQUILIBRIUM 249 per cent, the other of only one per cent. At the first boiling point, addition of heat causes the solution to evaporate, liquid changing into solid and vapor. At the boiling point at higher temperature, called by Roozeboom, who discovered it, the second boiling point, the solution boils on cooling. At the second boiling point, the liquid changes into solid and vapor with evolution of heat. If a melt of KNO3, saturated at its melting point with water, be quickly cooled, it will be seen to boil suddenly and violently, and at the same time to solidify. This second boiling point has been observed in many systems,* including silicate systems at high temperatures, and the phe- nomenon has been made the basis of a theory of volcanism,t which has been applied successfully to the activity of Mt. Lassen, California.! 10. The Equilibrium, Ice + Solution + Vapor. Of the four univariant equilibria which proceed from the invariant point we have considered but one, namely, the univariant equilib- rium, solid KNO3 + solution + vapor. The univariant equilibrium, ice + solution + vapor, is a second one in which we have both liquid and vapor, and in this case solid and vapor have the same composition. Our equation (8) becomes ^V _^ /ytl X' — x' and, since x^ = x* = 0, dp ^ ^^' - ''^ - ^^ ^^' - ^'^ ^ r - v'^ dt , ,, 0 - a;' ,x 2^" - v"' (y* — y') — ; (V — y') U — X But this equation refers to the vapor-pressure curve of ice; all terms relating to the liquid have disappeared. This is a general * H. W. Bakhuis Roozeboom, Proc. Z2o?/. (Soc. Amsterdam, 4,371(1901). t G. W. Morey, J. Wash. Acad. Sci., 12, 219 (1922). t A. L. Day and E. T. Allen, Carnegie Inst. Wash., Publ. No. 360 (1925). A. L. Day, /. Franklin Inst., 200, 161 (1925). 250 MOREY AET. Q relation; whenever any two phases in a binary system have the same composition the pressure-temperature relations become those of these two phases, without reference to the composition of the other phase present. 11. The Equilibria, Ice + KNO^ + Vapor, and Ice + KNOz + Solution. The preceding univariant equilibria have been formed from the invariant equilibrium, ice + KNO3 + solution + vapor, by the disappearance of ice or of KNO3, respectively. Two others can be obtained, by the disappearance of liquid or of vapor. In case the liquid disappears, we have left ice + KNO3 + vapor, and the p-t curve of this equilibrium will coin- cide with the vapor-pressure curve of ice, and from the invariant point will go to lower pressure and lower temperature. In case the vapor disappears we have the condensed system, ice + KNO3 + liquid, and the curve gives the change in eutectic (cryohydrate) composition with pressure. The equation of this curve* is dp _ ^^ ^ ^ a:'^^"' - x' ^^ ^ ^ and since x'" = 0, a;"''**" = 1, and x^ = 0.021, this becomes („.ce _ I) t ^1^ („^NO. _ ,) dp ^^ ^ ^ ^ 0.979 ^^ ^ ^ Here again the entropy and volume changes of the water are the predominating factors; since the entropy difference is positive and the volume difference, in the exceptional case of water, negative, the p-t curve of this equilibrium has a negative slope. But in this case, as in all condensed systems, the slope is very steep; the numerator is of the order of magnitude of 0.3 cal. or 0.012 liter-atmospheres; the denominator is of the order of * This is the equation of the tangent to the curve; but it is convenient to refer to it as the equation of the curve itself, and need not cause confusion. HETEROGENEOUS EQUILIBRIUM 251 magnitude of 0.1 cc, or 0.0001 liters. The value of dp/dt is thus about —0.012/0.001, or 120 atmospheres per degree; the curve will be almost vertical. In other words, pressure, as com- pared with temperature, has, as a rule, but little effect on the equilibrium temperature and composition, 13. Derivation of an Equation in Which the Argument Is Pressure, Temperature, and Composition. It will be of interest to correlate the solubiUty (t-x) curve more closely with the p-t curve.* The p-t curve gives the change of vapor pressure with temperature along the three-phase curve, representing coexistence of vapor, liquid (saturated solution), and solid, and the equation used in its discussion contained pressure and temperature as expressed variables. The t-x curve repre- sents the change with temperature of the weight fraction x of the second component in the saturated solution along the same curve, and for its discussion it is useful to have an equa- tion containing temperature and composition as expressed variables. Applying (1) [97] in the form of equation (7) to two coexisting phases, denoted by single and double accents, and eliminating dm, gives [v'(l - x") - v"{\ - x')\dp = h'(l - x") - 'n"{l-x')]dt + (x' - x")diJL2. (9) But /x is a function of pressure, temperature, and composition, so we may write From the equation de = tdR — Vdp + midm + miduz . . . + w„c/ju„, (11) [12] it follows that dn2 dV dfjii 9H T~ = :; — . and "77 = — 7 — . dp dnh dt dm2 * Cf. footnote on page 257. 252 MOREY ART. G which give the rate of change of total volume and of total entropy, respectively, on addition of mj. Since V = {mi-\- mijv, —-= V - {\ - x) — drrii dx and, similarly, an dv - — = ^ — (1 — a;) — • dm2 dx djJLi dfi2 Substituting these values of — - and Trin (10), inserting this O^ 01/ value of diJi2 in (9) and rearranging, gives ^y' _ ," _ (^' _ ^") ^£^ dp = [v - n" - {x' - x") ~\^ dt x' - x" dfji2 „ , , + 1 T,^ndx". 12) 1 — X dx This is a general equation* for the equilibrium between two dfJL2 phases in a binary system. The term r-j, can in general be OJu evaluated only from experimental data; indeed, the whole of chemical equilibrium is contained in the evaluation of this term. Gibbs has indicated the form it takes for dilute solutions, and has shown that it is necessarilyt positive for stable phases. 13. Derivation of an Equation Applying to the Solubility (t-x) Curve. Equation (12) can be written in the form x' — x" du.2 Av^' dp = AV^ dt + j^^ ^, dx", (13) * This equation can be derived in a number of different ways; the introduction of equation (1) [97] is not necessary nor is it the most convenient way. It is used here as being more in harmony with the general mode of treatment. Cf. E. D. Williamson and G. W. Morey, J. Am. Chem. Soc, 40, 49 (1918). dfX2 t Gibbs, I, 112. The proof refers to - — but it is easily shown that al7l2 if this is positive — — , must be positive also. dx" HETEROGENEOUS EQUILIBRIUM 253 in which Av^^ and At?^^ have been substituted for ^,' _ ," _ (^' _ :,-) ^, j and ^v' - -n" - ix' - x") ^^,], respectively. This appHes to any two-phase equiUbrium ; if we have in addition a third phase, denoted by triple accents, we have another equation of the same form. Elimination of dy , dt between the two equations and solving for t7/ gives ^ _ 1 a/x2 Av^'' {x' - X") - Ai;^^ {x'" - X") dx" ~ ~ I - X" dx" At;32 ^^12 _ ^yl2 ^^32 ^ ^ This is a general equation which applies to any three-phase equilibrium in a two-component system. r \ dV'l The terms of the form v' - v" - {x' - x") -^, requu-e some discussion. In equation (6) [129] the volume and en- tropy terms represent difference in specific volume and specific entropy, and, taken as a whole, represent the volume and entropy changes taking place along the three-phase curve. Equation (12) refers to two phases in a two-component system, and hence to a divariant equilibrium. The coefficients of dp and di in this case refer to the volume and entropy changes which take place when one gram of the first phase separates from a large quantity of the second, a type of change called "differential," "partial," or "fictive." 11^.. Correlation of the i-x and p-t Curves. Consider the application of equation (14) to the t-x curve of KNO3 in the binary system, H2O-KNO3, and let the phases with single, double, and triple accents be vapor, liquid (saturated solution), and solid, respectively. The equation then becomes dt 1 dfi2 Av'^ (x" - x^) - AV^ jx' - x^) dx'' ^ ~ 1 - x" dx" Av'^ At;"' - Aw"^ Arj'^ 1 9/i2 The terms :j 77 and —y, are necessarily positive. In the denominator, Av^^ is usually negative, Ar;"' always positive, hence the first term is usually negative. In the second term, 254 MOREY ART. Q At;"' is positive, Ar]'^ negative, making the second term always negative. Because of the preponderance of Av"^ the second term is greater than the first and, as this term has a negative sign, the denominator is always positive. In the numerator, Av*' is usually negative and (x" — x^) negative, so the first term is positive in the usual case. The quantity Av"' is dominant in the numerator also; its product with the term {x' — re') is always positive, but as it bears a negative sign, the dt numerator is usually negative. This makes j-j, positive, and the t-x curve has a positive slope. When, however, the composition of the solution has become very close to that of the solid, the negative second term becomes equal to the positive first term, and the t-x curve has a horizontal tangent, followed by a negative slope. In such cases as H2O-KNO3 this detail of the solubility curve is not detectable experimentally, but that it is necessarily present follows from the correlation with the 'p-t curve. The 'p-t curve passes first through a point of maximum pressure, then one of maximum temperature, and at its end-point coincides with the melting-point curve of KNO3, the univariant equilibrium (solid + liquid) in the unary system, KNO3. 15. Equilibrium Involving Solid Solutions. It was mentioned above that solid KNO3 exists in two enantiotropic modifications, but that consideration of this was not pertinent to the discus- sion. The two forms are both pure KNO3, there is no solid solution, and the inversion point extends across the diagram at constant temperature. It will, however, cause an abrupt change in slope on both the t-x and p-t curves of the equilib- rium, vapor -\- liquid + solid. In the not unusual case in other systems in which one or both of two enantiotropic forms takes into solid solution some of the other component, the equilibrium becomes univariant, and the inversion temperature is either raised or lowered, depending on which of the two forms contains the greater quantity of the other component. It will be interesting to apply equation (14) to this case. Let the phases with single, double, and triple accents be vapor, the high-temperature (a) form, and the low-temperature 03) form. The equation becomes HETEROGENEOUS EQUILIBRIUM 255 dt 1 dixj Av^" jx'' — X") — AV" {x» — x") d7' " ~ l-x"'dx" A/" At?'" - Ay'^Aij^" As before, :; ;; and —77 are necessarily positive. In the 1 — X ox denominator, Av^" is small and may be either positive or nega- tive; Arj"" is positive. In the second term, Av"" is large and positive; Atj"" negative, since by hypothesis the a-form is the high-temperature phase, and hence has greater entropy. The product is negative ; because of the large numerical value of the term Av"", the second term in the denominator predominates, and, being affected by a negative sign, the resultant denomina- tor is always positive. In the numerator the first term is of uncertain sign, but is smaller than the second term. The second term is the dominant one; Av"" is large and positive, and the sign of the numerator, and hence of the entire expres- sion, is determined by, and is the same as, that of the composi- tion difference (x^ — x"). When the high-temperature, or a-form, takes more of the other component into solid solution, (x^ — X") is positive, -77; is positive, and the inversion tempera- ture is lowered by solid solution. When the low temperature, or /3-form, takes the greater quantity of the other component into solid solution, the inversion temperature is raised. A well-known example of the second case is the raising of the inversion temperature of the low-temperature form of CaO • SiOj, woUastonite, by solid solution of MgO-Si02. The further treatment of equilibria in which there is solid solution is a simple extension of the above methods. The composition of the solid phase is no longer constant, but variable, a circumstance for which allowance is readily made in the discussion. In addition, the entropy and volume are no longer independent of the composition, but this again rarely leads to complications. In the case of solid solution in systems in which both components are volatile all of the coexisting phases in a uni variant equilibrium may be of variable composi- tion, but since compositions come into the equations as differ- ences the detailed application of the equations above presents no difficulty. 256 MOREY ART. G 16. Application of Equation [97] to a System in Which Com- pounds Are Formed. HiO-CaCk. We have considered the appUcation of equation (8) to the simplest type of system, that in which there is but one phase of variable composition, and no compounds are formed. It will be of interest to see what additional complications are introduced by the formation of compounds, and as illustration the system, H20-CaCl2, will be chosen. Projections of the solid pressure-temperature-com- position model are shown in Fig. 3.* The invariant point, ice + CaCla-GHaO + solution + vapor, is at — 55°, and the pressure is but a fraction of a milhmeter. The compound, CaCl2-6H20, contains 50.66 per cent CaCl2, and the cryohydrate solution, 29.8 per cent. The equation of the pressure-temperature curve of the solutions saturated with CaCl2-6H20is (v^ i\ \ ^ ft -V) dp - ^^ + 0.5066 - x^ ^" dt iv" 1\ 1 1 H^n -v^) - '^ + 0.5066 - x^ ^' As in the preceding case the volume change of the water vapor is the dominating factor at low temperatures, causing the curve to be concave upward (Fig. 3). As the temperature is raised the fractional coefficient of the second term becomes of increas- ing importance, as before, and again a point of inflection of the p-t curve is reached at 18°; the solution at this temperature contains 42 per cent CaCl2, so the coefficient of the second term is now 0.42/(0.5066-0.42), or about 4.2. The curvature falls off rapidly with increase in the CaCl2 content, and becomes zero at 28° and 48.5 per cent CaCl2. Since at this point X^ rj" — ry' 0.5066 - x^ n' - V^ the ratio of the entropy of vaporization to the entropy of solu- tion is 0.485/(0.506 - 0.885), or about 23 to 1. With further * H. W. Bakhuis Roozeboom, Z. physik. Chem., 4, 31 (1889). HETEROGENEOUS EQUILIBRIUM 257 increase in the CaCl2 content the slope of the y-t curve becomes negative, and the pressure falls with increasing temperature. 1 7. The Minimum Melting Point of a Dissociating Compound. It will be remembered that in the discussion of the system, H2O-KNO3, it was stated that when the liquid phase was very- close in composition to the solid phase, the coefficient of the second term would become large enough for the small negative volume difference (solid — liquid), multiplied by the large coeffi- cient, to equal the very much larger and positive volume difference (vapor — liquid), but that the effect would be difficult to detect in such a system. When that is the case, the denomi- nator approaches zero, the slope* of the p-t curve, dp/dt, becomes infinite, the curve has a vertical tangent, and hence a point of maximum temperature. This is shown clearly in this system. On further increase in the CaCl2 content of the solu- tion, a maximum temperature is found, after which both tem- perature and pressure fall. Two effects take place very close together here; first, the liquid approaches the solid so closely that the denominator becomes zero, then the two compositions become identical. When the two phases, solid and liquid, have the same composition, the equation of the p-t curve becomes dp ri' — 17' dt V' — v^ which is the equation of the melting-point curve of the hexa- hydrate. The condensed system, liquid CaCl2-6H20 + solid CaCla -61120, is one of the great majority of cases where melting causes expansion; both the specific entropy and the specific volume of the liquid are greater than those of the solid phase. This melting point of the hydrate is called the "minimum melting point" because it is the lowest temperature at which solid and liquid of the same composition can exist together in equilibrium; a whole series of such melting points can be obtained at higher pressures in the absence of vapor along the melting-point curve of the hydrate, the curve of the condensed * Cf . footnote on page 251 ; t is represented by the axis of x, p by . tiy . . dp the axis of y, hence ~ is equivalent to -j-. 258 MOREY ART. a system, liquid-solid. It should be pointed out that this mini- mum melting point is not at the point of maximum tempera- ture, but at a lower temperature. The point of maximum temperature is found at such a salt content that the denominator becomes zero, as previously stated, while the minimum melting point lies at a slightly higher salt content, and a lower tempera- ture and pressure. In a system containing a volatile component the point of maximum temperature is not at the composition of the compound, as is the case in systems of non-volatile com- ponents or in condensed systems, but at a composition slightly displaced toward the volatile component. In the case of CaCl2 -61120 the difference is very small, and the two points have never been separated, but at higher temperatures and pressures the difference is no longer negligible. After the minimum melting point has been passed, the coeffi- cient of the second term in the denominator becomes negative, so that in both numerator and denominator the second term, the entropy and volume differences (solid-liquid), in themselves negative, are multiplied by a negative coefficient, hence the second term in both becomes positive, and is to be added to the positive first terms. The slope of the p-t curve is then posi- tive, and remains so until the invariant point, CaCl2 -61120 -f CaCl2 - 4H2O + solution + vapor, is reached, at which a new solid phase, calcium chloride tetrahydrate, makes its appearance. The p-t curves that proceed from this invariant point when dif- ferent phases disappear present some novel features, and are considered in detail below. 18. Correlation of the t-x and p-t Curves. The sequence of the points of maximum temperature and minimum melting point on the three-phase curve, vapor + liquid (saturated solu- tion) + CaCl2-6H20, is brought out especially well by the appli- cation of equation (14), which in this case becomes d^ 1 dfxi Av'' (0 - x^) - Av'-^ (0.5066 - a:0 dx^ ~ 1 — x'- dx^ Av'^ At;"' — Ay"' Atj*' As before, the denominator is positive, and the sign of the numerator is determined by the sign of (x* — x^) = (0.5066 — x^). When the difference (x' — x^) is large and positive, the HETEROGENEOUS EQUILIBRIUM 259 second term predominates, the numerator is negative, and dijdx^ is positive; as {x* — x^) approaches zero, the numerator first approaches zero, and both the p-t and t-x curves show a point of maximum temperature. The numerator remains positive when x* = x^, at the minimum melting point, which is no special point on the i-x curve except when dealing with condensed systems, in which the vapor phase is absent. In the case in which Av''^ is positive, the numerator is still negative, hence dt/dx^ still positive, when x* = x^, and at the point of maximum temperature x' < xK In systems in which both components are volatile, complications arise from the varying composition of the vapor phase, and interesting special cases arise when the vapor-pressure curve of the liquid shows either maximum or minimum points, and also in connection with the location of the maximum sublimation temperature, es- pecially with dissociating compounds.* 19. The Equilibrium between a Dissociating Hydrate and Its Products of Dissociation. From the invariant point, CaCl2 • 6H2O + CaCl2 -41120 + solution + vapor (Fig. 3), four uni- variant equilibria are obtained by the disappearance of each, separately, of these four phases. If the liquid phase dis- appears we have the three phases, hexahydrate, tetrahydrate, and vapor; since all of these phases are of constant composition the pressure is a function of the temperature only; there is no concomitant change in composition of one of the phases. Our equation becomes ^ ^ ("• - "') - t^S^' - "•[ dt , ^ x" — x\ ^ {v" — v') — — -iv'' — v') x'' — x' in which the superscripts h and t represent the hexahydrate and the tetrahydrate, respectively. Substituting the numerical values of X', ^tetrahydrate ^^^ ^hexahydrate^ q^ O.QOQS, and 0.5066, * J. D. van der Waals, Verslag. Akad. Wetenschappen Amsterdam, 6, 482 (1897). A. Smits, Z. physik. Chem., 64, 5 (1906). 260 MOREY ART. G respectively, gives the value of 6.06 as the constant coefficient of the second term. The equation now becomes dp ^ (t?" - V) - 6.06 {-n^ - 7?0 lit ~ {V - vO - 6.06 (v'' - v'Y The numerator of this is always positive. The entropy differ- ence (vapor — tetrahydrate) is always positive. The entropy difference (hexahydrate — tetrahydrate) is negative, since the ^^ iy In ^ «r 5- /^\ 1 ^ I ;<4 < s '^m 1 1 V< «: \ / * \ / S2 \. / lu >t^ / ?: ^s,„^ / 0. o ■ >^'jy , , , ./ .2 .3 ^ .S £ COMPOSITION W H'£l6//rPe/rC£fT -•Kc? -20 o 20 ao TeMeeKATUKE /N DEGPeSS CENTIGRADE Fig. 3. The binary system, H20-CaCl2. Diagrams .4, B, and C are the projections of the curves representing univariant equilibria in the solid f-i-x model on the p-x, p-i, and t-x planes, respectively. decomposition of hexahydrate into tetrahydrate and solution, to be considered later, absorbs heat, and this negative term is multiplied by a negative coefficient, making the second term positive. The denominator is large and positive, because of the very large specific volume of the vapor. The value of dyjdt is consequently positive, and the pressure increases with the temperature, as is the case with the dissociation pressure of the hexahydrate. It is to be observed that this equilib- HETEROGENEOUS EQUILIBRIUM 261 rium requires the presence of both soUd phases, calcium chloride hexahydrate and calcium chloride tetrahydrate, which, together with the vapor, make three phases, hence three equations. The common name, dissociation-pressure curve of the hexahydrate, is misleading; it is the univariant equilib- rium involving all three phases. The invariant point is the high temperature termination of the stable portion of this curve ; when a mixture of these two solids, together with vapor, is heated, at the invariant point some solution is formed; some of the solid melts to form the eutectic liquid. 20. The Equilibrium, Two Solids -\- Liquid. A second uni- variant equilibrium is that formed by the disappearance of vapor. This is the condensed system composed of the two hydrates and the eutectic liquid ; the composition of the eutectic liquid and the eutectic temperature both change as the pressure is increased, but the change is small, and will not be considered further. SI . The Equilibrium, Solid -\- Solution -\- Vapor. Two univari- ant equilibria between solid, liquid, and vapor can be formed, the solubility curves of the hexahydrate and the tetrahydrate. The first of these, the equilibrium vapor + solution -t- CaCl2 • 6H2O, has already been considered; both temperature and pres- sure increase from the invariant point with increase in water content of the solution. At the minimum melting point solid hexahydrate melts to form a liquid of the same composition; this is called a congruent melting point. The other equilibrium between solid, liquid, and vapor is the solubility curve of the tetrahydrate. Application of equation (8) to this brings out no novel features; temperature and pressure both increase as the solution becomes richer in CaCl2, and this portion of the y-t curve is concave downward over its entire course. It differs from the preceding, however, because of the circumstance that, before the point at which the y-t curve has a horizontal tangent, a new solid phase appears, calcium chloride dihydrate. This gives rise to another invariant point, at which the four phases are tetra- hydrate, dihydrate, solution, and vapor. In the case of the hexahydrate the invariant solution was richer in CaCl2 than 262 MOREY ART. G the compound disappearing, the solution was a eutectic, and the compound had a congruent melting point. The solution at this invariant point contains 56.4 per cent CaCl2, while the tetrahydrate contains 60.6 per cent CaCl2; substitution of these values in equation (8) gives dp _ (v" - V) + 0.606 - 0.564 ^^' ~ ^^ dt (t;" - I'O 4- 13.4 (v - v') The positive entropy of vaporization is larger than the negative entropy of fusion multiplied by its coefficient, dp/dt is still positive, and both temperature and pressure are increasing along the solubility curve of the tetrahydrate at the invariant point. This solubility curve differs from the preceding in that solid and liquid do not have the same composition at any point ; calcium chloride tetrahydrate has an incongruent melting point and the invariant point is not a eutectic but a transition point. Pure hexahydrate, when heated, melts to form a liquid of its own composition ; pure tetrahydrate decomposes into dihydrate and saturated solution of the composition of the solution at the invariant point. From this invariant point three other univariant equilibria can be obtained. One of them is the condensed system, whose p-t curve is almost vertical; a second is the dissociation- pressure curve of the tetrahydrate, the univariant equilibrium, tetrahydrate + dihydrate + vapor; the third is the solubility curve of the dihydrate. The curves representing these equilib- ria are shown in Fig. 3. 22. Types of Invariant Points and Univariant Systems. While the preceding discussion has dealt primarily with the application of the Phase Rule to simple systems having only one phase of variable composition, with especial reference to the direct application of equation (1) [97], the modifications necessary to include additional phases of variable composition have been indicated. In a binary system, coexistence of three phases constitutes a univariant system, of four phases, an invariant system, and the possible types of such equilibria are the possible permutations of solid, liquid, and vapor, with the HETEROGENEOUS EQUILIBRIUM 263 additional empirical restrictions that there can be but one vapor phase, and, in a binary system, but two liquid phases. The possible types, representing vapor, liquid, and soUd by V, L, and S, are as follows: Types of Invariant Points; Four Coexisting Phases No. Solid Liquid Vapor 1 01D2O304 — — 2 S1S2S3 L — 3 O102OJ — V 4 S1S2 L1L2 — 5 S1S2 L V 6 s L1L2 V Types of Univariant Systems; Three Coexisting Phases, and the Invariant Types from Which They May Be Derived Derived from 1 blb203 — — 1,2,3 2 S1S2 L — 2,4,5 3 S1S2 — V 3,5 4 s L1L2 — 4,6 5 s L V 5,6 6 — L1L2 V 6 In these various types of univariant systems, one, two, or three of the phases may be of variable composition. Type 1, S1S2S3, is only of interest where there is solid solution. Type 2, S1S2L, is the "condensed" equilibrium, giving the change with pressure of the temperature and composition of a eutectic or an incongruent melting point. The most common example of type 3 is the "dissociation pressure" curve of a salt hydrate; and of type 5, the solubility curve of a salt in water, or the melting-point curve of a fused salt or metal system. Examples of all of the types have been discussed, except those containing two hquid layers, types 4 and 6. Systems in which two Uquid layers are formed are of both theoretical and practical interest, and water-phenol is an excellent example. 23. Equilibrium Involving Two Immiscible Liquids. Water- phenol. In the discussion of the system, water-phenol,* the * F. H. Rhodes and A. L. Markley, J. Phys. Chem., 25, 527 (1921). 264 MOREY ART. G compound formed between the two components will not be con- sidered. It is not readily formed; metastable equilibria be- tween phenol and water in which it is not formed are more easUy realized than the stable ones, with formation of the compound; and its consideration would involve no new prin- ciples. On addition of phenol to water, the ice curve is first traced, down to the eutectic between ice and phenol crystals. The invariant point at which both ice and phenol can coexist, 7^ -^flO . ^^^■v^^^ ^ ^^■~~^^ =8 1 1 1 1 ■ K^ 1 t L C Fig. 4. The binary system, H20-phenol. Diagrams A, J5, and C are the projections of the curves representing univariant equilibria in the solid "p-i-x model on the -p-x, p-t, and t-x planes, respectively. together with solution and vapor, is at —1.2° (Fig. 4) and at a concentration of phenol of less than one per cent. As the temperature is raised above this point, the solubility of phenol increases slightly, until at 1.7° the saturated solution contains about 1.8 per cent phenol. At this temperature the solid phenol in equilibrium with the solution melts, taking up water, and forming a second liquid layer. We have then four phases, solid phenol, a liquid containing 1.8 per cent phenol, a second HETEROGENEOUS EQUILIBRIUM 265 liquid immiscible with the first and containing about 36 per cent of phenol, and a vapor phase containing so small an amount of phenol that we may consider it as pure water. Four uni variant equilibria proceed from this invariant point. The equilibrium, solid phenol + solution + vapor, the solubil- ity curve of solid phenol; and the equilibrium, solid phenol + two liquids, a condensed system giving the change with pressure in the composition of the two layers in equilibrium with solid; present no new features, and will not be considered. The equilibrium between vapor, the water-rich liquid, and the phenol-rich liquid is of greater interest. At the invariant point equation (8) becomes dp _ rc'^ — x'' (^« — v^^) — (y'^ — v^') x^' — x^' Substituting the values 0, 0.018 and 0.36 for the composition of the vapor, the water-rich hquid and the phenol-rich hquid, respectively, gives us (t?" - tjO - W'-v^') dp ^ 0.36 - 0.018 dt (v'' - v^) - 0.053 (y'^ - i;'') and in this case also the entropy and volume of the water are the dominating factors. The p-f curve accordingly is concave upward. As the temperature is increased, the two liquids approach each other in composition, the water-rich layer chang- ing less than the phenol-rich layer. But at the same time their specific entropies and specific volumes approach each other, since both are liquids composed of the same components and increasingly close to each other in composition. For this reason the increasing value of the coefficient of the second term is offset by the decrease in the second term itself, and no maximum pressure is found. Finally, the two phases becomiC identical in composition and properties. At the same time that the differ- ence in composition becomes zero the difference in entropy and 266 MOREY ART. Q the difference in volume become zero, and the equation becomes indeterminate. This is as should be expected; the three-phase system was univariant because there were three equations between the three quantities, pressure, temperature, and com- position. When the two liquid phases become identical, not only in composition but also in properties, there are no longer three phases, but two only, and the system is no longer uni- variant but divariant. In the case of calcium chloride hexa- hydrate, when the liquid and solid phases had the same com- position at the minimum melting point, there was still an entropy difference, since it takes heat to melt a solid, and a volume difference. At the temperature at which the two liquids merge into one another, all distinctions between the phases disappear, and there are but two phases, liquid and vapor. At this temperature there may be not only the critical solution, but also any other mixture of liquid phenol and water; the composition of the solution or the vapor pressure must be fixed in order to completely determine the system. The critical Hquid itself is, however, completely determined. At a temperature very near to the critical solution temperature of the mixture, there are still three equations, and the critical solution is determined by the additional condition that the two phases become identical. We have, then, four equations; three of the type of (1) [97], and the additional equation expressing the condition of identity between the two liquids, so this solu- tion is uniquely determined. If from the invariant point, solid phenol + two liquids -\- vapor, the water-rich layer disappears, we have the univariant equilibrium, solid phenol + a phenol-rich Uquid + vapor. This equilibrium will be realized if the total phenol content of the mixture be greater than that of the phenol-rich liquid, and constitutes another branch of the solubility curve of phenol in water, or of the melting-point curve of phenol-water mixtures along which the solubility of phenol in water increases uni- formly, until the melting point of phenol is reached. This curve does not differ in any important respect from the upper portion of the H2O-KNO3 curve, except that the melting point HETEROGENEOUS EQUILIBRIUM 267 of phenol is so much lower than that of KNO3 that the vapor pressure of the solutions probably decreases, without first rising to a maximum. V. Application of Equation [97] to Systems of Three Components 24. Transformation and Interpretation of Equations. Prob- lems involving a greater number of components may be solved by the same analytical method of treatment, but it will not be possible to elaborate the discussion for systems of more than three components, or to give a complete treatment of ternary systems. *When equation (6) [129] is applied to a three- component system it becomes H' mi m2 mz dp dt H" mi" m^" W W2'" ms'" V mi' V" mi" Y"' m^" mi mz W2" mz" nh"' mr IV IV vrh mz in which the composition of the phases is represented by the actual masses of the components, mi, m^, and W3, and the volume and entropy refer to the total mass. By setting mi + m2 + mz = \, X = mi/inii + W2 -|- W3), y = mn/{mi + 7^2 + mz), we getj * From this point to the end of section (28), and again from (30), third paragraph (p. 281), to the bottom of p. 291, the text is taken, with some omissions, alterations and additions, from the article of G. W. Morey and E. D. Williamson, Jour. Am. Chem. Soc, 40, 59-84 (1917). t This equation has been used in the form of a determinant because of the great convenience of that form of notation. For those not familiar with determinants it may be said that this constitutes a shorthand method of indicating the familiar operation of elimination by cross multiplication. When dealing with systems of more than three com- ponents such a notation becomes almost indispensable. 268 MOREY ART G dp _ dt 7,' 1 X' v" 1 y x" y" ■n'" 1 x'" y'" ly ^ jy r,jy T) i. X y v' 1 v" 1 '"I y V V ly X x" y" x'" y'" 1 x^'^y'"' in which composition is represented by the weight fractions a:, ?/, and \ — x — y oi the three components. Expansion of the right-hand side of this equation gives (15) r?' 1 x" y" 1 x'" y'" -v" Ix' y' lx"'y"' 1 :r^^/^ + v"' Ix' y' 1 x" y" I X y -r Ix' y' 1 x" y" lx"'y"' v' 1 x" y" 1 x'" y'" ix'^'y"' -v" Ix' y' lx"'y"' ix^^'y'"' + v"' Ix' y' 1 x" y" 1 x^^'y'"' -/^ Ix' y' 1 x" y" lx"'y"' The coefficients of -q', 77", v' , v", etc., represent the areas of the triangles p"p"'p^^, prpr„piv^ P'P"P^^, and P'P"P"', re- spectively. It is important to bear in mind the direction in which a given triangle is circumscribed, since, if the area of the triangle P'P"P"' is positive, that of the triangle P"P'P"' is negative. Since the above coefficients represent areas, we will denote the determinants by the letter A, followed by subscripts indicat- ing which triangle is meant, and the direction in which it is circumscribed is given by the order of the subscripts. Thus A 123 represents the determinant 1 x' y' 1 x" y 1 x"'y'' the area of the triangle P'P"P"' II The equation becomes dy dt AiSiV ~ A 134 17" + Ai2iV"' — A 123 V A23iV' — Amv" -\- AmV AmV IV HETEROGENEOUS EQUILIBRIUM It is easy to show that 1 x' y' 269 1 x" y" 1 x'" y'" + 1 X y 1 X y 1 x^^'y'"' 1 x' y' 1 x'" y'" + 1 x'^'y'"' 1 x' y' 1 x" y" 1 x"'y"' or, expressed in areas, that A234 + ^124 = -4i34 + -4i23. Hence we can ehminate any one of the above coefficients,* and cast the equation into the form dp dt (V" -ri'n + iv^ , ^' (,' _ ,-) _ 4^^ (," - ,-) ^23 -123 iv'" -v")-\- IV^ , 4!i4(j;' v'") - 4^' iv" - v'") .(16) 1-123 ■123 S6. Equilibrium, KiO-SiOi-^H^O + Solution + Vapor. A systematic apphcation of this equation to the numerous types of equihbria that may arise in ternary systems will not be possible, and the discussion will be confined to one system, the ternary system, H20-K20-Si02-Si02,t which contains examples of several common types of uni variant equilibria. The experi- mental details are given in the first of the papers just cited; the phase relationships are shown in Figs. 5 to 8. Figure 5 shows the isothermal polybaric saturation curves; Fig. 6, the boundary curves and invariant points ;t Fig. 7, the experimentally deter- * In a 2-component system the corresponding determinant coefficients represent the lengths of lines; in a 4-component system, volumes of solids; in an n-component system, the supervolumes of n-dimensional supersolids. t G. W. Morey and C. N. Fenner, /. Am. Chem. Soc, 39, 1173 (1917). G. W. Morey and E. D. Williamson, /. Am. Chem. Soc, 40, 59 (1918). F. C. Kracek, N. L. Bowen and G. W. Morey, /, Phys. Chem., 33, 1857 (1929). t In the original, a eutectic between K2O -28102 and Si02 is indicated, but later studies (Kracek, Bowen and Morey, op. cit.) have shown that K2O -48102 is formed, and the compound, K2O- 48102 -H2O, may be con- sidered as a hydrate of the former. The necessary changes in the diagrams have been made. 270 MOREY ART. G mined pressure-temperature curves; and Fig. 8, a diagrammatic representation of the same curves. When equation (16) is applied to the ternary equilibrium K2O • SiOj • 5H2O + KzOSiOz ^20Si<^y2^2 /fsOSiO^H^ H20 2SfOg K20-4Si'02 HzO SiOp Fig. 5. The ternary system, H2O-K2O • SiOz-SiOa. The full lines are the isothermal polybaric saturation curves at the temperatures indicated. The broken curves are the boundary curves between the various fields. K2O -28102 + L + V (curve 6c, Figs. 5-8), the curve which pro- ceeds from the quintuple point Q2to quintuple point Qx, it becomes di (t;' - V) -\- \ — W - v^) - - — {v" - v^) i-121 U2I in which S' and S" represent the compounds K2O • Si02 • ^H20 and K20-2Si02- At Q2, the terms (n^ - 17") and (v^ - V), both of which are negative and much larger than the other terms, preponderate; dp/dt is positive. As with increasing tem- HETEROGENEOUS EQUILIBRIUM 271 A^syci\^/io /^OS/Cjr/VpO fe02Si0a t^O^Qt-MiO /^gO s/a, Fig. 6. The ternary system, H2O-K2O -8102-8102. This diagram shows the various boundary curves, which give the locus of the com- position of the liquid phase in the various univariant equilibria. The mvariant (quintuple) points are designated by the letter Q; the numbers on the curves are the same in Figs. 6, 7, and 8. Following is a list of phases stable along each curve. Curve 2. V -f L -f- K20-48i02-H20 + SiOj -I- K20-28i02-H20 + KjO- 48102 -HiO -I- K20-2Si02 -I- K20-4S102-H20 + KjOSiOi-HjO + K20-2S102H20 -I- K20-S102-^H20 + K20-28i02-H20 + K20-8i02-^H20 + K20-28102 -I- KjO-SlOj + KjO- 28102 + K20Si02 4H20 + KjO- 28102 HjO -|- K20-2810, 7b. V + L -f- K20- 28102 •H2O + K20-28i0j Curve Curve Curve Curve Curve Curve Curve Curve 4a. 4b. 6a. 6b. 6c. 6d. 7a. 7a V V V V V V V 4- L L L L L L -1- KjO^SiOjHsO Curve 7b + 7c. V + L + K20-2Si02H20 + K20- 28102 Curve 7a -I- 7b 4- 7c. V -f- K20-28102-H20 -|- K20- 28102, in binary system, HjO-KzO- 28102 Curve 8a. V -f K20- 8102- H2O -f K20- 8102- §H20 -|- KjO- 28102 -HzO Curve 8b. V + L -f- K20-8i02H20 + K20-8102-^H20 Curve 8a -|- 8b. V -f- K20-8i02-H20 -}- K20-8102-^H20, in binary system, H20-K20-8102 Curve 9. V + KzO- 48102 •H2O -|- K2O -48102 -|- 8iOj Curve 10a. V -|- K20-8102-§H20 + K2O-8IO2 -f K20-28i02 Curve 10b. V -i- L + K20-8i02-§H20 + K2O-8IO2 Curve 10a -|- 10b. V -|- K20-8102-iH20 + K2O-SIO2, in binary sys- tem, H20-K20-8i02 Curve 11. V + L -h K2O-28IO2 -|- K20-48102 Curve 12. V + L -f- KjO-48102 + 810, 272 MOREY AKT. G perature the liquid traces the curve Q2Q1, the triangle A^i becomes smaller, while the triangles A21V and Anv become larger. The values of the coefficients of (7?' — 7/O and (r?" — v^) in the I7S ISO i 1 i 1 2 2 us 1 i \ \ioo \ i' /i I j i i \ \ so 1 1 1 1 1 1 - 1 1 i i / \ \ \ x \ *0 30 - ;W \ 20 - /' ;' 1 1 A 0 200 400 600 000 fooa TeMP£f)AruR£ Fig. 7. The ternary system, H2O-K2O -8102-8102. This diagram shows the experimentally determined p-< curves for the various uni- variant equilibria. The dot-dash curves represent univariant equilibria in the binary systems, HjO-KjO-SiOz and HjO-KaO- 28102; the full curves the ternary univariant equilibria, V -|- L -f- 2 solids; the dotted curves the ternary univariant equilibria, V + 3 solids. The invariant points Qsa and Qsb are shown as point Qs, and the curves 11 and 12 are not shown. numerator and (v' — v') and {v" — v^) va. the denominator thus increase rapidly. Since the value oi {v^ — V) is comparatively- large, this increase in the coefficients at first affects materially HETEROGENEOUS EQUILIBRIUM 273 the value of the numerator only. As the Uquid follows the curve Q2Q1 the value of the last two terms of the numerator soon becomes equal to the value of the first term. The numer- ator then becomes zero, dp/dt becomes zero, and the curve has a horizontal* tangent. It will be observed that such a point of maxunum pressure is found on many of the p-t curves Fig. 8. The ternary system, H20-K20-Si02-Si02. A diagrammatic representation of the p-l curves shown in Fig. 7; the numbers on the curves are the same in Figs. 6, 7, and 8. The invariant points Q^a. and Qih are shown as point Qs, and the curves 11 and 12 are not shown. representing univariant equilibrium between two soUds, liquid and vapor in the system. It is most pronounced in the uni- variant equilibrium, K2O • 4Si02 • H2O + SiOa + L -1- V. On further increase in temperature the numerator becomes * Cf. footnote, page 257: -— takes the place of -7- of analytical at dx geometry. 274 MOREY AKT. G positive, the denominator remains negative, hence dp/dt is negative. This continues until, in the case we are considering, the phase K20-Si02 makes its appearance at the quintuple point Qi. Consider the metastable continuation of the curve, KaO-SiOs-^HaO + K2O -28102 + L + V (curve 6c). Beyond Qi, on further increase in temperature the triangle Am approaches zero, the coefficients of (y' — v^) and (v" — v^ in the denominator increase rapidly, reaching such a value that the sum of the last two terms in the denominator becomes numerically equal to the first, in spite of the large value of (v' — «"). The denominator then approaches zero, and dp/dt becomes infinite. At this point the p-t curve has a vertical tangent. Beyond this point dp/dt again becomes positive. An illustration of this case is found in the p-t curves of the univariant systems, K2O -28102 + K2O - 48102 - H2O + L + V (curve 46), and 8i02 + K2O - 48102 • H2O + L + V (curve 2), which proceed from Qs to higher temperature and pressure. 26. Coincidence Theorem. On further increase in tempera- ture the hquid will He on the fine, K2O • 8102 - ^H20-K20 - 28102, the area Ani becomes zero, and equation (16) becomes ^ _ A21V iv' - 7?') - Ally iv" - ■>?0 dt ~ A21V W - uO - Aiiv iv" - v^) ' At this point the curve has the same slope as the common melting-point curve of (K2O • 8102 • IH2O + K2O -28102), an illustration of the general relation that when a linear relation exists between the composition of n or fewer phases, the p-t curves of all univariant systems containing these phases coin- cide. When all the reacting phases have a constant composi- tion, the curves will coincide throughout their course; when the compositions of some or all of them are variable, and they only casually have such a composition that the above linear relation is possible, then the curves are tangent.* Let us prove this in detail for three phases lying on a straight line in a three-component system. Consider the p-t curves * F. A. H. Schreinemakers (Proc. Acad. Sci. Amsterdam, 19, 514-27, (1916) and subsequent papers in the same journal) mentions some special cases of this general theorem. HETEROGENEOUS EQUILIBRIUM 275 of the univariant equilibria, P' + P" + P'" + P^^ and P' + P" + -P^^ + P^ , which proceed from the quintuple point, P' + P" + P'" + P^^ + P^. The equation of the first of these is H' m/ W/i niz H" m/' W2" ms" dp _ dt F' w/ m^' rriz V" my" m," m," V" m,'" m^'" m,"' IV IV IV IV mi 7712 W3 F Now assume that P', P", P'" lie on a straight line in the com- position diagram,* We then have the relation and hence also and A'P' = A"P" + A"'P"', nt in nil > AW = A'W + A AW = A'W' + A"W'\ AW = A'W' + A By substituting these values of mi ', ma', W3' in the above deter- minants, and subtracting A" times the second row and A'" times the third row from A' times the first row, we get A'R' - A"}i" - A"'R"' 0 0 0 H" mi" ms" mz" dp jj/// m/" mz'" mr mz dt A'V -A"V" _ A"'V"' 0 0 0 Y" mi m2 mz" yiit ylV mr mz ml'' * An example of this is found in Fig. 5. Here the phases are K20-2Si02, K20-2Si02-H20 and V; the vapor phase contains only H2O, and its composition is represented by the apex of the component triangle. 276 which reduces to MOREY ART, G iA'R' - A"R" - A"'R"') dp dt mi" rrii" m" m{" m^" m,'" IV IV IV mi Mi W3 (A'V'-A"V"-A"'V"') mi" m-l' m" m{" mr mz"' IV IV IV mi m2 mz or ^ A'R' - A"R" - A"'R"' dt ~ A'V - A"V" - A"'V"' ' Similarly, the relation between the variations of p and t in the second of the above univariant equilibria, P' + P" + P"' + P^, reduces to the same expression. It will be observed that the coefficients A', A", A"' are those that occur in the reaction equation A'P' = A"P" + A"'P"'. Hence we see that whenever three phases lie on a straight line in the composition diagram, the p-t curves of all ternary equilibria containing these three phases coincide with each other and with the p-t curve of the univariant binary equihbrium between the three phases alone. 27. Equilibrium, K20-2Si02-H20 + KiO-SSiO^ + Solution + Vapor. We will now consider the application of our equation to a different type of equilibrium between two soUds, liquid and vapor. Consider the equilibrium, K2O • 2Si02 • H2O + K2O • 2Si02 + L + V (curve 76 + 7c). In the concentration diagram the course of this equilibrium is the curve Q2Q4, the boundary curve between the fields of K2O -28102 and K2O -28102 -1120. Since the two solid phases and vapor lie on a straight line, the equation becomes dp^ _ Aivi iv' - v") - Aui (v" - 77") dt ~ A2VI W - 2;") - Am {v" - v")' in which P' and P" represent K20-2Si02 and K2O -28102 •H2O, respectively. This is the equation of the dissociation-pressure HETEROGENEOUS EQUILIBRIUM 277 curve of K20-2Si02-H20- Hence, as we saw before, the p-t curves of the equUibrium, K2O • 2Si02 • H2O + K2O -28102 + L + V, coincide with the dissociation-pressure curve of K2O • 2Si02 • H2O, The slope of this curve will remain positive as we go along the boundary curve, K20-2Si02-K20 -28102 -1120, and will not show anything special until the liquid phase falls on the line, V-K2O-28IO2. But here the two triangles A234 and A 134 become zero at the same time, and the equation becomes meaningless. This point corresponds to the termination of the curve at the quadruple point, K20- 28102 + K2O - 28102 • H2O •f L + V in the binary system, H2O-K2O • 28IO2. When the liquid has crossed the line, H2O-K2O • 28102 the areas of all the triangles change sign, hence dp/dt remains positive, and with decreasing temperature we retrace the same p-t curve to the quintuple point Q4. This portion of the curve also corre- sponds to the equilibrium, K2O - 4SIO2 • H2O + K2O -28102 + K20-28102-H20 -f V. In the first equihbrium considered, the univariant equilib- rium, K20-8i02-^H20 -\- K2O-8IO2 + L + V, the assumption that the vapor phase is pure H2O was practically without Influence; the vapor phase might contain appreciable quantities of either K2O or SIO2 or both without appreciably affecting the course of the p-t curve. The only effect would be a slight diminution of the areas Auv and A21V, the coefficients of (77" — 77O and (v" — v^), and of {-q' - v^) and {v' - v^), respectively. In the second case, however, the assumption is of Importance; only in the Improbable case that the ratio of 8102 /K2O in the vapor is the same as in the solid, i.e., 2/1, would it still be true that the equilibrium, K2O -28102 + K2O • 28IO2 - H2O + L -f V, coincides with the equilibria K2O-28IO2 + K2O - 28IO2 - H2O + K20-8i02-|H20 + V and K2O -28102 + K2O • 28102 • H2O ■\- K2O- 48102 -1120 + V, and with the dissociation-pressure curve of K2O -28102-1120. In case the vapor contained a small amount of K2O, the curve, K2O-2SIO2 + K2O • 28IO2 - H2O + L -f V, would consist of two parts, one on one side, the other on the other side, of the dissociation-pressure curve, and the two parts would join at the top in a smooth curve, whose point of maximum temperature would be found at the point where 278 MOREY ART. G the entropy change in the reaction passes through zero, hence on the K2O side of the hne, K20-2Si02 - K20-2Si02-H20. But unless the K2O content of the vapor is large, which is improbable, the effect will be small; the area, K2O -28102 - K2O • 2Si02 • H2O - V, instead of being zero, will be a very- small quantity which will have but a shght influence on the above relations; the curves, instead of coinciding, would lie very close to each other. 28. Equilibrium, KiO-SiOi-^H^O + KiO-SiO^ + Solution + Vapor. All the p-t curves so far discussed have had their end-points inside the component triangle ; all of them have gone from one quintuple point to another. Let us now consider one which goes from a quintuple point to a quadruple point in one of the limiting binary systems, e.g., the curve, K20-Si02 + K2O • Si02 • ^H20 + L + V (curve 106), which goes from quintuple point Qi to the quadruple point, K20-Si02 + K2O • Si02 • ^H20 + L + V, in the binary system, H2O-K2O • Si02. Since the phases, V, K2O -8102 -^1120, and K20Si02, lie on a straight line, the area of the triangle, V-K2O • Si02 • IH2O-K2O • Si02, is zero, and the equation of the p-t curve reduces to dp _ A,„i (V - v") - Au, (V - v") dt A^viiv' -v") - A,,i {v' - v") ' in which the accents (') and (") refer to the solid phases, K2O • Si02 and K2O • Si02 • IH2O, respectively. This is evidently the dissociation-pressure curve of K2O • Si02 • ^H20 ; in harmony with our previous conclusions, the slope of the curve, K2O • Si02 -|- K20-Si02-^H20 + L + V (106), is the same as that of the dissociation-pressure curve of K2O • Si02 • IH2O (10a + 106). At the quintuple point it is evident that both numerator and denominator are negative, dp/dt therefore positive. Also, the denominator being much larger than the numerator, the numerical value of dp/dt is less than unity. As the liquid approaches the side of the component triangle along the bound- ary curve, both the triangles A2VI and Aivi diminish in size in about the same proportion, and the value of dp/dt will not change materially. When the liquid gets on the line, H2O- HETEROGENEOUS EQUILIBRIUM 279 K20-Si02, both triangles become zero simultaneously, and the equation becomes indeterminate; the curve is at its end point at the quadruple point in the binary system. It is evident that when the phases have the composition indicated above, no maximum is possible in the p-t curve of the univariant equilibrium. However, if the vapor phase, in- stead of being pure H2O, contained a small amount of Si02, the curve would have a horizontal tangent before the phases, L, K2O -8102 41120, and K20-Si02 fell on a straight line, as can readily be seen from the equation of the curve. 29. Equilibrium, KiO-^SiOi + K^O-J^SiOi-H^O + Solution + Vapor. In the discussion of binary systems, it was seen that when a volatile component is considered, the maximum temperature is not at the composition of a compound, as in condensed systems, but is displaced in the direction of the more volatile component. A similar condition is found in the general case; an example in a ternary system is found along the curve, K2O -28102 + K2O - 48102 - H2O + L + V (curve 46), which goes from Q4 to Qsa. The equation of this curve is dp Am Ani dt , , . , Aiiv , , . A.\\.o . {v^ - v^) + -r- {v' - v'-) - -r~ (^ " " ) A\2i Am in which the accents (') and (") refer to the phases, K2O- 28102 and K2O- 48102 -H2O, respectively. The condition for a temperature maximum is that the denominator of this expres- sion shall approach zero as a limit; dp/dt becomes infinite. Since the volume difference between vapor and liquid is far greater than that between solid and liquid, the denominator will approach zero as a limit only when the coefficients of the last two volume differences become very large, hence when the area of the triangle, K2O - 2Si02-K20 • 48102 • H2O-L, becomes very small. This point will be reached slightly before the liquid phase lies on the line, K2O - 28i02-K20 - 48102 • H2O, hence the point of maximum temperature has been displaced sUghtly in the direction of the volatile component. 280 MOREY ART. G 30. The Order of p-t Curves around an Invariant Point. In the general consideration of phase equihbria it is convenient to proceed from a consideration of the invariant points to the various univarlant equihbria which proceed therefrom, and to consider the sequence of the p-t curves around the invariant point. Such a course is often of great value in determining the stable phases in an investigation of complex systems. The order* of the p-t curves may be deduced from the theorem that whenever a linear relation exists between n of the n -f 1 phases in a univariant equilibrium, the p-t curves of all the univariant systems containing these phases coincide. But these curves extend in both directions from the invariant point ; in one direc- tion the equilibrium under consideration will be stable, in the other, metastable, and to tell the actual position of any curve, or to distinguish between the stable and metastable portions of any one curve, a knowledge of the entropy and volume changes is necessary. However, it will be shown that two adjoining curves, i.e., curves that are not separated by either the stable or metastable portions of other curves, e.g., the p-t curves of the univariant ternary equilibria, P' + P" + P'" + P^^ and pi _|_ pii _|_ pni _|_ pv ^ ^^jj coincide in their stable portions, that is, are stable in the same direction from the invariant point, when the phases P^^ and P^ lie on opposite sides of the straight hne P'P"P"', and vice versa. With the aid of these theorems and general considerations to be discussed later the actual position of the p-t curves may be fixed within certain limits. The above theorem may be proved as follows. From the definition of the chemical potential n, if the ^i of a substance in a given phase is greater than the n of the same substance in another phase, the two phases are not in equilibrium with respect to that substance and it will tend to pass from the phase in which its chemical potential is the greater into that phase in which its chemical potential is the less. At the triple point, ice + water + vapor in the one-component system, * By "the order of the p-i curves" is meant the sequence in which we shall cut the curves as we circle around the invariant point, with the stipulation that reversing the direction of rotation reverses the sequence but not the order. HETEROGENEOUS EQUILIBRIUM 281 H2O, the chemical potential of H2O in all three phases is the same. If we simultaneously change the pressure and tem- perature so as to proceed along any one of the three 'p-t curves that intersect at the triple point, one of the phases will dis- appear. By making these changes we have given greater incre- ments to the chemical potential of the phase that disappears than to the chemical potentials of the other two phases; the chemical potential of water remains equal in these two phases since we, by hypothesis, have made such changes of pressure and temperature as to proceed along the -p-t curve of stable coincidence of these phases. The fundamental equations of the form of (1) [97] for the three phases that coexist at the triple point are Vdj) = Wdt + m\l^\ V'dp = H'rfi + w'^m', V'dp = R'dt + m'dij.% in which the indices v, I, s refer to the vapor, liquid, and solid phases. Each of these equations may be divided by the mass m of the phase; in the resulting equations v^dp = rj^dt -f- dn", v^dp = rj^dt + djjL^, v'dp = ri'dt + dn', the volume and entropy terms refer to the specific volume and entropy of each phase. Now if, as stated above, we proceed along the p-t curve of the condensed system, ice-liquid, which is one of the p-t curves that intersect at the triple point, we can obtain a value for dn, the differential of the chemical potential, from the two equations of the type of (1) [97] referring to the liquid and solid phases, by solving the two equations for dt in terms of dp, which will give us yl _ y» dt = -j , dp, and substituting this value of dt in one of the original equations 282 MOREY ART. G Substituting in the equation referring to the Uquid phase, we get [v'- — v'~\ Similarly, the value of d^y in the stable direction of the curve, is given by r v^-v'l dp. Now since, by hypothesis, we have proceeded in the direction of the stable portion of the curve, ice + Hquid, (Zm" > dyiK Hence which reduces to dp [{V - v^)W - V') - (v^ - v')(v'' - v^)] 7}^ — t]' >0, one form of the condition for stability of the equilibrium solid + liquid. When we consider the actual magnitude of the various terms in this equation we see that the coefficient of dp in the numer- ator is necessarily positive. All the individual terms {v" — v^), W ~ v'), iff — V^) a-iid (^' ~ V') are of necessity positive except the last one, the volume change of melting of ice, which is negative. But the last term is affected by the negative sign, hence the term as a whole is positive, and the coefficient of dp has a positive sign.* The equilibrium in question will then be stable as the pressure is increased from the invariant point * The case that (v^ — v') is negative is, of course, exceptional. But in any case, the coefficient of dp is positive, since the two entropy changes are of the same order of magnitude, while the volume change on evaporation is many times larger than the volume change on melting. HETEROGENEOUS EQUILIBRIUM 283 when the denominator is positive; (tj' — rj') is of necessity- positive, hence the equilibrium, ice + Hquid, is stable with increasing pressure from the invariant point; on decreasing the pressure we pass on to the metastable portion of the curve, into a region where vapor is stable. By solving for dp in the above equations of the type of (1) [97] referring to the solid and liquid phases, v/e get a similar in- equality, Jjl _ J^,» >o, which gives the condition for stability with change in tem- perature. It will be observed that the condition for tempera- ture stability differs from the condition for pressure stability in having dt in place of dp in the numerator, and in having (v^ — V) in place of (r?' — rj*) in the denominator. Since the coefficient in the numerator is unchanged, it is always positive ; the equilibrium, solid -\- liquid, is stable with increasing tempera- ture when the denominator is positive, and is stable with de- creasing temperature when the denominator is negative. In the exceptional case of H2O, this volume change is negative, hence the equilibrium, ice + liquid, is stable with decreasing temperature from the triple point; on increasing the temperature we pass on to the metastable portion of the curve, into a region in which vapor is stable. SI. Generalized Theorem Concerning the Order of p-t Curves around an Invariant Point. The above reasoning may be generalized as follows. At an invariant point, if the differentials satisfy the (n + 1) equations of the type of (1) [97] for the univariant equilibrium, P' + P'" -\- P^^ ... + P"+i + pn+2 (jjj which phase P" is missing), we will move along the p-t curve of this equilibrium. In one direction from the in- variant point the missing phase P" will be stable, in the other direction phase P" will be unstable. In the first case, we will be on the metastable prolongation of the p-t curve, in the second case, we will be on the stable portion of the p-t curve. The condition that a given phase in a one-component system is unstable was found to be that its chemical potential is greater 284 MOREY ART, G than the chemical potential of the stable coexisting set of phases, which condition is represented by the inequality Vdjp — 'S.dt > midiii + niidni . . . + nindfXn. Similarly, the condition that the equilibrium P" + P'" + P^^ . . . + P" + i + P"+' is stable is that the missing phase P' is unstable. By solving the (n + 1) equations of the type of (1) [97], referring to the (n + 1) coexisting phases of the equilibrium in which P" is the missing phase, for dm, dixz, dm, and dt in terms of djp, and substituting in the above inequahty, (the quantities F, H, Wi, mz, . . . rUn referring to phase P") the stability is found to depend upon the sign of the following ex- pression : dp H" V" H'" V'" jjIV ylV mi mi II IV mi m m2, as < sb, so that s is closer to point a; when mi < nh, s is situated closer to point b. If we imagine a mass mi in point a and a mass m2 in point b, then it follows from (3) that point s is the centre of gravity of these masses. If we denote the f 's of the phases A and B by f] and ^2, then the total ^ of system (1) is yriiti + m2^2- If we call the i' of a unit quantity of this system ^s, then we have mi Ti + m2 ^2 ,_>, ts = T (^) m.i + m2 We now take aa' = fi and bb' = ^2 (see Fig. 2). Then f, = ss'. This can easily be proved. For ss' ^ sp + ps' = f 1 + ps\ (6) But from the similarity of the triangles a'ps', a'qb' it follows that ps' a'p as m2 ,„. qb a q ab mi + m2 and from (7) follows m2 , m2 , , ps' = — r~ X qb' = — —- X (r2 - ri). ^ mi + m2 mi + m2 Substituting this value of ps' in (6), mi Ti + ^2 12 ,„x ss = , Co; mi + m2 From (5) and (8) we see that f « = 8s' . REPRESENTATION BY ZETA FUNCTION 299 If we now call s' the f-point of the system, then we can state that the f -point of system (1) is represented by the centre of gravity of masses nii and m2 at the f-points a' and h'. From this it appears that each point of the line a'b' represents the f-point of a system (1) ; the closer this point lies to a' the greater the value of Wi:w2, the closer to h' the smaller the value of mi : rrii. For this reason we shall call a'b' the f-line of the two- phase system or phase complex A -}- B. 4. According to a theorem of Gibbs, at constant t and p a given quantity of substance arranges itself in such a way that the total ^ is a minimum. Or, of all systems (phases) at con- stant t and p with the same total composition (in regard to the independent components), that is the most stable one which /K Fig. 3 has the smallest f . In order to apply this in the graphical repre- sentation, we take a point e (Fig. 3). This point e may repre- sent a single phase, e.g., a liquid, a vapor, a mixed crystal, or possibly a compound. The point e may represent also various phase-complexes or systems, e.g., of the phases a and h, or z and u (see Fig. 4). We shall represent all these possible or conceivable phases and systems, which have the same composition e, by El, E2, Ez etc., and their ^-points by e', e" , e'" etc. It is clear that all these ^-points are situated on a vertical line (ordinate) through the point e. Since each of the phases or phase- complexes denoted by Ei, E2, Ez etc. contains in toto one mol of the components W and X and has the same composition with respect to these components, it foUows that each of these phases 300 SCHREINEMAKERS ART. H or phase-complexes (systems) contains the same amounts of the components W and X. As we have taken ee' < ee" < ee'", and consequently Ei has the smallest f , Ei is the most stable, according to the theorem of Gibbs mentioned above. Therefore Es and E2 may change into Ei, but the opposite transformation, i.e., of El into E2 or Es, is not possible. So in general we may say: of all phases and systems, the f-points of which are situated perpendicularly above one another in the (f, a;)-diagram at constant temperature and pressure, that one is the most stable l¥ z Fig. 4 which possesses the lowest ^-point. In the following con- siderations we shall make frequent use of this principle. 5. We now assume that the curve W'X' of Figs. 4 and 5 represents the f-curve of a series of liquids. This curve may be, as in Fig. 4, at all points convex towards the composition axis, or, as in Fig. 5, partly convex and partly concave. A point e of Fig. 4 may represent not only the single liquid phase e but also an infinite number of systems of two liquids, e.g., of the Hquids a and 6, or of z and u, etc. We call these the systems L(a) + L{b), or L(z) + L(u), etc. The ^point of liquid e is represented by the point e' of the ^--curve, that of L(a) + L(6) by the point e" of the hne a'b\ and that of L(z) + L{u) by the REPRESENTATION BY ZETA FUNCTION 301 point e'" of the line z'u'. So the transformations L{a) + L{b) -^ L{e) Liz) + Liu) -> Lie) are possible, namely a mixing of the liquids a and b or of z and u to give e. But the opposite changes, i.e., a separation of the liquid e into liquids a and h or into liquids 2 and u, are not possible. Since these considerations apply equally to every liquid e of Fig. 4, it follows that: when the ^-curve is wholly W w Fig. 5 convex towards the composition axis, all the liquids are stable and miscible with one another in all proportions. 6. In Fig. 5 we can draw a double tangent line, touching the f-curve in points a' and b'. Since the f-point e" of the system Lia) + L{b) now lies below the f -point e' of the liquid phase e, the conversion L(e) —^ Lia) + L{b) may occur, i.e., a separation of liquid e into the liquids a and b. Conversely, the liquids a and b cannot mix to give the liquid e. Hence we have the following result for Fig. 5. All the liquids of Wa and bX are stable; all the liquids between a and b are metastable or un- stable, and separate or tend to separate into the stable system Lia) + Lib). Let us take at ordinary temperature and pressure W = water, X = ether. If we now add so little ether to the 302 SCHREINEMAKERS art. h water that the former is completely dissolved, we get a solution of ether in water represented by a point of Wa. If we add so little water to ether that the water completely dissolves, we get a solution of water in ether represented by a point of bX. If, however, we bring ether and water together in such a propor- tion that their mixture is represented by a point between a and b, then no homogeneous liquid is formed, but on the contrary the system, or phase-complex, L(a) + L(b), i.e., a liquid a containing much water and little ether, and a liquid b containing much ether and httle water. 7. In relation to the further discussion we shall deduce the foregoing results also in the following way. Every chord we may draw in Figs. 4 and 5 is also the ^-line of a conceivable two-phase system. Thus each point of a'b' represents the f -point of a system L{a) -\- L{b), each point of z'u' the ^-point of a system L{z) -{- L{u), etc. So we may imagine an infinite number of ^-points on every arbitrary vertical line; the lowest f-point of every vertical line represents a stable state. Of all the f-points we can imagine in Fig. 4 on a vertical line, the point of intersection with the f-curve is lowest, and hence it follows that of all conceivable ^-points of Fig. 4 only those of the f-curve represent stable states. Of all chords which we may imagine to be drawn in Fig. 5, one, a'b', touches the ^-curve in two points. The part a'e'b' of the f-curve Ues above this chord a'b'. If we now imagine vertical lines drawn through the points between W and a, between a and 6, and between b and X, we see that of all conceivable ^-points of Fig. 5 only those of the parts Wa' and b'X' of the f-curve and those of the double tangent a'b' represent stable states. This means that only the liquids of Wa and bX and the system L{a) -\- L{b) are stable. 8. We now assume that the points of WX represent mixed crystals. Then their f-curve may also have the form shown in Fig. 4 or Fig. 5. When Fig. 4 obtains, it follows that the two solid components W and X are miscible with each other in all proportions and form an unbroken series of mixed crystals. When Fig. 5 obtains, then only the mixed crystals of Wa and bX are stable; all others (namely between a and 6) are meta- stable or unstable, and separate or tend to separate into the REPRESENTATION BY ZETA FUNCTION 303 stable system M{a) + M{b), i.e., into a mixture of the mixed crystals M(a) and M{h). In this case no continuous series of mixed crystals exists and consequently the two solid components W and X are not miscible with each other in all proportions. 9. Since vapors (gases) are miscible with one another in all proportions their f-curve always has the form shown in Fig. 4. 10. If we represent the entropy and volume of a phase by Tj and V respectively, then we have in accordance with Gibbs the following relations: d{^)p = -ndt, d(Ot = vdp, (9) for de = tdr] — pdv, and differentiation oi ^ = e — t-q -\- pv gives d^ = de — tdr] — 7]dt + pdv + vdp, whence d^ = vdp — -qdt. This means that the f of a phase decreases when the temperature (at constant pressure) increases, and increases when the pressure (at constant temperature) increases.* If we apply this to every point of a f-curve in our diagrams we see that every point of a f-curve sinks towards the a:-axis with increase of t. As, however, all phases do not possess the same entropy and consequently all f-points do not sink at the same rate, it follows that with increase of temperature the ^-curve sinks, with decrease of temperature it rises, its form changing at the same time. If we represent the f-points of solid W and solid X by (W) and (X) respectively, then they also will sink with rise of tem- perature and rise with fall of temperature. Since the liquids W and X have greater entropies (at a given temperature) than the corresponding solid substances W and X, the points W and X' sink with rise of temperature and rise with fall of tempera- ture, but in each case at a faster rate than the corresponding points (T^') and (X). * When the phases are closed and the components independent, 'Lfidm = 0. 304 SCHREINEMAKERS art. h III. Binary Systems in Which Besides Liquids Only the Solid Components W and X Can Occur 11. In a system formed from the components W and X, liquids, vapors and solid substances may occur, viz.: the pure substances W and X and their compounds or mixed crystals. It depends on the values of t and p, and on the nature of the com- ponents, which of these phases are formed. At first we take a system in which neither compounds nor mixed crystals occur. If now we make the pressure so high that no vapor can be formed, then the only types of phases possible will be liquids and solids W and X. We have therefore only to deal with the f-curve and the points (W) and (X). Furthermore, we shall assume in the first place that the f-curve is wholly convex towards the composition axis (Fig. 4, Figs. 6-9). If we lower the temperature for which Fig. 4 obtains, then, as we have seen, the points (W) and (X) and the whole f-curve will rise. Since X' rises more rapidly than (X), these points will first become coincident, after which X' will rise above (X). When this is the case, but W is still below (W), we get Fig. 6. With further fall of temperature W also rises above (W) and we get Fig. 7. Thus with continued decrease of temperature we have the succession of diagrams: Fig. 4 — Fig. 6 — Fig. 7 — Fig. 8— Fig. 9. We now represent the melting-points* of solids W and X (under a definite pressure) by T{W) and T{X), and for the sake of definiteness we take T{X) > T(W), e.g., X = a salt and W = water. We call the T for which Fig. 8 holds good T{e). Later on we shall see that this is the eutectic temperature of the system. We can now distinguish the following cases for the temperature T: (i) T > TiX) > T(W) > T(e). As T now is higher than the melting-points of each of the components X and W, these are stable only in the liquid state and hence W is lower than (W), X' lower than (Z), (case of Fig. 4). (ii) T{X) > T > T(JV) > T{e). The stable state of X is *From this point onwards in the present article, and in the corre- sponding figures, temperature is denoted by T. REPRESENTATION BY ZETA FUNCTION 305 the solid state, hence (X) is lower than X'. The point W is, however, still below (W) (case of Fig. 6). Fig. 7 (iii) T(X) > T(W) > T > T{e). Since now, by simUar reasoning, (X) lies below X' and (TF) below W, we have one of 306 SCHREINEMA KERS ART. H the Figs. 7, 8, and 9. As we take T > T{e), we get the case of Fig. 7. Fig. 9 (iv) TiX) > T(W) > T = T(e) (case of Fig. 8). (v) T(X) > T(W) > Tie) > T (case of Fig. 9). 12. We shall now deduce which phases and systems (phase- REPRESENTATION BY ZETA FUNCTION 307 complexes) are stable in each of these five cases. We shall represent them in Fig. 10, in which temperature has been taken as the ordinate (isobaric T-x diagram). The points T(W) and T(X) in this figure represent the respective melting-points of the substances W and X. (i) T > T{X) > T{W) > T(e) (Fig. 4). We have already seen that in this case the stable states for W and X are the liquid state, and that all liquids are stable. We represent these liquids in Fig. 10 by the points of a line 1.1' situated above T{X). (ii) T{X) > T > T{W) > Tie) (Fig. 6). Every straight line uniting an arbitrary point z' of the f-curve with the point (X) is the f-line of a system L{z) + solid X, (10) consisting of the two phases, liquid z and solid X. If we take, for example, the line a'{X), then every point of this line (e.g., h", c", etc.) represents the f-point of a system, L{a) + solid X. Similarly every point of the fine c'{X) represents the f -point of a system L{c) -f solid X. So we may imagine an infinite number of lines z'iX), of which in Fig. 6 only a'{X), c'{X) and d'{X) have been drawn. Of all these conceivable lines, the line c'{X), touching the f -curve in c', plays a great part. It is clear from the diagram that the f-points of all phase-complexes whose compositions lie between W and c lie above the corresponding points of the f-curve (f-points of the hquids of corresponding composition), whilst the f-points of all hquids whose composi- tions lie between c and X lie above the corresponding ^-points of the phase-complex L{c) + solid X. Hence of all conceivable f-points of Fig. 6 only those of the part W'a'h'c' of the ^-curve and those of the tangent c'{X) represent stable states. Thus of all conceivable systems of the type (10) only the system L{c) + solid X (11) is stable. Thus L(c) represents the liquid saturated with respect to solid X and therefore in equilibrium with it. All liquids between c and X are supersaturated and tend to pass into (11) with separation of solid X, whilst all liquids between W and c 308 SCHREINEMAKERS AKT. H are unsaturated. If we imagine the liquid c represented by point c in Fig. 10, then the points of 2-c represent unsaturated hquids, whilst the points of c-2' represent supersaturated liquids which pass into the system (11). (iii) T{X) > T(W) > T > T{e) (case of Fig. 7). Since Fig. 10 both the substances W and X are now solid we may imagine the systems L{u) + solid W, L(z) + solid X,] solid W + sohd X. (12) Besides the lines z'iX) discussed above, we must now imagine in Fig. 7 also the lines u'(W) and {W)(X), and we can now draw a tangent to the f -curve through each of the points (W) and (X). If g' and h' are the respective points of contact, we see that of all conceivable f-points of Fig. 7 only those of the tangents (W)g' and h'(X), and those of the part g'h' of the ^-curve, represent stable states. From this it follows that of all REPRESENTATION BY ZETA FUNCTION 309 conceivable systems (12), only L(g) + solid W and L(h) + solid X are stable. Liquid g is saturated with respect to solid W, and liquid h with respect to solid X. All liquids between W and g are supersaturated with respect to solid W, all liquids between h and X with respect to solid X. All liquids between g and h are unsaturated. In Fig. 10 the liquids g and h are repre- sented by the points g and h of the line 3.3'. (iv) T{X) > T{W) > T = T{e) (case of Fig. 8). When the points of contact g' and h' of Fig. 7 coincide we obtain Fig. 8, in which the f-curve and the straight line (W)(X) touch one another in the point e'. In this case we see that of all conceivable ^-points of Fig. 8 only those of the line (W)e'{X) represent stable states. Since the point e' lies not only on this straight line but also on the f-curve, the point e' may now represent not only solid W + solid X but also the liquid of composition e. We now have a f-line of which not only the two end points but also a third point e' represent stable phases. Every point of the hne {W)(X) can represent therefore a system soHd W + solid X, whilst each point of the part (W)e' can represent also a system L{e) + solid W, and each point of the part e'(X) also a system L(e) + soUd X. From this it follows that of all liquids only the liquid e is now stable, whilst of all conceivable systems (12) only the systems : L(e) + solid W, L{e) + solid X, solid W + solid X, (13) are stable. Since L(e) is saturated with respect both to W and X, therefore also the three-phase system L(e) + solid W + solid X (14) can exist, in which the reaction solid W + solid X ^ L(e) (15) can occur. For we have already seen that the liquid e has the same f as a system, solid W -f- soHd X, with the composition e 310 SCHREINEMAKERS art. h (i.e., f = ee'). The f of the three-phase system (14) remains unchanged, therefore, whether the reaction (15) occurs in the one or the other direction. When this reaction proceeds from left to right, heat is absorbed; when it proceeds from right to left, heat is produced. Given a unit system of composition e at temperature T(e) (and the given pressure) we cannot predict its phase structure without further information (e.g., concerning its past history, or its behavior on adding or abstracting heat energy, etc.). The hquid e is represented in Fig. 10 by the point e, and the systems discussed by points on the line 4 • e • 4'. (v) T{X) > TiW) > T{e) > T (case of Fig. 9). Since the line {W){X) now lies wholly below the f -curve (the free energy liquidus curve), all the liquids are metastable and tend to pass into the mixture, solid W + solid X. From this discussion it follows that T{e) is the lowest temperature for the existence of a stable liquid phase. T{e) is therefore the eutectic tem- perature and L{e) the eutectic liquid of the (W, X) system. If we take W = water, so that the three-phase system (14) be- comes L(e) -\r ice + solid X, then we call T{e) also the cryo- hydrate temperature. 13. From the preceding considerations we can now make the following statements about Fig. 10. The liquids saturated with solid W are represented by the points of a curve eT{W), the saturation curve of W, whilst the liquids saturated with solid X are represented by the points of a curve eT(X), the saturation curve of X. These two curves and the line 4-e-4' divide Fig. 10 into four fields. Each point of field I represents an unsaturated liquid. Each point of field II represents a system L(z) + solid X, or alternatively a liquid which is super- saturated with respect to solid A^. Similarly each point of field III represents a system L(u) + solid W, or a liquid super- saturated with respect to solid W, whilst finally each point of field IV represents a mixture of solid W and solid X. The two saturation curves do not terminate in e but are prolonged into field IV, in which they represent metastable states. We find the points of these prolongations, and we see also that they represent metastable states, when we imagine REPRESENTATION BY ZETA FUNCTION 311 tangents to the f-curve drawn from the points (W) and (X) of Fig. 9 (and similar figures). 14. When the sohd substance X can exist in the two modifi- cations a and /?, we may suppose the f-point of soHd a in Fig. 6 represented by (X) and that of soHd /3 by ^', so that the modifica- tion )8 is metastable with respect to a. If we draw a tangent to the f-curve from 13', the point of contact, which is situated somewhere between c' and X', represents the f-point of the liquid saturated with respect to solid /?, whilst the liquid itself lies somewhere between c and X. From this it follows that, fr 1' w e Fig. 11 u when a substance X exists in two or more modifications, the most stable form has the smallest solubility. 15. In Fig. 11, in which the f-curve has a part concave to the composition axis, the point of intersection of the double tangent z'u' with the line XX' has been represented by the point s. If we take T = T{X), then (Z), i.e., the f-point of solid X, coin- cides with X'. If we lower the temperature, then the point {X) and the f-curve rise, whilst the latter also changes its form. Since, however, X' rises more rapidly than (Z), the point {X) comes to fall below X', and the lower the temperature the lower 312 SCHREINEMAKERS art. h it becomes. Hence the point (X) lies at first between X' and s; then it coincides with s at a definite temperature, which we shall call T{s), and afterwards it lies below s. If we leave out of consideration the occurrence of solid W, we may now dis- tinguish the following three cases. (i) T{X) > T > T(s). We imagine the point (X), which is now situated between X' and s, represented by p' in Fig. 5. If we now draw the tangent p'd', we see that of all conceivable f-points of Fig. 5, only those of the parts TF'a' and h'd' of the f-curve and those of the lines a'b' and d'p' represent stable states. From this follows: all liquids of Wa and hd (Fig. 5) are stable; all liquids between a and h separate into the system L{a) + L{b); all liquids between d and X are supersaturated and pass into the system L(d) + solid X. Consequently, of all conceivable systems, only L(a) + L(b) and L(d) + solid X can occur in a stable state. We imagine these liquids a, h, and d represented by the points a, b and d of the line 1.1' in Fig. 12. (ii) TiX) > T = T{s). Now we imagine the point (X) at the point s of Fig. 11. We see that, of all conceivable f -points of Fig. 11, only those of the part W'z' of the ^-curve and those of the line z'u's represent stable states. This line z'u's, just like the line {W)e'(X) of Fig. 8, has a special property, namely that not two but three of its points represent stable phases, i.e., z' and u' represent the liquids z and u, and s the solid sub- stance X. From this follows: of all liquids, only those of Wz and the liquid u are stable (Fig. 11). Of all conceivable systems, only Liz) -f- solid X, L{u) + solid X, L(z) + L{u), (16) and the three-phase system L{z) + L{u) + solid X (17) are stable. We see that two liquids now exist, namely z and u, both of which are saturated with respect to solid X. In the same way that we deduced reaction (15) for the three- REPRESENTATION BY ZETA FUNCTION 313 phase system (14) of Fig. 8, we now find that in the three-phase system (17) the reaction L{z) + solid X :^ L(u) (18) can occur. On addition of heat L(z) passes into L(u) with solution of sohd X, whilst on removal of heat L{u) breaks up into L{z) and solid X. If in Fig. 12 we represent the Hquids z and u by the points z and u, then the systems discussed above are all represented by the points of the portion zu2' of the line 2.2'. (iii) T(X) > T{s) > T. The point (X) must now be situated below the point s. Although the f-curve has now a somewhat different form and is also situated higher than in Fig. 11, nevertheless we may imagine it as represented in this figure, and call the latter now Fig. 11a. We suppose the point {X) to be at q'. Imagine a line through q' touching the f -curve in a point h' between W and z'. It is then clear that of all conceiv- able f-points of Fig. 11a only those of the part W'h' of the f-curve and those of the tangent h'q' represent stable states. From this follows for Fig. 11a: all liquids of Wh are stable, whilst all other liquids, i.e., those of hX, pass into the system LQi) + solid X (19) with separation of solid X. If in Fig. 11 we imagine z' and u' substituted by m' and n', we see that the system L{m) + L{n) (20) also exists, but only in a metastable state. When the stable state is attained, these two liquids disappear, with formation of the system (19). In Fig. 12 the liquids h, m and n, are represented by points of the line 3.3'. When we raise the temperature, the f-curve not only shifts downwards but also changes its form. As the points of contact a' and h' in Fig. 5 are moved with respect to one another the liquids a and h also change their composition. When a' and h' coincide in a point c' at a definite temperature T{c), the liquids become identical in composition. We call c a critical liquid and 314 SCHREINEMAKERS ART. H T(c) a critical solution temperature. This temperature may be higher or lower than T{X). 16. The line zu2' and the curves hz, zcu and uT{X) divide Fig. 12 into fields, the meaning of which follows from the preceding considerations. At the same time it is apparent that the field zcu, i.e., the heterogeneous two-liquid phase field, does not end at the line zu but extends farther downwards, although in a metastable condition. As the liquids saturated Fig. 12 with respect to X are represented by the curves hz and uT(X), the solubihty of X at T(s) does not change continuously but jumps from z to u. If, however, we also consider metastable and unstable states, then a continuous transition from z to u exists. The saturation curve of X consists, as we shall presently show, of a curve hzgekuT{X) having a maximum temperature in g and a minimum temperature in k. In order to prove this, we at first imagine T = T(s), so that (X) in Fig. 11 coincides with s. Besides the two coincident REPRESENTATION BY ZETA FUNCTION 315 tangents z'(X) and u'{X) we may also draw a third tangent e'(X). Consequently, besides the liquids z and u there exists also a third liquid e which is saturated with respect to X. So in Fig. 12 there is possible, between z and u, a liquid e saturated with respect to X which is not stable (as appears from Fig. 11). We now take a temperature somewhat higher than T(s), so that (X) in Fig. 11 is situated a little above s. We may now draw three tangents through (X), which we shall call zi{X), ei(X) and Ui{X). Then point Zi is situated a little to the right of z', ex a little to the left of e' and w/ a little to the right of u' . Of the three liquids saturated with respect to X, which we call 2i, ei and U\, now only Wi is stable, as appears from Fig. 11. In Fig. 12 we represent them by the points 2i, ei and d (i.e., d = u-). If we raise the temperature still higher, then, as follows from Fig. 11, the pomts z^ and ex of Fig. 12 coincide finally in a point g. In a corresponding manner we may prove that in Fig. 12 there exists also the metastable-unstable branch eku. From this it appears that the saturation curve of X is a continuous curve with a maximum and a minimum temperature. Only the parts hz and uT{X) which lie outside the heterogeneous two-Hquid field represent stable liquids. The other liquids are metastable (viz., zg and ku) or unstable (viz., gh). IV. Binary Systems in Which Besides Liquids Only the Solid Components W and X and a Solid Compound May Occur. n. When W and X form a compound fl", we may imagine the systems : solid W + solid X, (21) solid W + solid R, solid X + sohd H, (22) solid W + solid X + solid H, (23) when we leave liquid phases out of account. The compound and its f-point are represented by B. and (//) in Figs. 13, 14, and 16. If in Fig. 14 we imagine the curves omitted and consider only the f-points (W), (//) and (X), together with their conjugation lines, we may distinguish three cases. 316 SCHREINEMAKERS ART. H REPRESENTATION BY ZETA FUNCTION 317 (i) Point (H) is situated below {W){X) (as in Fig. 14). It is clear that only the points of (W){H) and of {H){X) represent stable states, so that both the systems (22) are stable whilst (21) is metastable. From this it follows that the solid sub- stances W and X cannot exist next to each other in stable equilibrium, and that the reaction solid W + sohd X -^ solid H (24) will tend to occur. (ii) Point (H) is situated above {W)iX). It is clear that now only the points of (W){X) represent stable states; in other words, system (21) is stable, whilst both the systems (22) are metastable. Thus the compound H is now metastable and tends to separate into its components according to the reaction solid W + solid X ^ solid H. . (25) (iii) Point {H) is situated on the line {W){X). We have now again the special case that three points of a line represent stable phases (compare also {W)e'{X) in Fig. 8 and z'u's in Fig. 11). It is clear that all the systems (21), (22) and (23) are now stable and that the reaction solid W + solid X :f± solid H (26) can occur. The direction of the reaction on addition of heat will depend on whether the compound is endothermic or exothermic. It depends on the temperature and the pressure which of the three cases mentioned above will occur. In the considerations that follow we shall suppose that (H) always lies below (W){X). 18. In Fig. 13 the point H' of the f-curve is the f-point of a liquid which has the same composition as the solid compound, i.e., H' is the f-point of liquid H. Denoting the melting-point of H (under the pressure p) by T(H), then T < T{H). If we draw the two tangents z'{H) and u'{H) we see that they repre- sent more stable systems than the points on the part z'H'u' of the f-curve. From this follows: liquids between z and u (Fig. 13) are supersaturated; those between z and H separate into 318 SCHREINEMAKERS ART. H L(z) + solid H, those between u and H into L(u) + solid H, whilst liquid H solidifies to solid H. Thus two liquids, z and u, exist, both saturated with respect to solid H; z has a smaller, u a greater amount of X than the compound. In Fig. 15 these liquids are represented by the points z and u. As {H) and H' approach one another with increase of temperature and finally coincide at T' = T(H), so also z' and u' coincide at this tem- perature. Consequently the saturation curve of H will have the shape amq, shown in Figs. 15 and 17, with a temperature maximum at T{H), shown at point m. Fig. 15 We now imagine the f-curve of Figs. 14 and 16 at first totally above the lines {W){H) and {H){X). Since with increase of temperature the ^-curve approaches the composition axis WX more rapidly than these lines, it will lie totally below them at a sufficiently high temperature. Consequently the f-curve will touch the line {W){H) in a point a' at a definite temperature T{a), and will touch the line {H){X) in a point h' at a definite temperature T{h). If we take T{a) < T{h), then a' lies between (W) and (//); the point h', however, may then be REPRESENTATION BY ZETA FUNCTION 319 situated as in Fig. 14 or as in Fig. 16. We shall now deduce that the equilibria resulting from Fig. 14 may be represented by Fig. 15, and those resulting from Fig. 16 by Fig. 17. 19. AtT = T(a) three points of the line (TF)a'(i/) of Fig. 14 represent stable phases. So at T = T(a) the reaction solid X + solid H ^ L{a) (27) can occur. We represent L{a) in Fig. 15 by the point a. At a temperature a little higher than T{a) the f-curve intersects the line {W){H)', we may now draw tangents from {W) and (//), the points of contact representing liquids saturated with respect to W and H respectively. At a temperature a little lower than T{a) the f -curve lies above {W){H), so that only solid W and solid H exist as stable states. The tangents drawn from {W) and (H) now represent metastable systems only. From Fig. 14 we may therefore make the following deductions regarding Fig. 15. A field, solid W + solid H, must be situated below point a (Field I) ; two saturation curves, namely those of W and H, must run through the point a, their parts proceeding towards higher temperatures representing stable liquids, whilst the parts situated in Field I represent metastable liquids. In a corresponding manner it is apparent that at T = T(b) the reaction solid H + solid X ^ L(6) (28) can occur. If in Fig. 15 we represent L(6) by point 6, we find that the saturation curves running through h must be situated as shown, whilst Field II represents solid H + solid X. Since we have already proved that the saturation curve of H must have a maximum at T = T{H) in point m, it follows that we can represent by Fig. 15 all the equilibria resulting from Fig. 14. 20. ki T = T{a) in Fig. 16 the same obtains for the line {W)a'{H) as in Fig. 14. ki T = T{h), however, in Fig. 16 the point (H) is situated between b' and (X). Instead of reaction (28) we must now have sohd H ^ L(6) + solid X. (29) 320 SCHREINEMAKERS ABT. H If we represent, in Fig. 17, L{b) by b, then this point must now He to the left of Une Hm and not to the right, as in Fig. 15. iW) w H Fig. 16 fX) Fig. 17 REPRESENTATION BY ZETA FUNCTION 321 At a temperature a little higher than T{b) the f-curve inter- sects the line {H){X) (Fig. 18). We may now draw the lines h'{H) and x'(X) which touch the f-curve in the points h' and x' (not shown). Hence point h' is the f-point of a liquid h, saturated with respect to H and x' that of a liquid saturated with respect to X. Thus at this temperature the systems L(h) + solid H, L{x) + solid X, (30) exist. It appears from the position of these points of contact in Fig. 18 that h'{H) and (H){X) are situated above x'{X). Therefore the first one of the systems (30) is metastable, the second one stable. From this it follows that at T > T(h) the saturation curve of H is metastable, that of X stable. Fig. 18 If we take T < T(h), the f-curve lies above (H)(X) (Fig. 18). If we now also imagine the tangents h'(H) and x'(X) drawn, then we see that h'(H) and {H){X) now lie below x'{X). From this follows: at 7^ < T(b) the saturation curve of H is stable, but that of X metastable; also solid H + solid X (Field II) is a stable system. We can now make the following deduc- tions from Fig. 16 as regards Fig. 17. Two saturation curves, namely those of H and X, must go through point h of Fig. 17. Towards higher temperatures that of H is metastable and that of X stable, whilst towards lower temperatures the reverse holds good. In Fig. 15, at r = T(b), reaction (28) occurs, so that T{b) is the common melting point or the eutectic temperature of H and X. In Fig. 17, at r = T{h) reaction (29) occurs. Then T(b) is, as appears also from Fig. 17, the highest temperature at which solid // can exist, or the temperature at which solid H decomposes with formation of a liquid and separation of solid X. 322 SCHREINEMAKERS art. h V. Note by F. G. Donnan. (Analytical Addendum to the Geometry) It can be proved in the following manner that the f-curve touches the lines WW and XX' at the points W and X' respectively (see page 296 of Professor Schreinemakers' article). Denoting by f„ the zeta function (free energy) for a liquid phase containing ni mols of X and 712 mols of W, where rii -{- rii = n, then it follows from Euler's theorem that /afA , /afn\ tn = ni[-—] + ^2 I r~ I , since f „ is a homogeneous function of the first degree in rii and 712. This expression may be written in the convenient form tn = W]fi + 722^2, when f 1 and ^2 are termed the partial molar free energies of X and W respectively. Since fi = ni, ^2 = M2, we shall follow the notation of Gibbs and write f„ = n^ui + n2iU2, where /xi and 1x2 are the 'potentials (per mol) of the com- ponents A" and W respectively. For unit (molar) phase we must divide by rii + n2, and write therefore — — — = f = a:/ii + (1 - x) 112, Hi ~X~ 102 where X = ; ' 1 — a; = ni + W2 ni -j- 712 This expresses the f of unit phase in terms of the composition parameter x and the potentials. At constant temperature and pressure jui and ju2 are functions of x only. Differentiating the expression f „ = 7i\ni -\- 7121x2 for a change of rii and 712 at constant temperature and pressure (change of composition), d^n = Uidni + /i]fZn] + 'n2C?yU2 + ii2d7i2. But d^n — (JildTli + IJi2d7l2 REPRESENTATION BY ZETA FUNCTION 323 under like conditions. Hence, nidni -\- UidfXi = 0, or x j- -\- [l — x) — = 0. Differentiation of f = Xfxi + (1 — x)n2 with respect to x (at constant temperature and pressure) gives d^ dfjLi diJL2 Tx= ''d^ + ^' -^ ^'^ - ""^ dx - ^' = ^' - ^" from the preceding result. Thus at any x-point of the f-curve, we can determine both ni and ^2 by means of the two equations f = a^Mi + (1 — x) fjL2, dX ^ = ^^ - '^^' whence we deduce the results Ml = fi = r + (1 - x) -, ^^ = ^^ = f-^^' Consider now the state of affairs for x = 0 (pure W). From the preceding results we have (mi)x = o= (f)i = 0 + \dz/x^i It is clear that (r)x = o is the f (free energy) of 1 mol of pure W. Now fxi is the increase of free energy of an inj&nite phase of composition x on the addition (at constant pressure and tem- perature) of one mol of X, whilst (jui)x = o is the limiting value to which Ml approaches as x approaches zero. Let pi denote the partial vapor pressure of X in equilibrium with the liquid phase of composition x at the given pressure and temperature, and let (pi)o denote the vapor pressure of X in equilibrium with pure liquid X at the same temperature 324 SCHREINEMAKERS art. h and pressure. Also let (mi)o denote the free energy (poten- tial) of 1 mol of pure liquid A" under the same conditions. Then (/i:)o — Mi = total diminution of free energy resulting from the transference of 1 mol of X from the pure liquid state (as above defined) to an infinite mass of liquid of composition x (as above defined). It is easy to show that /•(pOo (mi)o — Ml = / vdp, where v = volume of one mol of the vapor J pi of X at the given temperature. Now y is a function of p, and for X = 0, pi = 0, and v = + co . Hence when x = 0 the ripih value of / vdp becomes + oo , so that (mi)x=o = — °o. From J pi the preceding results it follows therefore that \dx/t = — 00. Hence the f-curve touches the line WW at the point W. Sim- ilarly the f-curve touches the line XX' at the point X'. From the preceding analysis it is also evident that at the minimum point of the f-curve, mi = M2 = (f)inin. An analytical and a graphical treatment of solid-liquid phase equilibria in binary systems was given by A. C, van Rijn van Alkemade {Verhand. Akad. Wetensch. Amsterdam, 1, 1 Sec, No. 5, (1892); Zeitsch. f. physikal. Chemie, 11, 289 (1893)), who based his discussion on the properties of Gibbs' f -function. In his graphical treatment van Alkemade employed a ratio instead of a fractional composition parameter, so that the part of the dia- gram referring to one pure component is situated at infinity. The method employed by Schreinemakers avoids this defect, and is therefore much more general. It may be remarked in conclusion that the preceding analysis establishes very simply the geometrical method for determining the point on the f-curve which corresponds to a liquid in equilibrium with a pure solid phase, say pure solid W, for example. Let Piiti, ^1) and ^2(^2, X2) be two points on the ^-curve. The equation of the straight line P1P2 is ^2 ~ r _ ^2 ~ Ti Xz — X X2 — X]' REPRESENTATION BY ZETA FUNCTION 325 Suppose this line cuts the WW axis in the point Po(fo,0). Then ^2 "To f 2 ~ f 1 X2 X2 ~ Xi Allow the points Pi and P2 to coalesce in the tangent point Qmi^m, Xm), the tangcut line passing through Pq. Then we get or U/. fo — r»n ~" ^"i I J ) — (M2)x = z^. This result shows that the pure solid phase corresponding to the point Po on WW is in equilibrium with the liquid x^ determined by the tangent from Po to the f -curve. It is to be observed that Po is {W) in the notation of Schreinemakers. THE CONDITIONS OF EQUILIBRIUM FOR HET- EROGENEOUS MASSES UNDER THE INFLU- ENCE OF GRAVITY AND OF CENTRIFUGAL FORCE [Gibbs, I, pp. lU-150] DONALD H. ANDREWS The effect of gravity on the equilibrium of fluids has interested physicists and chemists for many hundreds of years. A Hst of those who have contributed observation and theory to this field includes many famous names such as Galileo, Laplace and Boltzmann. It is Gibbs' characteristic role to have shown how these special relations of gravity and fluid equilibrium fit into the general scheme of thermodynamics in a way that permits of the widest sort of application. Little comment is needed on the actual derivation of the equations.* The usual thermodynamic system is postulated, including in this case the force of gravity. The laws of thermo- dynamics and the various equations of condition then lead to the equations which define the state of the system. Temperature must be constant throughout, i.e., t = const.; [228] and the pressure must vary with the height,i.e., dp = -gydh. [233] The chemical potentials (mi, . . . m^) of the individual com- ponents (essentially the partial pressures if the system is not far from ideal) must satisfy the equations * Compare Section XIII of Article L of this volume. 327 328 ANDREWS art. i Hi -{• gh = const. Mm + 9'A = const. [234] It is emphasized in the text that we must distinguish the /xi, ... f^m, intr-insic potentials, from the general potentials of the components which include the action of gravity and are anal- ogous to the partial molal free energies. These latter are of course constant throughout the system. In the second part of this section (Gibbs, I, 147-150), Method of treating the preceding problem, in which the elements of volume are regarded as fixed, more detailed attention is given to the fac- tors introduced by the discontinuities between phases in a sys- tem under the influence of gravity. The condition of equilib- rium is found to be that "the pressure at any point must be as great as that of any phase of the same components for which the temperature and the potentials have the same values as at the point." The deduction which has had the widest application is that summarized in equation [233]. If we apply this to a component which is obeying the laws for an ideal gas we can relate density to pressure as follows pv = nRT, *(1) nM , ^ M being the molecular weight of the component, so that 1=V^' (3) If po be the pressure at some horizontal plane, the reference zero point from which we measure the height h, we can sub- stitute in equation [233], integrate and obtain the famous * Since the temperature which appears explicitly in equations (1) to (10) of this article is in all cases the absolute temperature it seems best to conform to current usage by representing it by T . GRAVITY AND CENTRIFUGAL FORCE 329 hypsometric or barometric formula _Mg_ p = Poe «^ ' (4) which gives us pressure as a variable depending only on height. The most famous application of this equation is in the study of variations in pressure of the earth's atmosphere with height, Galileo first pointed out that the atmosphere created pressure, and P^rier proved that the pressure varied with height by means of his famous ascent of the Puy de Dome, barometer in hand. Laplace^ deduced the correct formula for the varia- tion of pressure with height in his celebrated Mecanique Celeste and Gibbs showed that it took its place as part of the gen- eral thermodynamic scheme. As an example, substituting the numerical values M = 29 gm/mol, g = 980 cm/sec^, 72 = 8.31 X 107 erg/mol deg, T = 300°K, we find that at a height of 5000 meters the pressure has dropped to 56.5% of its value at the earth's surface. It was also appreciated at rather an early date that the con- centration of solute in a solution should vary with the height because of the influence of gravity. In the early part of the last century Beudant^ claimed experimental evidence of this effect. Gay Lussac,^ however, definitely proved that it was too small to be observed. He placed cylinders of various solu- tions in the cellar of the Paris observatory, and after a year's time analyzed the top and bottom portions, finding no differ- ences in concentration. Many years later Gouy and Chaperon^ showed by calculations that for solutes of ordinary molecular weight the effect is negligibly small. Though ordinary solutions failed to show the effect, the advent of colloidal solutions opened up new possibilities in this dir- ection. Einstein^ pointed out that a colloidal suspension should obey the same kinetic laws as an ordinary solute, and a starthng experimental confirmation was provided by Perrin.^ He al- lowed a suspension of gamboge to come to equilibrium after settling for some time and then actually counted the number of particles of a given radius (i.e., similar molecular weight) occurring at different levels. In order to test his result it is 330 ANDREWS ART. I convenient to modify equation (4) slightly. Since the osmotic pressure p will be related to the number of particles per cu. cm n by RT (5) in which N is Avogadro's number, we may substitute n for p, and no for po- We must also bear in mind that in this case the force of gravity enters because of the difference in density of the particles and the solvent. The depressant force will therefore be not Mg but f irr^Nipp - Ps)g, where r is the ra- dius of the particle and Pp and p<, the densities of the particle TABLE I Sedimentation Equilibrium in a Gamboge Suspension X n Obs. Calc. Xo 100 ... Xo — 25ju 116 119 Xo — 50/x 146 142 Xo — 75ju 170 169 Xo - 100m 200 201 and solvent. Equation (4) then becomes N 4 n = noe "^ ^ • vo; Table I shows the variation in the number of particles over a microscopic range as determined by actual counting and as calculated from equation (6). Westgren^ made similar measurements with gold sols and obtained even better agree- ment. His results are given in Table II. It is evident from an examination of the derivation of equa- tions [233] and (4) that the force involved does not neces- sarily have to be that of gravity. A system of particles acting under any uniform field of force will obey the same laws. For example, the distribution of particles under a centrifugal force provides a means of studying this sort of phenomenon. GRAVITY AND CENTRIFUGAL FORCE 331 Bredig^ was the first to show that centrifugal force does produce changes in pressure. By centrifuging gases in a tube containing several chambers joined by capillary tubes, he showed that the pressure in the outermost chamber was greatest. Lobry de Bruyn and van Calcar^ produced the same sort of effect in solutions, showing that solute is driven away from the axis of rotation. They were able by centrifuging to crystallize out a third of the solute from a saturated solution of sodium TABLE II Sedimentation Equilibrium in a Gold Sol Radius of Particles: 21m/i Radius of Particles: 26m;u n X n Obs. Calc. Obs. Calc. Om 100 200 300 400 500 600 700 800 900 1000 1100 889 692 572 426 357 253 217 185 152 125 108 78 886 712 572 460 369 297 239 192 154 124 100 80 On 50 100 150 200 250 300 350 400 450 500 1431 1053 779 532 408 324 254 189 148 112 93 1176 909 702 555 419 324 250 193 149 115 89 sulfate. It was not possible however to get a quantitative confirmation of the thermodynamic equation. A series of brilliant experiments of this sort has recently been performed by The Svedberg and his associates in connec- tion with the development of the ultra-centrifuge. While the major part of the work has been concerned with diffusion rather than equilibrium, certain aspects illustrate in a beautiful manner the relations which we have been considering. In the first place it is very important to know the relative distribution of the particles in equilibrium even if the study is mainly concerned with diffusion which will not be continued 332 ANDREWS ART. I long enough to bring about equilibrium. In calculating their distribution in the ultra-centrifuge where forces 5000 times that of gravity are encountered, one cannot consider the force as constant but must take into account the variation of force with distance from the axis of rotation. Using concentration c instead of pressure, the distance x from the axis of rota- tion instead of height, and the force due to the difference in density between particle and solvent instead of gy, equation [233] becomes N dc = — r— i irr^ (pp — ps) co^c xdx, (7) where co represents the angular velocity. If we wish to get the concentration at different points in a tube such as might be placed in the ultra-centrifuge, we may let x^ represent the end of the tube furthest from the axis, i.e., the bottom of the cell. Then on integrating we obtain ,. = ,,, -S I '■<—>-(^) (8) Figure 1 shows the distribution for various particle sizes as calculated by Svedberg from this equation, letting x^ = 5.2 cm. and co = IQOtt per sec. We may write equation (7) also in the form — = - ^ — ^ ^2 x dx, (9) where V is the partial specific volume of the solute. Integrat- ing and solving for M, we get 2 RT In (ci/c2) . ^, CO-'il — Vps) {Xi — X2) In this way the measurements of concentration at equihbrium may serve as a means of calculating the molecular weight of the particles. Svedberg and Fahraeus'" made observations of this sort on hemoglobin. The solution of hemoglobin was placed in the GRAVITY AND CENTRIFUGAL FORCE 333 centrifuge tube and photographs were made after various intervals of time showing the density of the solute at various distances from the axis of rotation. By analyzing these photo- graphs with a photo-densitometer very accurate measurements of concentration were secured. Table III shows how the molecular weight was calculated from the change in concentra- tion with distance for one set of experiments. During the course of the investigation the initial concentra- tion was varied from 0.5 to 3.0 gm. of hemoglobin per 100 cc. of solution, the length of the column from 0.25 cm. to 0.8 cm. and the speed of revolution from 7200 to 10,000 r.p.m. without 0/ OZ 03 O.* OS 06 01 O.B 0? J.O CfTt r= radius of particles in millimicrons (10-' cm). Fig. 1 producing any marked change in the calculated molecular weight.* * An important contribution to this subject has recently been made by Kai O. Pedersen, Z. physik. Chem. 170A,41 (1934). It consists of a study of the radial variation of the concentration of salts in aqueous solution at equilibrium in a centrifugal field of force of the order of 2 X 10^ times the earth's gravitational field. The change in concentra- tion is measured by photographing the distortion of the image of a scale observed through the column of liquid rotated at a speed of 55000 r.p.m. in the usual manner. From the displacement of the scale lines due to the change in the index of refraction, one can calculate the radial varia- tion in concentration due to the force field. A thorough discussion is given of the thermodynamic relations involved, and an equation is derived relating the molecular weight to the concentration changes observed and the activity coefficients. The average error of the molec- ular weights so determined is about ten per cent. If it is possible to obtain accurate values of the absolute concentration changes this may be a valuable means of calculating activity coefficients. 334 ANDREWS ART. I In addition to these experiments, which have involved true equihbrium, mention should be made of the interesting deter- minations of the effect of gravity on the electromotive force of cells. Tolman^^ has shown that much valuable information on the nature of solutions can be obtained by studying the electro- motive force which is produced when a solution of uniform con- centration is placed in a centrifugal force field. This e.m.f. is due, of course, to the fact that the concentration is uniform, and would disappear if diffusion were allowed to bring the concentration to the equilibrium values, such as we have been calculating from the above equations. The same principles have also been applied to particles in TABLE III The Molecular Weight of Hemoglobin as Determined by Sedi- mentation Equilibrium Xl X2 Cl C2 M X 10-3 cm. cm. gm. per 100 cc. gm. per 100 cc. 4.61 4.56 1.220 1.061 71.30 4.56 4.51 1.061 .930 67.67 4.51 4.46 .930 .832 58.33 4.46 4.41 .832 .732 67.22 4.41 4,36 .732 .639 72.95 4.36 4.31 .639 .564 60.99 4.31 4.26 .564 .496 76.57 4.26 4.21 .496 .437 69.42 4.21 4.16 .437 .388 66.40 electric and magnetic fields, notably in the work of Langevin'^ on the nature of paramagnetism. REFERENCES 1. Laplace, Mecanique Celeste, Book I, Chap. VIII, Paris 1799. 2. Beudant, Ann. chim. phys., 8, 15 (1815). 3. Gay-Lussac, Ann. chim. phys., 11, 306 (1819). 4. GouY and Chaperon, Ann. chim. phys., [6] 12, 384 (1887). 5. Einstein, Annal. Phys., [4] 17, 549 (1905). 6. Perrin, Comples rendus, 146, 967 (1908); Ann. chim. phys., [8] 18, 53 (1909). GRAVITY AND CENTRIFUGAL FORCE 335 7. Westgren, Z. phijsik. Chem. 89, 63 (1914); Arkiv for Matematik (Stockholm) 9, No. 5 (1913). 8. Bredig, Z. physik. Chem., 17, 459 (1895). 9. LoBRY DE Brutn AND VAN Calcar, Rcc. trav. chim., 23, 218 (1904). 10. SvEDBERG AND Fahraeus, J . Am. Chem. Soc, 48, 431 (1926). 11. ToLMAN, J. Am. Chem. Soc, 33, 121 (1911). 12. Langevin, Ann. chim. phys., [8] 5, 70 (1905). FUNDAMENTAL EQUATIONS OF IDEAL GASES AND GAS MIXTURES [Gibbs, I, pp. 150-184; 372-403] F. G. KEYES I. General Considerations {Gihhs, I, 150-164) 1. Pure Ideal Gases. The response of gases to changes of pressure, temperature and volume was a subject of the greatest interest during the latter half of the 17th century and con- tinuing through the 18th and 19th centuries. Boyle's work, appearing in 1660, and Mariotte's investigations (1676) estab- lished as a property of several gases the constancy of the pres- sure-volume product at constant temperature. Not until the beginning of the 19th century, however, was definite and sufficiently exact information secured regarding the volume- expansion law with temperature for constant pressure, and the pressure-increase law with temperature for constant volume. A knowledge of the latter laws, now known under the name of Gay-Lussac^'2 as well as the Boyle-Mariotte law, was necessary to understand experiments on the relations of the volumes of chemically combining gases, — experiments the interpretation of which proved of such incisive importance to chemistry as a whole. It remained for Amed^o Avogadro^ to draw the important inference from these investigations that the number of particles or molecules is the same for different gases of equal volume, the temperature and pressure being the same for all. There results then the remarkably simple expres- sion for the physical behavior of pure gases — = universal constant, (1) Q 337 338 KEYES ART. J where v is the volume of a "gram molecule" and 0 would have referred in the first half of the last century, to the absolute tem- perature as measured by a mercury thermometer. The upper limit of pressures was low and the precision of measurement, moreover, hardly sufficient to make evident the limits of vahdity of the relation (I) for describing the behavior of actual gases. The extraordinarily ingenious and precise measurements of Reg- nault were the first which showed the degree of inexactness which must be accepted. Thus for the gases air, nitrogen, carbon dioxide and hydrogen, compressed to a twentieth of the volume at zero degrees and one atmosphere, the following pressures were found : Air N2 CO2 H2 Vn Pressure at — atm 19.72 19.79 16.71 20.27 Percent deviation from Equation (I) -1.4 -1.1 -16.45 +1.4 At one-fifth of the volume, however, the magnitudes of the deviations reduce to —0.4, —0.3, —3.4 and +0.24 percent, respectively. Thus with respect to pressures at constant tem- perature Regnault's classical investigations, of which the fore- going is but a fragment, make it clear that equation (I) is to be regarded strictly as the expression of a limiting law to which actual gases may be expected to conform as the pressure is indefinitely reduced. The gas-thermometric investigations of Regnault^ and subsequently others'^ showed that the volume- temperature coefficient at constant pressure, and similarly the pressure-temperature coefficient at constant volume, tend to an identical constant with diminishing pressure, thereby estab- lishing the universality of the temperature scale definable by equation (I) for p -^ 0. In addition, researches of Joule and later of Joule and Thomson proved that the internal energy of a gas at very low pressures is a temperature function only. The investigations of the heat capacities of gases had, moreover, shown in many cases, particularly for the gases whose critical temperatures were low, that the temperature coefficients were very small indeed. FUNDAMENTAL EQUATIONS OF IDEAL GASES 339 The complete concept, therefore, of the perfect gas, accepted by Clausius and here taken by Gibbs, is defined by the first three equations of this section. For convenience of reference they will be designated as follows : pv = at, (II) de = c dt, (III) € = ct + E. (IV) It is noted that the heat capacity employed is that at constant volume rather than that at constant pressure. There is wisdom in the choice, for the former is the simpler quantity, and while it must usually be derived from measurements at constant pressure in default of direct measurements at constant volume, nevertheless this reduction may be carried out once for all as a special operation in preparing heat capacity data for use in the applications of thermodynamics where gases are involved. It is, moreover, not difficult to show that many applications of thermodynamics involving liquids and solids proceed very advantageously where the constant-volume heat capacity is employed. 2. Mixtures of Ideal Gases. The question of greatest impor- tance in all detailed applications of thermodynamics is that of determining the laws to be employed in representing the physical behavior of mixtures of gases. Until the various aspects of this problem are resolved no real progress with applications of the general theory becomes possible, and it is for this reason that Gibbs took the greatest care to investigate all ramifica- tions of this far from simple matter. It also seems evident from the statements and form of this section that Gibbs was seeking for a principle which would carry further than the popularly phrased statement of Dalton's law or rule for mixtures of gases. Indeed he found a statement of Dalton's law ("Gibbs- Dalton law") which he showed to be "consistent and possible" for mixtures of gases which are not ideal.* S. Ideal Gas Concept as Related to the Behavior of Actual Gases under Diminishing Pressure. Because (II), (III), (IV) A test of this law has recently been made. See reference (6). I 340 KEYES ART. J are believed to be limiting laws valid for infinitely extended volumes it is desirable to review briefly the circumstances surrounding the behavior of important functions along the path by which reduction of pressure to zero takes place. Con- sider in this connection, for example, the Joule-Thomson ex- periment. The effect is given by the thermodynamic equation, where Cp designates the constant-pressure heat capacity, x the "heat content" (e + pv) and t = t~^. The existing data show that the right hand member does not vanish as p goes to zero but on the contrary becomes constant and independent of the pressure. Joule and Thomson deduced, however, that the effect varied inversely as f at low pressures, which requires the following relation between p, v, and t: V = fip)t - J, (VI) or p)t -y^ th) fiv)t Clearly the condition that (II) be applicable at every tem- perature is that /(p), as is possible, may be taken to be R/p for t; -^ 00 . On the other hand, the change of energy with volume, \dv/t \ Bt /v has been shown in the case of one substance'' to vary as the density squared (at low pressures), which may be regarded as a verification by experiment of equation (IV) since {de/dv)t — > 0 as the density diminishes. The consequence of this is that 6 = f{t) and that p = f(v)t. Taking into account the validity of Boyle's law as an exact expression of physical behavior for p -^ 0 the latter relation leads to equation (II). The quantity FUNDAMENTAL EQUATIONS OF IDEAL GASES 341 (-f) is also well known ^-^ to proceed to a finite limit for p — > 0. The quantity is in fact never zero except at a unique temperature, characteristic of each pure substance (Boyle — point). It follows, therefore, that (pv — Rt) vanishes at all temperatures when p — * 0.* 4. Constancy of Specific Heat. The justification for defining a perfect gas by means of equations (II), (III) and (IV) is complete except as regards the absolute constancy of specific heat. Experiment has proved to a high degree of precision that the constant-volume heat capacities of monatomic gases, at low pressures, are independent of temperature. Thus c for argon is very closely 2.98 from below zero degrees to about 2000°C. However, in the case of diatomic gases the tem- perature dependence, while small at ordinary temperatures, is significant and the modern quantum theory is eminently satis- factory in the account it provides of the course of c for hydrogen from a value of 2.98 at low temperatures to a value of 4.98 at room temperatures. Molecules of a higher order of complexity have a correspondingly large positive temperature coefficient above zero centigrade. 6. Concluding Statement. We may therefore sum up the present position with respect to the validity of the relations (II), (III) and (IV) by stating that (II) may be assumed to have been abundantly shown by experiment to correspond with reality as a limiting law for computing pressures for all pure gases. The independence of c with respect to tempera- ture is, however, only true on the basis of present experience for monatomic gases, and the magnitude of the temperature coef- ficient of the heat capacity for all higher order molecules is large according to the order of complexity. 6. Comment on Gas Law for Real Gases. A discussion of the section might be carried forward from this point without explicit reference to an equation of state of greater complexity than (II). Gibbs has, however, adopted a definite hypothesis. * It should be understood that temperatures greater than absolute zero are referred to throughout in the considerations above. 342 KEYES ART. J the Gibbs-Dalton law (Gibbs, I, 155, beginning line 7), the implications of which can only be fully developed by using an equation connecting p, v, t and the mass, which is valid at sensible pressures (one atm. for example). Such an equation may be readily obtained by the use of equation [92] of Gibbs' Statistical Mechanics^'^, viz., V = ^7^7^' B = -2x71 I (e-'^'' - 1) rW. (VII) Employing the van der Waals' model," for example, there is obtained the following simple equation for B at low pressures ^ = ^-^ / aiA atA" \ It is true that the van der Waals model is often inadequate (case of helium, neon) but it gives results sufficiently in accord with fact for the purposes of this section to make it unnecessary to deal with the considerably more involved expres- sion following from a model more in accord with contemporary ideas of atomic and molecular structure '- i3. i4, 15, le, 17. is 'pjjg quantity B of (Vila) is a pure temperature function in which /?, -A, ai and ai are constants. Gases, it is apropos to state, may be sorted into two classes, those which have a permanent electric moment in the sense of the dielectric constant theory and those which have not. In the former class^^ are found water, ammonia, the hydro- halogen acids, sulphur dioxide, the alcohols, etc., while the noble gases, nitrogen, hydrogen, oxygen, methane have no moments. The simpler more symmetrical structure of the latter substances is reflected in their physical and quasi-chemical behavior (adsorption for example). Thus the departure from relation (II) for the latter gases is less, and it is not necessary to retain many terms of the bracketed part of (Vila). Mole- cules having permanent moments exhibit on the contrary great departure from relation (II).* * At zero degrees and one atmosphere nitrogen has a pressure less than that calculable from (II) by about one twentieth of one percent. Am- FUNDAMENTAL EQUATIONS OF IDEAL GASES 343 In many cases of interest in the application of Gibbs' theory to gaseous equiUbria, the temperature of measureable reaction rate and practically significant concentrations of the products of the reaction are sufficiently high to enable an equation of essentially the type of (VII), (Vila), to be used without involv- ing too serious error -•'• ^^' ^^- ^^. Every purpose will be served in what follows by omitting all terms in the brackets in (Vila) following the one having the coefficient ai. 7. Choice of Units of Mass arid Energy. The equations (II) to (IV) of Gibbs refer to "a unit" of gas and the gram or gram mol might equally well be employed. We will consider one gram as the unit quantity in what immediately follows and the gram mol in those instances where convenience is thereby better served. The unit of energy will be the mean gram-calorie equal to 4.186 abs. joules where practical applications require specification of the unit. The temperature scale will be that of the centigrade scale given by the platinum resistance ther- mometer plus 273.16, and the pressure unit the international atmosphere, volumes being taken in cubic centimeters per gram or gram mol. 8. Definition of Temperature. It is noted that the tempera- ture is defined by the perfect gas (Gibbs, 1, 12-15) or quite simply, if the heat capacity c is assumed an invariable constant, by the energy equation. Taking equation [11] (Gibbs, I, 63) for the energy, de = tdrj — pdv, temperature and pressure may be expressed in terms of the energy e, the volume, and the appro- priate constants. From (IV) and (II) there result € - E t = —^> (1) [257] V = ' (2) 258 V c monia under the same conditions of temperature and pressure has a pressure less than that given by (II) by one and one-half percent, and in conformity with the modern theory of cohesive and repulsive forces the bracketed expression on the basis of a van der Waals model be- comes more complicated. However, in the case of dipole gases at ever higher temperatures (VII) tends to a simpler form on account of the diminishing relative importance of those terms arising from the presence of the permanent dipole. 344 KEYES ART. J and substitution in [11] leads to a relation in which the variables separate. Integration then results in equation [255]. Evidently since c, except for the monatomic gases, is in general a quite complex function of the temperature it is not practical to write t as a function of the energy in a fundamental equation* in the variables energy, entropy and volume. f If c is taken as a function of temperature, f{t), the equation for the entropy may be readily obtained from [11] for de + pdv fit)dt + pdv <'" = — r~ = — i — or fit) -j + alogv + H. (3) The forms of f{t) which are known, as for hydrogen, make it practically impossible to eliminate t to give an equation in the variables e, rj and v. 9. Constants of Energy and Entropy. The remarks following equation [255] are important, for the assigning of the constants of entropy, H, and of energy, E, is a matter of importance in all cases of chemically interacting components. The conventions which have been used are, however, somewhat varied; thus Lewis and Randall ^^ define a standard state in terms of unit fugacity of the elements; and 0° on the absolute or Kelvin scale and one atmosphere ^^ has also been proposed. There is much advantage ^^ in adopting the actual state of the gas at 0° and one atmosphere, but any of the proposed systems is a possible one so long as interest centers on the treatment of ordinary chemical reactions by the two empirical principles of thermo- dynamics. J * See footnote, Gibbs, I, 88. t Gibbs has discussed the advantages of volume and entropy as inde- pendent variables (Gibbs, I, 20). t The statistical mechanics analogue of the entropy may for example be easily computed from equation [92] of Gibbs' Statistical Mechanics (Gibbs, II, Part 1, 33) for the simple case of a gas assumed to be composed of structureless mass points. Before making the computation, note should be taken of the fact that equation [92] may be dimensionally satisfied by dividing the right hand member under the logarithm by Planck's constant h raised to the 3nth power. FUNDAMENTAL EQUATIONS OF IDEAL GASES 345 10. \p Function for an Ideal Gas. On substituting its equiva- e — Em lent t for • in [255], and solving for rj there results, cm m r} = mc log t — ma log — + mH, (4) [255] 07-/-" ^ = — Q log { — 1 I . . . j e ' dxi dyi dzi, dxi dy\ dzi. (a) TTl If 6 IS given by ~ (i^ + y* + 2^) there results v^ = -elogf-^ j»-t;». (b) Applying the operation ——at constant volume and assuming n0 given ot by at the following analogue of the entropy results : 17 = I a log i + o log y + I a log ,, . (c) Here a definite value of the constant of entropy appears which bears a direct relation to the Nernst Heat Theorem and the so-called chemical constant ^^■^'■^^. Differentiation of equation (b) with respect to the volume at constant temperature and changing the sign gives the fol- lowing expression for the pressure: \ovJ V V which is equation (II). Again, forming the energy by the operation in- where t represents kQ~^ — t~^, k being the Boltzmann constant (1.37 X 10-« ergs/deg.) we obtain ©.-' = = -' = I n/c< = I c't. (e) Here no constant of energy is assigned nor should a constant appear in view of the properties of a system of structureless mass points treated by classical statistical mechanics. 346 KEYES ART. J as the expression for the entropy of a mass m of the pure gas. Using this entropy equation and (IV) and substituting in [87] there is obtained V yp = md + mE — met log t — mat log — — mUt, (5) [260] which is identical except for slight rearrangements with [260]. Differentiation with respect to t at constant volume and applying a change of sign gives /9A V I — I = mc log f + ma log — + mH = n, \at/v,m ^ (6) [262] which is the entropy of the pure gas. The pressure is given likewise by changing the sign and differentiating with respect to volume at constant temperature, i.e., \dV/t,m "^ = p. (7) 12631 V The energy and heat capacity are formed by operating on ^f-i = xpT, where r represents reciprocal temperature, as follows : c = r-f^) = md + mE, (8) \ OT /v.m \ OT^ /v.m t2 Finally the chemical potential may be found by differentia- tion with respect to m, keeping v and t constant, ( aA , , ^ — • ) = u = d — dlogt — atiog- dm/v. t m -}- at - Ht-\- E. (10) [264] Thus every quantity of thermodynamic interest may be obtained from the Helmholtz free energy function (\J/ = e — trj) FUNDAMENTAL EQUATIONS OF IDEAL GASES 347 by simple differentiation. Gibbs has obtained the same result by comparing the terms of the total differential of ip, drp = ( -^ ) dt + ( 4- ) dv + ( ^ ) dm, \dt/v,m \dv/t.rn \dm/v.t ' and # = - vdt - pdv + fidm, (11) [88] with equation [261]. 11. f Function for an Ideal Gas. Turning to the zeta function* [91], f = e + pv — trj, we may form the function in terms of pressure, temperature and the mass of a pure perfect gas with the following result : f = met + mE + "inat — met log t — mat log — - mHt. (12) [265] By differentiation the following equations are obtained: /9f\ , .at , r , - h;7 = V = mclogt -\- ma log— + mH, (13) [266] \ot/p,m V if) " mat , . r (14) [267] P I ~- I = met + niE + mat = we + mat, (IVb) \ OT /pm ~ ( ^ ) = c + a , m \dtdT/p^m /d^\ , , at I 7~~ I = n = ct — ct log t — at log — \dm/p, t ^ *^ p -\- at - Ht + E. (15) [268] * This function is called the "Free Energy" by Lewis and Randall in their treatise Thermodynamics and the Free Energy of Chemical Sub- stances. 348 KEYES ART. J The latter equation for /x is, as it should be,* identical with (10) [264], since at/p is equal to v/m. /H - c \ , /c\ By setting I ~ ^) ^^^ V / ~^ ■'" ^^^^^ ^^ ^^^ constants Ki and K2, (15) [268] may be written n - E p = a ■ e' f' e "' , (16) [270] or the density p is given by p = e'^t^-'e "' . (17) [270] 12. X Function for an Ideal Gas. The equation for xt is likewise readily formed from equations (II) and (IV). Thus X = e + py = m(c + a)« + mE, (18) [89] and on differentiating this equation there results, using [86], dx = tdv -\-vdp-^Z udm, (19) [90] showing that the independent variables are the entropy, pressure X — fnE and mass. From (18) [89] there is obtained t = —, — ; — r, and w(c + a) using the total differential of [89], with tdr\ replacing de. + pdv, we have X — mE X — mE adp dx = "7 I ^ • d'O + mic + a) (c + a) P or m(c + a) ;; = ^77 + am — , (20) X - mE p' which on integration, and using the entropy constant H, gives [271], or * See equations [104], Gibbs, I, 89. t This quantity is frequently referred to as the "total heat," a somewhat misleading term. It is also often designated by the symbol, H. FUNDAMENTAL EQUATIONS OF IDEAL GASES 349 77 — mH a X = mE + mic + a)e ""^^^ ( ?Y~", (21) mH /dx\ _ »"(= + «) /p\ ma 7' (22) but 1]— mH e (2) =Z7rT^.=<, (23) which gives ( m{c + a) dx\ fnat dP/r,, m V = V. (24) It is also easily shown that ( ~ ) = t, while { ~- ) gives an equation for /x identical with [268], 13. Vapor Pressures of Liquids and Solids. The footnote (Gibbs, I, 152) concerning the general problem of vapor pres- sures is important, for not only is a relation between pressure and temperature often required for pure liquids or solutions in equilibrium with a vapor phase, but equally important is the large class of compounds of solids with volatile components, as for example the salt hydrates, salt compounds with ammonia, sulphur dioxide, and numerous similar compounds. Innu- merable formulae for the vapor pressure of liquids have been suggested since the middle of the last century. Those that do not have a purely empirical origin may be obtained from the Clapeyron equation dp using various assumptions. Thus if the specific volume of the liquid Vo is neglected, the vapor, Vi assumed a perfect gas, and the heat of evaporation, X supposed a linear function of the temperature, there results dp at , ^ Xo + a< = i ir • -' (25) at p 350 KEYES ART. J where X, the heat of evaporation, is expressed in terms of a constant Xo and a. One obtains on solving (25) Xo a log p — — ~ + ~ log t + constant, at a (26) which is of the same form as Gibbs' equation [269]. The procedure adopted in the footnote, however, brings to the fore the precise nature of the assumptions upon which the resulting vapor pressure formula rests. Moreover, it is more direct than the above treatment, as may be easily shown. For the single accent phase (vapor) and the double accent phase (condensed substance) we have* -v' dp -\- ri' dt + m' dtx' = 0,1 -v"dp + r,"dt + m"d^" = 0. (27) Gibbs proceeds to solve these equations and, from the equilib- rium condition d/j,' = dn", to extract the pressure as a function of t. But on solving the above pair of equations subject to the same equilibrium condition there results v' m! v" m" dp = -(]' m' r," m" dt. (28) Expanding the determinants gives Wm" - v" m') ^ = {-n'm" - v"m'). dt (29) If m' = 1 = m", and rj' — r\" is set equal to -, the entropy of transfer from the first to the second phase, we have the Clapey- ron equation See equation [124], Gibbs, I, 97. FUNDAMENTAL EQUATIONS OF IDEAL GASES 351 from which the vapor pressure equation was obtained above. Gibbs preferred to proceed directly with the /x equations in estabhshing his vapor pressure relation. It will be noted that Gibbs has assumed that the heat capac- ity k of the liquid is independent of the temperature. In addition it is assumed that the internal energy is a constant. It is in this way that the simple expression for the entropy 1] = log t -\- H' is obtained. These assumptions are, however, far from being true if a range of temperature is considered, as a glance at the data for the heat capacities of liquids shows. As compared with the vapor at moderate pressures most of the internal energy of a liquid is molecular potential energy and f ( — ■ ) — p is very large. Ether, for example, at — 50 has a \ot/v /dp\ specific volume of 1.265c.c.per gm., and t{—j ~ p, equivalent to ( — 1 , amounts to 2780 atmospheres. The same quantity for the vapor in equilibrium with the liquid at — 50 is not far from 1.5 X 10~^ atm. For short ranges of temperature along the saturation curve the Gibbs' assumption is in many cases admissible where only modest accuracy is required. The subject of vapor pressure representations on the lines of Gibbs' treatment has recently been fully developed by L. J. Gillespie.^^ It is worth pointing out that Gibbs' treatment indicates the role played by the entropy constants in the constant of the vapor pressure relation. The heat theorem of Nernst is also closely related to the constants of the vapor pressure-tem- perature equation. To obtain, however, constants which are really characteristic of pure substances requires very reliable data at low pressures and skillful treatment of the data in formulating an equation ^"^ ^^' ^^^ ^^' ^^- ^^ The treatment of the case where a gas is dissolved in a liquid is also touched upon by Gibbs in the latter part of the footnote. It is assumed that the vapor pressure of the liquid absorbing the gas is small enough to be neglected. However, while the latter approximation may be satisfactory, as for example with carbon dioxide at one atmosphere dissolving in 352 KEYES ART. J water at zero degrees (vapor pressure of water 0.006 atm.), in many cases the solubility may be large enough to affect the vapor pressure considerably. The solubility of carbon dioxide in fact is sufficient to change the thermodynamic potential of the water considerably as the pressure of the carbon dioxide rises. There are several other factors to be considered if the case is to be treated with some degree of completeness, but for this a more extensive knowledge would be required than is at present available of the potentials of the components in the liquid mixture, and of the gas phase. Nothing is very definitely known about the energy of mixtures of liquids or the entropy of a liquid mixture as a function of the entropies of the components. It may be assumed, however, that f for a mixture of liquids is of the same general form as that for the separate components. Moreover, if one or several components are present in small quantity the coefficients of the f equation of the mixture may be confidently assumed to be linear in the masses of the soluble constituents, on the ground that any continuous and differentiable function of a variable is linear in the limit of small values. It is in this sense that the second equation on p. 154 of the footnote should be understood in its practical applications. The remaining steps lead easily to the equation for the pressure of the dissolved gas as a func- tion of the temperature. The values of the constants A, B, C and D will be constant for an invariable composition of the liquid solution. Differentiating the log (p/a) equation with re- spect to temperature at constant composition, and neglecting the term Dp/t which is small at low pressures, there is obtained f C-^) = C-BL (30) (d log p\ \ dt J This quantity is proportional to the energy required to transfer unit mass of the dissolved gas to the gas phase under equilibrium conditions. It is clear from the discussion above that a basis is here indicated for a theory of dilute solutions, for the treatment is by no means restricted to the case of gaseous substances which dissolve. Moreover, it will be observed that the latter case is FUNDAMENTAL EQUATIONS OF IDEAL GASES 353 capable of a considerably more detailed treatment along the lines laid down by Gibbs. Thus it would be easy to include in the dis- cussion the effect of the dissolved gas, and the gas in the gas phase, on the vapor concentration of the vapor emitted by the solvent. For this purpose use would be made of the italicized statement (Gibbs, I, top of page 155) together with an equation for the gas and vapor, such, for example, as (Vila). 14. Effect of the Presence of a Neutral Gas on Vapor Pressure. The paragraph beginning on p. 154 discusses the old obser- vation that, for example, the vapor pressure of a mixture of water and benzene is about the sum of the vapor pressures of each pure liquid at the temperature of the mixture. Since, however, the pressure on the liquid phase is greater than if either were alone present the liquids must be compressed. The nature of the effect of a pressure applied to the hquid phase and its magnitude may be obtained by applying the equation [272] obtained from equation [92] (Gibbs, I, 87). Taking the tem- perature constant and assuming equilibrium conditions there results d^ = {vdp + nidmi)t. (31) But dt = (Pj dp + (f^) dm,, (32) and, since p and Wi are independent variables, (33) Comparing equations (31) and (32) the latter may be written (f^) =Cf) . (34) (2721 Similarly it may be shown from [88] that /a^A ^_(^) . (35) 354 KEYES ART. J The case of a pure liquid under pressure in excess of its vapor pressure at constant temperature can be treated quite simply- using equation [272], provided it is assumed that the neutral ideal gas exerting the pressure on the liquid phase dissolves to a negligible extent, and that it is at the same time completely indifferent with respect to the vapor of the liquid. The latter restriction means, of course, not only that there must be no chemical action but also that the neutral gas must exert no "solvent" action with respect to the vapor. For the vapor phase dv'\ (36) [272] (37) (38) [272] (39) But if equilibrium subsists, fx' = fx", and moreover for a single pure phase, neglecting any possible complication due to the dissolved neutral gas, and for the liquid phase \dm/p. dm [{v"sat. + ap)m] = v"sat. + ap, (40) where a is the compressibility of the liquid. Substituting — for I -— and mtegratmg from the normal saturation pres- p \dm/p,t sure to the vapor pressure arising as a consequence of the changed potential of the compressed liquid in the case of the vapor, and from the normal saturation pressure to the pressure p of the neutral gas in the case of the right hand member, there is obtained log^ = ?^- (P - p,^,) + "1^ (P2 _ p2^„j . (41) Psat. at 2a t FUNDAMENTAL EQUATIONS OF IDEAL GASES 355 Clearly p > p,at. for P > psat- In the case of water at zero degrees under a pressure of 100 atm. there is obtained from (41) P/Psat. = 1.084. The effect (Poynting effect) is small, but in exact determina- tions of vapor pressure, as by the "streaming" method, the effect must be considered (the vapor pressure of water at zero degrees is altered by roughly one tenth percent per atmosphere pressure).* 15. Defect in the Sum Rule for Vapor Pressures. The rule that the total pressure over a liquid phase mixture of mutually immiscible substances is given by summing the separate vapor pressures suffers from the fact that the gases are actually not ideal. Thus ammonia deviates at one atmosphere and zero degrees by 1.6 per cent from the ideal pressure. A mixture of nitrogen and ammonia in equal molal proportions, however, exerts a pressure, at zero degrees and about one atmosphere. * The method of passing a neutral gas over liquids and subsequently absorbing the vapor out of a known volume of the gas mixture has been much employed in determinations of vapor pressures where the latter are small. In utilizing such data to compute vapor pressures the relation of the mass of the vapor to the mass of the neutral gas must be accurately known. Frequently the perfect gas laws have been invoked to compute the pressure of the vapor in the neutral-gas-vapor mixture. If, however, precise results are desired this procedure is inexact owing to the fact that Dalton's rule of mixtures may not be as close an approxi- mation as desirable. See Eli Lurie and L. J. Gillespie, J. Am. Chem. Soc, 49, 1146, (1927), also Phys. Rev., 34, 1605, (1929) and Phijs. Rev., 36, 121, (1930). The disability of the method, due to the failure of Dalton's law, might be avoided by passing the neutral gas through a saturation apparatus containing pure water and then through a similar apparatus in series with the first but containing the solution of interest. The temperature of the latter could then be raised until suitable tests showed that the content of water in the neutral gas was the same after each saturation apparatus. Determinations at several temperatures would then establish the vapor pressures of the solution from the known values for pure water. It can be shown that strictly the "Dalton defect" is not precisely the same in both saturations because of the temperature difference, but the error thus made can be shown to be exceedingly small. 356 KEYES ART. J not far from that calculated by the ideal gas law for mixtures. At higher or lower temperatures, nevertheless, the differences may be greater or less than that given by the latter law. As a general and approximate statement present knowledge warrants the conclusion that as far as low pressures are concerned, the order of accord of the actual behavior of pure gases and mixtures with the prediction of the perfect gas laws does not often exceed two percent from zero degrees to higher temperatures. Below zero the actual behavior of gases may show larger depar- ture from the idealized state in special cases. 16. Gihhs' Generalized Dalton's Law. The rule of pressures stated in italics (Gibbs, 1, 155, 7th line) is one of very great inclu- siveness.* It leads, for example, to a proposition relative to the entropy of a gas in a mixture which is of very far reaching theoretical significance and practical importance. It contains and is also far more inclusive than Dalton's rule of partial pressures as commonly stated, since its consequences involve the proposition that the energy and all the thermodynamic functions of gases in a mixture are of the same value as though each gas alone occupied the same volume as the mixture, the temperature remaining unchanged. In the formulation there is incorporated also the idea of equilibrium, which does not appear to be associated with the usual statement of Dalton's Law. The significance of the equilibrium idea, both thermal and mechani- cal, must be emphasized because of its extensive importance in every application to which thermodynamics lends itself. The Gibbs rule may be written, where the constants — ^- -^ and -—^ — ^ are represented by hi and Ci. Ml - ^1' aieH% "' , (42) [273] * Gillespie (P/iys. Rev., 36, 121, (1930)) has recently discussed in con- siderable detail the implications contained in Gibbs' italicized state- ment. It is shown that Gibbs' statement is, as would be expected, an approximation. It is, however, a useful rule, and is analogous to the Lewis and Randall rule of fugacities (Lewis and Randall, Thermo- dynamics, p. 226, 1923). The Gibbs rule and the fugacity rule often show deviations of opposite sign from the true pressures of binary mixtures. FUNDAMENTAL EQUATIONS OF IDEAL GASES 357 but (/ii — El) /ait may be formed from [268] and expressed as Jmi - El) /ait Pi aieH^' (43) whence or iV = 2pi), amit (44) [277] The former may apply even when the gases are not ideal. 17. Entropy of an Ideal Gas Mixture. Differentiating (42), [273] and rearranging gives the following equations: dp = 2 Ml -El aieh'^'e "'' (r ^iLJzZA 1 r" a^t ) dt + s Ml — El hi4Ci„ ait aie'T'e ait dm, (45) but by [98] dt + S[S] ^^" (46) dp = - dt ■{- / , — dfiu (47) whence using the value of Ml — El ait = - hi + log ('-9 from [269] there results ' = S [S {^' + <"■ + "^ '»« ' - "' "^ f}] ^''^ ■'''" 358 KEYES ART. J mi ^ Pi V a\t m-i p2 — = "~ ' etc. (49) [275] and r? = ^ (miHi + mi(ci + ai) log t + miai log — j- Where v is the volume of the mixture the entropy becomes ry = /, (miHi + wiCi log t + miai log — j- (50) [278] The latter equation requires that the entropy of a gas in a mixture of volume v and temperature t be the same as though it existed alone at the volume v, the temperature remaining unchanged. The result may be exhibited in another form. The total volume v is given by the expression - 2aimi where y is the total pressure of the mixture. Substituting in (50) [278] there is obtained rj = 2 ( ^1^1 + ^1 ^1 ^og t + m,a, log — ^^ y (51) [278] \ HaimJ but V is a quantity which is called the partial pressure for ZaiTWi the gas with subscript (1), i.e., pi, and 2pi = p, which is equa- tion [273]. It follows then that if a gas exists in the pure state at pressure p and temperature t its entropy in the gas mixture of pressure p will differ from that in the pure state by — miai log z , which is the same thing as — ri/C log Xi, Zttimi where Xi = —, the mol fraction, and C"^ = Miai (see equation [298], Gibbs, I, 168), where Mi is the molecular weight. 18. Implications of Gihhs' Generalized Dalion's Laio Apart from Ideal Gas Behavior. The discussion, Gibbs I, 156-157, FUNDAMENTAL EQUATIONS OF IDEAL GASES 359 beginning eleven lines from the bottom of 156 and ending at the corresponding point on 157 comprises material and inferences following quite directly and simply from equations [273] to [278]. The last sentence is significant. "It is in this sense, (equations [282], [283]) that we should understand the law of Dalton, that every gas is as a vacuum to every other gas." The statement that Gibbs' relations [282] and [283] are "con- sistent and possible" for other than ideal gases refers evidently to the belief that the relations in question, taken quite generally and without reference to the idealized gas laws, might lead to better accord with fact than would be possible with the latter. Thus the pressure of the individual gases composing the sum in the first of equations [282] may be any function of volume and temperature. By the use of (VII) for example, the total pressure would be written, Saitnii The energy, entropy and i/' function then become + Y^m.E,, (53) V = 2j'^i= 2jm^ J^ ci* - + 2j^,a, log ^^ \l/ = //Wi / Ci*dt + / jTUiEi — f /.mi / Ci* dt/t •^-\ V — Binii -^^ — t / jMiai log — t / jViiHi. (55) Equation (53) may be established by starting with either of the equations \dv)t \dt). p, (56) 360 KEYES ART. J Taking the first we find, using (VII), /*" amiH^ /dBi\ ^ , , €l = where f(t) is a pure temperature function. The integral may be taken from v = oo to y, resulting"^, if 5 is a pure temperature function, in n amiH^ /dBA ., = »,j_^o.'d<-„-^^_(-) + mA, (59) where Ei is a constant of reference for energy, and c* is the heat capacity for constant volume at infinitely low pressures, — a pure temperature function. The other equation of the pair gives for e «i = mi I Ci*dt + / J to J to since J ' — tfao (S) n '" "^ '"'^" '®*" (g)/v, m = c*, where c* is the heat capacity of a gas at infinitely low pressure and is known to be a pure temperature function. But '(S). = mKI).-p]' whence the second integral above becomes FUNDAMENTAL EQUATIONS OF IDEAL GASES 361 using equation (VII). Finally the equation for e becomes .. = ™, |_ c..d(- ^-3^ (^) + ».£.. (62) This equation is, as it should be, identical with the energy /de\ equation obtained by starting directly with the ( — j differential equation. The entropy may be computed by solving the equations ©. = &).• @). =r ^^^^ The entropy expression, using (VII) in connection with the first differential equation becomes, after adding and subtracting V — Bitrhi . aiTUi log ' mi V — Bitrii + m,/i(0 + m,Hx. (64) Integration gives finally m = rmMt) + a^m, log ^^^ - ^^T^^) Yt + ''''^'- ^^^^ Starting with the second differential equation there results, again using (VII), •ni = wi I 1 dt -\- mifiiv) + miHi = nii j ci* — + mi / / ti— 1 fit' y + mi/i(t;) + mj/fi = mi / ci* y + / f — j - y dy + mi/i(i;) + mi^i P dt ai miH dBi , ^ , ,^^, = ^^ 1 ''* 7 - (. - 5imi) ~^ + ^^-^^^^^ + ^'^^- ^^^^ 362 KEYES ART. J Comparing the two entropy expressions gives for the final entropy equation f dt V — Bimi Tji = mi / Ci* — + aiWi log "^"^^^ '^^ + m./7. (67) (v - Bimi) ai The f function ei + piWi — ^771 may now be formed by sub- stituting the energy and entropy, with the result f 1 = mi / ci* dt + miEi + aimit + miBipi — mii / ^* T ~" ^1^1^ log — — miHit, (68) and for a mixture, employing the rule of Gibbs, f = 2 f 1 ^ 2 *"'^ / ci* (^f + ^ mi^Ji + 2j ^1^1^ + 2j ^1-^iPi — ^ mit j ttii — /, miOii log — — 2j 'f^iHit. (69) The equations for ^ui, m, ... and Ci, C2, ... can be readily obtained from the last equation by differentiation, i.e., /•' r dt Ml = / ci* d^ + ^1 + pifii + ait - t ci* J ait - ait log — - Hit, (70) Pi mici = mici* + (^ _ 5^^^)^^^^^,' (71) \_dt \ dt J J using (Vila) and neglecting higher terms in the reciprocal of FUNDAMENTAL EQUATIONS OF IDEAL GASES 363 (v — BiiTii). Equation [280] now becomes c = 11 + higher terms in 7 and ~' (72) t V 19. Ideal Gas Mixture in a Potential Field. The paragraph beginning Gibbs, I, 158, last line, is introduced to emphasize the fact that in a mixture of gases, as in the atmosphere, each gas may be assumed to react to the gravitational field inde- pendently of the presence of the other gases". The point is made use of by Lord Rayleigh to investigate the work of separating gas mixtures and the reader is referred to Vol, I p. 242 of Scientific Papers, Lord Rayleigh, Camb. Univ. Press, 1899; Phil. Mag., 49,311, (1875). SO. Vapor Pressure of a Liquid under Pressure from a Neutral Gas. The subject of the effect of an insoluble and neutral gas on the vapor pressure of a liquid has been discussed earlier, making use of [272] in connection with the comments on the additive law of vapor pressures. The treatment taking account of a finite solubility of the neutral gas in the liquid is given in Gibbs, I, beginning p. 160, last paragraph. It will be seen that the phenomena connected with Henry's law con- stitute a special case of a binary mixture. Thus with carbon dioxide at zero degrees the pressure may be increased to 34.4 atm. at which point carbonic acid would liquefy since this is the saturation pressure. The temperature of the system may also be above the critical temperature of the neutral gas as with carbon dioxide above 31°, and in the process for separating helium from the natural gas in Texas. The general equations for the case of a two-phase binary mixture are — v' dp + r]' dt + mi 'dtii ' -j- m^ 'd^i ' = 0,1 -v"dp -f i)"dt -H mi"dMi" + m^'dii - ^) dp = (— , - ^) dt + (r' - r") dM2'. (75) \mi mi / ^ \mi mi / When r' is equal to r" the ratios of the components in both vapor and Hquid phases are identical, and the system resembles a pure substance in its thermodynamic behavior (mixture of constant boiling point). To show this, add equations (74) and (75), put (m/ + m2') = M' = 1, {mi" + m2") = M" = 1, and since (76) r' = r" L'+ m27 ^ (1 + ry r' (1 + r"Y r" Ui" + 1 ' ma", There is obtained finally iv' - v' '): = ^^'- ■ V). (77) The v' in this formula is the volume of one gram of the vapor mixture in equilibrium with the liquid mixture of constant boiling point t, and v" the volume of a gram of the latter liquid at t. The heat required to evaporate one gram of the special composition is, therefore, X = i f (.' - v"). (78) The heat of evaporation generally, and other quantities per- taining to a binary mixture may be obtained from the equations (73) when dm' and dn2 are known. A convenient trans- formation of form is the following, whereby the potentials are expressed in terms of the quantities a', a", dr', and dr". To carry out the transformation use is made of the following rela- tionships obtained from [92] by cross differentiation, tempera- ture and pressure being kept constant. (79) (6) a' = — wi' a" = - mi' (80) {() FUNDAMENTAL EQUATIONS OF IDEAL GASES 365 (a) {—) = (—\ Kdmi'/p, I, mj' \dm-i' ) p, t, mi' \dnii/p, t, TBj" \9w2 / p, t, mi" ' f— -\' 1 \dmi / p, t, m^' \dmi" ) p, t. tnj" , ~; I dm\ l\ J p, t, mj' (:; — -, ) dm2', drrh /p. t, Tn,' \a7n2 / p, t, m,' The following equations may now be written, where Xi, X2 are the quantities of heat required to evaporate a unit quantity of constituent 1 or 2 from the mixture, and Aiv, 1^20 are the corre- sponding changes in volume of a unit of components 1 or 2 in passing into vapor: dni = dyL\ + \dmi (81) Xi t h t dt = Aivdp - a'dr' + a"dr", dt , dr' „ dr" L^vdrt + a' — - Vi" —^^ r r (82) (83) 21. Application to "Gas-Streaming" Method of Measuring Vapor Pressures. An instance of some practical importance in the application of these equations will now be discussed. The determination of vapor pressures by the "streaming method" was referred to earlier in connection with the Poynting effect, ' but a fuller discussion was postponed until the Gibbs-Dalton 366 KEYES ART. J rule and some of its consequences were developed. There are essentially three effects which it is necessary to consider in order to use the method for the exact determination of vapor pres- sures. First, the effect of the pressure of the neutral gas on the vapor pressure of the liquid must be determined. This is the Poynting effect and has already been sufficiently discussed. Second, the depression of the vapor pressure of the liquid due to the dissolved gas must be computed. If, as usual, the solubility is slight, as with water at zero degrees saturated with air at atmospheric pressure, the change in vapor pressure due to solubility is neghgible. Third, Dal ton's law in the form usually applied, pi = Xip or pi = - — p (Gibbs' notation, c.f . [298]), where x is the mol fraction, is inexact. The example to follow will illustrate the use of the Gibbs-Dalton rule, p = 2pi. The third correction may be made by using the latter rule, or we require actual experimental data relative to the p, v, t behavior for the mixtures of interest and the neutral gas. Equivalent to the latter data is a knowledge of the constants of the equation of state for the two gases (gas emitted by liquid and neutral gas) together with the law of combination of the constants of the equation of state^^ to give the properties of mixtures. Enough knowledge of the latter sort is available to be useful in many cases. As a concrete problem, suppose an aqueous salt solution at the fixed temperature 21.2° is in equilibrium with nitrogen, the total pressure of the gaseous mixture being one atmosphere. Let the water vapor be absorbed and weighed while the nitrogen is passed along to be measured for pressure and volume at 25°C. The weight of the water is 0.45 gram or 0.02498 mols, and the nitrogen has a volume of 24000 c.c. at 1 atm., or 0.98111 mols. The perfect gas law is suitable for computing the latter since nitrogen is very nearly a perfect gas at 25° and 1 atm. The constants jS and A of the equation of state (Vila) for water and nitrogen* are * The constants given for water are only approximate. Those for nitrogen are valid for low pressures at ordinary temperatures. This is not the place for a complete and exact exposition of the theory of reduc- FUNDAMENTAL EQUATIONS OF IDEAL GASES 367 ^H.o = 81, ^H.o = 57 X 10«, |3n, = 47.6, ^N: = 1.255 X 10«, the units being c.c. per mol and atmospheres. Using the Gibbs- Dalton rule that the total pressure is equal to the sum of the pressures which each of the separate gases would manifest if alone present in the total volume of the mixture we find 82.06 X 294.3 X 0.02482 82.06 X 294.3 X 0.97516 ^ " F + 56.6 "^ 7 + 4.2 A few trials will be found to give 24144.4 c.c. as the volume for the pressure of one atmosphere. The first term of the right hand side becomes 0.02477 and the second 0.97523. But these terms are the equilibrium pressures according to the Gibbs- Dalton rule and hence the pressure of the water vapor is 18.825 mm. The application of the Dalton rule as usually applied (pi = pxi) gives on the other hand 18.866 mm. ; a difference of one part in 460. The actual vapor pressure of the solution is 18.820 mm. A similar computation may be made using the fugacity function^^'^''''*^'^. In the latter case the equilibrium fugacity, as proposed by Lewis and Randall, is given by the rule /« = fpXi, where fp is the fugacity of the gas of interest at the pressure p of the mixture. Finally the equilibrium pressure may be computed using the equation of state constants for the gases of interest and computing the equation of state constants for the mixtures by combination rules for the constants known to hold for mixtures of nitrogen and methane'*^. The latter method has met with success in a number of applications. S2. Heat of Evaporation of a Liquid under Constant Pressure. The discussion (Gibbs, I) beginning at the bottom of page 161 and continuing to the top of page 163 contains an elegant proof of the impossibility of an uncompensated change in ing "gas-current" observations, especially since the procedure has been given in detail recently by H. T. Gerry and L. J. Gillespie (Phys. Rev., 40, 269 (1932)) for the case of the vapor pressures of iodine. 368 KEYES ART. J vapor pressure when the emitting soHd or Hquid is compressed. It will be recognized that the proof depends on the use of [272] by which the change in vapor pressure with pressure on the liquid or solid phases was computed. It may be well to remark that the energy equation corresponding to this case may be easily deduced from the general equations (73) applied to one component. Thus, u' dp = n' dt + m/ d^ii', \ (84) v"dP = ■q"dt + mi"dni".j Here dp refers to the vapor pressure change of the pure substance (single accent), but if the pressure P is maintained constant on the liquid phase and equilibrium subsists we have or dp . , ^ \p = t-^ v'. (85) at The latent heat of evaporation under conditions of constant pressure on the liquid phase accordingly differs from the normal heat under saturation conditions. In a similar manner if a pressure P is applied to the solid phase but not the liquid phase we find Xp = < ^ v", (86) dt dt where v" is the volume of the liquid. Evidently — , the change in melting point with pressure, will be large compared with the ordinary change of melting point with pressure where the same pressure is applied to both phases. The equation aids inci- dentally in understanding the extruding of metals, made possible no doubt because of actual instantaneous creation of liquid phases under the enormous pressures applied to the solid. FUNDAMENTAL EQUATIONS OF IDEAL GASES 369 £3. Fundamental Equations from Gibhs-Dalton Law. The fundamental equations in the form given in [291], [292] and [293] are easily obtained. The latter equation may also, how- ever, be expressed in the form: r = 2 ^'^^^^ '^ mit(ci + ai - Hi)] - 2 ci^i^ log f - 2 «i^i^ log ^> (87) [293] where Xi, the mol fraction, is equal to r The content of the paragraph following [293] should be carefully noted. 24. Case of Gas Mixtures Whose Components are Chemically Reactive. Thus far only gas mixtures with independently variable components have been considered. The material following [293] (Gibbs, 1, 163) therefore emphasizes the distinction which must be made between gas mixtures of the former kind, and those with convertible or chemically reactive components. The characteristic of the latter is of course that chemical changes proceed by whole numbers or fixed ratios. Two molecules of hydrogen always require one molecule of oxygen, never more nor less, to form one molecule of water, and three molecules disappear when two water molecules are formed. As a consequence we need only be concerned, in our equations of thermodynamics for chemically combining gases, with these whole number ratios and not with actual masses. Thus it is clear that, in so far as convenience is served, our equations for gas mixtures could be expressed in units of mass proportional to the masses of the molecules of the separate and distinct chemical species. This, of course, is the almost universal custom in chemistry at present, and in all the preceding formulae it is merely required that n, the number of mols, be substituted for m the masses. The constants ai, 02, . . . must also be expressed in terms of the mol as the unit of mass. Thus (87) [293] would be written f = 2^^ r^i + tic + R- i7i)] Rt - 2 ^^1^1^ log f - 2 ^1^^ log '^' (88) ^293] 370 KEYES ART. J where R, the universal gas constant, is equal to the product of tti, 02, ... and the corresponding molecular weights. Here El, Ci and Hi are also assumed to have been multiplied by the corresponding molecular weights. II. Inferences in Regard to the Potentials in Liquids and Solids (Gihbs, I, 164, 165) There might be included under this heading a large portion of the principles and doctrine which have found application in physical chemistry in the last half-century. The fact that a comparatively simple basis of fact could have such general applicability was well known to Gibbs, as is indicated by the last sentence of the section (7th line from bottom, p. 165). Indeed a few empirically discovered facts interrelated thermodynamically suffice to form the theory of those liquid mixtures wherein the masses of one or several constituents are very small relative to the mass of one of the components*^. The principle of the equality of the potentials of a component in equilibrium in the coexisting gaseous and liquid or solid phases affords the means of deter- mining the potentials of the condensed phases. Because of this a full knowledge of the properties of pure gases and their mixtures is of fundamental importance in extending the range of applica- bility of the general theory. Thus it becomes clear that great im- portance attaches to a knowledge of the constants of the equation of state for different substances, and the rules for combining these constants, in order that the constants for the equations for mixtures may become available. On the other hand" given sufficient data for pure substances and their mixtures, the required thermodynamic quantities may be accurately com- puted empirically, using the assumption that the ideal gas laws hold rigorously in the limit of low pressures. It is evident, however, that on this basis an almost prohibitive amount of experimental data would be required to satisfy the needs of the science, and therefore continuous effort should be made to develop a rational form of equation of state with the aid of statistical mechanics. It is, indeed, apropos to add that the correlations of physico-chemical facts by thermodynamics can FUNDAMENTAL EQUATIONS OF IDEAL GASES 371 receive much independent assistance and support from the theorems and results deducible from statistical mechanics. It is also evident of course that, outside of the field of equilib- rium states, thermodynamics is of no service and progress in the theory of non-equilibrium states depends on the perfection of statistical theory. Modern atomic and molecular theories likewise have an important part to play in leading to an improved knowledge of molecular constants and molecular encounters, which is indispensable to the future progress of physical chem- istry. So. Henry's Law. The law that the concentration of the dis- solved constituent is proportional to the pressure of the gaseous constituent is to be regarded as applying strictly only in the limit where the amount of dissolved gas is vanishingly small. The deviation in the case of carbon dioxide and water, for example, where it amounts over the interval 30 atm. to 37 percent at zero degrees and 29 percent at 12.43 degrees^* is typical. The pressure of the gas phase, in this case, increases more rapidly than the amount of gas dissolved. By way of accounting for the deviations from Henry's law it may be noted that the gaseous mixture over a liquid is now known to be far from a perfect gas. This particular aspect of the problem has received recent attention, and the changes in volume on formation of the mixture, together with the signifi- cant thermodynamic formulae, have been developed '^^-^^'^ using the fugacity function introduced by G.N. Lewis^^-^^-^^'^''^ This convenient function in the case of a pure gas is related to the n function of Gibbs as follows : '• [h ^' - ^'-^l /= pexp.\ — (n - Mi) I' (89) where ^ is the potential at pressure p and temperature t, and /x« is the potential at the same pressure and temperature assuming the ideal gas laws to hold. From the equation it is evident that f —> p in the limit when the pressure approaches zero. The equilibrium fugacity, /«, of one of the gases, 1, in a mixture of gases, is given by the equation ^^' ^2 /.= "''' '^^- [h r('' ~ f ) *]' ^'^°^ 372 KEYES where vi is the partial volume (: dv\ dmj p, t, m ART. J , and Xi the mol fraction. The analogue of Henry's law in terms of fugacity becomes for dilute solutions /« = kmi", where m/' is the mass of the dissolved gas in the liquid phase. A glance at the expres- sion above for/e makes evident that a part of the deviations from Henry's law will be found in the failure of the equihbrium gas mixture to conform to the ideal gas laws. £6. RaoulVs Law of Vapor Pressure and the Thermodynamic Theory of Dilute Solutions. Another principle in the same class with Henry's law is Raoult's law, according to which the ratio of the vapor pressure of a solution to the normal saturation pressure is equal to the ratio of the number of molecules of the solvent to the sum of those of the dissolved substance and the solvent. Designate the salt with subscript 2 and the solvent with subscript s. V Psat. n. Ua + n2 or Psat. — P _ n2 Psat. Psat. P P Us + ^2 W2 ns (91) The relation of this result to the general Gibbs theory is easily established for dilute salt solutions. A salt solution may be regarded as a special case of a binary mixture in which the component in smallest amount is non-volatile. The second of the pair of equations in a, equation (83), vanishes and there remains, since m^' = 0, Xi dp „ dr" 7 = ^^^^+^'V- (92) Note in the first place that if m-l' jm-i' = r" is constant, and we let Xo = ^ "^ (vi - v^ FUNDAMENTAL EQUATIONS OF IDEAL GASES 373 represent the heat of vaporization of the pure solvent, the heat of dilution is obtained at once for the case where the vapor, of volume vi, may be taken to be an ideal gas, and the liquid volume V2 is negligible. We find X. - X. = AX = a.^.P ^-^^\,: (93) Taking the temperature as constant in the general equation, n TYi f assuming that v = — — — {m,' is the mass of vapor of solvent), V we drop the accent in a" and r". This gives dr AiU aamst Integrating the last equation there is obtained /, — = log = - 7, r. (95) p.at. V P'at. asMst But psat. — p may be put equal to Ap, and w/ may be taken to be numerically equal to ilf / the molecular weight of the vapor, whence ^=i^'- (96) p,ai. at nis Raoult's law in dilute solution may be expressed in the form ^p/Ps = Ui/n, when Ui is small relative to n«. By comparison we find \dmjp, t. which is constant at constant temperature and depends only on ilf, _ „ , . the molecular weight ratio vr. Fmally we obtam M.2 [2 Ms M2 /X2 = —-^Rt log rris + /(p, t, nh) for the relation between ^2 and the masses of solvent and dis- solved substance. 374 KEYES ART. J Again for constant pressure there is obtained from the general equation ^dt\ t a * (97) (; dr/p Xi From the previous inference it is clear that a is a positive quantity, hence dr and dt change in the same sense or for increased concentration there is a proportionate rise in tem- perature. Inserting the value of a found in the preceding paragraph we find on integrating : i - fo = -7- -' 98 which is the usual equation for the elevation of the boiling point. A similar equation of corresponding form gives the depression of the freezing point for dilute solutions. If Xi is assumed given by [Xo + ci\r we obtain i — t{s , t n-2, , ^ Xo— — + clog- = R— (99) to' to IT'S Expanding log t/to in a series of powers of — - — leads, as a first fo approximation, to equation (98); retaining however the second term leads to the equation Rtot 712 r. c^o~l Xo Us L Xo J From the foregoing discussion the nature of the deficiencies in the formulae arising from the approximations used will be clear. A more complete theory may be constructed in various ways, but up to the present time no very systematic coordination of the theoretical development and exact experimentation has been undertaken. Recently a method has been discussed * Note that Xi is the heat required to remove unit mass of solvent vapor from the salt solution. We may assume that Xi is equal to the heat of evaporation of the pure solvent, or better, that it is a function of temperature of the form [Xi = Xo + cit]r where ci is a constant. FUNDAMENTAL EQUATIONS OF IDEAL GASES 375 by G. van Lerberghe^^ which has as a basis the develop- ment of the function p = f(ti, Vi, mj, W2, . . . ) by Taylor's theorem. That it is possible to develop a consistent and rational system for the discussion of the properties of solutions on such a basis has, in fact, been pointed out by Planck ^^ The method is equivalent in some respects to the system of treating solutions developed by G. N. Lewis and systematically presented by Lewis and Randall in their Thermodynamics. Methods of treating solutions along these lines have, however, the limitations of procedures whose foundation is entirely empirical. On the other hand any other procedure requires much detailed knowledge pertaining to molecular interaction and the surmounting of formidable mathematical difficulties*^. Although the initial steps have been taken in acquiring the requisite knowledge of the attractive and repulsive fields of molecules, very much ground remains to be won before a complete molecular statistical theory of solutions can be achieved. The mathematical difficulties, forming an important part of the problem, remain at the moment practically unsolved^^ except for the case of infinitely dilute solutions*'^. The case of electrolytes at infinite dilution has been treated by Debye and Hiickel ^^- *^, and the accord of their theory with the facts is astonishingly good in spite of important fundamental limitations. III. Considerations Relating to the Increase of Entropy Due to the Mixture of Gases by Diffusion (Gihbs, I, 165-168) The entropy change on mixing gases has already been mentioned with reference to the difference in entropy which arises when pure gases mix at temperature, t, and constant pressure, p. Thus we may imagine two perfect gases 1 and 2, contained in the apparatus indicated in the diagram, Fig. 1. Suppose that the pistons are permeable to the gases as indicated and the usual assumptions made with regard to the absence of frictional effects. Each gas is assumed to occupy its portion of the cylinder at the same pressure and temperature when the pistons are in contact. As the pistons are slowly moved out each gas passes through its respective semi-per- 376 KEYES ART. J meable membrane into the space between the pistons, constitut- ing finally a mixture of the two gases originally in the pure state. By moving the pistons together the separation can be effected. With the gases in the pure state we have, rji = ruiCi log t + Wiai log — + miHi, Till 772 = W2C2 log t + wi2a2 log — + m2i/2. W2 (101) [278] But = Vi and = F2, while aimi + 02^2 = ( k 1 + 1^2) 7 p p t pV = —-, and after mixing each gas will occupy the total volume V F = Fi + F2, or F t;/ = viiCi log t + miai log — + rriiHi, V ri2 = W2C2 log t + m2a2 log — + niiHi. 7VL2 (102) The difference between the respective entropies after and before mixing is given, therefore, by the following equations: Fi aimi ,,-,, = -m,a, log - = -rma^ log ^^^^ ^ ^^^; F2 , ^2^2 172 — •'72 = —nhai log — = — ?W2a2 log ; — ■' (103) since Fi/F = aiiui and F2/F = aiVii aitni + a2m2 by the relations following (101) [278] above. Each difference is positive since the mol fractions are neces- sarily each less than unity, and therefore an increase of entropy has attended the mixing. If each gas is present in equal amount the total increase becomes vV {ami + a^rrii) log 2 = y log 2. (104) [297] FUNDAMENTAL EQUATIONS OF IDEAL GASES 377 The generalization of the above result follows easily, and if Xi, Xi, ... Xi are the mol fractions we find ^ - V 1 = r^-iV Sri 2J (vi - vx) = 2j «i^i ^°g - = C'-i Zy '' l^s ' ^^^^^ ^^^^1 Xi Ti where C~^ in Gibbs' notation is equal to the universal gas constant, usually designated by R. The discussion following equation [297] is too complete to require comment other than to draw attention to the remark which admirably sums up the import of the Gibbs theorem on entropies: "the impossibility of an uncompensated decrease of entropy seems to be reduced to improbability" (15th line from bottom p. 167). It is of addi- tional interest to note that an entirely analogous theorem may P/STOf^ 1 PERMEABLE TO &AS1 GASl V z GAS 1 a/x/ GAS Z V A GAS a P/STOA/Z PERMEABLE TO GASZ Fig. 1 be deduced by starting with equation [92] of Gibbs' Statistical Mechanics (Gibbs, II, Part I, 33) and extending the equation to include two or more molecular species. IV. The Phases of Dissipated Energy of an Ideal Gas Mixture with Components Which Are Chemically Related (Gihhs, I, 168-172) Before reading this section, the section on "Certain Points relating to the Molecular Constitution of Bodies," pp. 138-144, should be consulted. The immediate goal is to provide the basis for treating the phenomena exhibited by mixtures of gases which are capable of chemical interaction. What is sought is a scheme whereby the equilibrium amounts of the different distinct molecular species may be correlated as a function of 378 KEYES ART. J the energy of interaction, the pressure or volume, and the tem- perature. At least this is the goal which is of chief interest to the chemist using thermodynamics as a means of correlating equilibrium data, and some conceptions of a molecular nature are required in practice notwithstanding the often repeated statement that thermodynamics has no need of molecular hypotheses. The latter dictum is really true only in a restricted sense in the field of the applications of thermodynamics to the extensive and varied phenomena of chemistry. The term phases of dissipated energy may be assumed equiva- lent to what is now generally called the equilibrium state. It is for this state alone that the energy is a minimum and the entropy a maximum (see Gibbs, I, 56, "Criteria of Equilibrium and Stability' ' ) . Of course equilibrium states are not always easy to realize, but in every case of doubt as to the establishment of equilibrium in the case of chemically interacting components the usual test in practice is to vary the independent variables, pressures or temperature or both, at the supposed state of equilibrium and to observe the displacement, finally verifying the possibility of reproducing the original condition of true equilibrium at the point in question. Gibbs' treatment involves the masses of the components instead of the mols now used. Equation [299] in the concrete case of the formation of water from the elements would be written, 1 g. (H2O) = 8/9 g. (O2) + 1/9 g. (H2). (106) [299] But for the condition of equilibrium it has been proved that Zfii8mi ^ 0, and our knowledge of the principles of chemical combination allows us to identify the variations 5wi, 8m2, ... as proportional to the X coefficients as in (106) [299]. In equation [300], 8ms may be replaced by —1 if water is assumed to disappear in the reaction, whence 5w2 becomes 8/9 and 8mi 1/9, both reckoned plus, i.e., ^ Ml + I M2 = M3, (107) [301] FUNDAMENTAL EQUATIONS OF IDEAL GASES 379 In terms of v and t as independent variables [276] gives 1 mi 8 m2 m3 , , , , - ai log — + - a2 log - - as log - (108) ]302] = A+Blogt- c/t, in which the values of ^, 5 and C are given by [303], [304], [305]. The mass law is contained in the left-hand member of (108) [302]. For, on multiplying and dividing each term by the respective molecular weights, there results (1 , wi 8 , 1712 1 , wisN ,^^^^ rrr log — + rrr log — - — log — )• (109) 9ilf 1 ^ V QMz * y Ms V / ^ ' Multiplying and dividing the bracketed member by ilf 3 = 18, and taking Mx = 2, M2 = 32, gives -|_log-+-log--log-j (110) but — ' etc., become — :' — :' ~~:' Using Dalton's law of par- V Qit a2t azt tial pressures in its usual form pi = pxi, we jQnd The term in the partial pressures is the usual mass law expres- sion, or Kp as the quantity is commonly designated, while the remaining term in the a's is a constant. The case where /3i -|- /32 — 1 is zero corresponds to the case where the sum of the exponents of the partial pressures vanishes. An example exists in the case of the union of H2 and I2 to form 2HI, where the total pressure does not enter the reaction equation. 27. Restatement of the Above in Different Notation. Em- ploying mols as the unit of mass, and recognizing from the foregoing that the variations of mass 5wi, bnii, . . . need only be considered as ratios equal in value to the coefficients in the chemical reaction, we write [300] as Smii'i ^ 0, (112) [300] 380 KEYES ART. J where v represents the coefficients, for example — 1, 1/2 and 1 in the decomposition of water. Here the minus sign signifies that a component vanishes while the positive sign signifies the appearance of components formed from those having the minus sign. Assume also that the heat capacities Ci, c^, ... are not constants but functions of the temperature. Starting with equations [265] and [283] there is finally obtained 2^= S'^i / ^1*^^ + ^n,Ex -h^n, Rt -Y^Uit \ J to J to - ^niRtlog—^ - ^nitHi, whence f r dt Rt Ml = / ci*dt +E^-t \ c*-r -Rt\og-- + Rt- H,t. (113) The equivalent of equation (2) [300] may now be easily formed, and on rearrangement there results 2jVi log pxi = - + Zj""' ^°S Rt - ^ Rt ' Z-V—-^"" Rt 't ^U'^*^'^' S^^^^-S^^^ + -^ + -]f^ (114) [309] This equation is perfectly general within the limits of appli- cability of the perfect gas laws, and [282] and [283] apply. The energy constants and the entropy constants may be adjusted to suit practical convenience, but this has already been referred to earlier and need not detain us here. The case of the dissociation of water vapor and of the decom- position of hydriodic acid will illustrate in detail the points raised by Gibbs. For the former we have H2O =^02 + H2, 1 Vz = —\, J/2 = 2' "1 ^ -^• (115) FUNDAMENTAL EQUATIONS OF IDEAL GASES 381 In general the heat capacities are known over a Hmited range of temperature, for H2 is the only gas whose heat capacity is known at low temperatures. The question of whether the heat capacity approaches 3/2 R or vanishes at zero Kelvin is, moreover, not yet settled. In the case of water vapor values of C3 are available to temperatures where water vapor is detectably dissociated. Such values must, however, be corrected for heat absorbed due to dissociation; a correction evidently impossible to obtain until the dissociation data can be correlated, and then a final and exact result is only possible by successive approxima- tion. Above zero degrees the heat capacities of most gases increase rather slowly, and in the absence of a generally appli- cable theory of heat capacities of gases linear expressions, or at most quadratic expansions, may be used. On this basis the heat capacity terms become, when the linear form is used, 2^1 / c,*dt = ^v,a, {t - to) + SV (^' - ^0')' (116) J to 2)"! / ci*dt/t = ^via,\og{t/to) + ^vA (t - to). (117) J to The present custom is often to integrate the linear terms between zero Kelvin and t, but such practice, as is frequently the case, had its origin in the earlier erroneous belief that the heat capacity dependence on temperature was as simple below the ice point as it appeared to be above. Note should be taken also of Gibbs' decision to express the reaction pressure- temperature function in terms of the energy constant £"1, a choice very likely induced by the somewhat simpler treatment possible when non-ideal gases are involved. When Zi'i vanishes in (114) [309] the mol fraction function Si'i log xi becomes a function of temperature alone, and thus pressure is without influence on the numbers of the different kinds of molecules so long as the gases are ideal. A further simplification would result if the terms / J vi I Ci*dt and 2j vx \ Ci*dt/t 382 KEYES ART. J vanished, and this assumption is sometimes made when, as is often the case, there is a practically complete lack of heat capacity data. The leading term is of course ZvEi/Rt and is very large in the usual case of gas reactions. The equation (114) [309] contains the generalization set forth in equations [311] to [318]. It includes also the case referred to in the sentence following [318]; "graded" dissocia- tion illustrated by the reaction HI ^ H2 + I2 -^ 2H + 21. It is clear also that the presence of a neutral gas in the reaction mixture is without influence on the value of the equilibrium constant (114) [309] provided p is understood to be the total pressure diminished by the pressure the neutral gas would exert if it alone occupied the volume of the mixture. The influence of a gravitational field of the magnitude available on the earth is exceedingly small and equation [234], Gibbs, I, 146 provides the basis for investigating such effects. V. Gas Mixtures with Convertible Components {Gibbs, I, 172-184) The equation (114) [309] of the previous section includes the case of interest here developed. The term convertible com- ponents refers to the formation of multiple molecules such as (N02)2; a case which would also be included under the term reversible polymerization or association. The painstaking justification of the application of the principles established for the treatment of mixtures of chemically related components to the present case may seem unnecessary. On the other hand it should be recalled that one of the former axioms of chemistry was that substances of the same qualitative and quantitative composition must possess the same physical properties. Reference may be made to Liebig's discovery of the identity of composition of silver fulminate and silver cyanate as the first definite fact invalidating the axiom. Had NO2 been colorless the explanation of the considerable change in density of the gas with pressure would probably not have been ascribed to association and dissociation for a long time. As a matter of fact it was the change in color on change of pressure FUNDAMENTAL EQUATIONS OF IDEAL GASES 383 and temperature which prompted the supposition of a change in molecular species, and the measurements of density were then used as confirmatory evidence to establish the fact of the con- version of NO2 into colorless N2O4 as the pressure increased or the temperature diminished. The assumption has often been made that the departure of gases from the ideal state is to be ascribed generally to the tendency to polymerization. The same idea appeared later in modified form in the attempt to explain all departures from Van der Waals' equation as due to an association collapse of the molecular system, and again in the idea that the formation of the liquid phase was conditioned upon such a collapse. It is clear however that a distinct molecular species of the associated type such as (N02)2 occurs comparatively rarely, and that the formation of the liquid phase and the departure of gases from the ideal state must in general be ascribed to quite different causes. The case of convertible components offers one point of contrast with that of chemically related components, for the latter is as a rule subject to passive resistance (Gibbs, I, 58) whereas the former appears not to be limited in the rapidity with which the ratio of the molecular species can adjust itself to follow the fluctuations of pressure and temperature.^'' The test, that equation [309] be applicable to the case of con- vertible components, rests on its successful application in inter- preting the densities of N2O4 observed under various conditions of temperature and pressure. Admittedly the dissociation of the latter substance into two molecules, and similar chemical reactions, form ideal examples to which the thermodynamic principles of chemical interaction may be expected to apply. Reactions of this class in the gaseous phase appear to be free from the effects of passive resistance and are subject unquestion- ably to the conditions of equilibrium discussed by Gibbs from page 56 on. They present a problem exemplifying a wide range of the interpretative possibilities latent in thermody- namics. Evidently it is difficult to provide specific heat data to use in the reaction equation (114) [309] since the freedom of con- 384 KEYES ART. J vertibility of the simple and complex molecules cannot be arrested. The apparent heat capacity of the gas mixture will therefore consist of the sum of the heat capacities of quantities of the NO2 and N2O4 molecules dependent on the temperature and pressure and on the heat absorbed in the shift of the molecular species while the mixture is being changed in temperature. An exact knowledge of the ratio of the number of mols of NO2 and N2O4 as a function of temperature and pressure would of course enable such apparent heat capacities to be operated upon with a view to extracting the heat capacities of the separate molecular species, but it is quite impossible to evaluate the terms of equation (114) [309], for example, without the heat capacity data. It might be supposed that (114) [309] could be evalu- ated omitting the heat capacity terms as a first approximation, and that with such a provisional relation between the amounts of NO2 to N2O4 as a function of p and t one could treat the apparent heat capacity data. The provisional values of the heat capacities could then be used to secure a second approxi- mation of the reaction equation, and this in turn would permit a further refinement in computing the true heat capacities. But this tedious process could not lead to an exact result since in the treatment the perfect gas laws would be involved. Of course, sufficiently precise measurements of the actual density of the mixture would conceivably permit a semi- empirical formulation with (114) [309] as a basis, provided the composition of the mixture could be exactly determined. This is, however, a matter of the greatest difficulty because of the great reaction mobility so that, generally considered, the exact interpretation of density data for mutually convertible com- ponents in terms of the numbers of the reacting molecules, the pressure and the temperature, must be admitted to be sur- rounded with difficulties. We proceed with the application of equation (114) [309] by omitting all the heat capacity terms and writing for ZviEi ^viH\ — "EviR AE, and for the symbol I, giving K l«S^r^= - ^+^- (118) [309] Kt x^^Q lit FUNDAMENTAL EQUATIONS OF IDEAL GASES 385 This is the form adopted by Gibbs.* We proceed to examine a few properties of this equation. The equation of state of the gas mixture is assumed to be pv = Rt(ni + 712), where ni is the number of mols of NO2 and ria the number of N2O4, which permits the equation to be expressed as rii AE log — = - — + /. n^v Rt (119) [309] Setting p — equal to kp, and — equal to kc, and differentiat- X2 ThP ing (118) [309] with respect to t at constant pressure gives the equation 'd log kp\ AE + Rt c- dt /p Rf But equation [89] on differentiation and substitution of (120) '(|),* + 'va(/. dp + Cpdt for de + pdv, where Cp is the heat capacity at constant pressure, gives dx = Cpdt — .dt, — V dp, (121) and ®r'- ©. = -['©.-"] 'dVT — ' (122) where r = t"^. The summation principle [283] leads to the con- clusion, however, using the first of the above pair of equations, that X = [S I'lXi + 2 viCp,dt]p. (123) In (118) [309] the heat capacity terms have been assumed to * See paragraph beginning line 4, Gibbs, I, 180. 386 KEYES ART. J vanish, and application of the same condition to the last equation leads to y^ = 2^ix = Axi = A^ + i:viRt = AE -i- m. (124) But this is the numerator of the expression (120) for the derivative with respect to t of log kp, which is to be identified as the heat of reaction at constant pressure subject to the condi- tion that the specific heat capacities of the reacting gases are all equal (i.e., 2viCi = 0). The temperature derivative of log kc, taken for constant volume, is /8 log k,\ AE and AE is the heat of reaction at constant volume. From [86] we find { — ) = c and integrating at constant volume using [283] \ot/v we have ' ^ (126) = \aE+ jY^ViCidt which is the general equation for the energy at constant volume. The above is the equivalent, with some elaboration of detail, of the material of Gibbs, I, 180 and the first third of 181. It remains to note that since we have defined log kp and log kc as equal to 2j ^^ ^^^ P^i ^^^ Zj ^^ ^^^ ~' V — ^ — ~ ) ^^ ^^^° and (d log fcA ^ ^ from (114) [309]. If, however, we set S vi log xi equal to log kx, then from (114) [309] it follows that \ 8p 7, p \ dv ), V FUNDAMENTAL EQUATIONS OF IDEAL GASES 387 31. A More General Application of the Gibhs-Dalton Rule. A more general reaction equation than (114) [309] may be readily obtained by applying the Gibbs-Dalton rule in the form p = 2pi using the equation (VII) to compute the pi's. The equations for energy (53), entropy (54), and \p (55), have already been given, and from these the equation for 2 vim may be formed and the equilibrium equation found, i.e., (129) [309] 2 ^1^1 = 0' 2j vi log kp = 2^vi log poXi - 2 y "-'bIhx where Sj'i log pnXi is given by equation (114) [309]. The second term of the right hand member of (129) [309] may be written, using (Vila) and omitting ai, 0:2, • • . -is ''^^^^^ = i [S ''^^^ Rt-1^ '^''^^^ ^^30) Substituting in (129) [309] there is obtained 2j vi log pxi — 2j^i log poXi "^1^1X1 Ai Si'i.riiSi Rt V- (131) Thus it is seen that at constant temperature the left hand mem- ber, or the quantity log K^/Kq should vary with the pressure. For the reaction N2O4 -^ 2NO2 we may write log KJK, = - ■(2^1 + ^2) (2Ai + A,) Rt (Rty ] Xip + '§2 Rt (132) where /3i, (32, Ai, A2, are the constants of the equation of state for the gases NO2 (mol fraction Xi) and N2O4 (mol fraction x^). At constant temperature and low pressure, Xi the mol fraction of the simple species is small, and log Kp/Ko depends more largely on the second term of the right hand member, which is independent of Xi but proportional to pressure. The coefficient 388 KEYES ART. J of p, it should be noted, can be positive, negative or zero de- pending on the temperature, and of course the coefficient of xip has the same property although the temperature at which each coefficient vanishes will not in general be the same. Certain considerations may be shown to make plausible the assumption that 2/3i = ^2, ^Ai = A2; where /3i, Ai, ^2, A2, are the constants in mols of the equations of state. Under such an assumption the last equation reduces to log K,/Ko = [I - ^J (x. - X,) V '^2 A2 [ Rt (my 1 - 3« 1 + a V, (133) where a is the fraction of N2O4 dissociated. A recent paper by Verhoek and Daniels " contains material which affords a test of the formulation above. The measure- ments show that the values of log Kp/K^ do actually vary linearly with pressure over a range of pressure which however does not exceed one atm. The data have been used to pre- pare Fig. 2 illustrating the course of the experiments at three temperatures. The slopes of the lines do not appear to be in regular order as would be expected from the equation above. However, if the equation above were capable of representing the data, a line would start from the origin for every isothermal series of experiments forming a "fan" composed of lines in both the positive or upper part of the diagram and the lower or nega- tive part. Eventually Kp will equal Kq independent of the pressure but, as P increases, the sign of the right hand member would come to depend upon {x2 — Xi). A continuation of the exact investigation of this reaction evidently holds much of interest. The reformulations of the data '^2, 63 q^ ^^ig reac- tion, using the ideal gas laws, which have appeared since the publication of Gibbs' papers, can add nothing to the thermo- dynamic theory as applied to cases of convertible components. 29. General Conclusions and the Equation of State of an Ideal Gas Mixture Having Convertible Components. The heat capacity at constant volume for a real gas possessing a coefficient {dp/dt)v FUNDAMENTAL EQUATIONS OF IDEAL GASES 389 which is constant and independent of temperature is the same as it would be for the gas in the ideal state at infinitely low pressure. This may be proved by considering the two general equations and /M ^ /dp\ ^ \dv)t \dtjj ). = © [337] ♦aos 0.00 -0.05 -0.10 0.5 1.0 ATMOSPHERES PRESSURE Fig. 2 390 KEYES ART. J Performing the operations indicated in [338] the following equation is deduced : Accordingly the right hand member of the latter vanishes for a substance whose (dp/dt)v coefficient is constant, and the con- clusion follows that Cy is a function of temperature only. But no restriction has been put upon whether (dp/dt).^ is to be taken at high pressures or low, for perfect or imperfect gases, and therefore c^ is the same whether the fluid is of great density or of vanishing density. A fluid following van der Waals' equation would possess the latter quality. Comparison of the heat capacity c» of ether, for example, in the liquid phase and the gaseous phase will show that the heat capacities are equal for the substance in the two phases. This, however, is not to be taken as an indication that ether follows van der Waals' equa- tion. As a matter of fact, however, {dp/dt)v is remarkably independent of temperature in the case of many substances, (in both the gaseous and liquid phases) •^■* particularly non- polar substances in the dielectric constant sense of the term. Assuming the gases NO2 and N2O4 to be ideal the equation of state may be written pv = Rt (ni + ^2) where rii and ^2 denote the number of mols of the two gases. Assume that one mol of N2O4 is dissociated to the extent a, the fraction dissociated. The quantity Ui will be then given by 2a and 712 by (1 — a) whence pv = Rt(l + a). On the other hand [333] in terms of a becomes or log p : ^ Ao+ Bologt - i — (X I Ao' t^o e « -' (136) 1 — a^ p where Ao, Bo and Co are constants related to similar ones appearing in [333]. By means of the latter an expression for p FUNDAMENTAL EQUATIONS OF IDEAL GASES 391 as a function of a and t is found and, using the equation for pv, another equation giving v in terms of a and t. These are 1 - a2 _Co p = — Ao't^ e ' ' (137) a a 1 - 2 Co 1 -B a Ao" r - ^° e ' ' (138) R where Aq" is -j-,. From the equations it is clear that (dp/dt)v cannot be independent of temperature except in the strict hmit oi p = 0 or t = CO , for /dp\ R ^ ^ Rt /da\ [Vtl = ; (1 + «) + 7 [m): Equation [342] is the Gibbs-Dalton rule, p = 2pi, applied to the case of binary mixtures assuming equilibrium to subsist at Rt , all times. It is equivalent to the equation p = — (1 + a) where mols are used instead of masses. The equation for v above corresponds to [345]. Since the entropy and energy conform to the summation rules, [282], [283] may be easily formed in terms of mols from the foregoing, while the calcula- tion of the specific heat capacity of the equilibrium mixture may be carried out by differentiating the energy equation [346] of Gibbs with respect to temperature at constant volume. VI. On the Vapor-densities of Peroxide of Nitrogen, Formic Acid, Acetic Acid, and Perchloride of Phosphorus (Gihhs, /, 373-403) This section comprises material examined with a view to demonstrating the applicability of [309] or (114) [309]. Since 1879 a quantity of new density data for these substances has appeared, but no new facts or inferences can be gleaned by repeating Gibbs' treatment. In the case of the N2O4 —> 2NO2 reaction Verhoek and Daniels' work, already referred to, has shown that the perfect gas laws are not sufficiently valid to 392 KEYES ART. J warrant attempting a refined correlation on the usual basis. There is no doubt whatever that the same statement will hold true for the other gases or vapors listed in the heading of the section. REFERENCES (1) Dalton, Mem. Lit. and Phil. Soc. of Manchester, 5, 595, (1802). (2) Gay-Lussac, Annates de chimie, 43, 137, (1802). (3) Journ. de phys., 53, 58, (1811). (4) Henri Victor Regnault, Ann. chitn. phys., 4, 5, (1842). (5) For example, P. Chappuis, Archives des sciences (Geneve), 20, 5, 153, 248, (1888). (6) Louis J. Gillespie, Phys. Rev., 36, 121, (1930). (7) J. R. Roebuck, Proc. Am. Acad, of Sci., 60, 537, (1925); 64, 287, (1930). (8) W. E. Deming and Lola Schupe, Phys. Rev., 37, 638, (1931). (9) G. TuNELL, Journ. Phys. Chem., 35, 2885, (1931). (10) Collected Works of J. W. Gibbs, Vol. II, Part I, Statistical Me- chanics, Longmans Green & Co. (11) F. G. Keyes, Chem. Rev., 6, 175, (1929). (12) Heitler and London, Zeit. f. Phys., 44, 455, (1927). (13) Sugiura, Phil. Mag., 4, 498, (1927). (14) J. C. Slater, Phys. Rev., 32, 349, (1928). (15) Eisenschitz and London, Zeit. f. Phys., 60, 491, (1930). (16) London, Zeit.f. Phys., 63, 245, (1930). (17) Slater and Kirkwood, Phys. Rev., 37, 682, (1931). (18) Kirkwood and Keyes, Phijs. Rev., 37, 832, (1931). (19) P. Debye, Polar Molecules, Chem. Catalog Co., p. 40, (1929). (20) F. G. Keyes, /. Am. Chem. Soc, 49, 1393, (1927). (21) L. J. Gillespie and J. A. Beattie, Phys. Rev., 36, 743, (1930). (22) L. J. Gillespie, /. Am. Chem. Soc, 48, 28, (1926). (23) L. J. Gillespie and J. A. Beattie, Phys. Rev., 36, 1008, (1930); 37, 655, (1931). (24) Lewis and Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill Book Co., New York, 1923. (25) Int. Crit. Tables, 5, 84, (1929). (26) R. C. ToLMAN, Statistical Mechanics with Applications to Physics and Chemistry, Chem. Cat. Co., New York, p. 138. (27) R. H. Fowler, Statistical Mechanics, Camb. Univ. Press, 1929, p. 144. (28) Lewis and Randall, Thermodynamics, Chapter 30. (29) L. J. Gillespie, Proc. Amer. Acad. Arts and Sci., 66, 153, (1930). (30) Nernst, Die theoretischen und experimentellen Grundlagen des neuen Wdrmesatzes, 1918. (31) Egerton, Phil. Mag., 39, 1, (1920); Proc. Phys. Soc London, 37, 75, (1925). FUNDAMENTAL EQUATIONS OF IDEAL GASES 393 (32) Edmonson and Egerton, Proc. Roy. Soc, 113, 533, (1927), (33) Zeidler, Zeit. physikal. Chem., 123, 383, (1926). (34) Etjcken and Fried, Zeit.f. Phys., 29, 36, (1924). Eucken, Kar- MAN AND Fried, ibid., 29, 1, (1924). (35) WoHL, Zeit. Elektrochem., 30, 37, (1924). (36) J. A. Beattie, Phijs. Rev., 31, 680, (1928); 36, 132, (1930). (37) J. H. Jeans, Dynamical Theory of Gases, Camb. Univ. Press, 1921. (38) F. G. Keyes and H. G. Burks, /. Am. Chem. Soc., 50, 1100, (1928). (39) Lewis and Randall, Thermodynamics, pp. 191, 226. (40) Gibson and Sosnick, /. Ajh. Chem. Soc, 49, 2172, (1927). (41) Merz and Whittaker, /. Am. Chem. Soc, 50, 1522, (1928). (42) L. J. Gillespie, Phys. Rev., 34, 1605, (1929). (43) Lewis and Randall, Thermodynamics , p. 232. (44) Wroblewski, Wied. Ann., 18, 302, (1883) ; and Winkelmann, Hand- buch der Physik, Vol. 1, p. 1513, (1908). Sander, Zeit. phys- ikal. Chem., 78, 513, (1911). (45) Larson and Black, /. Am. Chem. Soc, 47, 1015, (1925). (46) Pollitzer and Strebel, Zeit. physikal. Chem., 110, 768, (1924). (47) L. J. Gillespie, Phijs. Rev., 34, 352, 1605, (1929). (48) G. Van Lerberghe, Bull, de I'acad. roy. Belgique, 14, 349, (1928). (49) G. N. Lewis, Proc Amer. Acad., 37, 49, (1901). (50) Lewis and Randall, Thermodynamics, p. 190. (51) L. J. Gillespie, /. Am. Chem. Soc, 47, 305, (1925); 48, 28, (1926). De Donder, Comptes rendus, 22, 1922, (1925) . (52) G. Van Lerberghe, Comptes rendus, 181, 851, (1925). (53) G. van Lerberghe, Bull, de I'acad. roy. de Belgique, 14, 349 (1928); 15, 488 (1929); 16, 94 (1930); Calcul des affinites physico-chim- iques, Gauthier-Villars, Paris, (1931). G. van Lerberghe and G. Schodls, Bull, de I'acad. roy. de Belgique, 15, 1 (1929). (54) M. Planck, Treatise on Thermodynainics, Trans. Alex. Ogg, Longmans Green, p. 225 (1903). (55) Kramers, Proc. Acad. Sci. Amsterdam, 30, 145, (1927). See J. H. Hildebrand, Solubility, Chem. Cat. Co., New York, (1924). (56) MiLNER, Phil. Mag., 23, 551, (1912); 25, 742, (1913). (57) Debye AND HtJcKEL, Phys. Zeit., 24, 185, (1923). . (58) Gronwall, Proc. Nat. Acad. Sc, 13, 198, (1927). (59) Gronwall, La Mer, Sandved, Phxjs. Zeit., 29, 358, (1928). (60) KiSTiAKOWSKY AND RicHARDS, /. Aju. Chem. Soc, 52, 4661 (1930); Richards and Reis, /. Chem. Physics, 1, 114, 737, 863, (1933). C. E. Teeter, /. Am. Chem. Soc, 54, 4111 (1932); /. Chein. Physics, 1, 251, (1933). (61) Verhoek and F. Daniels, /. Am. Chem. Soc, 53, 1250, (1931). (62) M. Bodenstein and M. Katayama, Zeit. Elektrochem., 15, 244, (1909); M. Bodenstein, Zeit. physikal. Chem., 100, 69, (1922). (63) A. CoLSON, Comptes rendus, 154, 428, (1912). (64) F. G. Keyes, Am. Soc Refrig. Eng., 1, 9, (1914). K THE THERMODYNAMICS OF STRAINED ELASTIC SOLIDS The Conditions of Internal and External Equilib- rium FOR Solids in Contact with Fluids with Regard to all Possible States of Strain of the Solids [Gibbs, I, pp. m-218] JAMES RICE Note. In order to follow this part of Gibbs' work the reader must know Bomething about the mathematical treatment of the relations which exist between the stresses set up in an elastic medium bj the action of external forces on it, and the strains which accompany these stresses. In the study of the thermodynamics of these media, such relations take the place of the equation of state in the thermodynamics of a fluid medium. The treatment of Gibbs is formally somewhat more compli- cated than that usually employed, by reason of his desire at the outset to make use of two sets of axes of reference which need not be regarded as identical, although they are similar, i.e., capable of superposition (p. 185). It will therefore be advisable to deal with these matters in a less complicated manner at first. In consequence we shall have to prefix to the commentary proper a rather long exposition of the analy- sis of strain and stress, with some account of the thermodynamics of a single strained body. I. Exposition of Elastic Solid Theory So Far As Needed for Following Gibbs' Treatment of the Contact of Fluids and Solids 1. Analysis of Strain. When a body is deformed or strained, its parts undergo a change of relative position. In order to deal with this in the classical mathematical way, we conceive the body to be constituted of particles each of which has in any assigned state of strain definite coordinates with regard to assigned axes of reference; and yet we compromise with these 395 396 RICE ART. K notions of molecular structure and also conceive that the material of the body is "smoothed out" to become a continuous medium. We picture a "physically small" element of the body around a particle, i.e., an element of volume small enough to be beyond our powers of handling experimentally and yet large enough to contain a very great number of molecules; the quotient of the mass of the molecules contained within this element by its volume being regarded as the density at the point. If a body is strained, obviously some of its particles must be displaced from the position previously occupied in the system of reference. Yet displacement may not produce strain. Clearly there is no strain if each particle receives a displacement equal in magnitude and direction to that to which all the other particles are subject. Again a simple rotation, or a motion compounded of a simple translation and a simple rotation, will produce no strain. In short, strain involves not only displace- ment but also a difference of displacement for neighboring particles (which is not compatible with a simple rotation), and the business of the mathematician is to determine the most convenient mathematical way of stating how this difference of displacement varies for two neighboring particles P and Q supposing that one of them, P, is kept in mind all the time while the other one, Q, is conceived to be in turn any one of the other particles in an element of volume around P. If this statement when formulated turns out to be quantitatively the same for all the elements of volume, we call the strain "homogeneous;" otherwise it is "heterogeneous." We will consider (with Gibbs) that the body is first in a "com- pletely determined state of strain," which we shall call the ^' state of reference." Let P' be the position of a point or particle of the body in this state. It is then strained from this state, and we denote by P the position of the same particle. Consider another particle, near to the former, whose position in the state of reference is Q' and after the strain is Q. The mathematical formulation of the nature of this strain will summarize all the essential information concerning the elongation of the element of length P'Q' and also its change of orientation when it is dis- STRAINED ELASTIC SOLIDS 397 placed to PQ, and this for all possible positions of Q' in the neighborhood of P'; and this again, if the strain is heterogene- ous, for all possible positions of P' in the body. The use of the words "homogeneous" and "heterogeneous" in connection with strain must not lead to confusion with their use as referring to substances. A homogeneous material may- very readily be subjected to a heterogeneous strain, as will appear presently. It is as well also at this point to reahze what is meant by an elastically isotropic material as distinct from one which is elastically anisotropic (or aeolotropic). Thus we suppose that the body is deformed from its state of reference by a completely defined set of external forces acting on each element of volume (gravitational, for example; or definite mechanical pulls applied to definite elements of volume in the periphery of the body). Each element of length P'Q' in the body is subject to a definite change in length and direction. Suppose now that all the external forces remain unchanged in magnitude but all are changed by the same amount in direction, then the strain in the linear element P'Q', i.e., its change in magnitude and direction from the state of reference, will not in general remain as before; but if the body is isotropic a linear element P'R' which bears the same relation of direction to the directionally changed forces as did P'Q' to the external forces formerly, will experience the same strain as that to which P'Q' was subject in the first case. But for an anisotropic (crystal- line) body even this statement is not in general true. These definitions in general terms will be more clearly stated in precise mathematical form presently; but the fact mentioned embodies the essence of the distinction between anisotropy and isotropy. Before proceeding to a general mathematical treatment of strain it may be advisable to consider one or two special cases where there are certain simplifying conditions. Imagine for example that all points are displaced in one direction, parallel to the axis OX' say, and that the displacement of the point P'ix', y', z') is a function of x' only. Representing this dis- placement by u{x') (or briefly by u), we have X = x' -{■ u{x'), y = y', z = z'y 398 RICE ART. K where the coordinates of the point P after the strain are x, y, z. Let Q' be a point adjacent to P' whose coordinates in the state of reference are x' + ^', y' , z'\ the coordinates of Q, i.e., the position after the strain, are a;' + £' + u{x' + r), 2/, 2, where u{x' + ^') is the same function of the argument x' + ^ that w(x') is of x' . Hence the hnear element P'Q' has been altered from a length |' to a length ^ + u{x' + ^') — w(a;'), besides of course experiencing a bodily translation which is of no importance in discussing the strain. Thus the alteration in length of the linear element is u{x' + r) - u{x'), which by Taylor's theorem is equal to du ^ ^ d^u „/ ^ + 2 j„/2 ^ "T If the differential coefficient du/dx' does not vary in value appreciably over a range within which we choose the value of ^', we may neglect the terms in ^'^ etc. (Thus if P'Q' is a range of length extending over a few molecules in the actual body this proviso is the same as that referred to by Gibbs on page 185, line 20.) Under these circumstances the length of P'Q', viz., ^', is altered to ^' (1 + du/dx'), and hence du/dx' is the fraction of elongation of the body at P', viz., the ratio of the change in length to the original length. Gibbs in his discussion actually uses the differential coefficient dx/dx', but it is readily seen that this is just 1 + du/dx', i.e., the ratio of elongation, or the "variation" of the length in the strict meaning of "variation," viz., the ratio of the varied value of a quantity to its previous value. If u(x') is a linear function of x' so that du/dx' is con- stant over the whole body, the elongation has the same value everywhere, and the strain is homogeneous. Otherwise du/dx' varies from element to element of the body, and is in fact a function of x' itself, so that the value of du/dx' depends on where the point P' of the element is situated in the body, STRAINED ELASTIC SOLIDS 399 and the strain is "heterogeneous." Nevertheless, on account of the proviso mentioned above, we can regard the strain as being homogeneous throughout any assigned physically small element of volume. If the length actually contracts, the extension du/dx' is negative. As another simple example consider again the case in which all particles are displaced parallel to OX', but now taking the displacement to be a function of y', the distance of the particle from a plane parallel to which the displacement takes place. Now choose Q', the neighbor of P', to be a point such that P'Q' is perpendicular to the direction of the displacements. M Q 0 Fig. 1 Thus if x', y', z' are the coordinates of P' and x, y, z are the coordinates of its displaced position P, X = x' + u(y'), y = y', z = z'. Also if x', y' + t]', z' are the coordinates of the undisplaced position Q' of the "neighbor," its displaced coordinates are x' + u{y' + 7,0, y' + -n', z'- The displacement P'P is u{y') and the displacement Q'Q is u{y' + r]') or u{y') + (du/dy'W. Hence MQ in Fig. 1 is 400 RICE ART. K {du/dy')r}' and the angle QPM has for its trigonometrical tangent the value du/dy'. The figure shows that this strain is what is called a "shear." A bar shaped element of volume which is extended parallel to the axis OZ' (perpendicular to the plane of the paper) and whose section by the plane OX'Y' is P'Q'R'S' (Fig. 2), is displaced to a position whose section is PQRS. This is equivalent to a simple displacement of the bar as a whole from P'Q'R'S' to PMNS and a real strain or change of shape from PMNS to PQRS. This latter is the "shear" and its magnitude is measured by the tangent of the angle QPM (or simply by the angle itself when the strain is so small that the tangent of the angle and its radian measure are practically identical), i.e., by du/dy'. If w is a linear function of y', the O'MO R'NR Fig. 2 shear is homogeneous throughout the body; otherwise it is heterogeneous and the amount of shearing varies from point to point of the body. When we undertake a general analysis of strain these special cases give us a hint how to proceed. The point P' whose co- ordinates are x', y', z' experiences a displacement whose com- ponents we represent by u{x' ^ y', z'), v{x', y', z'), w{x', y', z'), for the displacement must have some functional relationship with the position of P' if analysis is to be possible at all.* * Will the reader please note that we are, for the time being, referring the body before and after the strain to the same axes OX', OY', OZ'. Formally Gibbs' procedure is a little wider since he refers the body after STRAINED ELASTIC SOLIDS 401 Hence the coordinates of the point in its displaced position, viz., P", are given by x" =x' + u{x', y', z'), y" = y' + v{x' , y' , z'), z" = 2' + w{x', y', z'). (1) Consider a neighboring point whose undisplaced position is Q' with the coordinates x' + r, y' + V, z' + r. After the displacement, the coordinates (of Q") are a:' + r + u{x' + r, y' + V, 2' + r), and two similar expressions. Neglecting as before and for the same reason the differential coefficients higher than the first, these become x" + ^", y" + r,", z" + f ", where du , du , du , dv , dv , dv , dw dw dw , (2) (For convenience and brevity we drop the bracketed coordinates after the symbols u, v, w; but it must not be forgotten that u is to be understood as the function u{x', y', z'), etc). It will be convenient to introduce single letter symbols to the strain to a different set of axes OX, OY, OZ. The two sets of axes are not necessarily identical, but he regards them as "similar, i.e., capable of superposition" ; so that if one set is orthogonal, then also is the other. At the outset, however, there is an element of simplification in keeping the same set of axes; but in order that there may be no confusion later when we adopt Gibbs' wider analysis we are now referring to the co- ordinates of the displaced point as x", y", z" instead of x, xj, z, thus keeping the latter triad of letters to represent, as Gibbs does, the coor- dinates of the displaced point with reference to a second system of axes. 402 RICE ART. K replace the differential coefficients, so we shall write these equations as r' = enr + eW + Cut',] v" = 621^ + 6227?' + e23r,(' (3) r" = 631^ + 632^' + e33f'J where* du ^^^ - ^ + dx' - dx" dx' du en - \ f - dy dx" dy' du 6l3 - „ , dz dv dy" ''' - dx' = dx'' 622 dv dy dy" dy' dv ''' ~ dz' dw dz" ''' = dx'^ dx'' 632 dw dz" dy' dy' 633 dw dx" ^ dz'' djT dz'' dz^ dz'' (4) 2. Homogeneous Strain. In order to grasp most readily the physical interpretation of these "strain coefficients" which are denoted by the symbols e„, let us consider the case in which u, V, w (and therefore x", y", z") are linear functions of x' , y', z'. Under such a limitation, the quantities e^ are uniform in value throughout the body; in other words the strain is homogeneous. Now it is very important to remember at this juncture that it is not so much the actual displacements of the various points which determine the strain, as the differences between the dis- placements of the various points. In Fig. 3 P' is displaced to P" and Q' to Q" ; but to obtain a clear idea of the strain in the part of the body surrounding P' , we must imagine the whole body translated without change of shape and without rotation, i.e., as a rigid body, so as to bring the point P" back to its * We are of course using the well-known notation of the "curly" d for partial differentiation. When Gibbs wrote his paper this device for indicating a partial differential coefficient had not established itself universally, and many writers used the ordinary italic d to indicate total and partial differential alike, relying on the reader's own knowledge to make the necessary distinction in each situation. But as, of course, the differential coefficients in [354] and in subsequent equations are partial, we venture to make this small change in Gibbs' notation in view of the universal practice adopted in these matters nowadays. STRAINED ELASTIC SOLIDS 403 former position P'. This will bring the point Q" to R", where Q"R" is parallel and equal to P"P'. The magnitude and direction of the line Q'R" is the vector which, when estimated for all Q' points in the neighborhood of P', would give us the necessary information for calculating the strain. Now the components of the vector length Q'R", the "differential dis- placement" of Q' with reference to P', are ^" — ^', rj" — 7/', ^" — f ' and are therefore equal to the expressions (en - l)r + e:2rj' + e^t',^ 621^' + (622 - 1)^7' + e23r',[ (5) 631^' + 63271' + (633 - l)i'',, which are linear functions of ^', t]' , f ' if en, 612, 613, ... 633 are constants. H P' P Fig. 3 Let us impose for a moment a simplifying condition with regard to these nine strain constants and assume that 612 = 621, 623 = 632, esi = ei3. It will be very convenient for a moment to write a for en — \,h for 622 — 1, c for 633 — 1, / for 623 or 632, g for 631 or en, h for 612 or 621. Thus r' - r = ar + h' + gf',1 r," -v' = H' + hr,' -^n',} (6) Taking P' as a local origin, and axes of reference through P' parallel to OX', OY', OZ' ("local axes" at P'), let us suppose the family of similar and similarly placed quadric surfaces con- 404 RICE ART. K I structed, which are represented, in the "local" coordinates ^', r\ , f ', by the equation where fc is a constant which has a definite value for each member of the family. One member of this family will pass through Q' and, if we recall the statements made concerning quadric sur- faces in the author's Mathematical Note (this volume. Article B, p. 15), it will be seen by reference to (6) that the dif- ferential displacement Q'R" of the point Q! is normal to this surface at this point. The result of this will be that points originally on a straight line will still lie on a straight line after the strain. (The expressions in (6) are linear in ^ , t]' , f '.) But in general the angle between two lines will be altered in value; in particular two lines at right angles to each other before the strain will not be at right angles after it. However, there is an exception to this general statement. There are three mutually orthogonal directions and any lines which are parallel to these before the strain remain at right angles to each other after the strain. These directions are in fact the directions of the three principal axes of the quadric surface; for if Q! is on one of these, then, since Q'W is normal to the surface at Q', R" is on the axis too, and the lines F'Q! and F'R" are coincident. But by construction F"Q," is parallel to P'R"; therefore it is parallel to P'Q'. Hence the three principal axes are displaced into three lines parallel to them respectively, and so are at right angles to each other as before. To prove this we apparently had to restrict our reasoning by assuming that 623 = ez2, etc. We can remove this restriction however and still arrive at the same result. To show this we must resort to a simple artifice. Take the first expression in (5), and treat it thus: {en - 1) r + e,W + eisf ' = (eu - D ^' + 612 + 621 V I ^31 + ei3 , 612 — 621 , 631 — en , 2 631 — ei3 2 ei2 — 621 STRAINED ELASTIC SOLIDS 405 Treat the remaining two in a similar fashion and for temporary convenience put . , , , g23 + 632 - 623 — 632 a for 611 — 1, / for — r ' p for 6 for 622 — 1, <7for r — > g for C ^ -LC ^'2 + ^21 , c for 633 — 1, Ai for — - — ' r for We then have r' - r = ar + u + ^r' + r-n' - gr',1 77" - V = h^ + 6V +/r + pf - rrl (7) r" - f ' = g^ + /V + cf ' + q^ - pv'.j If we take the first three terms on the right hand side of each equation in (7), it is clear that they represent, as before, a differ- ential displacement which at each point is normal to the corre- sponding member of a family of similar quadric surfaces. As we have seen, this part of the whole differential displacement still leaves three certain lines orthogonal and unaffected in direction. Now consider the last two terms. They represent a displace- ment due to a small rotation about a line whose direction cosines are proportional to p, q, r. This is readily seen by observing that Virr)' - qn + q{p^' - r^ + r{q^' - Pv') = 0 and ^'(rv' - qn + l(pt' - rn + ^'(q^' " pV) = 0; thus the small displacement of which the components are '''v' — Qt'> P^' — f^', q^' — PV, is at right angles not only to the line whose direction cosines are proportional to p, q, r, but also to the line P'Q', whose direction cosines are of course propor- tional to ^', 7/', f'. But a rotation does not disturb the angles between two lines. Hence the result follows as before, so that there are in every case of a small strain three particular lines. 406 RICE ART. K the so called "principal axes of strain," which are not only mutually orthogonal before the strain, but remain so after it, although in general they are not pointing in the same directions after as before. This is a result used by Gibbs and demon- strated by him in a different manner (Gibbs, I, 205 et seq). On page 204 also occurs the sentence: "We have already had occasion to remark that the state of strain of an element con- sidered without reference to directions in space is capable of only six independent variations." This remark is illustrated by the result which we have just obtained, since although there are nine strain-coefficients, the strain, apart from the rotation which produces no relative displacement of neighboring parts, depends on the six quantities €l\, 622, 633, ^23 + ^32 631 + ei3 ei2 + 621. , , Gibbs then continues: "Hence it must be possible to express the state of strain of an element by six functions of dx/dx', . . . dz/dz', which are independent of the position of the element." The functions chosen by Gibbs are not so formally simple as those written above and have a certain appearance of arbitra- riness about them. So we will address ourselves to the task of explaining how the six functions defined in [418] and [419] naturally arise in a further discussion of strain. Indeed, the whole of the material treated in Gibbs, I, 205-211 may prove troublesome to follow without some help over analytical difficulties, which will now be given. The treatment which follows will present the matter from a somewhat different angle and at the same time bring out the physical nature of the er» coefficients. Let us revert to equations (3) and use them to determine the length of P"Q" as a function of the local coordinates of Q', the original position of Q", with reference to the axes through P', the original position of P". It is easy to see that p"Q" = r" + v'" + r' = e,^" + e^v" + esf'^ + 2e,v't + 2e,^'^' + 266^^?', (8) STRAINED ELASTIC SOLIDS 407 where ei = en^ + 621^ + e3l^ 62 = 612^ + 622^ + e32S 63 = ei3^ + 623^ + essS 64 = 612613 + 622623 + 632633, 6b = 613611 + 623621 + 633631, 66 = 611612 + 621622 + 631632. (9) Choose for the moment a special case, letting the point Q' be placed on the local axis of x' at P', so that its local coordinates are ^', 0, 0. It follows from (8) that P"Q" = ei^" = eiP'Q' • Thus (61) i is the "ratio of elongation" parallel to OX', and (62)^ and (63)* can be interpreted in a similar manner. It was men- tioned above that two lines at right angles to each other before the strain will not remain so after it. We shall show how this fact is connected with the 64, 65, e^ quantities. For let us con- sider Q' to be a point in the local plane of x' y' at P', its local coordinates being ^' , r\ , 0. Drop perpendiculars Q'M' , Q'N' on the local axes of x' and y' at P'. Let Q", M", N" be the posi- tions of these points after the strain. From the result obtained just above ,2 P"M" = eiP'M'^, P"N"^" = e2P'N'\ From (8) we obtain PW' = ei^' + 6277'2 + 2e,^'v', and so (6162)^ But by the application of elementary trigonometry to the parallelogram P"M"Q"N" P"Q" = p"M" + P"N" + 2P"M"-P"N"-co& {N"P"M"). 408 Hence RICE cos {N"P"M") = 66 (6162)" AET. K (10) and similar results can be obtained for the other pairs of axes. A glance at Fig. 4 shows that the rectangle P'M'Q'N' has suffered a shear to the shape P"M"Q"N". (It is in general also subject to a rotation.) The shear is measured by the angle L"P"N" whose sine is by equation (10) equal to 66/(^162)^ If the strains are sufficiently small we can simplify this. Recalhng the original definitions of the era coefficients in (4), we see that ^11 — 1, 622 — 1, 633 — 1, 623, 632, 631, ei3. 612, 621 X' Fig. 4 are small compared to unity if the relative displacement of two points is a small fraction of their distance apart. Hence, by (9), ei, 62, ez each differ from unity by a small amount. Also in the definition of ee the third term is the product of two small quantities, the second term differs from 621 by a small fraction of 621, and the first term differs from e^ by a small fraction of 612. Thus, apart from a neghgible error, the sine of L"P"N" is equal to 612 + 621. The angle being also small in this case, its value, that is the shear of the lines originally parallel to OX' and OY', is practically 612 + 621," this in fact measures very closely the amount by which the angle between these lines has changed STRAINED ELASTIC SOLIDS 409 from a right angle. The shears of lines parallel originally to the axes OY' and OZ', and of those parallel to the axes OZ' and OX', are likewise given to a close approximation by 64 and e&, re- spectively, or practically 623 + 632 and esi + 613. Now we know that there is one set of axes of reference, for which there is no shear. Suppose we had chosen them at the outset and carried through the analysis just finished, then three of six strain functions calculated as in (9) would be zero, viz. the three indicated by the suffixes 4, 5, 6, To make this as definite as possible let us indicate these three principal axes of strain by OL', OM', ON', and let the coordinates of Q', relative to three local axes through P' parallel to these, be denoted by the letters X', ij.', v'. We should arrive at a result similar to (8) viz., ,2 P"Q'> = e,x'2 + e2^'2 +63 /2 + 2un'v' + 265/X' +2e,\'n', where ei, €2, cs, etc., would be six strain functions such that (ei)i would be the ratio of elongation parallel to OL', etc., and also such that the cosine of the angle between two lines originally parallel to OL' and OM' would be ee/Ceieo)*. But as this angle still remains a right angle, ee would have to be zero and simi- larly for €4 and €5. Hence we would arrive at the result '2 P"Q"" = e{K" + 62^'^ + e^v In his discussion Gibbs indicates the three "principal ratios of elongation" by the letters n, r2, n, so that his notation and ours are connected by ei = ri^, €2 = ra^, ea = n'^. Certain relations, very necessary to our progress, between the €r and the e^ symbols can now be obtained very elegantly by an artifice depending on a theorem concerning quadric surfaces quoted in the Mathematical Note. Keeping P' as our local origin, allow Q' to move about on a locus of such a nature that the corresponding positions of Q" lie on a sphere of radius h around P" as centre. By (8) we see that the equation of this locus in the ^', 77', f ' coordinates is eir^ + eov" + e3f'- + 2eW^' + 2e,^'^' + 2e,^'rj' = h\ 410 RICE ART. K It is an ellipsoid, and its position in the body is entirely independ- ent of what axes of reference we choose. So the same surface referred to the principal axes as axes of coordinates has the equation By a theorem on quadric surfaces quoted in the Mathematical Note, observing that ^', T]' f ' correspond to x, y, z in the note, X', n', v' correspond to x' , y', z' in the note, ei, 62, 63, 64, 65, 66 correspond to a, 6, c, /, g, h in the note, ei, €2, €3 correspond to a', b', c' in the note, we arrive at these three results: 6263 + 6361 + 6162 61 + 62 + 63 = €1 + C2 + f3, 64- — 66^ — 66^ = 6263 + €361 + eie2, 61 66 65 66 62 64 = cicaes. 6b 64 63 } (11) Now let the reader look at the equations (9) which give 6i, 62, etc., in terms of the squares and products of the Crs coefficients, and refer to the well-known rule for multiplying determinants which will be found in any text of algebra. He will find that the determinant in (11) is the square of the determinant 611 612 613 621 622 623 631 632 633 (12) Thus the last of the equations in (11), on extracting the square root, is equivalent to 611 612 613 621 622 623 631 632 633 = rir2r3. (13) which is essentially equation [442], the third equation of (11) STRAINED ELASTIC SOLIDS 411 being essentially the third equation of [439]. Our equations dif- fer from those of Gibbs in the greater generality which he adopts concerning axes of reference before and after strain. But this restriction we shall be able to eliminate presently, with no great trouble. In the meantime let us continue with the other two equations in (11). A glance at (9) shows that the first is just en^ + ei2^ + ei3^ + 621^ + 622^ + ^23^ + eai^ + 632^ + 633^ = ri2 + Ti" + n\ (14) The second of (11) gives a little more trouble; but the reader may take it on faith, if he does not care to go through the straightforward algebraic operations, that the following result can be verified. If one squares the nine first minors of the determinant (12) and adds them then the sum is equal to €263 + 6361 + 61^2 — €4"^ — 65^ — e^,^. (A less tedious method of showing this would have involved us rather too deeply in the theory of determinants.) Hence, by the second equation of (11), En' + £"22' + i?33' + £"21' + £"22' + Eiz' + En' + ^32^ + £33' = raVs^ + nW + nW, (15) where we are representing the first minor of en in the determi- nant of the ers by En, that of 612 by £'12, and so on. (The use of this double suffix notation is obviously of great convenience at the moment. The Ers used here must not be confused by the reader with the symbol E used by Gibbs without any suffix, to which we will be referring presently.) Equations (14) and (15) are essentially the first two of the equations [439]. If we consider a rectangular parallelopiped whose sides are parallel to the principal axes and each of unit length, we know that it remains a parallelopiped after the strain (although it may be rotated) and its sides become n, r^, rs, respectively. Hence nriTs is the ratio of enlargement of volume, and so we see that this is a physical interpretation of the determinant (12), while the determinant in (11) is of course equal to the square of that ratio. Further, the sum of the squares of the nine first 412 RICE ART. K minors of (12) is equal to the sum of the squares of the ratios of enlargement of three bounded plane surfaces, respectively- parallel to the three principal planes of the strain. Of course the sum of the squares of the nine Crs coefficients is equal to the sum of the squares of the three principal ratios of elongation. The interpretation of these results in terms of ratios of en- largement is of some importance. Equation (13), which is really the third equation of (11), is an especially useful result and is involved in Gibbs' equation [464]. The first equation of (11) is perhaps the least important of the three for our purpose, but the second result in the form of equation (15) plays a part at one or two points of Gibbs' treatment, e.g., at equation [463] and still earlier on pages 192, 193. It will be well to pause a moment to consider the geometrical significance of the nine minor determinants £"11, £"12, etc. To this end let us imagine a triangle P'Qi'Qi in the unstrained state such that the local coordinates of Qi, Q2, with reference to the local axes at P', are ^i, r;/, n' and y, 772', ^2- After the strain the triangle will assume the position P"Qi"Q2". If ki", Vi", Ti" and ^2", V2", h" are the co- ordinates of Qi" and Q2" with reference to local axes at P" parallel to the original axes we have by (3) the following rela- tions: ki" = en^i' + enm' + eM, y = eM + 612^72' + ei3f2',] Vl" = €21^/ + 6227?/ + 623^/, 772" = 621^2' + e22r?2' + ^23^2', \ (16) fi" = 631^/ + 63217/ + e33fi', h" = ez^y + e32i?2' + 633^2'.] Denote the area of the triangle P'Qi'Qi by K' and that of P"Qi"Q2" by K". The projection of the triangle P'Qx'Qi' on the local plane of reference perpendicular to the axis of x' is a triangle whose corners have the 77, f coordinates 0, 0; 7?/, f/; 772', ^2'. By a well known rule its area is livi^i — f]2^i), and similar expressions hold for other projections. Now the area of a projection is equal to the product of the projected area and the cosine of the angle between the original plane and the plane of the projection, which is the angle between the normals to the planes. So if a, /3', 7' are the direction cosines of the normal STRAINED ELASTIC SOLIDS 413 to the plane of P'QiQi, and a", /3", 7" those of the normal to the plane of P"Qi"Q2", we have the following results: K'a' = KVf/ - 172'f/), K"a" = Km"r2" - ^2"h"l K'p' = Kf:'^2' - f2'^/), K"^" = Kfi"e/' - r2"^/'),[ (17) K't' = Ka'ri2' - ^2'm'), i^"7" = m"V2" - ^2"vn.j If one now uses equations (16), and is careful to keep to the convention about the signs of the first minors as explained in the note, it is not very troublesome to prove that m"^2" - W'^x" = Enim'h' - ^2'fi') + ^i2(fi'^2' - r2'^/) + £'13(^/^2' - ^2'm'), and two similar results which can be succinctly written K"cx" = K'(Ena' + E,ol3' + Enl'V, K"fi" = K'iEW + ^22/3' + EnV),\ (18) K"y" = K'(Ez,a' + ^32/3' + ^33t').. These are essentially the steps by which one passes from equation [381] to equation [382], K' and K" being the Ds' and Ds" of Gibbs. (There is of course at the moment some restriction on our Brs and E^ symbols, i.e., our differential coefficients and the determinants constructed from them, due to our restriction as to the axes chosen in the strained system; we have already referred to this and it will be removed shortly; for the moment it involves us in the use of doubly accented symbols such as ^", K", a", etc., so as to avoid confusion later when we widen our choice of axes.) The interpretation of the quantities En as determining super- ficial enlargement caused by the strain is very clearly indicated in (18), and a very elegant analogy can be exhibited between equations (18) and the equations (3) in which the ers quantities obviously determine finear enlargement. To this end we remind ourselves that an oriented plane area is a vector quantity, and is therefore representable by a point such that the radius vector to it is proportional to the area and is parallel to the normal. Thus the triangle P'QiQ/ can be represented in orientation and magnitude by a point whose coordinates are 414 RICE ART. K X', Y', Z' where X' = K'a', Y' = K'^', Z' = K'y'. Similarly a point whose coordinates are X", Y", Z", where X" = K"a" , etc., can represent the triangle P"Q]"Qi". The equations (18) can then be written X" = EnX' + EnY' + ^i3Z',1 Y" = EnX' + EnY' + EnZ',)- (19) Z' = EziX' -\- E32Y' + EzzZ' .^ The reader will probably feel intuitively that, as can be estab- lished by definite proof, by choosing the principal axes of strain as the axes of reference, we can reduce the nine coefficients to a form in which £'23 + £'32, -£^31 + E^, En + £'21 are zero, and En, E21, E33 become the principal ratios of superficial enlarge- ment, i.e., TiTs, rsn, viVi. Squaring and adding the equalities in (19) we obtain K"^ = EiX" + E2Y'^-\-EzZ'^-\-2EiY'Z' + 2E,Z'X'-\-2EeX'Y', where £1 = En' + £21^^ + £31^ and two similar equations, Ei = Eiibjiz ~r E^itjiz "T Ezitiizz and two similar equations.^ (20) An application of the theorem in the Mathematical Note already used would lead to the result that the value of Ei-\- E2-\- E3 is independent of the choice of axes (just as was ei + 62 + es in the discussion of equations (3) and its results). Since, with the choice of the principal axes of strain, the values of the Er, are as stated above, it follows that £1 + £2 + £3 = (r^ny + (nny + (nr^y, which is just equation (15). The details of the proof of these statements are not difficult to supply, but for our purpose it is the result (18) which is important. As a final step in the elucidation of Gibbs, I, pages 205-211 we shall now adopt Gibbs' plan of allowing the axes to which we refer the system in its strained state to be any set of STRAINED ELASTIC SOLIDS 415 orthogonal axes OX, OY, OZ, not necessarily coincident with OX', OY', OZ'. Referred to these axes the coordinates of P" are x, y, z and those of Q" are x -\- ^, y -{- tj, z -\- ^,so that the local coordinates of Q" in a set of local axes through P" parallel to OX, OY, OZ are ^, r/, f. The procedure now can be practi- cally copied from the previous pages. Let us use a symbolism similar to that employed above, and write dx dx «ii for — / ai2 for — ' ox ay , dy a,, for -' etc. Then we find that ^ = aii^' + aW + an^', ] V = «21^' + 0227?' + 023^', f r = 031^' + 03217' + flsar'- j (21) It follows that P"Q" - ail ' + a,r,'^ + ast' + 2airi'^' + 2a,t^' + 2a,^'n', (22) where Oi = au^ + ^21^ + a3l^ a2 = ai2^ + «22^ + a32^ 03 = ai3^ + 023^ + a33S . Gi = a.i2ai3 + 022^23 + «32a33, ( 06 = aisfln + O23O21 + ^33031, de = CinCl'12 ~\~ ^21^22 4" a3ia32. (23) Now although an, a^, a^, etc., are not respectively the same as Cii, ^12, 621 etc. (unless of course OX, OY, OZ should coincide with OX', OY', OZ'), nevertheless a comparison of (8) and (22), which are true for any values of ^', t] , ^' , shows that fli = ei, 02 = ei, az = €3, tti = Bi, In consequence of (11), therefore, 05 = 65, Oe = 66. Oi + 02 + 03 = ri^ + Ti^ + rs^, a^as + 0301 + aitti — 04^ — 05^ — Oe^ r2V3^ + rsVi^ + riV2^ Ol Oe 06 06 02 05 04 03 . (24) = r^T'^r^. 416 RICE ART, K «11 ai2 an «21 ^22 a23 flsi 032 a33 Just as before, we recognize that the determinant in (24) is the square of the determinant (25) and this is actually the determinant indicated by H in Gibbs, while the one in (24) is there indicated by G. Hence equations [437] and [442] are included in (24) and (25). We have been using a double suffix and single suffix notation as the most convenient to follow in this exposition and the most consis- tent with present day practice, but for comparison with Gibbs' treatment the reader will observe that A, B, C, a, h, c defined by him in [418] and [419] are respectively ai, a2, az, ai, as, a^ in this exposition. A glance shows that the first of equations (24) is the first of the equations [439]. The second of (24) is, as before, a little more troublesome to deal with by straightforward algebra, but it can be verified that the expression on the left hand side is the sum of the squares of the nine first minors of the determi- nant (25) . A similar notation for these minors can be introduced as before, viz., An for the minor of an, A^ for that of ai2, A^i for that of 021, etc. Thus the equations (24) can be written / J 2j ^ra^ = ^1^ + ^2^ + ^3^ r s au ^12 ^13 O21 Ct22 023 O31 O32 033 2 = r^r^r-^. (26) The left hand side of the first of these is the expression denoted by E in Gibbs; the expression on the left hand side of the second is referred to as F (see [432] and [434]), and, as already mentioned, H is used for the determinant in (25) and G for the determinant in (24). Thus equations (26) are just the STRAINED ELASTIC SOLIDS 417 set [439]. Again pursuing a line of argument such as led to (18) we obtain Ka = K'iAncx' + ^12/3' + A,,y'),] K^ = K'iA^icc' + A22^' + A2,7'),\ (27) Ky = K'iAncc' + ^32/3' + AW),] where K and a, j8, y are the area and direction cosines of the normal after the strain for a bounded plane surface (referred to OX, OY, OZ) whose area and direction cosines are given hy K', a, j8', y' in the unstrained state (referred to OX', OY', OZ'). As already stated these results are of importance on pages 192, 193 of Gibbs' discussion, 3. Heterogeneous Strain. In the discussion just completed X, y, z have been considered as linear functions of x' , y', z' , with the result that the Ors quantities (i.e., dx/dx', etc.) are constants throughout the system, and the same remark applies to the Ars quantities (viz., (dy/dy') (dz/dz') - (dy/dz') (dz/dy'), etc.). If, however, the displacements of the points from the un- strained to the strained states have such values that x, y, z are not linear functions of x', y', z', then the quantities denoted by Gts are functions of x', y', z' varying from point to point, and the same is true for the quantities denoted by Ars and also for the determinant denoted by the symbol H in Gibbs. (The flexure or the torsion of a bar are examples of heterogeneous strain.) As far as interpretation is concerned these functions still determine the various ratios of enlargement, with the understanding that the values of these functions at a given point give the necessary data for calculating the conditions of strain in a physically small element of volume surrounding the point. In short, we regard the strain as homogeneous through- out any physically small element of volume, giving the various Qri and Ars quantities the values throughout this element which they have at its central point. 4. Analysis of Stress. In using such a phrase as "the system in its unstrained state" we implicitly assume that we shall take this state as one in which the internal actions and reactions between any two parts of the body shall be regarded as vanish- ing. When we begin to consider if such actions are really zero, 418 RICE ART. K we are facing the very difficult physical problem of explaining by what mechanism such actions are exerted. We may imagine that an elastic medium is free from everything in the nature of external force, even gravity; we can hardly say, in view of the customary notions of molecules and intermolecular forces, that across the surface which separates two parts of the medium no forces are exerted. Therefore in using the word "stress" as a general term for the actions and reactions across dividing surfaces which accompany strain and vanish when the strain vanishes, we must regard stress as referring to change in the integral of the intermolecular forces exerted across some finite portion of such a surface, if we adopt a molecular theory of the constitution of matter. However, in thermodynamical reasoning we avoid the use of such conceptions, and we take it as a fundamental assumption, well backed by experience, that there is for any solid or fluid medium a condition of equilibrium to which the system can be brought which can be termed conventionally the unstrained state, and from which the medium can be strained by the application of external forces, this process giving rise to reciprocal internal forces across any conceptual surface dividing the medium into two parts. Of such external forces the most obvious example is gravity. This is sometimes referred to as a "body force," being proportional to the mass of each element of volume considered as pulled by the earth, moon, sun, etc. Other types of external forces are the thrusts on the surface of a body exerted by some liquid or gaseous medium surrounding it, or on certain parts of the surface by a solid body in contact with it. The pulls exerted by chains, ropes, etc., may be con- sidered as body forces exerted throughout small parts of the body; e.g., if a pull is exerted by means of a string fastened to a nail embedded in the body, we can regard the medium as actually existing throughout the small hole made by the nail, and a body force existing in that small volume. Or alterna- tively they might be regarded as surface pulls exerted across a definite small portion of the bounding surface of the body. If a body is electrified or magnetized the forces exerted by external magnets and conductors, charged or conveying current, are also external forces. Such external forces must be clearly dis- STRAINED ELASTIC SOLIDS 419 tinguished from the stresses which are occasioned by them. To give a definition of the "stress at a point," we must conceive a surface, on which the point Hes, dividing the body into two parts. We also conceive a small element of this surface sur- rounding this point. Of the total force which we imagine one portion of the body to exert on the other across this surface, a certain small part is considered to be exerted across this element and, when the element is small enough in size, to be practically proportional in magnitude to the area of the element and unchanged in direction as the element is made smaller and smaller. The quotient of this force by the area is assumed to have a limiting value as both are indefinitely diminished in magnitude. The reader is certainly acquainted with this con- ception in the case of liquids and gases; but in such a case there is a special simplification. For one thing the force is almost always in the nature of a thrust in a fluid medium; in a solid medium it may be a thrust or a pull. Moreover, in the case of a fluid at rest, the force is normal to the element of the con- ceptual surface. That is not in general the case for solid media. The limiting value of the quotient of force by area referred to above is called the stress across the surface at the point, and, as stated, it is not as a rule directed along the normal to the surface at the point. Another important distinction should be noted here. In the case of a fluid not only is the pressure always normal to the element, but it retains the same value as the element assumes different orientations. (If the reader has forgotten the proof of this it would do no harm if he refreshed his memory, as the proof involves some considerations of value to us presently). But in the case of a solid medium the stress generally alters in value, as well as direction, as the orientation of the element of surface is changed. In the technical language of the vector calculus, the stress is a vector function of the unit vector which is normal to the element and changes in magnitude and direction as the unit vector is turned to be in different directions. In the case of a fluid medium at rest one numerical magnitude is obviously all that is required to specffy the pressure at a point, and the physi- cal problems raised involve the functional dependence of this pres- 420 RICE ART. K sure on the position of the point. But for a sohd medium the conditions are more complex, and we must consider carefully- just how many numerical magnitudes must be given in order to specify the stress at a point, i.e., to indicate what is the stress at the point across any assigned element of surface. We shall see presently that there are six, and, as is readily suggested by the example of a fluid, each of these may vary in value with the position of the point, i.e., be a function of the coordinates of the point. The analysis of the stress at a point proceeds as follows. Consider the point P, the displaced position of a point P' in the unstrained state, and let its coordinates referred to the axes OX, OY, OZ (chosen for the strained state) be x, y, z* First let the conceptual dividing surface be parallel to OYZ, i.e., a plane surface at right angles to OX. We can resolve the postulated force across the element of area at P into three components parallel to the axes, and these when divided hy the area of the element we denote by Xx, Yx, Zx, the suffix indicating clearly that the plane surface under consideration is normal to OX. Xx is of the nature of a tension or pressure, while Yx and Zx are "shearing tractions," their directions lying in the dividing surface. Of course each of these in general varies in value with the position of P and so should strictly be written as ^x{x, y, z), Yx(x, y, z), Zx(x, y, z) to indicate their functional dependence on the values of x, y, z; however, for brevity, we drop the bracketed letters, but this point should never be lost sight of. By considering plane surfaces containing P normal to OF and OZ we can introduce components of the forces at P across these surfaces, when divided by the area of the element, as Xy, Yy, Zy and Xz, Yz, Zz. By the aid of these nine quantities we can now express the stress at P across any element of surface containing P whose direction cosines are given, say «, /3, 7. To do so, draw local axes at P (Fig. 5) and let a plane surface whose direction cosines * We may from this point onwards drop double accents in symbols for gtrained positions and coordinates as no longer necessary. STRAINED ELASTIC SOLIDS 421 are a, /?, 7 cut them in the points Q, R, S. Let K be the area of the triangle QRS; then Ka is the area of the triangle PRS, Kfi of PSQ and Ky of PQR. The portion of the medium within the tetrahedron PQRS is in equilibrium under the body forces on it and the stress actions on it across the four triangles mentioned. Let us enumerate the latter first. Parallel to OX we have a force across PRS of amount —KaXx. (We are assuming that Xx is positive if it is a tension, and negative if a pressure; also that the tetrahedron PQRS lies in the positive octant, i.e., the octant for which the local coordinates ^, tj, f are all positive). Also parallel to OX we have a force —K^Xy (a tangential shear- ing force) across PSQ, and across PQR a force —KyXz (also KaX^ 4 Y <, 9 KiaK^-^px^i-yX^) Fig. 5 shearing). In considering the equilibrium we can, if we gradually reduce the size of the tetrahedron, neglect the body forces on it in comparison with the surface forces just enumer- ated. The point involved is the same as that introduced in elementary treatises on hydrostatics when proving the uni- formity of fluid pressure in all directions, and will doubtless be known to the reader, or easily looked up. (Actually it only requires us to remember that the body forces involve the product of a finite quantity and the volume, while a surface action involves the product of a finite quantity and an area. As the size of the tetrahedron diminishes, the magnitude of the volume becomes very small in comparison with the magnitude of the surface, since the former involves the cube of a small 422 RICE ART. K length and the latter the square.) It follows that if equilibrium exists the component of force across the surface QRS parallel to OX is, for a small value of K, practically equal to K{aXx + pXr + yXz). The quotient of this force by the area K is the a:-component of the stress at P across the plane (a, /3, 7) (meaning the plane whose normal has these direction cosines). Similar results can be obtained for the other components, and we arrive at the result that the stress across the plane {a, /S, 7) has the components aXx + pXy + yXz, aYx + ^Yr + yVz, aZx + /3Zk + yZz. (28) We know that in fluid media in equilibrium the pressure varies with the depth owing to the action of gravity, and in general the pressure at a point varies with the position of the point when body forces are exerted on the fluid. The reader may be acquainted with the relation between the "gradient of the pressure" (i.e., the rate of variation of pressure per unit of dis- tance in a given direction) and the body force. It is dealt with in works on hydromechanics and is given by the equations dx dy dz where Fx, Fy, Fz are the components of the force F on unit volume of the fluid. Moreover, if at any point on the surface of the fluid there is an external force in the nature of a thrust or pull on the surface, and if F is the value of it per unit surface at the point, then the value of the pressure at that point of the surface is given by ap = -F,, /3p = -Fy, 7p = -F^, where a, /3, 7 are the direction cosines of the outwardly directed normal to the surface at the point. By exactly the same type of reasoning which leads to this result, we can find relations between the body forces on a solid body and the space differ- ential coefficients of the "stress constituents" Xx, Xy, . . • Zz. STRAINED ELASTIC SOLIDS 423 To obtain them we visualize a very small rectangular parallel- opiped (Fig. 6) of the medium in the state of strain which has the point P at its center. It is bounded by six rectangular faces parallel in pairs to the planes of reference OYZ, OZX, OXY. The local axis of x through P cuts one face parallel to OYZ in a point Q and the other in a point U, such that PQ = PU = ^, the coordinates of Q being x -\- ^,y,z and oiU,x — ^, y, z. The local axes of y and z each cut two faces, in the points R, V and S, W, respectively, RV being equal to 2??, and SW to 2^. Thus the volume of the parallelopiped is 8^??^ , its sides being 2^, 2?/, 2f and its faces having the areas 477^, 4f^, 4^??. Let Xx, ... ^z be the values of the "stress-constituents" at P. At Q they are S ■^ R u : V 0 * W / '' / At U they are ^^ ~ bx ^' Fig. 6 dXy dXy dZz dx dZz and similar formulae give the values at R, V, S, W. If we assume the values at Q to be the average values over the face containing Q, then the medium outside the parallelopiped exerts a pull on it across this face in the direction of OX of amount 477f, 424 RICE ART. K since 47?f is the area of this face. Across the face containing U there will be a pull in the opposite direction XO of amount The difference of these, viz., dXx dx Hvt is the resultant of these two in the direction OX. To proceed, we also have a shearing force on the parallelopiped in the direction OX of amount / dXy \ across the face containing R, and one of amount / dXy \ across the face containing V in the direction XO. These two forces yield a resultant dXr dy 8^^r in the direction OX. The remaining pair of faces contribute a resultant force in the direction of OX of amount dXz dz Thus the stress actions exerted by the surrounding medium on the parallelopiped are equivalent to a force whose x-component is (■ dXx dXy dXz\ ^ dx dy dz The resultant body force arising from external influences on the STRAINED ELASTIC SOLIDS 425 parallelopiped we represent by the symbol F, estimated per unit volume, so that the a;-component of this on the element of volume we are considering is Fx • S^v^. Since the medium is in equilib- rium, the sum of the components in any direction of all the forces on an element of volume (including those due to influences external to the medium and those arising from the part of the medium surrounding the element) is zero, and therefore dXx dXy dXz dx dy dz In just the same manner we can prove that dYx dYy dYz \ (29) dx dy dz dZx dZy dZz — +— +— +Fz = 0. dx dy dz The equations [377] constitute a particular case of these; for the forces arising from gravity have no horizontal components and, since in Gibbs OZ is in the vertically upward direction, Fz is his —gT. If at the surface there are external forces in the nature of thrusts or pulls on it, and if at any point such an external force is represented by F estimated per unit area (regarded as positive if it is a pull), then at the surface we also have the equations aXx + iSXr + yXz = F., aVx + ^Yy + yYz = Fy, aZx -h pZy -j-yZz = F,,j (29a) where a, jS, y are the direction cosines of the outwardly directed normal to the surface at the point. This follows from the consideration that a thin layer of matter at the surface of the body exerts on the matter in the interior a stress-action per unit area, whose component parallel to OX is aXx + fiXy + yXz, etc. Hence the interior matter exerts on this thin layer an action whose a;-component per unit area is — {aXx + ^Xy + 426 RICE ART. K 7X2). For equilibrium the sum of this and Fx, the external sur- face force-component per unit area, must be zero. It was stated that the stress at a point was determined by six independent quantities, but so far we seem to have reduced it to a representation by nine. So we shall now turn our atten- tion to three relations which exist between these nine constit- uents, and which are given in [375] and [376], proving these, however, by a more direct and more easily grasped method than that employed by Gibbs. To this end let us once more give our attention to the conditions controlling the equilibrium of the parallelopiped (Fig. 6), and recall the fact that not only must the total resultant force on the parallelopiped vanish, but also the total couple as well. This couple is obtained by taking moments about the point P, and has three components, one around the local axis of x through P, one around the local axis of y', and one around the local axis of z. Consider the contri- butions made by each influence on the parallelopiped to the component of the total couple round the local axis of x. The pulls across the faces involving the constituents Xx, Yy, Zz are symmetrical with regard to P and contribute nothing to the couple. On the other hand the individual shearing forces obviously tend to produce twists. Those that tend to twist the element around the local axis of x are the shearing forces parallel to the local axes of y and z, and they are the following four: / dZy \ 4f ^ across the face containing R, ( dZr \ ~ [Zy — T~ V ) 4f ^ across the face containing V, dYz \ „ Yz + ~r~ r ) 4^77 across the face containing S, az J ( / dYz \ — [Yz — "r~ r ) 4^77 across the face containing W . The moment of the first about the local axis of x is / dZy \ STRAINED ELASTIC SOLIDS 427 in a right-handed sense; that of the second is ( aZr \ also in a right-handed sense. That of the third is / dYz \ in a left-handed sense and that of the fourth is also in a left-handed sense. Thus the four shearing tractions yield a couple around the local axis of x in the right-handed sense of amount S^vUZy - Yz). Turning now to the body forces we see that even if their action on the element is not symmetrical about P (as would be the case for example with gravity forces) they can yield in com- parison with the moments arising from the shearing forces only a vanishingly small couple, since about the local axis of x, for instance, this couple must have an order of magnitude which cannot be greater than the product of Fy, 8^r/f and i;, or Fz, 8^7?^ and 77. Since ^ry^^ or ^tj^^ is small compared to ^i)^ when ^, rj and ^ are small, these contributions are evanescent in comparison with that written above, when the volume con- sidered is small. Thus the total couple on the parallelopiped has components around the three axes given by {Zy - F^)8^r,r, {Xz - ZxM-n^, {Yx - Xy)S^-n^. But in equilibrium these components must be zero, and so Yz = Zy, Zx = Xz, Xy = Yx' (30) This demonstrates that there are only six independent strain- constituents, as already stated. 428 RICE ART. K It must not be forgotten that this analysis relates to any arbitrary choice of axes of reference. Actually it is possible, by selecting a special triad of orthogonal lines as axes, to intro- duce a diminution in the number of stress-constituents required for the formulation of the stress across any given plane at a given point. A proof of this statement appears in Gibbs, I, 194, 195, but it is not so famihar and not so easy to grasp as the usual proof given in works on elasticity, which follows a line of reasoning similar to that adopted earlier to indicate the existence of three principal axes of strain, and is here outlined. Conceive that a quadric surface whose equation is is constructed with P as center and with any local axes of ref- erence at P; Xx, Xy, . . . Zz being the values of the stress con- stituents at the point P. Let a line whose direction cosines are a, 13, 7 be drawn from P cutting this quadric in the point Q; denote the length of PQ by r so that the local coordinates of Q are ra, r^, ry. Now draw the tangent plane at Q to the quadric surface and drop PN perpendicular to this plane. By the theorem already used we know that the equation of this tangent plane is (Xxra + AVr/3 -f X^ry)^ + {Y^ra + YyVlS -^ Yzry)r, -f (Z^ra + ZyrlS + Z^ry)^ = k (remembering that Yz = Zy, etc.), and so the direction cosines of PA'' are proportional to aXx + fiXy + yXz, aYx + ^Yy + yY z, O^Zx + (3Zy + yZz. Thus a glance at (28) shows us that the stress action at P across a plane normal to PQ is itself parallel to PN. In general PN is not coincident with PQ, i.e., the stress action across any plane is in general not normal to the plane, as we know already; but the information now before us about its direction indicates that there are three special orientations of the plane for which this happens to be true and for which PA^ lies along PQ. They are STRAINED ELASTIC SOLIDS 429 clearly the three principal planes ol the quadric surface whose equation has been written down above. Were we to choose as axes of reference the three principal axes of this quadric, we know that the equation would only involve terms in ^^, 7?^, f ^, but not in Tjf , f^, ^r). In short, with such a choice of axes of reference only three of the stress-components would have a finite value, viz., those corresponding to Xx, Yy, Zz. The remaining six (actually only three) would be zero, and as Gibbs states in equation [392] the stress action across any plane (a, /?, 7) would have as its components aXx, ^Yy, yZz. These three special axes are called the principal axes of stress, and their existence is a point of considerable importance in the discussion in Gibbs, I, 195 et seq. Special cases arise if the quadric surface at a point referred to above is one of revolution, i.e., if the section by one of the principal planes is a circle. In this event, assuming that it is the plane perpendicular to that one of the principal axes of stress designated as OX, it is clear that Yy = Zz, and the stress action across any plane containing the local axis of x at P is normal to this plane. Or it may happen that the "stress- quadric" is actually a sphere, so that Xx = Yy = Zz. Any triad of perpendicular lines will serve as principal axes of stress if this be so, and the stress-components which do not vanish have one numerical value, the stress across any plane being normal to it and having a value independent of direction. This is in fact the general state of affairs for a fluid at rest and Xx = Yy = Zy = —p where p is the fluid pressure. It is clear that the equations of equilibrium (29) then degenerate to those for a fluid quoted on page 422. 5. Stress-Strain Relations and Strain-Energy. We have now considered at some length the mathematical methods by which the strains and stresses in a body are analyzed into their most convenient constituents, and it is clear that the differences of behavior observed in various elastic media when subject to given external forces arise from the different "constitutive" re- lations which exist between the constituents of stress and the co- efficients of strain in these different media. We know for instance that the same pull will elongate a wire of brass of given section 430 RICE ART. K and one of steel of the same section in different ratios; in both cases the Xx stress constituent is the same, but the en strain coefficient is different (the axis of x being supposed to be directed along the length of the wire). Obviously any complete theory would place at the disposal of the investigator the means of calculating in any given case, the strains which result from the imposition of definite external forces. Equations (29) are differential equations which connect the external forces with the stresses, so that with sufficient knowledge of these forces and of the state of stress at the surface of a body we can in theory determine the stress at any other point of the body. But this will not lead to a knowledge of the strains at each point unless we have a sufficient number of algebraic equations connecting the stress-constituents with the strain-coefficients. So far we have relied on the mathematician to develop the right conceptions and deduce the correct differential equations; we now have to turn to the experimenter who by subjecting each material to suitable tests determines the various "elastic con- stants" of any given substance. This is a matter on which little can be said here, but provided the tests do not strain a body beyond the limits from which it will return to its former condition without any "set" on removing the external forces, it is found, as a matter of experience, that there is approximately a linear relation between strain-coefficients and stress-constit- uents. Under these conditions the deformation of solid media is relatively so small that, although a rectangular element is in general after the strain deformed to an oblique parallelopipcd, the various angles have been sheared from a right angle by relatively small amounts, and we can use the coefficients en, en, . . . 633, referring the system to the same axes before and after the strain. As we have seen above, the pure strains depend actually on six quantities, en, e^, 633, 623 + ^32, esi + en, en + 621, as the rotations are not a matter of importance; furthermore there are only six numerically different values involved in the nine quantities Xx, . . Zz Let us therefore introduce for convenience a small modification of the sym- bolism, and write STRAINED ELASTIC SOLIDS 431 Zi for Xx, /i for en - 1, X2 for Yy, J2 for 622 — 1, X3 for Zz, fz for 633 - 1, Xi for Fz or Zy, /4 for 633 + 632, Xi for Zx or Xz, ft, for 631 + e^, Xe for Xy or Yx, /e for 612 + 621- (fh h} h are the fractions of elongation along the axes and fi, fh, /e are the shears or changes in the angles between the axes.) A complete experimental knowledge of the elastic properties of any material would therefore be embodied in the ascertained values of the 36 elastic constants Crs in six consti- tutive ''stress-strain" equations such as Xi = Cu/i + C12/2 + C13/3 + C14/4 + C15/5 + Cifi/e,! j (31) X2 = C21/1 + C22/2 + C23/3 + C24/4 + C25/5 + C26/6,J and four similar equations. These equations are the expression of a general Hooke's law, a natural extension of the famous law concerning extension of strings and wires due to that English natural philosopher. This apparently presents an appallingly complex problem for the experimental physicist; however, there are important simplifications in practice. To begin with, it will appear from energy considerations to be discussed presently, that even in the most general case the 36 constants must only involve 21 different numerical values at most, and actually for a great variety of materials still further reductions are involved. Indeed, for isotropic bodies all the elastic constants of such a material are calculable from the numerical values of two "elastic moduh," the well-known "bulk modulus" (or "elasticity of volume") and the "modulus of rigidity." For various crys- talline bodies conditions of symmetry also involve a material reduction of the number of independent constants below the number 21. The two moduli for isotropic bodies are referred to by 432 RICE ART. K Gibbs and perhaps merit a brief remark here. When a body is subject to a uniform stress in all directions we have Xx = Yy = Zz and Xy — Yx = Yz = Zy = Zx = Xz = 0. If the body is isotropic, then referred to any axes en = 622 = 633 and ei2 = 621 = 623 = 632 = 631 = 6i3 = 0. Thus along any line there is a fraction of elongation /, where f = e — 1, e being the common value of en, 622, 633. Hence the fraction of dilatation of volume is e* — 1 or practically 3/. The quotient of the common value of Xy, Yy, Zz by 3/ is called the bulk-modulus. (Gibbs calls it "elasticity of volume" on page 213.) The conception is most important in the case of a fluid. Here a variation of external thrust on the surface pro- duces a variation of pressure from p to p -\- 8p; there results from this an alteration of volume from v to v -{- 8v (8v is essen- tially negative if 8p is essentially positive), i.e., a fraction of 8v dilatation 8v/v. The bulk-modulus is the limit of — 8p/— ; V i.e., it is dp(v, t) — V — - — ' dv where p{v, t) is the function connecting pressure with volume and temperature. (See [448].) This definition is synonymous with the previous one, since for a liquid p = — Xx = —Yy = —Zz and the shearing stresses vanish. (In fact the state of stress uniform in all directions, mentioned above, is often referred to as the case of "hydrostatic stress".) We can have a state of stress also in which the six constituents STRAINED ELASTIC SOLIDS 433 vanish except (say) Yz{or Zy). In this case, for an isotropic body, /i = /2 = /s = 0 and also /s = /e = 0. Only fi is finite and for the case of Hooke's law varies directly as Yz. The quotient of Yz by fi is called the "modulus of rigidity," or simply the "rigidity" of the material. Of course one should bear in mind that the strains must be small if the physical facts are to be consistent with these definitions We thus see that a given system of external forces on a body involves a determinate set of stress-constituents when the body is in equilibrium under the forces, and these in their turn by reason of the stress-strain relations (hnear or otherwise) determine a definite condition of strain. Infinitesimal va- riations in the external forces change the stress infinitesimally to Zi + dXi, etc. in the new state of equilibrium, and the strain coefficients are altered to/i + dfi, etc., where Xi + dXi, etc. are connected with /i + dfi, etc. by the same six equations as before. Actually we can conceive that "in the neighborhood" of a given state of equilibrium involving a definite condition of strain there are an infinite number of other states, which are not necessarily equilibrium states, characterized by values /i + 8fi, etc. of the coefficients where the 8fr are entirely arbitrary, so that /i + dfi, etc. are not connected with the external forces by means of the stress-strain relations. For further information on these matters the reader is referred to standard texts on elasticity and to R. W. Goranson's "Thermodynamic Rela- tions in Multi-component Srjstems" (Carnegie Institution of Washington, Pub. No. 408, 1930).* Our ultimate object in what has preceded is to lead up to the expression which represents the change in the energy of strain when the condition of strain has been altered by a change from a state of equilibrium to a neighboring state. This must be included in the expression for the total change of energy when we are formulating the first and second laws of thermo- dynamics. It is in fact the expression which is to replace * The reader must be careful to remember that the author's symbol- ism, which has been chosen to diverge as little as possible from that of Gibbs, differs in some details from that used in these references. 434 RICE ART. K — pdv in the law for a fluid medium 8e = tdr] — p8v. The natural method of procedure would be to consider the movements of the points of application of the external forces involved in the change of strain and, combining these with the forces themselves, to determine the work of the external forces; this work, if there is no exchange of heat, will be equal to the change in internal energy. Unfortunately this method involves the use of certain general theorems of mathematical analysis which may be unfamihar to some readers and the writer will therefore make shift with a more elementary, if less rigorous, method. We revert to our picture of an element of volume surrounding the point P in the state of strain determined by the values en, ... 633 of the strain-coefficients (see Fig. 6). The element is assumed to be strictly rectangular in this state (although not necessarily so in the state of reference); its sides are parallel to the axes OX, OY, OZ and have the elementary lengths h, k, I respectively. We conceive that this medium receives a further strain to the condition determined by en + Sen, etc., and this involves infinitesimal elongations and shears in the rectangular element. We now imagine the element to be isolated and to experience the same movements under a set of external forces which are equal to the forces which we assume to exist across its faces when in situ. The work of these hypothetical forces we take to be the increase in strain-energy of the element. In the circumstances of the case en, ^22, 633 are near to unity in value, so that in comparison with them en — 1, 622 — 1, 633 — 1, 623, ^32, esi, ei3, ei2, 621, as we noted earlier, are small. The rectangular element has had its side h elongated by a fraction 5fi. The matter surrounding the element is exerting on it forces across the kl faces equal to klXx. Hence work is done which can be calculated by conceiving one of the kl faces fixed and the other moving a distance h8fi in the direction of the force klXx. (The shearing forces klYx and klZx across these faces are at right angles to the elongation and so this movement involves no work on their part.) This work is hklXx8fi, and this is therefore STRAINED ELASTIC SOLIDS 435 one part of the increase of energy in the element of volume. The other pairs of faces when treated similarly yield further parts of the energy increase, viz. hklYybfo,, and hklZ^bfz. Now let us turn to the shears and fix our attention for the moment on the faces of the element which are parallel to the plane OXY and are separated by the distance I in the direction of OZ. A little thought will show that one of these faces has moved in a shearing manner relatively to the other by an amount which is the vector sum of a component U{ezi + en) parallel to OX and a com- ponent Z5(e23 + 632) parallel to OY. (A glance at equation (10) will remind the reader that the "shear" of hues parallel origi- nally to OX and OZ is 5[e5/(e3ei)'] which is substantially 6(e3i + eia) ; the "shear" practically measures the small change in the (right) angle between OZ and OX.) We can again simplify our argument by conceiving one of the hk faces fixed and the other slipping over it by amount Uf^ in the direction of OX. The shearing pull across this face by the surrounding matter in the element is hkXz in this direction. (The face is perpendicular to OZ and the pull is in the direction OX.) Thus the work done on this account is hklXz^fh- Similar reasoning yields hklYz^fi for the other component. Each of the other pairs of faces treated in a similar manner would yield similar terms ; the faces parallel to OYZ would yield hklYxdfe and hklZxSf^, and the faces parallel to OZX would yield hklZydfi and hklXr^fe.. It would seem that in order to obtain the increase of energy asso- ciated with the shearing movements, we ought to add these six terms. This is, however, one of the pitfalls of this simple method which we are using so as to evade advanced analytical operations. If we adopted this procedure we should obtain twice the correct increase associated with the shears, and it is not difficult to realize that this is so. For a shear of one Z-face past the other Z-face (meaning the faces perpendicular to the direction OZ) in the direction parallel to OX involves of necessity a shear of an X-face past the other X-face in the direction parallel to OZ. Either shear is one of two alternative ways of describing the resulting distortion. Now our method of cal- culating the work done in this case really requires us to conceive the element of volume as isolated and sheared either by a shear- 436 RICE I ART. K ing pull hkXz across a Z-face or a shearing pull klZx across an X-face. One way yields hklXzdfi for the work done; the other yields hklZxBf^ for it; these are the same quantity since Zx = Xz, but we must not count both or we shall obtain twice the correct value, and this is just what we would be doing if we added all the terms obtained above. In this comparatively simple way we can reasonably assume a result which can be more rigorously established by other methods, viz., that when the strain of a solid is varied from a state in which the strain coefficients are en, • . . ^33, to one in which the coefficients are en + 5en, ■ . . 633 + 8633, the increase in energy in an element of volume is the product of the volume of the element and Xi8f, + X25/2 + X35/3 + X45/4 + X,8f, + X,5U (32) This expression takes the place of the expression —p8v for a fluid in the formulation of the variation of the internal energy of a solid body in any general change of temperature and state. That the expression (32) degenerates to this in the case of a fluid can be readily demonstrated, for we have seen earlier that in the case of a fluid X4, X5, X 6 are zero, and Xi = X2 = X3 = —p; hence (32) becomes -p5(/i+/2+/3), and, since unit volume expands in this case to (1 + 6/0 (1 + 5/2) (1 + 5/3), or practically l+6(/i+/2+/3), it follows that 8v is equal to the original volume of the element multiplied by 8(fi + /2 + /3). The whole of the argument so far has avoided any considera- tion of changes of temperature arising from strain and assumes all the energy to be mechanical. In so far as this is allowable the expression X16/1 ... + X&Sfe must be regarded as the variation of a function of the six quantities fi, ... /e, so that STRAINED ELASTIC SOLIDS 437 if we denote this "strain-energy function" by W(fi, . . . /e) it follows that _dW dW If then each Xr is a linear function of /i, ... /e, as experiment shows to be approximately the case for isothermal small changes, it follows that W must be a quadratic function of the six variables /i, . . . /e- Now such a quadratic can only involve 21 numerically different coefficients; thus W = hCufi" +^66/6^ + C12/1/2 + Cie/i/e + C23/2/3 + C2G/2/6 + C34/3/4 . . . + Cirjaf^ + C45/4/5 + Cicfif^ + C^efhfe, and so it appears in assuming that the various stress-constitu- ents satisfy equations such as Xr = Crlfl . . . + Criflj , that This justifies the statement made above that in the cases where there are linear isothermal stress-strain relations, there are at most 21 elastic constants. In the arguments that follow, however, we shall require no such restriction as to the nature of the relations between stress- constituents and the strain-coefficients. Actually these relations also involve the temperature. Moreover, if we are going to follow Gibbs' reasoning we shall have to realize his somewhat different treatment of the stress-constituents from that outlined 438 RICE ART. K above, which is the usual treatment. It arises from his en- deavor to make the foundation of his arguments as wide as possible. He lays down no restriction that the state of reference shall be so near to that of the state of strain that a rectangular element is but little strained from that form in the changes which take place between the two states. His only- proviso is that the differential coefficients dx/dx', etc. shall not alter appreciably over molecular distances, i.e., that the strain is homogeneous within a physically small element of volume. Let us retrace the ground covered by the argument which we followed when deahng with the energy of strain. The rectangular element of volume in the state of strain has its center at a point P whose coordinates are x, y, z with reference to the OX, OY, OZ axes; this element was, in the state of reference, an obhque parallelopiped whose centre was at the point P' whose coordinates are x', y', z' with reference to the OX', OY', OZ' axes. Let the edges of the element in the state of strain be parallel to OX, OY, OZ, and following the course we used earlier let us call the mid -points of the faces perpendicular to OX, Q and U, so that the local coordinates of Q with reference to local axes of x, y, z at P are ^, 0, 0, and of U are — ^, 0, 0. Those of Q', the center of the corresponding face of the un- strained element, for the local axes of x', y', z' at P' are ^', -q', f ' where, by equations (21), k = an^' + anv' + Qisf', 0 = a.i^' + 0227?' + a23^', 0 = 031^' + ^321?' + Ossr'., (33) Now let the slight increase of strain take place which we considered above when we treated this problem in a more restricted manner; the point P is displaced to a neighboring point Ps, say, while Q and U are displaced to neighboring points Qs and f/g. The strain-coefficients are now an -\- 8an, etc. The local coordinates of Qs with reference to local axes of X, y, z at P5 are ^ + b^, 8r}, 8^ where STRAINED ELASTIC SOLIDS 439 ^ + 5^ = (an + danW + (ai2 + Ba^iW + (ai3 + Sa^)^', 5rj = (rt2i + 5a2i)^' + (a22 + 5022)77' + (023 + 5a23).C', 5f = (a^l + Sflsi)^' + (032 + ^a^^W + {a^s + 6033)^'.* Hence 5^ = dan-^' + 5ai2-77' + 5ai3-f', 67? = 6021-^' + 5022-77' + 5a23-r', 5r = Sasi-^' + 5a32-77' + 5a33-f'. Now we need to express these variations in terms of ^, and this is easily done; for, on solving equations (33) for ^', 17', f' in terms of ^, we find that ^ H ^* , A,, where Ara is the first minor (vv^th its correct sign) of ars in the determinant H. We write for convenience hrs for Ars/H, and in consequence we have the following three results 8^ = (hnSan + 6i25ai2 + hsdais)^,) 8r] = (6ii5a2i + 6120022 + &i35a23)^, f (34) 5f = (6ii5a3i + 6i25a32 + 6135033)^.] It is easy to see that the coordinates of f/5 for the local axes at Ps are just — (^ + 5^), —677, —8^. Thus it appears that the rectangular element has had its edge parallel to OX elongated * Observe that Pd and Qs are positions in the slightly altered state of strain of the same original points P', Q' in the state of reference. 440 RICE ART, K by 25^ i.e., bj' the fraction (6ii6aii + hnda^ + bisSan) of its lengtli 2^. In short, bndan + hnban + hnban is just the infini- tesimal quantity 5/i or ben which occurred in the previous treat- ment. Similarly the face containing Q has in this infinitesimal change of strain been sheared by an amount 2bri relatively to the opposite face containing U in the direction parallel to OF and by an amount 25^ parallel to OZ. But as we have seen in the earlier treatment these shearing displacements are be-n ■ 2^ and bez\ • 2| respectively. Hence we find that 5621 = hnba^i + 6i25a22 + hnba^z, ben = bnbasi + 6]25a32 + 6136033. The other faces can be treated similarly and we thus arrive at the nine equations ben = 6n5aii + 6i25ai2 ben = 0225ct22 "l~ 0235<223 5633 = 6335033 + 63l5a31 5623 = 6335^23 + 63i5a2i 5632 = 6225^32 "l~ 6235(233 5631 = 6ii5a3i + 6i25a32 56i3 = 6335013 + 63i5aii 56i2 = 6226012 + 6235013 5621 = 6ii5o2i + 6126022 + 6136013, + 6216021, + 6326032, + 6326022, + 6216031, - + 6136023, + 6326012, + 6216011, -1- 6136023.^ (35) By our previous result the increase in the energy of the element of volume 8^7?f is equal to the product of 8^7?^ and the expression Xi5/i . . . + XeS/e or Xx ben + Yy 6622 -\- Zz 6633 + Yz 6623 -\- Zy 6632 + Zx 6631 + Xz ben + Xy ben + Yx 6621. This by reason of the equations (35) becomes an expression such as TiiSoii + ri26oi2 + ri36oi3 + r2i5o2i + etc. . . . + T335033, (36) STRAINED ELASTIC SOLIDS 441 where th, ... T33 are nine linear functions of the stress-con- stituents Xx, • ■ • Zz, involving the quantities brs in the co- efficients. It will be found in fact that Til = bnXx + &2i^y + bziXz, T12 = bnXx ~\~ 622-^ r + O32XZ, Tl3 = blsXx 4" 023Ay + bszX z, and six similar equations. Now the expression (36) represents the change in the strain-energy caused by the infinitesimal increase of strain in the matter occupying unit of volume in the state of strain. But, as we have seen previously, this matter occcupies a volume H~^ in the state of reference, and so we must multiply the expression (36) by H in order to obtain the increase in strain energy of the matter which occupies unit volume in the state of reference. Now from the definition of brs given above we see that brsH is equal to Ars- Hence we arrive finally at the result that the infinitesimal increase in strain energy estimated per unit of volume in the state of reference is Xx'^dn ~\~ Xy'Sun ~\~ Xz'Sais + Fx'5a2i + Fy'5a22 + Yz'Sa^s + Zx'Sasi + Zy'dasi ~\~ Zz'dazz, (37) where Xx' = AnXx + ^21-X^y + ^31^ Z, X Y' = A^Xx + A22XY + .432X2, Xz' = AizXx + A23XY + AzzXz, Yy' = A22YY + Az2Yz + A,2Yx, Yz' = A23YY + AzzYz + A,zYx, Yx' = A21YY + AzxYz + AnYx, Zz> = AziZ z + AizZx + A2zZy, Zx' = AziZz + AnZx + A21ZY, Ztyi = ^32^2 + A12ZX + A22Zy- (38) The expression (37) occurs in Gibbs' equation [355]. It is essentially his notation with the convenient simplification of replacing dx/dx' by an, etc. It is really an important matter to realize that Gibbs' stress- 442 RICE ART. K constituents Xx, etc., are not to be confused with the stress- constituents Xx etc., of customary elastic sohd theory. Gibbs himself gives on page 186 a physical signification to his constit- uents, which brings home to the careful reader how essential it is to be on guard when it is a question of giving a measure of a physical quantity -per unit of length or area or volume. His own statement is so brief that for clarity it can be somewhat ex- panded. He asks us to consider an element of mass which in the reference state is rectangular (a "right parallelopiped" as he calls it) with its edges parallel to the axes OX', OY', OZ'. We shall adopt a method similar to that employed previously and regard the center of this at a point P', whose coordinates are x' , y', z'. The middle points of the faces perpendicular to OX' shall be named Q' and U', the coordinates of Q' being x' + ^', y', z', and of U', x' - ^', y' , z'; and so on. (The dx', dy', dz' of Gibbs are 2^', 2r]', 2^'.) In the strained state the element is in general an oblique parallelopiped the center of which is at P, whose coordinates are x, y, z with reference to the new axes OX, OY, OZ. The coordinates of Q, the displaced position of Q' , and still the center of one of the faces (now a parallelogram), are a: + ^, ?/ + 77, 2 + f , where k = ank', r = asir. (See equations (21), noting that the local coordinates of Q' in the local axes at P' are ^', 0, 0.) Now consider a further infini- tesimal displacement from this state in which only an varies, but not any of the other eight strain-coefficients. In such a varia- tion ^ will alter by ^ • 8an but 77 and f will not vary; i.e., the face we are considering will move further from the center of the element in the direction of OX (as Gibbs postulates in line 12 of page 186) by an amount ^' • 8an. Similarly the face opposite will move relatively to the element's center an equal distance in the opposite direction; in other words one face will have separated from the other face by an amount 2^'oaii. Hence the work done by the components of the force on the element across these faces parallel to OX is equal to the product of 2^'aaii and this force. STRAINED ELASTIC SOLIDS 443 But a glance at (37), or [355] of Gibbs, shows us that, if no heat is imparted and only an varies, the increase in energy of the element is Hence as work done is equal to energy increase the force just referred to is 4:r]'^'Xx', or Xx> per unit of area in the state of reference. The symbolism clearly indicates the physical signification; the accented x' in the suffix indicates that the force is estimated on an area which was perpendicular to OX' in the unstrained state and was equal to the unit of area in that state. The unaccented X, to which x- is the suffix attached, indicates that the force is a component in the direction OX. The force of course only exists in the strained state, since the reference state is assumed as an unstrained state, that is, one in which the stress-constituents vanish. (See the remarks on this on page 418.) It is clear from this (quite apart from the type of equations connecting Xx', ... Zz' with Xx, . . . Zz which are indicated above) that Xx is quite distinct from Xx'', for Xx is the force across a face which is perpendicular to OX in the state of strain estimated on an area which is equal to the unit area in that state; it is however, like Xx', a component in the direction OX. Similar differences can be drawn between the other com- ponents of stress in the two systems of coordinates. From this it can be perceived that because Yz = Zy it is not of necessity true that Yz' = Zy. It should be observed that these results do not depend on the fact that one may choose the axes OX, OY, OZ not to coincide with OX', OY', OZ'; for even if they were made to coincide the symbol Xx, for example, could not be made to do double service, on the one hand for a component parallel to OX of a force across an area which was unit area in size and was perpendicular to OX, and on the other hand across an area which is unit area in size and is perpendicular to OX. Thus the double naming of the axes is of service even when they are regarded as coincident. This is a justification for Gibbs' apparently pointless complication of procedure. Only if the state of strain is regarded as being little removed from the state of reference can we assume that an approximate equality may 444 RICE ART. K exist between Yx' and Xy, and so on, provided the two sets of axes are regarded as coincident. At the risk of appearing to be prohx on this matter, the writer would hke to point out that the equations (38) offer an alter- native method of giving the correct physical signification to Xx', etc. If we recall the arguments developed from equations (16) to (27) above, we will remember on looking at (27) that a unit area, which was in the state of reference perpendicular to OX' (so that for it a' = 1, jS' = 0, 7' = 0), is strained into an area whose projections on the planes perpendicular to OX, OY, OZ are An, A21, Asi, with similar results for unit areas originally normal to OY' or OZ'. In other words, if unit area which was in the state of reference perpendicular to OX' is strained into an area of size K with direction cosines a, j8, 7 with reference to OX, OY, OZ, then Ka = An, m = A21, Ky = A31. But by (28), the force across this surface in the state of strain in the direction OX has the value aXx + ^Xy + 7X2 per unit area, and so the actual force across the area Kin the state of strain is AnXx + A21XY + AsiXz, which by (38) is just Xx', thus giving us the physical inter- pretation of Xx' once more. In the same way we can demon- strate that Xy' is the force parallel to OX across an area in the state of strain, which in the state of reference was unit area in size and normal to OY' in orientation; and so on. 6. Thermodynamics of a Strained Homogeneous Solid. The treatment of heterogenous systems in the earlier parts of Gibbs' discussion of the subject is of course based on equation [12] which is a generalization from equation [11], the equation for a homogeneous body when uninfluenced by distortion of solid masses (among other physical changes). In the same way any treatment of heterogenous substances in which elastic effects STRAINED ELASTIC SOLIDS 445 must be taken into account will require a knowledge of how a homogeneous substance when strained must be dealt with in thermodynamical reasoning. The equation which is to replace [11] is now easily derived in view of what has just been accom- phshed in the previous parts of this exposition. Thus in [11] c and r] are regarded as functions determined completely by the state of the body. For a homogeneous fluid, we can regard them as functions of its temperature and volume, or of its tem- perature and pressure, and their differentials are connected by the equation de = td-q — pdv. (39) If we consider this as applying to the matter within a unit of volume, dv is actually the fraction of dilatation, essentially the one strain-function which plays any part in the case of a fluid, since the elongation in all directions is uniform and shears do not exist. For a strained solid e and r? are still functions of the state, and we can take as the variables the temperature and the strain-coefficients. There are nine of the latter, but we have seen that six quantities are sufficient. In equations (9) we have defined six such quantities d, 62, ... ee, and later in (23) and (24) we have seen that they are quantities which are entirely independent of the choice of the axes in the strained state, (of course, their particular values depend on what axes we choose for OX', OY', OZ', the axes to which the unstrained state is referred; in particular we can choose axes so that 64, 65, 6 6 vanish — the principal axes of the strain which are not sheared but merely rotated). For our immediate purpose it is more convenient to take the quantities /i, ... /e as our "thermodynamical variables," where /] = ei' — 1, ...... .; f^ = 64/(62^3)% ....... As we know, /i then represents the fraction of elongation parallel to OX', etc., and fi represents the shear of lines parallel to OY', OZ', etc. For a fluid body —p8v represents the change of internal energy of strain (compression) when the (unit) volume ex- periences a dilatation whose fraction is 8v. Similarly, when the strain-functions /i, ... /e are altered, the energy of strain of unit volume of the strained material alters by Xi8fi . . . + XeS/e. 446 RICE ART. K Here we make a natural generalization and assume that for any change of state of a homogeneous solid de = td-n + Xirf/i . . . + Xed/e. (40) Fully interpreted this means that we consider e and 17 to be functions of t, /i, ... /e. Strictly we should write them i{t, fi, ... /e) and r){t, fi, ... /e). If the state of the solid alters to another state of equilibrium in which the variables change to t + dt, /i + dfi, . . . /e + dfi, then equation (40) connects the various differentials. It will help us if we briefly recall how from equation (39) we derive the equations which connect those thermal and mechani- cal properties of fluids which can be observed and measured by experimental methods. Thus c,{t, v) lit, v) dr] = — - — di + — - — dv, (41) L If where c„ is the specific heat at constant volume, and U the so- called latent heat of change of volume at constant temperature. We are, at the moment, taking t and v as the variables and indicating this precisely by writing the symbols in brackets after each quantity to show that in each case we are considering the appropriate functional form which expresses that quantity in terms of these variables. This device will also indicate without any ambiguity what quantities are being regarded as constant when we write down any partial differential coefficient. From the equation deit, v) = tdr](t, v) — p{t, v)dVf we derive the differential equation of the Gibbs yf function (free energy at constant volume), viz., d^{t, v) = -7](t, v) dt - pit, v)dv, (42) where ip = € — trj. STRAINED ELASTIC SOLIDS 447 Thus dv(t, v) dp(t, v) But by (41) dv dt _ dyjt, v) '" ~ ^ dv (43) Therefore dp(t, v) h = i -^' (44) the well known relation connecting the latent heat of change of volume at constant temperature with the temperature coeffi- cient of pressure at constant volume. Also from (41) we derive dCyjt, v) _ ± ( dv(t, v)\ dv ~ dv { dt j = t dtdv But by (43) Hence d^vjt, v) _ d^pjt, v) dtdv ~ df^ dc,{t, v) ^ d'pjt, v) which is another well-known relation. If we choose we can take the temperature and pressure as the thermodynamical variables. We then write dv(.t, p) = — ^ — dt + — ^ — dp, (46) where Cp and Ip are the specific heat at constant pressure and the 448 RICE ART. K latent heat of change of pressure at constant temperature. An- other differential equation which we require now is that for the f function of Gibbs (the "free energy at constant pressure") dUt, P) = -v{t, p)dt + v{t, p)dp, (47) where ^ = € - tri -{- pv. From this we derive dy]{t, p) dv{t, p) But by (46) dp dt _ dv(t,p) In — I Therefore a well-known relation. Also, from (46), dCp(t, p) dp dS{t, p) dtdp But by (48) d'vit, p) dtdp d'vit, p) dt^ Hence dCp{t, p) dH(t, p) dp df" (48) dp l.= -i'^^ (49) (50) There remains one more well-known relation. If an infinitesimal change takes place at constant pressure, STRAINED ELASTIC SOLIDS 449 the change of entropy is equal, by equation (41), to - < Cv{t, v)dt + U{t, v) — ^ dt V It is also, by (46), equal to - Cp{t, v)dt. Equating these two expressions we obtain the result Cp{t, p) = Cv(t, v) + U{t, v) — ^^' and using (44) we arrive at In exactly the same manner we can derive the equations which connect the thermal and mechanical properties of a solid. For the sake of brevity we shall write eQ, f) and 7?(^ /) for e{t,fi, . . . /e) and 7?(i, /i, ... /e); so that when we write, for example, dyjt, f) drjjtj) or ' we mean the temperature variation of t? at constant strain or the rate of variation of r] with respect to /r, the temperature and the five strain functions other than /r being maintained constant. In analogy with (41) we write dv(t,f) = ^ * + S '-^ if- (52) The summation extends over six terms; c is the specific heat at constant strain of the solid (per unit volume as measured in the state of strain), which means that the solid is prevented from changing volume and shape. The six quantities Ir are various latent heats of change of strain; in each case the temperature 450 RICE ART. K and five strain-quantities are unchanged. A well-known illustration can be given of the idea involved here. When one extends a piece of rubber suddenly, it rises in temperature. Thus if one wished to maintain the temperature constant one would have to extend slowly and take heat from the solid, which shows that the Ir coefficients for rubber are negative. The en- ergy relation (40) is now written deitj) = tdriitj) + i:Xr{t, f)dfr, (53) and from it we derive the differential equation for Gibbs' \p function, viz., dKtJ) = -n{t,f)dt + XXr(t,f)dfr, (54) where \p = € — tr]. From (54) we derive driitj) dXritJ) But by (52) dfr dt a.(^/) (55) dfr Therefore lr= -t —^' (56) There are of course six equations of the type (56), and they connect the heat required to maintain the temperature constant when the strains are altered with the variations of stress re- quired to maintain the strains constant (i.e., to prevent expan- sion and change of shape) when the temperature alters. To continue, from (52) we derive dcjtj) ^ d^tj), dfr dtdfr STRAINED ELASTIC SOLIDS 451 But by (55) dtdfr dt^ Hence we obtain the six relations It is, of course, open to us to choose as thermodynamic variables the temperature and the six components of stress. The energy and entropy are then expressed in full by the symbols €{t, Xi, ... Xe) and r](t, Xi, ... Ze) or briefly €{t, X) and r](t, X). The entropy equation then becomes at, X) s;^ Lrjt, X) 7]{t, X) = — ^ — dt -]- 2j — ~t — ' ^ where C is the specific heat at constant stress, i.e., under prac- tically the usual conditions of measurement, where the external forces on the solid are unchanged. Li, ... Le are six latent heats of change of stress, each one at constant temperature and with five of the stress-components unaltered. The energy differential equation is once more adapted to the choice of variables by using Gibbs' f function, viz., € — ^77 — 2 XtSt. Thus d^{t, X) = —n{t, X)dt - Xfr{t,X)dXr. (59) From (59) we derive dr){t, X) ^ dfrjt, X) dXr dt But by (58) dv(t, X) (60) Lr = t dXr 452 RICE ART. K Therefore Lr-t ^^ > (61) giving us six equations connecting the heat required to maintain the temperature constant when the stresses are altered with the variations of strain which accompany changes of temperature when the stresses are maintained constant. In addition we derive from (58) the equation dC{U X) ^ d^r^it, X) and by (60) dXr dtdXr d^riit, X) b%{t, X) dtdX, ~ df^ Hence we obtain the six relations A relation analogous to (51) can also be derived, which connects the difference of the two specific heats with the temperature coefficients of the strain-functions and the stress-constituents. Thus let an infinitesimal change take place at constant stress; the change of entropy can be expressed in two ways. For by (52) it is equal to ]{c(u)dt + J;uu)'-^ and by (58) it is also equal to 7 at, X) dt. V Equating these two expressions we obtain the result C{t,X)=c{t,f)^^Ut,f)^-^^' STRAINED ELASTIC SOLIDS 453 and using (56) we reach, finally, Some further relations can be obtained from the differential equations for the entropy and various energy functions. Thus from (52) we see that l.{t,j) = t dfr dvitj) dfs Hence dlrjtj) ^ dhjtj) dfs ~ dfr (64) and there are fifteen such "reciprocal relations" between the latent heats and the strains. Similarly from (54) we obtain fifteen reciprocal relations between the stresses and strains, viz., By using equations (58) and (59) we can obtain two sets of reciprocal relations, one between the latent heats and stresses, one between the strains and stresses, viz, dLrjt, X) ^ dLsjt, X) ,QQ^ dXs dXr and dfr(t, X) _ dfs(t, X) dX, dXr (67) From the thermodynamic equations we can also give a more general signification to the elastic constants of a solid, which were 454 RICE ART. K introduced in equations (31) as purely mechanical conceptions. By means of equations (53) or (54) we can express the stress- constituents as functions of the temperature and the strains; thus Xr = dfr (68) Now suppose the body experiences a small variation of strain at constant temperature; the variations in the stresses are given by the six equations where 8Xt = Crl5/i . . . + Credfe, dXrjt, f) ] _ d'Ht, f) dfr dfs (69) (70) Equation (69) replaces (31). The elastic constants are of course functions of the temperature and the strains. If the xp function is quadratic in the strains, the quantities Crs are inde- pendent of the strains, and this leads to the generalized Hooke's law referred to earlier. In any case equation (70) shows that Cra = Csr aud that at the most there are only 21 elastic con- stants. For an isotropic material, we have as before essentially only two, the bulk modulus or elasticity of volume, defined as before, and the modulus of rigidity given by any one of the differential coefficients or a/4 a¥M), dX,{t, f)^ a/5 aVO/), a/52 aXeO/), a/e aVO/), a/e^ (71) which are equal for such a substance. For those interested to pursue these matters further, a short chapter on the thermodynamics of strain will be found in Poynting & Thomsons' Properties of Matter. For a very full STRAINED ELASTIC SOLIDS 455 treatment consult Geiger and Scheel's Handbuch der Physik, Vol. VI, Chap. 2, pp. 47-60 (Springer, Berlin). We have now completed this long exposition of elastic solid theory. It has been necessary to go into it in some detail, since without some modicum of knowledge concerning it, this section of Gibbs' treatment, brief as it is, would be utterly unintelhgible. Indeed its very brevity renders the task more difficult; for although Gibbs, in his treatment of heterogeneous phases con- sisting of solids and fluids, does not employ in every detail the analysis of stress and strain in a solid usual in the texts of to-day, every now and then he interposes a short remark which would puzzle a reader unacquainted with that analysis. The very first page of the section is a case in point. Moreover, this analysis usually forms part of one of the more specialized courses in the physics or mathematics department of a university, and even students of physics, not aiming at a highly specialized degree in that subject, might well find their knowledge of stress and strain too rudimentary to follow Gibbs at this point. We now take the section itself and give a commentary upon it page by page. II. Commentary 7. Commentary on Pages 184~190. Derivation of the Four Equations Which Are Necessary and Sufficient for the Complete Equililrium of the System. We have already in the preceding exposition dealt extensively with the introductory defini- tions and formulations of Gibbs, I, pp. 184-186. We would remind readers that in [354] the usual practice of to-day would replace a differential coefficient such as dx/dz' by dx/dz', since it is implied that x, regarded as a function of x' , y', z', is being differentiated ^partially with respect to z', with the condition that x' and y' do not change in value. Actually it will probably be more convenient if we keep the notation introduced above and refer to dx/dx' as an, dx/dy' as an, dy/dx' as a^i, etc. If the strain is homogeneous these ars strain-coefficients are independ- ent of the particular values of x', y', z'; they are constant throughout the soHd body. In general, however, the strain may be heterogeneous, and in that event any a^g is a function 456 RICE ART. K of x', ij', z', and a^, implies a functional form and is really a con- traction for a„ {x', y', z'). Care should be exercised also to retain a clear idea of the meaning of the variational symbol 5. We have already used it in the exposition in the sense in which it is employed by Gibbs; thus b{dx/dx') or, as we shall write it, 6an refers to an infini- tesimal variation of the strain-coefficient, at a given -point, i.e., in a given physically small element surrounding the point which was originally at x' , y', z'. The reader must guard himself carefully against the misconception that he is to think of a point neighboring to x' , y', z', say x' + 8x', y' + W, z' + hz' , and to regard han as short for 9aii 9aii ha^ , ^ ax' + ^ iy' + - &', i.e., as the difference between the strain-coefficient at a point and at a neighboring point. Such a blunder would be fatal to any understanding of [355] . Indeed it was to avoid giving the reader any unconscious bias toward such an idea, that the writer, in re- ferring in the exposition to a point near to x' , y' , z' employed the notation x' + ^', y' + t]',z' + f ' and not x' + bx' , etc. In the exposition we used e and -q as symbols for the energy and entropy of the amount of material which occupies the unit of volume in the state of strain from which an infinitesimal variation is made; there was no need for suffixes as there was no ambiguity involved at that point. It is, however, the general practice of Gibbs to refer the material to its state of reference when considering magnitudes of measured properties per unit length, area or volume. Hence his use of the suffix v to bring that clearly before the reader's mind. Occasionally when he wishes to make a statement concerning magnitudes measured per unit of volume in the state of strain he employs the suffix v without the accent. In the exposition we saw that dtv = td7}v + ^Xrdf. Now a unit of volume in the state of reference becomes the STRAINED ELASTIC SOLIDS 457 volume vv in the state of strain. (See Gibbs, I, 188, line 27.) This quantity is, as we proved in the exposition, the determinant of the Urs coefficients, which is denoted later in Gibbs' discussion by the symbol H. If we multiply the differential equation written above by vv we obtain dev' = tdijv + H ZXrdfr. Also, the fr coefficients are defined in the exposition as certain functions of ei, ... ee) i.e., of ai, ... ae which are in their turn functions of the nine coefficients an, so that any differential dfr can be expressed as a sum of the differentials dara, such as ndaii + 4>i2dai2 • • • + ^zzda^z, where ^n, n, ... ^33 are functions of an, a^, . . . a^. In this way we arrive at Gibbs' expression [355], where Xx', Xy', . . . Zz' are functions of Xx, ■ • • Zz, an, • ■ • 033- The actual func- tional forms we have already developed in the exposition and given the actual linear relations which connect Gibbs' stress- constituents with the usual stress-constituents. On page 187 we have an expression for the variation of the energy of the solid body if an infinitesimal amount of material is added to it. Again we must carefully distinguish between the variational symbol 8 and the differential symbol D, and interpret correctly the use of the accents. Thus an element of the surface of the body in the state of strain is represented by Ds. If by crystallization from a surrounding fluid, for example, the body increases in size, the surface is displaced normally outwards by an infinitesimal amount which we represent by 8N. This might be regarded as having a constant value every- where on the surface, giving a uniform thickness for the addi- tional layer. But this is not so of necessity; 8N in general is regarded as a function of the position of the center of the element Ds, a function obviously infinitesimally small in value. Indeed 8N could be regarded as some ordinary function (t>{x, y, z) of the coordinates of a point on the surface multiplied by an infinitesi- mal constant. A sign of integration, of course, refers to the differential Ds. For example f8NDs is the increase in volume 458 RICE ART. K of the solid as it is when the deposition of matter takes place, viz., in the state of strain. (Note lines 4 and 5, where Gibbs expressly indicates this.) We could, however, conceive the solid to be brought back to the unstrained state after the deposition, the additional matter following the same change. In consequence the solid would be larger in its unstrained state than the original solid (before the increment) in the unstrained state by an amount J'dN'Ds'; where 8N' now represents the thickness of the additional layer in the unstrained state and Ds' the size of the element of area which is Ds in the strained state. Since ev > refers to the quotient of the energy of strain of a small portion of the strained matter by its volume in the unstrained state, the expression J'evdN'Ds' is justified. (It could, of course, be just as well represented by J^evdNDs, but the former expression is the more convenient for Gibbs' argument.) In cases where the solid has in part dissolved, 8N and 8N' would be negative in value. Thus we arrive at expression [357] for the variation of the intrinsic energy of the solid. We are not however concerned with this energy alone, nor with the entropy and mass of the solid alone. The system is heterogeneous and involves fluid phases also, and so we are led to the considerations dealt with in the remainder of page 187. Again the form of [358] may puzzle readers not acquainted with the methods of the calculus of variations, although the content or meaning of it should not be very much in doubt. The passage of matter and heat to (or from) the solid from (or to) the liquid will change the entropy Dt] and the volume Dv of a given elementary mass of the fluid by amounts 8Dr} and 8Dv; and in addition will alter the masses of the constituents Dmi, Dm2, etc., composing it. The condition laid down towards the end of page 187, which obviates the necessity of dealing with the internal equilibrium of the fluid itself, involves as a natural result the simplification that the integrations through- out the narrow layers of fluid between rigid envelop and solid are free from any troubles concerning original and present states, and do not require the use of accents to avoid ambiguity. Expression [359] embodies the fact that the potential energy of an element of matter 7n, raised through a height 8z, acquires potential energy of an amount ing8z. STRAINED ELASTIC SOLIDS 459 The method of deaHng with the variational equation [360] is essentially the same as that of dealing with the variational equation [15] in the early pages of Gibbs' discussion, although the presence of integral signs and merely formal differences of appearance betweert [15] and [360] may mask the identity of the methods. It would have been quite legitimate to write in [15] f f ft'h-q'v'dx'dy'dz' for t'hri, the integration being throughout the phase indicated by one accent, and so on; but it was unnecessary, as the conditions were uniform throughout any given phase in equilibrium. But for a solid the strain may be heterogeneous, and so ■qv might well change in value from point to point of the solid body with the changing values of an, ai2, . . . flss. Hence the necessity for the integral. Also if the strain were homogeneous we could write the second term in [360] as F'ZS'Xx'San, Y' being the volume (unstrained) of the solid; but in general this is not possible. Reflection on this and similar considerations for the remaining terms will remove any difficulty in understanding raised by pure differences of form. Following this hint we see that [361], [362] and [363] are the additional equations arising from constancy of total entropy, from constancy of the total volume of the system within the envelop, and from constancy of total mass of an independent constituent of the system; they are entirely analogous to equations [16], [17] and [18] respectively. Con- dition [361] is straightforward. In [362] we consider any element of the fluid Dv in the form of a thin disc lying between an element of surface Ds of the solid and a similar element of the rigid envelop. First of all the variation of the strain in the solid involves displacements hx, by, 8z of the point x, y, z, the center of Ds; thus Ds is displaced normally towards the envelop by abx + ^by + 'ybz. This reduces the volume Dv by an amount {abx -\- ^by + ybz)Ds. In addition the accretion of new matter reduces it also by bNDs or vvbN'Ds' as we saw above. These two causes therefore bring about a change 8Dv in Dv which is given by [362]. Equation [363] offers no difficulty. The subsequent reasoning leading to equation [369] is based on an application of Lagrange's method of multipliers, referred to and used earlier in Gibbs' discussion. 460 RICE ART. K (See Gibbs, I, 71-74.) The object of the method is to ehminate certain of the variations from the condition of equihbrium so as to leave in it only those variations which are independent of each other and are therefore completely arbitrary in their relative values. Those variations which can be regarded as arbitrary are the displacements of the points in the solid and on the surface arising from the arbitrary variation of strain in the soHd, and also the thickness of the layer of material deposited on or dissolved off the soUd. The object is partly attained by the time we reach equation [367] and the steps are fairly obvious; but in addition to bx, by, bz and bN' we have also the nine variations ban, ba^t, . . . baas. But as we have seen these are not independent of each other since straining only depends on six functions of an, a^, . . . ass- The step from [367] to [369] actually eliminates them all and replaces them by varia- tions bx, by, bz for points in the solid and on its surface. Gibbs is very brief at this point, and to elucidate the step made in [368] we shall have to make a short digression. The point P'{x', y', z') in the reference state is displaced to P(x, y, z) during the strain an, ai2, . . . 033- The additional strain ban, bai2, . . . bas3 displaces it still further to Psix -\- bx, y -\- by, z + bz). Hence the variation in the value of an, i.e., ban or b(Jdx/dx'), is equal to b{x + bx) dx dx' dx' Thus \dx') ~ dx' bx. Similarly <5)= a —,bx. dy (Note that x, y, z are definite functions of x', y', z' and x + bx, y -\- by, z -{- bz are also definite functions of x', y', z' slightly different in value from the former; thus bx, by, bz are also defi- STRAINED ELASTIC SOLIDS 461 nite functions, small in value, of x', y', z'.) On this account •'(S) Xx' Sail = Xx' 51 , = Xx' ^ , ^x, dx which on integrating by parts is equal to 9 . dXx' -, (X., Sx) - ^ Sx. Hence Xx' dan dx'dy'dz' = — {Xx' 8x) dx'dy'dz' dXx' ——r 8x dx'dy'dz'. dx' ^ The first integral on the right hand side, which is an integral throughout the volume of the soHd, can be transformed by Green's theorem into an integral over its surface, viz., fa'Xx'dxDs', and in consequence we obtain the result [368]. (Will the reader accept the truth of this transformation for the moment so as not to interrupt the argument? We shall return in a moment to Green's theorem for the sake of those unacquainted with it.) In a similar manner /dx\ Xr'-5ai2 = Xy' 8[ p. / j d = Xy> —, 8x dy d . dXy' = -, (Xy> ox) - -^ SX, 462 RICE ART. K and therefore Xy ban dx'dy'dz' = — (Xy 8x) dx'dy'dz' dy' - ff 'dX Y' T 8x dx'dy'dz' J J dy = U'iXy 8x) Ds' - j I j-^ 8x dx'dy'dz', and so on. When we make the substitutions in the first integral of [367] justified by these transformations, we convert equation [367] into the form [369]. It might be as well to write the first integral in [369] in full for the sake of clarity; it is f f f ( /dXx' dXy dXz'\ /dYx' dYy dYz'\ -^\^ ^~By^^^F)^y , /dZx' dZy dZz'\ \ , , , where of course 5a:, dy, Sz are to be regarded as functions of x', y', z', infinitesimal in value. Similarly the third integral written in full is /{ (a'Xx' + ^'Xy. + y'X,,)8x -\-(a'Yx' + /3'Fk' + YYz')5y + (a'Zx' + ^'Zy + yZzO^z }Ds'. We shall neglect for the moment the point raised at the bottom of page 189 concerning surfaces of discontinuity, returning to it when we give a proof of Green's theorem, and proceed with the general fine of development. Taking the result [369] we shall rearrange it so as to collect all the terms involving 8x, all those involving dy, all those involving 8z and all those involving 8N'. It is then written in the form STRAINED ELASTIC SOLIDS 463 9 Ax' dXr' dXi JO 'dYx' 9Fy' dY + (dZx' dZy' dZz' A I . . . + (a'Xx' + ^'Xy' + t'Xz') + av D£ Ds' 8x + + Dsl + (a'7.v' + /3'Fk. + 7'FzO + pp j^A 8y (a'Zx' + ^'Zy> + t'-^z') + TP;^J 5z\ds' ev - tr]v' + pvv - 2 (mi^i) ^^' ^^' = ^■ This is equation [369] written in full. Since, in the volume integrals, 8x, by, 8z are arbitrary varia- tions, the expressions multiplying them must be zero at all points of the solid in order that [369] may be true for any rela- tive values of 8x, 8y, 8z. Thus we arrive at equations [374]. In the second integral of our rewritten [369] the expressions multiplying 8x, 8y, 8z respectively must also be zero at all points of the surface for the same reason. Thus we arrive at equations [381]. There remains only the third integral in the rewritten [369]. If 8N' is quite arbitrary, i.e., if crystal- lization on the solid and solution from it are both possible we must accept the truth of [383] ; but if the values of 8N' can only be chosen arbitrarily from infinitesimal negative numbers, i.e., if solution only is possible, we justify only the wider conclusion [384]. At the bottom of page 190, Gibbs makes a passing reference to the stress-constituents Ax, Xy, . . . Zz i.e., the constituents measured across faces perpendicular to the same axes as those which indicate the directions of the thrusts or pulls involved in the definitions of the constituents. His proof of the equality 464 RICE ART. K of Xy to Yx, Yz to Zy, Zx to Xz is one of those succinct, sweep- ing statements which he makes from time to time with complete justification, but with a whole array of intermediate steps in the reasoning omitted, to the bewilderment of the reader not so well versed in analytical processes. It was in \ iew of the awkward situation at this point that we have in our discussion introduced and defined Xx, Xy, . . . Zz first, treating them in a manner which will have been familiar to any reader acquainted with modern texts on elasticity, and have already proved the equality of Xy to Yx, etc. Later, it will be recalled, we intro- duced Gibbs' more general stress-constituents Xx', Xy', . . . Zz' and gave some care to their precise definition and to the equa- tions (38) which connect them with Xx, Xy, . . . Zz. It will be apparent from these equations that in general Zy is not equal to Yx', for example. Let us, however, make the two sets of axes coincide so that an becomes en, etc., and ^^s, the first minor of Urs in the determinant | a \ becomes Ers, the first minor of Crs in the determinant \e\. Equations (38) will be replaced by equations in which Ers is substituted for A rs. Even so, as we pointed out earlier, Xx' does not become identical with Xx, etc., unless the difference between the state of reference and the state of strain is so little that a rectangular parallelopiped in the one is but little distorted from that shape in the other. To elabo- rate this latter point a little more, it will be observed that in such a case the determinant en ei2 eis 621 622 623 631 632 633 approximates to the form 1 612 1 — 612 — ei3 — ^23 for en, 622, 633 are little different from unity, and 623 + 632, etc., ei3 623 1 STRAINED ELASTIC SOLIDS 465 from zero. It appears that in such case £"11 approximates to unity since 623 is small and 1 + 623^ differs but little from unity. Similar statements are true of jE'22 and £'33, while E23, E32, etc., all approximate to zero for similar reasons. On examining the modified equations (38) it will appear that in the event of such coincidences Xx' approaches to Xx, Xy' to Xy, Xz' to Xz. We thus illustrate in another manner Gibbs' conception of gradually bringing not only axes of reference but the two states into coin- cidence. But it will be realized on a little thought that even if we have the states approximating to coincidence, but not the axes, the considerations just raised do not hold; for then an, an, ... 033 involve not only the actual elongations and shears but also the direction cosines of the axes OX, OY, OZ with reference to OX', OY', OZ' which change with any reorientation of the former relative to the latter. In consequence an, an, ... ass do not approximate to unity in general even for slightly separated states, and An, An, ■ ■ • ^ss do not tend to the values which are the limits of £"11, £'12, . . . Ess. Gibbs' own proof may now be clearer to the reader. From [355] dev' dev' Xy' = ~ — and Yx' = ~ — oax2 0021 Under the conditions of coincidence assumed ai2 approaches en and a2i approaches 621 in value. Hence the limit of Xy is dev/ den and that of Yx' is 967/9621 since under these circumstances ev ' approaches ev. Now actually ev is a function of /e, and /e becomes in the limit 612 + 621- Since therefore in the limit and dev den dev 9/6 9/6 9ei2 dtv ~ 9/6 dev 9621 dev ~ 9/6 9/6 9621 dev ~ 9/6 it follows that Xy which is the limit of Xy is equal to Yx which is the limit of Yx'. The reference in Gibbs to the difference 466 RICE ART. K being equivalent to a rotation simply recalls the fact that in the analysis of strain the e^ and 621 coefficients involved the strain through their sum and a rotation around the axis OZ through their difference. (See equations (7) of this article.) The reader may at this point feel a little mystified about making the states of reference and of strain coincide ; for in such case he may well ask, how can one have stresses at all. If he will refer to the top of page 185, and read over the remarks on this point by Gibbs, he will feel once more that they are too brief to be very illuminating. The essential point is this. We are after all not treating the state of strain itself and its relation to a state of reference which is physically an unstrained state; we are treating other states of strain obtained by slight deformations from the state of strain in question, involving variations of an, etc.; and for that purpose it does not matter what particular state, strained or not, we take for a state of reference. The position is similar to the treatment of the geometry of a surface. There we are considering the relations of points on a given geometrical locus to some other geometri- cally relevant point (e.g., spherical surface to center, cone to apex, etc.) and it does not matter theoretically what particular set of axes we set up for assigning coordinates to the points in question. We choose in each case a set which is practically the most convenient. To give as wide a theoretical basis as possi- ble to his analysis, Gibbs does not confine himself to any partic- ular set of axes or any particular state of reference; but he does at this point make a passing reference to those axes and states which in practice are the most convenient by reason of the simplifications which they make possible, and to which we con- fined ourselves, for that reason, at the outset of our discussion of elastic solid theory. Before we go on to comment on pages 191-207 in which Gibbs goes into certain details connected with equations [374], [381] and [383], it will be as well to dispose of the question of discon- tinuity referred to at the bottom of page 189. We have already mentioned that in deriving [369] from [367] Green's theorem is used. This theorem states that, if <^(a:', y', z') is a function which is continuous, one-valued and finite throughout a region of STRAINED ELASTIC SOLIDS 467 space bounded by a surface s', then the three following rela- tions are true ^, dx' dy' dz' = \ a'4> Ds', ox I 30 dy -, dx' dy' dz' = / l3'(i> Ds', ^ dx' dy' dz' = / y'(}> Ds', dz I where the volume integrations are to be taken throughout the re- (i'K'\) Fig. 7 gion bounded by s' and the surface integrals over s' . Figure 7 illustrates the proof of the first equation. The region is divided by up into elementary columns parallel to OX' , whose sections by planes parallel to OY'Z' are elementary rectangles, bounded by sides parallel to OY' and OZ' . Let us integrate {d(f)/dx')dx'dy'dz' throughout that part of the region contained in one of the columns which intersects the surface in two elements of area Dsa and Dsb' at the points A and B; the result is in the limit equal to the product of the definite integral / {d(}}/dx')dx' by Jb the sectional area of the column. Now the definite integral is equal to <}>a — B, where (J)a and 0b are the values of 0(x', y', z') 468 RICE ART. K at the points A and B respectively. Also if a^', 13/, Ja and an', ^b', Jb' are the direction cosines of the outward normals to s' at A and B, respectively, then u/Dsa' and —cxb'Dsb' are each equal to the sectional area, since the sectional area is equal to the projection of either of these sections by the surface on the plane OY'Z', and a is the cosine of the angle between the normal to an element of the surface and OX', which is normal to OY'Z'. (The figure shows that the minus sign is necessary in one of the results, since in one case the normal directed outwards will make an obtuse angle with OX'.) Hence the result of integrat- ing (d(j)/dx')dx'dy'dz' throughout the part of the region within this column is equal to aA(i>ADSA + aBBDSB. Adding similar results for all such columns and passing to the limit we obtain the first of the relations given above. The re- maining two are obtained by employing columns parallel to OY' and to OZ'. In the derivation of [368] by means of this the- orem the function 4> is Xx'^x. Suppose, however, that in the above proof (i>{x', y', z') is dis- continuous at a certain surface s" which divides the region of integration into two parts, li AB (Fig. 8) intersects this sur- face s" in C then as we approach C in passing along BA from B the function {x', y', z') reaches as a limit a value ci which differs finitely from the limit /dx') dx' dy' dz' first along a column stretching from B to C taking 0ci as the value at C, and then along the column from C to ^ taking 0^2 as the value at C. In this way we ar- rive at the result —f dx' dy' dz' (throughout the column) = as' 4>B Dsb' + aci" ci DSc" + otc-l' 0c2 -DSc" + ola! <^a Ds/, where the direction cosines with the suffix 1 are for the normal to Dsc" directed outwards from the first part into which the STRAINED ELASTIC SOLIDS 469 region is divided by s", and those affected by the suffix 2 for the normal directed outwards from the second part. (Of course a/' = -ai",^i" = -182", 7i" = -72".) On adding results for all the columns we obtain the result 9^ dx -, dx' dy' dz' = j a> Ds' + j{a," 4>x + «2" .^2) Ds", and two similar results can be derived by using columns parallel to the axes OY' and OZ'. If considerations such as these are given their due weight when discontinuities in the nature and state of the solid exist, it Fig. 8 follows that in [369] a further term must be included on the left hand side, viz., the integral over such a surface of discontinuity, represented by where bx, by, 8z, whether in the terms affected by the suffix 1 or in those affected by 2, refer of course to the same variation, viz., the variation in position of a point on the surface of discontinuity arising from an arbitrary change of strain; since this is just as arbitrary as the variation of any other point in the interior of the solid or on the surface bounding the solid, we 470 RICE AHT. K must conclude that the three factors in the integrand multiply- ing 8x, 8y, 8z are severally zero, and so we arrive at [378]. (The doubly accented direction-cosine symbols used in the argument for the sake of distinction between s' and s" are, of course, not required any longer.) The expression referred to in [379], and the two similar expressions are of course the expressions in (29a) of this article, except that the former are the com- ponents of the stress-action at a surface on an area which was unit size in the state of reference, the latter on one which is unit size in the state of strain. The interpretation then put on [378] is obviously necessary for the equilibrium of an internal thin layer of the solid, bounded by two surfaces parallel and near to the surface of discontinuity, one in one part of the solid and one in the other. 8. Commentary on Pages 191-197. Discussion of the Four Equations of Equilibrium. Let us now resume the commen- tary on details in pages 191-197. The equations [377] are a particular case of (29) of this article in which the compo- nents Fx, Fy of the force per unit volume are zero and Fz = —gV. (Remember that OZ is directed upwards so that gravity is in the negative direction of OZ.) The meaning of the remarks which immediately follow concerning [375] and [376] may perhaps not be obvious to all readers at first sight. When we proved these equations in this exposition, we assumed that the solid was in equilibrium, but strictly this assumption was un- necessary. For if we refer once more to the proof leading to equation (30) and do not assume equilibrium, we must put the couple on the element of volume arising from the stresses of the surrounding matter and from the body forces on it equal, not to zero, but to the sum of the moments of the mass-acceleration products of the various particles of the element; i.e., to the product of the moment of inertia of the element and the angular acceleration. Now, without going into too much detail, this moment-sum, like the moment of the body forces, involves terms which have as a factor the product ^rjf and a length of the same order of magnitude as ^, 77 or f . In consequence it is evanescent, just as is the moment of the body forces, in comparison with the moment of the stress-actions, and the same result follows as STRAINED ELASTIC SOLIDS 471 before. In consequence [375] and [376] are true in conditions other than those of equiUbrium; they express in fact, as Gibbs says, "necessary relations," — necessary, that is, in the sense that otherwise there would be involved a contradiction with the laws of dynamics in situations more general than those con- sidered in the text. The equations [381] should be compared with (29a) of this article, in which the expression {aXx + fiXy + yXz)Ds is the stress-action across Ds in the direction OX of surface matter on interior matter, and — apDs is F^Ds, the a;-compo- nent of the external force on Ds. The difference here is purely formal, since (a'Xx' + ^'Xy' + y'Xz')Ds' is still the stress- action of surface matter on internal matter across the same element of area which was Ds' in the state of reference. The transformation of the equations to the form [382], which in- volves throughout the direction cosines a', ^', y' of the element in its state of reference, can be obtained at once without going through the argument in Gibbs, I, 192, 193; for we have already considered that argument in somewhat greater de- tail when proving equations (18) and (27). The notation we used in our discussion allows us to write equations [382] more fully, thus, a'Xx' + /3'Xr + y'Xz, + p{a'An + /8'^i2 + y'A,^} = 0, and two similar equations, since by (27) Ds ( Ka\ and An is the second minor of On in the determinant | a\, i.e., All = 0,22(133 — 023^32; dy dz dz dy ^ dy' dz' ~ dy' dz'' and so on. We pass on to the arguments based on equation [386] or [387]. The symbols p and mi refer of course to the surrounding 472 RICE ART. K fluid (ni being the potential of the sohd substance in the Hquid) ; €v,r]v and r, to the sohd. The subsequent discussion is Umited to the case of a sohd body which is not only homogeneous in nature, but also homogeneous in its state of strain. The first point considered by Gibbs is concerned with the conditions under which this latter proviso is compatible with a uniform normal pressure over any finite portion of the surface. (The effect of gravity, the only body force considered in the general discussion preceding, is disregarded as negligible in producing heterogeneity of strain or variation in the value of pressure at different points of the surface.) This leads at once to Gibbs' discussion concerning the three principal axes of stress on pages 194 and 195. We need not comment on this, as we have already proved the necessary propositions in our exposition, starting from an expression similar to [389]. Gibbs' proof is an analyti- cal one based on the methods of the calculus as applied to questions of maximum-minimum values of functions of several variables, and will be easily followed by those acquainted with these methods, whereas the method we have used, being based on the elementary geometrical properties of the stress- quadric will probably be intuitively perceived by those not so well versed in mathematical analysis. Actually, if we revert for a moment to the form of equations [382] which we have written above, the conclusions arrived at in the paragraph which includes the equations [393], [394], [395] can be obtained in a very direct and suggestive manner. Equations [382] in our form can be written thus : (Xx' + Anp)a' 4- (Xr> + Ay,p)l3'^ + (Xz' + A,sp)Y = 0, (Yx' + Anp)a + {Yy> + A,,p)^' + (Yz' + A2zp)y' = 0, {Zx' + A3ip)a' -t- {Zy + A32PW -f (Z^, + Anp)y' = 0. If the solid is in a given homogeneous state of strain, Xx', . ■ ■ Zz', > [382a] STRAINED ELASTIC SOLIDS 473 an, ... ass are all constant and given in value throughout the solid. The same is true of the first minors ^u, . . . ^33. In con- sequence [382a] combined with form a system of four equations to determine four "unknowns" a, /3', 7', p, which will thus yield not only definite values of the fluid pressure, but also definite orientations of the solid surface compatible with this assigned state of strain. To see how many definite values and orientations are involved we consider [382a] carefully. Suppose that a definite value is assigned to p ; this would give us three simultaneous equations to determine the values of the unknown a', /3', 7', at least apparently. In reality, however, we should have three equations to determine two unknowns, viz., a/j' and 13' /y'. In short we have one equation too many; values of a'/y' and jS'/t' which v»^ould satisfy the first two would not necessarily satisfy the third, unless a special relation existed between the nine coefficients. The relation embodies the fact that the determinant of the nine coefficients is zero, i.e., Xx' + Aiip Xy' + A12P Xz' + Anp Yx' + A21P Yy> + A22P Yz' + Aizp Zx' + A31P Zy' + A32P Zz' + Azzp = 0. Without actually multiplying this out, the reader will realize that the left-hand side is an expression involving p, p^ and p^. The equation is a cubic in p. Hence there are only three values of p which are compatible with the state of strain. They are the roots pi, p^, pz of this equation. If we insert one of these values, say pi, into the first two of [382a] we can solve for the ratios a'/y', ^' /y', and combining these with a'^ + /8'^ + 7'^ = 1, we obtain values of a, /3', 7', say a/, /S/, 7/. Actually, as is obvious, —a/, — jS/, —7/ will also satisfy the equations. (Not of course —a/, jS/, 7/ nor any triad with an arrange- ment of signs other than the two mentioned; for these would give ratios not satisfying [382a].) Inserting p^ and pz we find 474 RICE ART. K that once more only a pair of orientations, given by a^, fi-i, 72'; —OC2, —^2^ — ji' qm6. oii , ^2! , 73'; —0:3', — jSa', —73', are com- patible with these pressures respectively and the given state of strain. Furthermore, it can be proved from the equations that «!'«/ + iS/iSa' + 7/72' = 0, cii'az' + /32'/33' + 72'73' = 0, az'ai' + /33'/3i' + 73'7i' = 0, showing that the three directions are normal to each other; but the proof would lead us too far into the theory of such deter- minantal equations. Indeed, as doubtless many readers know, the analysis is quite similar to that employed in analytical geometry when determining the directions of the three principal axes of a quadric surface, and in fact Gibbs derives the result by a direct appeal to the existence of the three principal axes of stress which will, of course, have the same directions at all points of the solid if the strain is homogeneous. These directions are in fact the directions on', fii , 71'; 0:2', ^2, 72' and az, 183', 73'; and pi, P2, Ps are respectively —Xx, —Yy, —1z if the analysis of the stress-constituents has been referred to these principal axes as the axes of reference in the state of strain. (Xy, Y z, Zx, etc. are of course each zero in such case. In order to avoid con- fusion we have thus far had to use suffixed symbols for the three pressures instead of accented symbols; for the use of ac- cented symbols to indicate measurements in the state of refer- ence makes it awkward to use them for any other purpose, such as distinguishing three different values of a quantity. How- ever, as the subsequent treatment will not require the use of direction-cosine symbols, we shall revert to Gibbs' notation and substitute p', -p", jp'" for pi, p-i, pa.) In this way the important conclusion emerges that only three fluid pressures are compatible with an assigned homogeneous state of strain of the solid in contact with the fluid, and if one of these pressures is established in the fluid, the solid, if equilib- rium is to be preserved, can only be in contact with it at a pair of plane surfaces whose normals are opposite to one another in direction. Of course, this is a general statement; there are STRAINED ELASTIC SOLIDS 475 special cases where wider possibilities can exist. If, for instance, in the state of strain the three principal stresses are equal to one another, the "stress quadric" is a sphere; all sets of three axes are principal; there are no shearing stresses for any axes. (See case (3), Gibbs, I, bottom of page 195.) This is in fact the case of ''hydrostatic stress" referred to frequently in these pages by Gibbs. In such a state the form of the solid does not matter. Immersed in a fluid throughout which there exists a constant pressure a sohd will be in a homogeneous state of strain compatible with the condition of hydrostatic stress, that is, the condition in which there are no shears and the stress over any surface is normal to it and is of the pressure type. (The reader should not misconceive the phrase "homogeneous state of strain." This implies that an, an, • • • ass have values which are severally constant throughout the solid. But there is no implication, for instance, that an = a22 = 0,33- It should be clearly recognized that this is not necessarily the case even for a state compatible with hydrostatic stress. It would be so, no doubt, if the solid were isotropic in nature; in that event all linear contractions or extensions would be equal and no shears would exist, but for crystalline solids the more general nature of the stress-strain relations would permit of wider conditions of strain, even if for any set of axes Xx, Yy, Zz were equal to one another, and the remaining stress-constituents zero.) If, how- ever, one is to maintain the rectangular parallelopiped of solid material, imagined by Gibbs at this juncture, in equilibrium in a general homogeneous state of strain, one must arrange for different pressures on the different pairs of faces. So if the solid is in contact with a fluid of suitable pressure at one pair of opposite faces, it cannot be so at the other two pairs. It must be constrained by some other surface forces (pressural or tensional) on these faces to maintain the assigned state of strain. If these constraints are released and the fluid comes into contact with all six faces there will be an immediate change to another state of homogeneous strain compatible with the condition of hydrostatic stress. In such a change there will be a diminution of intrinsic energy of strain, since all release of constraints if followed by movement converts potential energy into kinetic 476 RICE ART. K energy of sensible masses, or heat. This justifies the brief statement of Gibbs on page 196 near the bottom: "This quantity is necessarily positive except, etc." The remarks so far have been concerned with mechanical equilibrium. Equation [388], rewritten for the three possible pressures in [393], [394], [395] involves equilibrium as regards solution of the sohd in the fluid, or crystallization on the solid from the fluid. This amplification of Gibbs' treatment of the mechanical relations will, it is hoped, render the task of master- ing these pages easier for the reader; there appears to be noth- ing of special difficulty in the deductions on page 197 concern- ing the supersaturation of the fluid. It should be carefully borne in mind that the argument has been confined to a homogeneous state of strain in the solid. Gibbs remarks on page 197 that "within certain limits the relations expressed by equations [393]-[395] must admit of realization." But even if it were hardly practicable to make the special arrangements conceived in these arguments, that does not invalidate the conclusions. We are all thoroughly familiar with "perfect engines," "perfectly smooth surfaces," "perfect gases" and other conceptual devices of the physicist and chemist which are the "stock in trade" of many mechan- ical and thermodjTiamical arguments. Of course in any prac- tical case, if a solid of any form immersed in a fluid were subject to distorting surface forces the strain would be hetero- geneous. Perhaps some readers, recalling equations (29) of this article or [377] of Gibbs, might wonder how a hetero- geneous state of strain can exist without body forces; for in such a case the equations referred to would become dXx dXy dXz dYx dYr dYz _ dx ~^ dy ~^ dz ~ ^' dZx dZr dZz ox dy dz (We are neglecting gravity.) One might rashly conclude from STRAINED ELASTIC SOLIDS 477 these that Xx, Xy, • • • Xz must individually maintain constant values throughout the solid, and that the strains, therefore, being definite functions of these, would also be uniform in value throughout ; but the conclusion is unwarranted, as the equations do not assert that each of the nine differential coefficients is zero. The torsion of a bar by gripping in the hands and twisting is an instance of heterogeneous strain under surface forces, which will be familar to all readers who have a special acquaintance with text-books of elasticity. 9. Commentary on Pages 1 97-201 . The Variations of the Tem- perature of Equilibrium with Respect to the Pressure and the Strains. The Variations of the Composition of the Fluid. At the bottom of page 197, Gibbs begins an argument leading to equations [407] and [411]. Equation [407] is the analogue of the well-known equation, first discovered by James Thom- son, giving the alteration in the melting point of a solid due to the increase of pressure on the surface. Perhaps if we put the analysis in a more general form than in the text it may assist the reader. We make no special arrangement about axes. The unit cube in the state of reference becomes in general, in the state of strain, an obhque parallelopiped whose volume has changed to y^/, which as we have seen is equal to the determinant an Ol2 ai3 an ^22 ^23 asi az2 flss A pair of opposite faces of the cube are in contact with the fluid in the state of reference and in the state of strain, so that one of the principal axes of stress is normal to this pair of faces of the oblique parallelopiped, the assigned homogeneous state of strain being maintained by suitable surface constraints on the remain- ing pairs of faces. Let there be an infinitesimal change to a new condition of equilibrium; this will involve changes of the strains to an + dan, an + dan, ■ ■ ■ ass + dazs, of the fluid pressure to p + dp, of the temperature to t + dt, of the potential ni to Ml + dni, and of the energy and entropy of the soHd to e + de 478 RICE ART. K and ri -\- dr). There is no change in the mass of the solid, but its volume will change by an amount given by dvv = Andan + ^i2 dp dp ) (dfll {t, P, nir) dm (t, p, Mr) + m< ~ — :; dm2 + 1 dnia + etc. ( dm2 drriz = {Xx' + An p) dan + {Xy + An p) dan ■ • • + {Zz' + ^33 p) dass. 480 RICE ART. K (In this iii(t, p, Wr) is a contraction for ni{t, p, mi, mo, ms, . . .) indicating the functional dependence of m on t, p, mi, m2, mz, . . .;m is of course the mass of the soHd.) The treatment by Gibbs on pages 198-201 is based on certam geometrical postu- lates. In the state of reference he chooses lines parallel to the edges of his unit cube as axes of reference. In the state of strain he takes OZ to be perpendicular to the faces in contact with the fluid, i.e., to be one of the principal axes of stress. The other two axes OX, OY are of course in the plane containing the other two principal axes of stress, and one of them, OX, is chosen so as to be parallel to one of the edges of the oblique parallelopiped. Thus all points which have the same s'-co- ordinates in the state of reference have the same s-coordinates in the state of strain; in consequence ^ is a function of z' alone being independent of x' and y', and so a^i and 032 are zero. (See [398].) Moreover all points which have the same y' and z' co- ordinates in the state of reference, i.e., lie on a line parallel to OX', have the same y and z coordinates in the state of strain. Thus yisa, function of y' and z' and is independent of x', and so 021 is also zero, (again see [398]). From this point on he pursues the analysis as above with the absence of certain terms which vanish on account of the conditions «21 = «31 = ^32 = 0. Thus the determinant of the ar, coefficients becomes an ai2 «13 0 ^22 «23 0 0 a33 which is just aiia22as3 as in [402]. The reader will find no difficulty now in following the steps in the remaining three pages, having had these postulates explained and having followed the argument already in a more general manner. Finally, before leaving this sub-section we shall refer to the remark at the top of page 199. The increase in the energy of STRAINED ELASTIC SOLIDS 481 the solid during the infinitesimal strain is as usual Xx'daii + XY'dai2 . . . + Zz-dazz. This is of course equal to the work of all the surface forces during the variation of strain. These surface forces may be regarded as due to the pressure p on all the faces (a hydrostatic pressure) together with additional forces on four of the faces. The work of the hydrostatic pressure is —'pdv which is equal to — p(Aii^aii + Avidan . . . + Azzdaz^. Hence by subtracting this from the increase of energy of strain we obtain the work of the additional forces and this is seen to be equal to the right hand member of our [404a], and becomes the right hand side of [404] when Gibbs' special geometrical con- ditions are assumed. 10. Commentary on Pages 201-211. Expression of the Energy of a Solid in Terms of the Entropy and Six Strain-Coefficients. Isotropy. Having discussed the conditions of equilibrium Gibbs proceeds in the subsection on the Fundamental Equations for Solids to consider the problem of expressing the functional re- lationship between the energy per unit volume, the entropy per unit volume and the nine strain-coefficients. If ck- is expressed as a function of -qv, an, an, . . . azz, or i/t' is expressed as a function of t, an, an, . . . azz, we can by differentiation obtain, as we have already pointed out in this article, the stress-strain relations, which will be nine of the eleven independent relations referred to by Gibbs on page 203 . He opens the subsection with some rather involved considerations on a special point, which we pass over for the moment, and then briefly touches on the fact that the energy or free energy functions must have a special form in the nine strain-coefficients, inasmuch as the strain of an element is capable of only six independent variations. This we have already explained in our discussion, where we chose the six quantities /i, f^, ... /e to represent the displacements arising from pure strain, as distinct from possible additional dis- placements involved in the nine coefficients an, an, . . ■ azz, which are the result of a pure rotation and produce no distortion of the 482 RICE ART. K material. The fr quantities are themselves functions of the six quantities ei, e-i, ... ee (or ai, a2, ... ae) which are the same as A, B, C, a, h, c defined in [418], [419]. Thus the energy or free- energy functions must be functions of these six quantities, or in other words "the determination of the fundamental equation for a solid is thus reduced to the determination of the relation between ev, riv, A, B, C, a, b, c, etc." (page 205). Having pointed this out Gibbs at once proceeds to discuss a further limitation on the form of these functions if the solid is isotropic, and this involves him at once in an appeal to the existence of three principal axes of strain for any kind of material, a fact to which we have already referred in this article. Thereafter he deals with approximations to the form of these functions and concludes this subsection on that topic. Let us proceed to the subject matter of pages 205-209 of the original which has been treated in our discussion in a somewhat different manner. The starting point of Gibbs' treatment is the equation [420] and this has already appeared implicitly in this article. For we know that if P' and Q' are the positions in the state of reference of two adjacent points, and P and Q are their positions in the state of strain, then PQ' = air' + a2v" + asf" + 2a4Vr' + 2a,^'^' + 2ae^'r,', where x', y', z' and x' + ^ , y' + tj', z' + f ' are the coordinates of P' and Q! and ai, ai, az, ai, as, ae are six functions of the strain coefficients defined in (23), or, as already stated, the same functions which Gibbs defines in [418] and [419] denoted by the symbols A, B,C, a, b, c, respectively. If a, ^', y' are the direc- tion-cosines of P'Q' with reference to the axes OX', OY', OZ' so that a' = ^'/P'Q', etc., it follows that PQ" aia'2 -f- a2)3'2 + asj'^ + 2a,^'y' + 2a,y'a' + 2a6a'^' = =^ = 7- P'Q' which is just Gibbs' equation [420]. The method pursued by Gibbs at this point to demonstrate the existence of the principal axes of strain employs the analyti- cal processes associated with the discovery of maximum- STRAINED ELASTIC SOLIDS 483 minimum conditions of a function of several variables, and resembles that employed by him on pages 194, 195 when demonstrating the existence of the principal axes of stress. It will be followed easily by those versed in such analytical methods, but for other readers not so well acquainted with mathematical technique we can give a geometrical flavor to the argument which may prove helpful. We saw in the previous discussion that is the equation of a locus drawn round the local origin P' which is strained into a sphere around the center P. This locus is an ellipsoid, and its actual form and the orientation of its principal axes in the body are of course dependent entirely on the magni- tude and nature of the strain and not at all on the particular choice of the axes of reference, OX', OY', OZ'. We have already seen in this article that the principal axes of this "elongation ellipsoid" experience no shear and so are the principal axes of strain, and we can therefore proceed at once to the deduction of equations [430] and [431] on page 207. The method is well known to students of analytical geometry. Suppose that R' is a point in which one of the principal axes of this elongation ellipsoid through its center P' cuts the surface, and let its local coordinates be ^Z, tji', f/. We know that the direction cosines of the normal at P' are proportional to But since P'R' is along a principal axis, the normal at R' coin- cides with P'R' and so the direction cosines are also proportional to ^i, r}i', fi'. Thus the three quantities fli^i' + aem' + ctBfi' fle^i' + a2Vi' + «4fi' ; ' ; ' F~' ' 484 RICE ART. K have the same value. So it appears that if a, /3', 7' are the direction cosines of any one of the three principal axes then aia + ae/S' + a^y' aea + a2/3' + 0*7' asa' + ttifi' + 037' pa', pt', where p is a multiplier still undetermined, but the same in all three equations. These, combined with the equation Q,'2 _j_ ^'2 _|_ y'2 = X, are sufficient to determine, first the value of p, and then the values of a', ^', y' in terms of the six strain-func- tions, tti, 02, ... a 6. The analysis is exactly similar to that which we employed earlier when explaining the conditions for the existence of a homogeneous strain in a solid in contact with a liquid. We write the preceding equations in the form (ai — p)a + ae/S' + 057' aea' + (a2 - p)l3' + 047' a^a' + a^jQ' + (as - p)7' [429a] (The reader will easily satisfy himself that these are the equa- tions [429] with p substituted for rl) Now, for reasons which we have already discussed in the place just referred to, these three equations are not consistent with one another unless the follow- ing determinantal equation is true: ttl - P tte ae 02 Ob tti as O3 — P = 0, and this is actually equation [430], with p substituted for r^. It is of course a cubic equation in p and can be written, on expanding the determinant, as Ep^ -\- Fp - G = 0, where E F ai + a2 + as, a2a3 + azai + aia2 ai^ — as^ a6^ fll fle as as ^2 a4 as tti as STRAINED ELASTIC SOLIDS 485 G = a6 di as = aia2a3 + 2a4a5a6 — aiQi"^ — a^aC" — aza^. (See equations [431], [432] [433], [435].) This equation in p has three roots pi, p2, ps, functions of course of E, F and G; if one of these roots is substituted for p in any two of the equations [429a] above we can solve for the ratios «Vt', fi'/y' and thus, using the condition a'^ + ^'^ + y"^ = 1, determine a , /3', 7' for one of the axes; the remaining two values p2, P3 determine similarly the other two axes. It remains to interpret the physical meanings of pi, p2, ps, and that offers no difficulty. We saw above that if r is the ratio of elongation parallel to any direction a, /S', 7' then ^2 = a^a'^ _|_ a2/3'2 + 037'^ + 2a4i8'7' + 2a57'a' + 2a,a'^' = {a,a' + ae/S' + a57')«' + («6a' + ag/S' + aa')^' + (asa + a4i8' + a37')7'. If now a, jS', 7' is the direction of the first principal axis, then, since aia + ae/S' + 057' = pia', etc., it follows that = Pi- Similarly p2 = r^"^, pz = ri^. The remaining steps now follow easily. By the well-known relations between the roots and coefficients of an equation of integral order in one unknown we have Pi + P2 + P3 = -E", P2P3 + psPi + P1P2 = F, P1P2P3 = Gf and these are just equations [439], which we obtained in this ART. K 486 RICE article by another method. (As mentioned at that point a straightforward, if tedious, piece of algebra will show that 0203 + ascti + aia2 — 04—05 — 0 6 = Al + Al^^... +A 2 33' where Apg is the first minor of Opg in the determinant of the coefficients, viz. H. This gives the alternative expression for F in [434]. Also, we have already seen that the rule for multi- plying determinants will verify that H^ = G.) A rather special point is raised and disposed of on pages 210, 211. It concerns the sign of the determinant H. It is clear from [439] that G is a positive quantity, but H may, of course, have a negative value instead of a positive one from a purely mathematical stand- point; but from a physical standpoint negative values of H are ruled out, provided we agree that the axes OX', OY', OZ' and OX, OY, OZ are capable of superposition, meaning that if the latter are turned so that OX points along OX', and OY along OY', then OZ will point along OZ' (not along Z'O). In short, if one set of axes is "right-handed" the other must be likewise, if one is ''left-handed," so also is the other. (A right-handed set of axes is one so oriented that to an observer looking in the direction OZ', a right-handed twist would turn OX' to OY', etc.) Gibbs illustrates this by considering a displacement of the particles which is represented by X = x', y = y', z = -z', the two sets of axes being regarded as identical. (If they were not they could easily be made so by a rotation.) Now the H determinant of this is 1 0 0 0 1 0 0 0 -1 whose value is —1. But such a displacement is one which moves every particle to the position of its "mirror image" with respect to a mirror imagined as located in the plane z' = 0, i.e, STRAINED ELASTIC SOLIDS 487 OX'Y'. This displacement cannot be effected by any simple rotation. (A rotation of the body for example round the axis of OX' through two right angles would be represented by the equations X = x', y = -y', z = -z' whose U determinant has the value +1.) Indeed, to produce the displacement indicated we would have to conceive a con- tinuous distortion of the body in which all the particles of the body would have to be gradually "squeezed" towards the plane OX'Y' , the body growing flatter and more "disc-like" until it is squeezed to a limiting volume zero; thereupon it would begin to swell again to the same size as before, but with all the particles previously on the positive side of the plane OX'Y' now on the negative, and vice-versa. Such a process while conceivable is hardly possible physically. It should be noted that in the course of such a conceptual continuous process the volume would pass through the value zero; also the determinant H, which is the ratio of volume dilatation, would pass through decreasing small values from unity to zero, then change to negative values and grow numerically (decreasing algebraically) to the limiting value —1, as we indicated above. This short discussion will perhaps help the reader while perusing pages 210, 211. We now revert to the short paragraph beginning near the top of page 205 with the words "In the case of isotropic bodies." Unless the reader is on his guard the position of this paragraph in the general argument might unconsciously incline his mind to the view that the subsequent discussion concerning principal axes of strain is only valid for isotropic solids, and this would be unfortunate. Nothing in Gibbs' own argument nor in that given earlier in this article warrants such a restriction. No mat- ter what the nature of the solid, any group of external forces will produce a distortion and a system of stresses such that there are in any element three principal axes of strain for which the shearing strain-coefficients d, Ch, ee vanish, and three principal axes of stress for which the stress-constituents Yz (or Zy), Zx (or Xz), Xy (or Yx) vanish. If the strain is homogeneous 488 RICE ART. K the principal axes of strain are oriented alike in all elements; that will also be true of the principal axes of stress if in addition the body is homogeneous in nature. But it will naturally occur to the reader to inquire whether the principal axes of strain are coincident with those of stress, and indeed this query and its answer is just the matter at issue at this point in Gibbs' text. A few lines before, Gibbs refers to the now familiar fact that the state of strain (as distinct from rotation) is given by six func- tions of the strain-coefficients an, a^, . . . ass, choosing, for reasons now fully discussed, ai, . . . ae as these functions (or A,B, C, a, b, c, as he styles them) and points out that for any material, homogeneous in nature or not, isotropic or not, the energy per unit volume will be a function of the entropy per unit volume and the six strain-functions. This we have already discussed in the present article. For isotropic materials, however, there is a certain simplification, three functions of the strain-coefficients being sufficient for this purpose. Gibbs derives this result from the sentence at the end of the short para- graph referred to above, namely the sentence: "If the unstrained element is isotropic" (the italics are the writer's) "the ratios of elongation for these three lines must with rjv determine the value of €v'." Now this is hardly obvious without some further consideration of the meaning of isotropy in this con- nection. Space does not permit us to discuss the matter fully, but the central idea can be indicated. The essential character of an elastically isotropic solid is embodied in two facts. 1. For any system of external forces the principal directions of stress in any element are identical with the principal direc- tions of strain. 2. The number of elastic constants required to express the relations between stress and strain for small strains is two. Thus if we take the axes of reference to be parallel to these principal directions, we have the extremely simple stress-strain relations (in the conventional text-book form) Xx = X3 + 2/xeii, Yy = X8 + 211622, Zz = X5 4" 2^1633. STRAINED ELASTIC SOLIDS 489 In these equations X and m represent the two elastic constants, 8 is the sum of en, 622, 633 being known as the "dilatation." (623, ^32, esi, ei3, 612, 621 as well as Yz, Zx, Xy are zero.) The various moduli can be expressed in terms of X and n. (In fact /x hap- pens to be the modulus of rigidity itself.) Indeed the idea of isotropy may be broadly indicated by reverting to an illustration which we gave in a rather vague form at the outset of our exposition. Imagine a system of forces to be exerted on a body, spfierical in shape, at definite points of the body. These will produce a system of strains and stresses. In a given element there will be a common triad of principal directions. Now conceive the body to be rotated round its center to another orientation, but conceive also that the same forces as before are acting, not at the same points in the body, but at the same points in the frame of reference, i.e., points with the same coordinates with respect to the axes of reference, which we regard as fixed. Exactly the same system of stresses and strains will be produced as before. This does not mean that the element referred to above (i.e., the element occupying the same situation in the body) will be strained just as before; but the element of the body occupying the same situa- tion in the frame of reference will experience the same strains and stresses as were experienced previously by the element originally in that situation, with the same orientation for the principal axes. (It must be carefully borne in mind that this is true for isotropic bodies only; in fact it constitutes a definition of isotropy in elastic properties.) The energy of the spherical body after the rotation is the same as before. This gives us the key to the situation. Such a rotation would be equivalent mathematically to referring a strained body first to any axes of reference (not necessarily principal axes of stress or strain) and then referring to another set; equivalent in fact to what the mathematician calls a "transformation of axes." The values of the strain- coefficients and strain-functions will change. In the first set of axes OX', OY', OZ', ai, o^, az, at, a^, ae are the strain-functions and ^', r]', f ' the local coordinates. The elongation-ellipsoid is ax^" + a,-n'^ + az^'^ + 2a,r]'^' -{- 2a,^'i' + 2ae^'r,' = k\ ( 490 RICE ART. K Now we rotate the axes of reference to OU, OM', ON'. Let the strain-functions for these axes now be cxi, a2, as, on, as, ae and the local coordinates X', yJ , v' . Of course ai is not in general equal to ai, nor a^ to a^., etc.; for ai is the ratio of elongation parallel to OU , while ai is that parallel to OX', etc.; and aii/{ocia2)^ is the shear of OL' and OM' while a6/(aia2)^ is the shear of OX' and OY', etc. But the equation aiX'2 + «2m" + oizv'^ + 2a4/x'''' + 2a5/X' + 2a6X'M' = ^' represents just the same elongation-ellipsoid as before, situated in the same way in the body. Let the function which expresses the strain energy in terms of ai, 02, ... a& be 0(ai, a^, ... aa). Exactly the same function of ax, ai, ... a a must also be equal to the strain energy. This must be so on account of the isoiropy. In the illustration above, assume the sphere to be strained homogeneously for simplicity, and refer to any axes of reference. Keeping the forces as it were "in situ," we rotate the sphere and axes. The energy is unchanged. But the mathematical con- s "derations leading us to a certain function of ai, 02, ... Oe which is equal in value to the energy will lead us in the second case to just the same function of ai, ai, ... ae; for the general oper- ations are unchanged by a change of axes and just the same re- lations exist between the stress-constituents and the strain-co- efficients for any one set of axes as for another. Once more that is the essence of isotropy. We are thus naturally led at once to the purely mathematical question of trying to solve the following problem : "An ellipsoid referred to OX', OY', OZ' has the equation air^ + ai-n" + azt" + 2a,v'^' + 2af,^'^' + 2a,^'rj' = k\ When referred to another set of axes OL', OM', ON' its equation is q:iX'2 + aofx'^ 4- aa/' + 2a4M'/ + 2ayX' + 2a6XV' = k\ What function of ai, 02, as, ai, a^, as is equal in value to the same function of ai, a2, as, ai, as, aa?" That problem we have implicitly solved in the note on STRAINED ELASTIC SOLIDS 491 quadric surfaces (see Article B of this volume) . For there we have mentioned, with references to sources, the fact that it can be proved that «i + ^2 + fls = ai + 0:2 + as, a2«3 + «3«1 + CLlCli — Cli — CI5 — Qq = azas + mai + aia2 — a^ — a^ — a^, ai as ae as a2 04 = 04 as «i a& 0C6 Oi2 CCb OCi as Thus we see that there are three fairly simple functions which enjoy the property referred to in the enunciation; and of course any given function of these three functions will also have the property. Thus the strain energy of an isotropic body per unit volume must be expressible in terms of the three functions writ- ten above on either side of the equality sign. These functions are in fact E, F, G of the text. The upshot of the argument is that, while for any material the strain-energy per unit volume is a function of the strain-functions ai, a^, aa, ^4, ob, a 6, it can be shown that for isotropic material the function has a special form, being a function of three special functions of the strain-func- tions. Gibbs' own argument, based, as we stated, on the sen- tence from page 205 quoted above, assumes that the strain- energy is solely dependent on n, 7-2, rs (and temperature), and of course by reason of [439] these are functions of E, F, G. As he himself remarks on page 209, although we could regard the strain-energy per unit volume as a function of n, ro, rs "it will be more simple to regard €f' as a function of r]v' and the quan- tities E, F,G." It seems therefore to the writer not out of place to have put the argument on grounds which do not directly in- volve the principal elongations and which appeal to general ideas of isotropy. The argument outlined above does not apply to an aeolotropic (anisotropic) body. We cannot afford space to go into this further but must refer the reader to standard texts on elasticity or to Goranson's book* on this matter. For one thing, * See p. 433 of this article. 492 RICE AHT. K in an aeolotropic body the principal direction of stress and those of strain do not in general coincide, and if we carried out the conceptual experiment suggested above of rotating a spherical body keeping the forces and their points of application "in situ" in the frame of reference, the strains and stresses would not in general be same in an element as they were previously in the element which originally was situated in the same place in the frame of reference ; for the orientation of the two elements would be different although their relation to the external forces would be the same, and that would be a significant change for an aeolotropic element, even although the two elements were homogeneous in nature. Hence the rotation would in general involve an entire alteration in the general state of stress and strain and a change of strain-energy. Thus one of the premises of the argument would collapse. We have already referred to the arguments by which Gibbs justifies the use of the determinant H (with a positive value) instead of G for expressing the energy of an isotropic material. 11. Commentary on Pages 211-214- Approximative Formulae for the Energy and Free Energy in the Case of an Isotropic Solid. The approximative formulae given by Gibbs in [443] and [444] are just examples of the expansion of a function in series by the use of Taylor's theorem, neglecting powers higher than the first. For small strains ri, r2, rz differ little from unity. By [439] E differs little from 3, F from 3, and G or H from unity. Writing E' for E - 3, F' for F - S, and H' for i^ - 1, we can express any function of E, F, H asa, function of E', F', H'. We can expand this function as a series by Taylor's theorem, say k-}-aE' + bF' + cH' + higher powers and products of E', F' , W . For small strains the higher powers and products are negligible compared to the terms involving the first power. So to the first approximation the function will be 1 + aE + hF -]- cH (where Z = fc — 3a — 36 — c), which has the form of [443] or [444]. STRAINED ELASTIC SOLIDS 493 The justification of [445] can be easily given as follows. Re- membering that i^F' is a function of E, F, H, say ^{E, F, H), it follows that dypv _d4> BE d4> dF d dH dri ~ dE dn dF dri dH dn dE dF dH Similarly ^ = 2r. % + 2r. (rl + r?) ^ + r^n -^• ara dE dF dH Obviously dxf'v' _ d\f/v' dri dri if ri = Ti = rs, and exactly similar arguments cover the other equations. The wording of the argument at this point on page 212 is a little confusing; for, as the text itself points out, this theorem is true "if i/^' is any function of t, E, F, Hj" not merely the approximative linear function of [444] ; then just lower down we have references to "proper" and "true" values of ^pv. It might be better therefore to introduce two functional symbols one o, xo be the values of <^ and x when ro is substituted for each of the quantities n, 7'2, ^3 in E, F, H. Let {d(j>/dr)o, STRAINED ELASTIC SOLIDS 495 (d'^o, >ar/o \a r/o \ar2/o \drdr'/o \ .drWo — V drdr'/o we have four simultaneous equations to determine the four quantities ^, e, f, h; these, as the text says, will give to the approximations x, dx/dn, 5x/9^2, dx/dr^, . . . d^x/dridrz their "proper," i.e., correct, values ^, d/dn, d4)/dr2, dcjy/drs, . . . d^(t)/dridr2 when n = r2 = n = ro, i.e., when the solid is in its unstressed state not at the original temperature of the state of reference but at the temperature for which it has expanded (or contracted) from that state in the ratio ro. But by Taylor's theorem, if we expand + higher powers \dridr2/o J = <^o + ( — 1 (ri + rg + rs - 3ro) \ar/o 496 RICE ART. K + 2 av v9r9r -, ) [(?'2 - ro) (rs - ro) + (rj - r^ (n - ro) + {n — To) (fi — ro)] > + higher powers, and similarly X = Xo + f — j (ri + r2 + rs - 3ro) ,1 f/9^X "^ 2! + / 9^ X \ 2 \7^f) K^2 - ro) (rg - ro) + (rg - ro) (n - n) + (ri — ro) (r2 — ro)] > + higher powers. Hence (f)(t, E, F, H) and x{t, E, F, H), the true and the approximative expansions of ^l/v agree to the terms of the second degree inclusive. The remaining statements on page 212 can be deduced similarly. The equations r^ + ra^ + rg^ = On^ + a^^ + a^^ + a^i^ + 022' + 023^ + agi^ + aga^ + agg^, ra^rg^ + rgV^^ + nVa^ = ^u^ + An^ + An'' + ^21' + ^22^ + ^23' + ^31^ + A322 + ^3g2, an ^12 ^13 rir2rg = 021 ^22 0,23 dzi Cli-i ^33 are equations [432], [434], [437] of the text. By partial differ- entiation with respect to an, we can, as Gibbs points out, STRAINED ELASTIC SOLIDS 497 regard the three quantities dn/dan, Qr^/dan, drs/dan as deter- mined by the resulting three simultaneous equations in these quantities (determined, i.e., in terms of the Upq coefficients). Similar statements are true for any of the partial differential coefficients dri/da„y, drt/dapq, dr^/dapq. These are of course correct values and have nothing to do with the approximation to \pv made in [444]. Now Xx' is determined as we know by the equation Xx' = 3 dan (See equation, bypv' = = 22(Xx'5apg P Q ), near the top of page 204.) Since d(ri — ro)"/dan = n(ri - roy~^dri/dan, etc., we can express Xx' as an ascending series in the quantities ri — ro, T2 — To, rs — To, and since the true and the approximative series for xpv' agree to the second degree, the true and approximative series for Xx' will agree to the first degree, and the error in Xx> involved in using the approximative series will be of the order of magnitude of the squares of n — ro, ^2 — ro, r^ — ro. On pages 213, 214, e, f, h are determined in terms of the bulk- modulus and the modulus of rigidity. These two moduli, as we have mentioned earlier, possess physical significance only in so far as Hooke's law is obeyed; and this, as experiment demon- strates, restricts the range of stress allowable from the unstressed state at a given temperature. Gibbs' calculations on page 213 are limited by this consideration, as he himself expressly admits; for he indicates that his moduli are determined for "states of vanishing stress," and in the final results he goes to the limit at which n = r2 = rs = ro; ro as before being the uniform ratio of elongation due to the change from the tem- perature for the state of reference (regarded as unstressed) to the temperature indicated by t. The formula for the bulk- modulus in [448] we have discussed earlier. To use it we must express p as a function of v and t. Consider a mass of the solid which has unit volume in the state of reference. It is subjected to the change of temperature which gives it the volume ro^ It is now subject to uniform pressure p which gives it a uniform 498 RICE ART. K elongation with the ratio Vi in all directions as compared with the state of reference at the original temperature, so that its volume is now ri^ (n = rz = ra). Thus E = Sri^ = Sv' ; P = 3ri* = Sv^; H = r^ = V, and so we arrive at [451]. By equation [88] from the earlier part of Gibbs' discussion we obtain the general expres- sion for p in [452] in any state of uniform stress small enough to be consistent with Hooke's law. Differentiation gives us [453], and an approach to the limit at which v = r^ gives us the result [454]. The writer is unable to justify the equation [449] as it stands; as far as he can judge it ought to read dXy' R = ro da 12 To see this, let us consider the matter from the point of view of the ordinary treatment of isotropic solids in the text-books of elasticity. Limiting ourselves to strains so small that Hooke's law applies, the modulus of rigidity is defined as the common value of the quotients Yz ^x Xy. fi U U The quantities fi, /e, /e are the shears of the lines parallel to axes of reference (the same axes for the state of strain as for the state of reference). As we saw in our discussion the value of /a, for example, is ee/(ele2)^ although it can be replaced by an approximation Cn, + 621 for very small strains. This, of course, implies that changes of temperature are not involved. Let us, however, consider the situation which arises when the state of strain is at a temperature t, different from the temperature of the state of reference. The definition of the modulus of rigidity at temperature t must of course involve the shears of the axes from an unstressed state also at that temperature, that is, a state in which all lengths are elongated in the ratio ro as com- pared with the state of reference. The definition of R is still Xy/fi (say), and /e is still 66/(6162)^ But we have to be careful about the approximation. Let us recall the definitions of STRAINED ELASTIC SOLIDS 499 ei, 62, ... et from this article or from [418], [419] of Gibbs: ee = 611^12 + 621622 + 631632, 61 = 611^ + 621 '^ + e3l^ 62 = 612^ + 622^^ + 632^ In making the approximations we take as usual 623, 632, 631, 613, 612, 621 to be very small compared to en, 622, 633; but the three latter quantities do not now approximate to unity, as formerly, but to 7*0, since in the unstressed state at temperature t, there exist elongations of amount ro as compared with the state of reference. Hence the approximations now must involve re- placing 66 by ro(6i2 + 621), 61 by ro^ 62 by ro^ Hence _^ 612 + 621 Thus ro Xy a j R = — #= J'o /e ' 612 + 621 As we are assuming that the range of stress and strain is covered by Hooke's law it is also true that Xy ~\~ 8Xy R = To 1 — ; : ' 612 + oei2 + 621 where SXy is a small change of shearing stress produced by a small change Sen in the coefficient 612, and thus 8Xy ^ = ^0 ^ ' 06X2 This corresponds to Gibbs' equation [449] but with the ro on the right hand side of the equation, not on the left. The symbol ro can be obtained on the left if 66 is taken as the approxi- mation to /e (which is the case when change of temperature is not involved since en and 622 are then approximately unity) ; for 500 RICE ART. K if this is done and we write ro(ei2 + 621) for/e we obtain Gibbs' result. But this amounts to putting en or 622 equal to unity in one part of the complete formula for /e and equal to r^ in another. We should obviously approximate from 66/(6162)* and not from ea. If the writer is correct, then we should write equation [449] as R = To - — [449a] with of course an = 022 = 033 = ro and the remaining apg coefficients put equal to zero; for we are considering the value of R for the state of vanishing stress. This will change equa- tions [455] and [457]. Thus and we have to differentiate this partially twice with respect to an. The term multiplied by e will yield 2e. In the term which is multiplied by /, four of the Ap^ minors involve an, viz., ^33^ Azi"^, Ais^, ^2^^ so that this term yields aAsa . , 3^31 . , 9^23 . dA 2/i^33 z~ + As, -^' + A,3 z-^ + A 21 5ai2 aai2 aai2 da 12 On passing to the limit when 023, 032, 031, ais, ai2, 021 are zero and On = ^22 = 033 = To it will be easily seen that the only surviving part of the derivations from this term is 2/A21 (9^21/9^12) which becomes 2/a332 or 2/rol Hence [449a] becomes R = 2ero + 2/ro^ [455a] which replaces [455]. It will then appear that in place of [457] we shall find J 6 -I ~ 2 > ro^ ro h = - i- - V. [457a] STRAINED ELASTIC SOLIDS 501 Similar chaDges will have to be made in [459] and [461], if the writer's emendation of [449] is correct. Before leaving this subsection we shall revert for a moment to the special point passed over at the beginning of the com- mentary on this part. Pages 201 and 202 are rather involved but the point appears to be as follows. It has been implied hitherto that no particular physical properties are imposed on the state of reference. In ordinary elementary discussions in the text-books it is taken as unstressed, i.e., without any strain energy. Thus if a relation is given between ev and rjv', duy ^12, . • • ^33, then ev is the intrinsic energy of the state of strain; but if no such restriction is imposed on the state of reference then, since the coefficients an, an, ... ass express a relation between the state of strain and the state of reference, the function ev will give the excess of energy in the former state over the latter for the material occupying unit volume in the latter. Provided the state of reference is at all events one of homogeneous strain, this introduces no difficulties since the energy in any element of the solid in the state of reference is the same as that in any other, and therefore ev differs from the intrinsic energy in the state of strain (per unit volume of the state of reference) by a constant amount, (i.e., the same for all elements of volume). But if, as Gibbs suggests, it happens that in some cases it is impossible to bring all ele- ments in the state of reference simultaneously into the same state of strain, this means that in the state of reference the energy in an element depends on its position in the state of reference, i.e., on the coordinates of the point which it surrounds. We can, however, take some particular element in the state of reference as being in what we may call a "standard state." The condition in any other element in the state of reference can be stated in terms of the strain-coefficients which give the relation between the state of this latter element and the standard state, and the energy in this element in the state of reference will, apart from a constant, be a function of these latter strain-coeffi- cients. Thus ev will now be a function not only of the strain- coefficients ail, ai2, ... ass (connecting the state of strain with the state of reference) but also of other strain-coefficients con- I 502 RICE ART. K necting the state of reference with the standard state (which will vary in value from point to point of the state of reference). IS. Commentaiy on Pages 215-219. Solids Which Absorb Fluids. Elucidation of Some Mathematical Operations. In the final four pages of the section, viz., pp. 215-219, the general argu- ment offers no difficulty and only a few comments need be made on the mathematical operations. Regarding the equations [463] and [464], we refer the reader to equations (38) of our exposition. If we are considering a state of hydrostatic stress, we know that Xx = Yy = Zz = -V and Yz ^^ ^Y ^^ ^x = A z = A K = Yx ^ 0- Hence by (38) Xx' = -Anp, Xy' = -A12P, Xz' = —Aisp, Yy' = —A22P, etc. which constitute [463] of Gibbs. Also Xx'^aii -(- Xr'5ai2 ...-]- Z z'^azi = —p{Aiiban + An^an . . . + Azzbazz). As we have already seen on several occasions, the bracketed expression on the right hand side is bH, and of course H is the ratio of enlargement of volume, i.e., the volume of an element divided by its volume in the state of reference ov vv. Thus we obtain [464]. The equations subsequent to [471] are obtained by the familiar device by means of which we obtain the yp and ^ func- tions from the e function. Thus since , dev = tdr]v -\- S2(Xx'6aii) + ^HadTa, STRAINED ELASTIC SOLIDS 503 we regard €r' as a function of riv, an, an, . . . ass, Va, Tb , . . . and the result just written embodies the equations dev' dev' dtv' ^y. = ^' a"^ = ^^'' "^"•' ^ = ^^ '*'• leading to [471] and other similar results. Also regarding \j/v'( = ev' — t-qv') as a function of t, an, a^, . . . flss, T/, Tb, etc., we can write d\pv' = d{ev' — triv) = -riY'dt + S2Zx'£/aii + 2/iaC?r„', and this is equivalent to the equations dypv' dypv' d\pv' ~^ = - ^'"' a"^ = ^^'' '^'•' ^ = ^- '^'•' which yield dr]v' dXx' dan dt and similar results. Also from either of these we obtain by repeated differentiation dXx' d^ey djXq dVa' ~ dVj dan ~ dan and so on, where Xx', etc. and Ha, etc. are regarded as func- tions of r]v' (or 0, an, an, . . . ass, Ta, Tb, etc. We can also introduce a function 0r' of t, an, an, . . . ass, Mo, fib, etc. defined by V' = €v' — trjv — HaTa' — jJ^bTb — etC, whose differential satisfies the equation d Xy', . . . Zz', tJ-a, y^b, etc., defined by will yield the first group of [473]. The function tv gives us the equation [471], viz.. or I.e., dt dXx' dan ~~ drjv dt dXx' tdan tdrjv dlogt dXx' dan dQv and so on, which is the first group of [474]. The function ypv' gives us dr]v' dXx' dan dt i.e., dQv' dXx' dXx' dan dt d log t and so on, which is the second group of [475]. The function €v' — 22Ax'flu regarded as a function of rjv', Xx', Xy', • • ■ Zz', Ta, Tb', etc. will yield, when treated similarly, the second group of [474]; while the first group of [475] can be derived from the function ^v in a similar manner. THE INFLUENCE OF SURFACES OF DISCONTI- NUITY UPON THE EQUILIBRIUM OF HET- EROGENEOUS MASSES. THEORY OF CAPILLARITY [Gibbs, I, pp. 219-331; 331-337] JAMES RICE I. Introductory Remarks This part of Gibbs' work can be broadly divided into two por- tions; the first of these, and much the longer of the two, deals with surfaces of discontinuity between fluid masses, while the second consists of a brief treatment of liquid films and surfaces of discontinuity between solids and liquids. The first portion itself falls broadly into three parts, one of which, after formulat- ing the general conditions of equilibrium in a surface phase between fluids, derives the famous "adsorption law" (a name not actually employed by Gibbs) and treats briefly the thermal and mechanical processes in such surface phases; another deals with the stability of surfaces of discontinuity; and the third part is concerned with the conditions relating to the formation of new phases and new surfaces of discontinuity. In addition, a few pages of the succeeding section on Electromotive Force are devoted to electrocapillarity, a commentary on which naturally belongs to this portion of the present volume. 1 . The Surface of Discontinuity and the Dividing Surface As Gibbs points out in the first paragraph of this section, the basic fact which necessitates a generalization of the results obtained in the preceding parts is the difference between the environment of a molecule situated well within a homogeneous mass and that of a molecule in the non-homogeneous region which separates two such homogeneous masses. In the sub- 505 506 RICE ART. L sequent pages he formulates in his customary careful and rigorous manner the fundamental differential equation for this region and gradually leads the reader to the abstract idea of a 'dividing surface" as a convenient geometrical fiction with which to represent the 'physical non-homogeneous region which has in reality extension in three dimensions, one however being very small. This region he frequently refers to as a "surface of dis- continuity" but is careful to point out that the term does not imply that "the discontinuity is absolute," or that it "dis- tinguishes any surface with mathematical precision." The term "dividing surface" does, however, refer to a surface in the strict geometrical sense and the reader is warned to keep this distinction well in mind. There is a certain latitude, as he will presently learn, in the precise position to be assigned to the dividing surface and in later developments of Gibbs' work this latitude has been the cause of some doubt concerning the validity of certain deductions. In this way a certain part of the whole energy of the system is associated with this dividing surface. Now this part is not actually the energy situated in the non-homogeneous region or "surface of discontinuity," but is the excess of this energy over and above another quantity of energy whose amount depends on the precise location of the dividing surface. The matter is carefully dealt with by Gibbs (I, 223, 224), in equations [485] to [492]. Thus there is a certain latitude in the quantity of energy which is to be associated with the dividing surface, and this lack of precision in the value of this energy must not be lost sight of. A similar lack of precision accompanies the amounts of entropy and of the various components which are to be associated with the dividing surface, and whose actual values will in any given system depend to some extent on where we conceive the dividing surface to be situated. Gibbs denotes a physically small element of the dividing surface by s, and the quantities of energy, entropy, etc. associated with this element by e'^, rf, nii^, w/, etc. As is the case for any of the homogeneous phases, the variables which determine the state of such an element of the surface of discontinuity include the quantities s, t]^, nh^, rui^, etc., just SURFACES OF DISCONTINUITY 507 referred to. The energy e^ associated with the dividing surface is of course a function of these variables. (Actually Gibbs introduces the curvatures of the element of the surface as further variables, but disposes of them as of negligible impor- tance, a point which we shall consider at a later stage, but shall ignore for the present.) The partial differential coefficient of e^ with regard to r?^ is of course the temperature of the dis- continuous region, and those with regard to rrii^, mi^, etc., are the chemical potentials of the various components in this region. In the first few pages of this section we are provided with a proof on exactly the same lines as that in Gibbs, I, 62 ei seq. that the temperature and potentials in the discontinuous region are equal to those in the homogeneous masses separated by this region, provided of course that the usual condition is satisfied, viz., that the components in the surface are actual components of the homogeneous masses; if some of them are not, the usual in- equalities hold. All this proceeds on familiar lines. There remains the partial differential coefficient of e^ with regard to the variable s; this is denoted by o-. It is clearly the analogue of the partial differential coefficient of the energy of an ordinary homogeneous mass with respect to its volume, i.e., the negative pressure, — p, which exists in that phase. Equation [493] (with the last two terms omitted for the present as explained above) or equation [497] gives the formulation of the ideas just outlined. The paragraphs between equations [493] and [497] may well be omitted at this stage. The reader will then find that the succeeding two paragraphs lead in a direct and simple manner to the extremely important result expressed in equations [499] or [500]. 2. The Mechanical Significance of the Quantity Denoted by a If the reader pauses to reflect he will observe that in the earher portion of Gibbs' treatment the quantity — p makes its appear- ance strictly as the partial differential coefficient of the energy with respect to the variable v. To be sure p has a mechanical significance which is always more or less consciously kept before us, but nevertheless in its original significance it is concerned with the quantity of energy which is passed into or out of a 508 RICE ART. L phase from or to its environment by reason of a simple volume change in the phase. Now it is to be observed that equation [500] opens up the possibility of giving a mechanical significance to a, despite the purely formal introduction of it in [493] or [497]. It is well known that if a non-rigid membrane or a liquid film, such as a soap bubble, separates two regions in which there exist two different pressures p' and p" then there exists a surface tension T uniform in all directions in the membrane or film, and moreover V' - v" = T{c, + C2) , where ci and c^ are the principal curvatures at any point of the membrane or film. The exact agreement of the form of this equation with [500] suggests a plausible mechanical interpreta- tion for (7 as a "superficial" or "surface" or "interfacial" tension. Actually in a converse fashion T, which is introduced as a tension in the membrane, can easily be given an interpretation in terms of energy. If the membrane, for instance, encloses a gas at pressure p' which receives (reversibly) an elementary amount of heat and expands by an amount bv, the increase of energy of the system, gas and membrane, is t b-q — p"bv , where p" is the external pressure, since p"bv is the amount of energy transferred by mechanical work from the internal gas- membrane system to the external gas system. Now, since p" = p' - T{c, + C2) , it can be proved (the proof is a familiar one and will be found in the standard texts, being just a reversal of the steps in Gibbs' treatment between [499] and [500]) that p"bv = p'8v - Tbs, where s is the area of the whole membrane; and thus the increase of energy of the system, gas and membrane, is tbri - p'bv + Tbs. SURFACES OF DISCONTINUITY 509 The analogy between the quantity a- for an interface between two Kquids or between a Hquid and a gas, and the quantity T tor a membrane in tension between two gases, is thus drawn once more from another standpoint. It is therefore quite natural for Gibbs at this point to say, as he does, that equation 1 499] or [500] "has evidently the same form as if a membrane without rigidity and having a tension a, uniform in all directions, existed at the dividing surface," and thereupon to suggest the name "surface of tension" for a specially selected position of the dividing surface and the name "superficial tension" for cr. The cautious nature of Gibbs' statement might easily be over- looked by the reader. It clearly does not commit him to the view that the interface between two fluid masses must be regarded actually as a membrane in a state of tension. This idea is certainly a prevalent one, and the treatment of "surface tension" in many of the elementary texts of physics fosters it. So it may be of some service to the reader if a short discussion of this much debated point is inserted here. This will require us to enter into a more detailed consideration of the molecular structure of the fluid phases than actually occurs in the original, but that is hardly avoidable in any case in view of the develop- ments of Gibbs' work by subsequent writers. In addition, later workers have availed themselves of the statistical calculations and results which are nowadays associated with molecular pictures of matter in order to give a deeper interpretation to some of Gibbs' results and to help to elucidate certain difficulties of the purely thermodynamical treatment. So it may prove serviceable to seize the opportunity at this point to give also a brief discussion of the fundamental statistical idea involved in such calculations. II. Surface Tension 3. Intrinsic Pressure and Cohesion in a Liquid The behavior of soap films, in which there may well be a strong lateral attraction between long-chain molecules such as those of the fatty acids, "anchored," as it were, side by side in the surfaces of the film (an attraction which may with some 510 RICE AKT. L justification be really considered as a surface tension since it resembles a tension in an elastic membrane in most respects), gives a bias towards an explanation of the phenomena at the free surface of a simple liquid, or at the interface between two such liquids, in terms of the same concept. As already hinted, most elementary texts of physics deal with the "surface tensions" of liquids as if there did exist in their surfaces lateral pulls, tan- gential in direction, between the surface molecules, of an order of magnitude much greater than that exerted between these molecules and those immediately under them in the interior. At times one reads accounts of suspended drops of water which imply that the main body of water in the drop is contained in an "elastic" bag made of molecules which cohere together very powerfully like the molecules in a rubber sheet. Now it is true that the mathematical form of the results de- duced from such an assumption is precisely the same as that which can be deduced from a physically more real picture of the situation at a liquid surface; and it is also true that this assumption provides an easier mathematical route to these results then does the alternative hypothesis, which when worked out in detail involves rather troublesome analysis of a type first developed by Laplace. However, the course of that analysis and its outcome can be quite easily indicated without going into the purely analytical steps. An analysis of the situation requires us first of all to be very careful concerning the interpretation of the word "pressure" in connection with a liquid. When we speak of the pressure of a gas we are thinking of the integral effect of the bombardment of the swiftly moving molecules on unit area of the enclosing vessel, or of the rate of transfer of normal momentum across unit area in the interior. The notion will be quite familiar to those who have some acquaintance with the kinetic theory of gases, and everyone recognizes that pressure arising from weight is usually an entirely evanescent quantity in a gas. Theoreti- cally, of course, the pressure at a point in a gas increases as the point descends in level, but the difference of pressure between the top and bottom of an ordinary-sized vessel is negligible. On the other hand, the pressure in a liquid arising from the SURFACES OF DISCONTINUITY 511 weight of a superincumbent column of liquid is in general the most important portion of the thrust on the enclosing vessel. Yet it only complicates the situation we are discussing to bring this in at all. It is best to conceive the liquid to be free from gravity, as Gibbs actually does in a great part of his treatise. We may, if we wish, consider it to be contained in a vessel which it touches everywhere, and which can be regarded as fitted with a piston so that a thrust can be applied if required, — a thrust which by Pascal's law is distributed at all parts of the surface in proportion to the size of each part, or is exerted normally across any conceptual dividing surface in the interior, again in pro- portion to its extent. Or we may think of the hquid as a spherical mass subject to the pressure of a surrounding gas and for the moment regard the sphere as so large that any small portion of the surface is practically plane. If now the pressure of the surrounding gas were zero the pressure would also vanish in the liquid. (Actually the pressure cannot be less than that of the saturated vapor.) The reader who has studied the earlier portion of Article K of this volume (pp. 395 to 429) will realize that this would be just a special case of an unstressed state of a body. Yet in the interior of the liquid there must be a relatively enormous pressure in the sense in which that word is used in connection with a gas; "kinetic" pressure we shall call it. In the liquid there exists a thermal motion of the molecules, and on account of the much larger density of the liquid the rate of transference of momentum across an interior conceptual surface is very great indeed. Clearly this internal kinetic pressure cannot be the quantity which is denoted by the symbol p in our equations; for that, as we have seen, would practically vanish when the stress in the liquid produced by the thrust of an external gas or piston in an enclosing vessel dis- appears. Of course at the surface there is the well known inward pull on each molecule in the layer whose thickness is equal to the radius of molecular attraction. This has the effect of turning inwards all but a small fraction of the molecules moving through this layer towards the surface, and in conse- quence the actual kinetic pressure at the surface is enormously reduced below the kinetic pressure which exists in the interior. 512 RICE ART. L We may look at this matter from another standpoint, a purely static one. We can assume a molecular configuration practically unchanging in average conditions and imagine a plane to be drawn in the interior of the liquid. Across this plane there will be exerted repulsions between molecules in very close proximity to one another and attractions between mole- cules rather more separated. These ideas resemble somewhat those of Laplace who regarded the liquid as a continuum whose neighboring elements attract one another, this attraction tend- ing to make the liquid contract; such contraction would be opposed by an internal pressure. These concepts of cohesion and intrinsic ^pressure are quite familiar. The molecular picture defines them a little more closely. The force between two molecules for distances greater than a certain critical amount is an attraction falling off in value very rapidly as the distance increases. At the critical distance, which must approximate in value to the size of a molecular diameter, the force is zero and changes to a repulsion when the distance apart is decreased; this repulsion must increase with very great rapidity as the distance apart is reduced below the critical separation. Van der Waals formulated these forces of cohesion and intrinsic pressure in his famous equation a Rt V + ~2 = 1)2 V — b for a/v"^ is nothing more than the cohesion varying directly as the square of the density, and Rt/{v — h) is the intrinsic pressure varying inversely as the excess of the volume of the fluid above its irreducible minimum volume 6. The symbol p represents the ordinary pressure with which we are concerned in the con- ditions of equilibrium. When p is small the cohesion and intrinsic pressure are nearly equal, which means that we have on the average a molecular configuration in which the repulsions and attractions across an internal plane nearly balance one another. The reader will recall in our discussion of the theory of elasticity (Article K) the warning that the stress-constituents Xx, Xy, etc. (which in the case of a fluid reduce to —p) are not to be confused with molecular attractions and repulsions, which SURFACES OF DISCONTINUITY 513 may readily exist even in the ''unstressed" state, when Xx, Xy, etc., vanish. Just as the stress-constituents in the case of a strained soHd arise from change of molecular configuration, i.e., strain, so the experimentally observable pressure p in a liquid arises from change Ln molecular repulsions and attractions due to the change in average molecular separation which we con- ceive to accompany compression. 4- Molecular Potential Energy in a Liquid Having disposed of these considerations concerning pressure, which will be of service presently, we turn our attention to a treatment of the energy of a liquid from the point of view of molecular dynamics. We shall not, of course, go into the de- tailed mathematical analysis (which can be found by the reader in the works of Laplace or Gauss, or in accounts such as that of Gyemant in Geiger and Scheel's Handhuch der Physik, Vol. 7, p. 345) but shall content ourselves with quoting certain impor- tant results. If we assume that there is a law of force between two molecules we can obtain in a familiar manner their mutual potential energy which we will represent by ^(r), where r is their distance apart. The magnitude of ^(r) increases as the mole- cules separate until r reaches a value at which the attractive force vanishes. For values of r greater than this the potential energy of the two molecules remains constant. In all expres- sions for potential energy there is an indefinite constant of integration and for purposes of calculation it is necessary to assign a definite value to this constant. In the present instance it is most convenient to choose the value of the integration constant in the function (r) to be such that the maximum value attained by (r) will be greatest at this distance.) Anj'" further decrease in r will produce an increase in {r) it is easy to express the mutual potential energy of one molecule with respect to all the molecules within its sphere of action; but, of course, the result will vary according to the situation of the selected molecule. Suppose in the first instance that it is well within the general body of the liquid, so that a sphere around this molecule as center with a radius equal to the radius of molecular action, denoted by /i, is com- pletely filled with liquid. It is easy to see that the potential energy in question is represented by 47rn / r^(f>(r)dr f (1) where n is the number of molecules per unit volume and I is the minimum distance between molecules, a distance which must approximate closely to the critical distance referred to above. Doubtless the integral form of this result should not be taken too seriously for purposes of actual calculation in view of our pres- ent-day knowledge of the properties of molecules, especially the fact that the radius of molecular action is not many times larger than a molecular diameter. But it will serve as a repre- sentative expression suitable for the purpose we have in mind, viz., the elucidation of the true nature of the "surface tension" of a simple liquid. Actually the numerical value of the expression (1) (we must bear in mind that it is an essentially negative quantity according to our conventions) is the amount by which the energy referred to is less than that for a molecule separated by relatively great distances from all others. It must also be noted that while this expression represents the potential energy of one molecule, this energy is nevertheless shared, as it were, with other molecules, so that when we wish to represent in a similar manner the potential energy of the group of molecules in unit volume we do not multiply the above expression by n but by n/2; otherwise we should be counting the energy of every pair of molecules twice. Thus the potential energy of the molecules in unit volume is 27r?i2 / r^(f>(r)dr. (2) SURFACES OF DISCONTINUITY 515 This expression is of course essentially negative by the con- vention stated above, which means that the numerical value of (2) is the amount by which the energy of these molecules is less than what it would be were they all widely separated from one another at the same temperature, i.e., in the gaseous state. If we now wish to obtain the potential energy of all the mole- cules in the body of liquid, we must not merely multiply the expression (2) by the volume. To do so would be to overlook the vital point that if a molecule lies in the layer of depth h at the surface, part of the sphere of molecular action lies outside the Uquid and the expression (1) is not correct for the potential energy of this molecule. For such a molecule the contribution to expression (2) is numericalUj smaller since n is zero* for certain elements of the spherical volume of radius h surrounding it; but as (r)dr*. (3) (Once more, since the definite integral in (3) is essentially negative, a itself is essentially positive.) The expression (3) represents the potential energy per unit area of surface. This is not the whole energy of the surface since in that we must also include the kinetic energy of the molecules in the surface layer. We have here a mechanical interpretation of the well-known division of the total surface energy into the surface "free energy" a, and the "bound energy" - tda/dt. 5. An Alternative Method of Treatment There is another method of approaching this question of surface energy which leads to the same result. In the interior of a liquid mass there is on a given molecule no force perma- nently acting in a given direction. As the molecule changes its relative position and suffers many more encounters with other molecules than it would meet in a gas in the same tune, the attractions and repulsions of its neighbors on it change in a fortuitous fashion. At the surface of a Hquid, within the layer of thickness h, there is an inward normal resultant force on a molecule which increases in value as the molecule approaches the surface. Also in a layer of the vapor outside the surface of the liquid this field of force also exists, reaching the value zero when the molecule is at a distance h from the surface. A molecule in such a situation possesses potential energy, just like a body raised above the ground against gravity. Just as a body under gravity tends to move downwards, so molecules in the surface tend to "fall inwards" towards the interior and so reduce the extent of the surface, thus producing the illusion of a surface contracting "under tension." But of course the truth is that the effective force on a molecule in the surface layer is * In arriving at (3) certain assumptions are made about the behavior of 0(r) and certain functions derived from it at the lower limit I of r. This, however, concerns mathematical details and does not concern physical interpretation. SURFACES OF DISCONTINUITY 517 not parallel to the surface but normal to it. As stated above, it is by reason of this that the enormous kinetic pressure in the interior (the intrinsic pressure) never manifests itself to our senses or our measuring instruments. Only a small fraction of the molecules, whose kinetic energy is sufficiently above the average and whose direction of motion is sufficiently near to the direction of the outward normal, will manage to effect their escape and impinge on an enclosing solid wall or enter into a vapor phase. Thus it is chat, apart from artificially produced thrusts on the surface of the liquid mass and the effects of gravity, the observed pressure of the liquid is just the saturated vapor pressure. This picture of the surface conditions enables us to make a calculation of the surface potential energy in a manner alterna- tive to that suggested earlier. The basic idea of it is just the same as that employed in calculating the potential energy of a body raised above the ground ; perhaps the potential energy of a wall of given height is a better analogy. The details are again too troublesome to reproduce here, but once more we reach the same result as before for this energy per unit area of surface, viz., the expression (3). This second method of analyzing the situation also enables us to obtain a formula for the "cohesion," i.e., the amount by which the intrinsic pressure of the liquid exceeds the observed pres- sure. It can be shown that the attraction of the interior liquid on all the molecules contained in the amount of surface layer which lies under unit area of surface is 4>(r)dr. (4) - 27rn2 (This happens to be expression (2) with the sign reversed.) This is the well-known result of Laplace, and this expression (4) for the "cohesion" is usually denoted by the letter K. It is, of course, as well to remember that this expression, like the previous results, is derived on the assumption of a liquid so fine- grained in structure as to be practically continuous, and there- fore these expressions can only be regarded as approximate representations of the proper formulae in the case of an actual 518 RICE ART. L liquid. This, however, does not invalidate the general tenor of the argument. The expression (4) for K represents the van der Waals' cohesion a/v"^. If the constant a is reckoned for unit mass of the liquid it is easy to see that a = — where m is the mass of a molecule. III. The Quasi-Tensional Effects at a Curved Surface 6. Modification of the Previous Analysis Hitherto we have regarded the surface of a liquid mass as plane. When we consider the situation in a surface layer at a curved surface we have to modify the calculation of the inward attraction on this layer. In the same broad manner as before we can indicate the modification and thereupon it will be clear how it comes about that the quantity represented by a, which is manifestly an energy per unit area, appears to take on the role of a surface tension, i.e., a force per unit length. (It is, of course, obvious that energy/area and force/length have the same physical dimensions.) To make this clear we shall have to indicate in a little more detail how the calculation which leads to (4) is effected. In Figure 1, ^ is a point in the surface (supposed plane) and C a point at the distance h below. If P represents the position of a molecule in the layer, we consider another point B such that AP = PB; it is then clear that the layer of liquid between the surface of the liquid mass and the parallel surface through B produces no resultant force on the molecule at P. Thus the inward attraction on P will arise from the layer of liquid between the surfaces through B and C, and a little thought will show how this attraction increases as P approaches A . This argument is made use of in the calculation of the entire force on all the molecules lying between the surface through A and that through C, — a calculation which, as stated, leads to (4). Supposing, however, that the surface of the liquid were spherical and convex, and that we were proceeding as before to determine the attraction inwards on a molecule SURFACES OF DISCONTINUITY 519 situated at P; we realize at once that the layer of liquid near the surface which has no resultant effect on the molecule is not bounded by a plane surface through B but by a concave one having the same curvature as the surface of the liquid mass. The net result of this will be that the inward attraction on the B C Fig. 1 '^^'^ 4 ■ ^^ P F 6 D B E c Fig. 2 molecule will be greater than for a similar situation beneath a plane surface, since in the latter case we determine the effect of the molecules under the plane surface DBE (see Figure 2), whereas in the former we include the effect of the molecules between the surfaces DBE and FBG as well When the analysis ( 520 RICE ART. L is carried out it yields the result that the inward attraction on a small prism of the liquid at the surface, whose depth is h and whose sectional area is bs, is equal to 6s < — 27rn2 / r^{r)dr — — j r^4>{r)dr > , where R is the radius of curvature of the spherical surface. A reference to (3) and (4) shows that this is just (5) Were the surface of the liquid mass concave, we could show in a similar manner that the attraction on a molecule situated at P would be less than for a plane surface and that the result for the total attraction on the prism would work out to be 8. {a- -I}- (6) The analysis is due to Laplace, and it is customary to denote the quantity 2o- by the letter H. (See, for example, Freundlich's Colloid and Capillary Chemistry, English translation of the third German edition, pp. 7-9, where K is called the internal pressure, an unfortunate term since i^ is a cohesional attraction and not a pressure, and H/R is referred to as a surface pressure, another unfortunate name for what is really an additional cohesion.) 7. Interpretation of a as a Tension We can now use this material to elucidate the apparent role of cr in this connection. In the first place, if we consider a plane surface we have the result Po- K = po, (7) where Po stands for the intrinsic pressure within a (weightless) liquid bounded by a plane surface,* and po stands for the external pressure on its surface which arises from its saturated * I.e., by a spherical surface of very large radius. SURFACES OF DISCONTINUITY 521 vapor (with the possible addition of effects arising from artificial thrusts). Actually, even for a liquid under gravity, we can regard Po as the intrinsic pressure just within the horizontal free surface. As the depth increases, the intrinsic pressure, just like the usual "hydrostatic pressure", will increase by the amount gpz, where p is the density of the liquid and z the depth. Now Pn arises from the momentum of the thermal motion of the molecules of the hquid, and Pq — K represents this kinetic pressure enormously reduced by the cohesion on the surface layer. We might therefore call Pq — K the internal pressure of the liquid at the surface, but care will have to be taken to avoid any confusion between this use of the term "internal pressure" and the use of it by Freundlich and others (erroneously in the writer's opinion) to refer to the cohesion K* On the other hand po is the external pressure on the surface of the liquid and is the pressure actually measured by a manom- eter; so that the result for a plane surface simply states that the external and internal pressures at the surface are equal. Turning now to a spherical surface of radius R (convex to the exterior), the expression (5) yields the result P - {k+^-~)=V, (8) where P is the intrinsic pressure inside the liquid mass (at any point if the liquid is weightless, or at the free surface if gravity is supposed to act) and p is the external (observable) pressure on the surface. As before, we may call P — K the internal pressure of the liquid at its surface, and denoting this by p' we have P'-P = |- (9) Now this result is identical in form with that which connects the gas pressure inside a membrane or liquid film and that external to it. This formal identity has led to the use of the * Or we might use the old-fashioned phrase "vapor-tension" for Pq — K, as distinct from "vapor-pressure" the term for po. 522 RICE ART. L term "surface tension" for the quantity denoted by a, with unfortunate results for the real understanding of certain phenomena by students reading elementary accounts of capil- lary rise, for example. In consequence vague notions are preva- lent that in some way a tight skin of water holds up the elevated column in the capillary tube and "pins it" to the inner wall, or, on the other hand, that a tight skin of mercury holds the mercury in a capillary tube down below the general level in the vessel outside. In the case of a spherical membrane under ten- sion enclosing one body of gas and surrounded by another, both pressures are available for observation, the inside as well as the outside. In the present instance the intrinsic pressure of the liquid is not open to observation, nor its cohesion; but we can infer from the result (9) that the internal pressure just within a spherical mass of liquid, subject to a definite external pressure, is greater than it would be under a plane surface, subject to the same external pressure, by the amount 2(t/R. In short the liquid in the sphere is a little more compressed than that under the plane surface, but tliis extra compression is not due to a "surface membrane" in tension, but to a small change in the inward attraction on the membrane due to the curvature. Indeed the elevations and depressions observed in capillary tubes are easily seen to arise indirectly from this cause. In the first instance, the curvature at the surface of water in a capillary tube dipping into a beaker of this liquid is caused by the strong molecular attraction of glass on water as compared to the attraction between the molecules of water (water "wets" glass and adheres powerfully to it). This concave curvature can only exist if the internal pressure just at the surface is less than the external pressure; this external pressure is practically the same as exists on the plane surface of the water in the beaker. Thus the internal pressure just under the curved surface in the tube is less than that under the plane surface in the beaker, and this cannot be so unless the level in the tube is higher than in the beaker; in short the column in the capillary tube is pushed up, not pulled up. For a liquid like mercury which adheres scarcely at all to glass, the absence of molecular attraction by the glass necessitates a convex curvature in the capillary tube, SURFACES OF DISCONTINUITY 523 and a similar argument demonstrates that the mercury must be pushed down in the tube, in order to preserve conditions of hydrostatic equilibrium. The writer feels that there exists so much misconception con- cerning the surface tension of Hquids that the preceding elemen- tary account may not be out of place at the outset of a commen- tary on a portion of Gibbs' work which is so vitally concerned with the concept of surface energy, with which the term ' 'surface tension" has come to be practically synonymous. Before proceeding, it may be desirable to take this opportunity to clear up a misconception about another matter which experience shows to occur often in this connection. Outside a spherical mass of liquid the vapor pressure is less than the internal pressure just inside the surface. It is quite easy, as the writer knows from teaching experience, for the unwary student to pick up the notion that the saturated vapor pressure outside a liquid with a convex surface is therefore less than that outside a plane surface; but, of course, the very reverse of this is true. The capillary tube phenomena actually demonstrate this, as well as the complementary fact that the saturated vapor pres- sure above a concave surface is less than that above a plane surface. The chapter on the vapor state in any good text of physics contains the necessary details on this point. Moreover, the matter can be argued out correctly from statistical con- siderations. In any case the equations (7) and (8) show that P - p > Po- Po, but unless we had some definite prior information concerning the equality or inequality of P and Po we could draw no in- ference from this as to the relation of p to po. Actually, as stated just above, capillary experiments or statistical arguments demonstrate that p > po, and so we can infer from this fact that P > Po also. IV. Statistical Considerations 8. The Finite Size of Molecules While the foregoing analysis is very instructive in giving some insight into the true nature of the conditions at the surface of a 524 RICE ART. L liquid, it is limited by the fact that implicitly it regards the liquid as divisible into elements infinitesimally small com- pared to the range of molecular attraction, and this is not the case in actual fluids. However, molecules although not mathematically infinitesimal in size are so small that great numbers of them exist even in any "physically small" volume of a gas. By "physically small" we mean small in so far as our capacity to deal with it experimentally is concerned. Under such conditions we can apply certain well-known statistical results which will prove of service to us later when we shall endeavor to supplement the thermodynamical arguments of Gibbs' treatment by considerations based on molecular structure. The previous discussion introduced us to an expression which represents the potential energy of one molecule with respect to its surrounding neighbors. It is given in (1), and ostensibly it is proportional to n, the numerical concentration of the molecules. We have already noted the hypothesis of infinite subdivision of the fluid on which this is based. But even if we waive that difficulty we must draw attention to the fact that the factor multiplying n is a function of the lower limit of the integral, viz., I. Now this limit is by no means so definite as the upper limit. Undoubtedly, if the concentration is not too great, we may take it to be a fixed quantity so that the expression in (1) may be regarded as varying directly with n;and as we have seen it then supplies the theoretical basis for van der Waals' cohesion term. But as the concentration increases, or as the temperature rises so that molecular impacts are on the average more violent and penetration of molecule into molecule more pronounced, the quantity I itself will become a function of concentration and temperature. Thus the linearity in n of the function expressing this mutual potential energy disappears at sufficiently high concentrations. We shall still require this conception of the potential energy of one molecule with respect to the others or, to put the definition in another form, the change of energy produced by introducing one more molecule into the system, and we shall consider it as some function of concentra- tion and temperature. Of course, one part of this change will SURFACES OF DISCONTINUITY 525 be the average kinetic energy of one molecule ; with that we are not^ seriously concerned; it is the average potential energy of a molecule with regard to all the others with which we wish to deal, and we shall represent it as a function of the concentra- tion, say 6(n). As stated, if n is sufficiently small d{n) is simply a multiple of n and is, according to our conventions, negative, approaching the value zero as n approaches zero. But at sufficiently large concentrations d{n) will reach a minimum (negative) value and as the effect of intermolecular repulsive force begins to make itself more marked in the great incompres- sibility of the fluid, 6{n) will increase in value with further increase in the value of n and must be considered as theoreti- cally capable of reaching any (positive) value, however large, unless density is to grow without limit. 9. Distribution of Molecules in Two Contiguous Phases Now suppose that we have two phases of the fluid in a system, represented by suffixes 1 and 2. The gain in energy of a molecule when it passes from the second phase to the first is d{ni) — d{n2). (We are assuming that the average kinetic energy of a molecule is the same in each phase.) It is a well- known result familiar to those acquainted with the elements of statistical mechanics that the concentrations in the two phases are related by the equation where k is the "gas constant per molecule," i.e., the quotient of the gas constant for any quantity of gas divided by the number of molecules in this quantity.* ♦ For a gram-molecule, ft = 8.4 X 10^; A^ = 6.03 X lO^^; so A; = R/N = 1.36 X 10"^^ Exp (x) is the exponentialfunctionofx, viz., the limit of the infinite convergent series X x^ x' ^'^ri'^21'^3!'*' ■••■' exp(a;) = e', where e is the Napierian base of logarithms. 526 RICE ART. L By taking logarithms we can write this in the form log ni + -^ = log n2 + -^ or, if we represent the gram-molecular gas constant by R and the number of molecules in a gram-molecule by N, we can write it thus: Rt log ni + Ne(ni) = Rt log n^ + Ndin^). (11) If the first phase is a vapor, so that 6(ni) approaches zero, the expression on the left-hand side approaches Rt log rii. Now, as is well known, the chemical potential of a gram- molecule of a dissolved substance, provided its concentration is small, is given by Rt log ni, where rii is the concentration. In seeking to discover how this formula must be generalized so as to embrace more concentrated states, statistical as well as thermodynamical argument may easily prove of service, and the equation (11) gives a hint of a possible line of attack. Equation (10) shows that the function Rt log n + Nd(n) is the same in both phases of the fluid. When we remember that the chemical potential of a given component is the same in all phases in equilibrium, and compare Rt log n with the formula for the chemical potential of a weakly concentrated component, we may well consider that the full expression just written might prove to be the pattern for a formula for the chemical potential under other conditions. We shall return to this point in the commentary. In conclusion, we may point out a phenomenon at the surface of a liquid which bears some resemblance to adsorption, and is explained by statistical considerations, When we were treating the field of force which exists at the surface separating liquid and vapor it was mentioned that the field exists in a layer of the vapor as well as in a layer of the liquid extending in both cases as far as the radius of molecular action. Now, just as the density of our atmosphere is greater the nearer we are to the SURFACES OF DISCONTINUITY 527 ground, so this field in the vapor will tend to retain molecules in this layer in greater number than exist in an equal volume elsewhere in the vapor; so that at the surface there is an excess concentration in the vapor phase. Furthermore this "ad- sorption" is accompanied by a decrease of the surface energy; for the reader will recall the fact that any concentration of mole- cules near the surface of the liquid tends to reduce the total potential energy, since the nearer one molecule is to another, outside the distance where repulsion begins, the smaller their mutual potential energy. Again there is an analogy with the mechanical conditions in the atmosphere, since any aggregation of molecules of air in the lower levels produces a diminution of potential energy as compared with a state of affairs in which the molecules are more uniformly distributed in the atmosphere. Indeed, when one is endeavoring to interpret thermodynamic phenomena in terms of mechanical laws, we may expect to find that any occurrence in which free energy tends to decrease is to be explained by the mechanical fact that, in the passage of an isolated dynamical system to a state of equilibrium, poten- tial energy always tends to a minimum. V. The Dividing Surface 10. Criterion for Locating the Surface of Tension We now return to the text of the treatise and consider one of the most troublesome features of the earlier pages of this section, viz., the location of the abstract dividing surface which in the course of the reasoning replaces the non-homogeneous film or region of discontinuity. The argument of Gibbs (I, 225- 228) leads to a criterion based on theoretical grounds for locat- ing this surface in a precise fashion; yet, as will appear, it is one which gives way in practice to other methods of placing the surface more suitable for comparing the deductions from the adsorption equation [508] with the results of experiments. Nevertheless, as there are one or two points in the argument which may require elucidation, we shall devote some considera- tion to it. Fig. 3 will help to illustrate Gibbs' reasoning. He chooses first an arbitrary position for the dividing surface which 528 RICE ART. L he calls S. In the figure, K represents the closed surface which cuts the surface S and includes part of the homogeneous masses on each side; the portion of K which cuts S and is within the non-homogeneous region is generated by a moving normal to S; the remaining parts of K in the homogeneous masses may be drawn in any convenient fashion. The portion of S referred to by the letter s (m Clarendon type) is indicated by ^5 in the figure, and its area is given by the italic s. CD and EF indi- cate portions of the other two surfaces mentioned at the top of page 220. The parts referred to in Gibbs' text by the letters, M, M', M" are also indicated in the figure. In the succeeding K v' ht c K v' M D K A v' M B E v' It F K Fig. 3 paragraph the difficulty of defining the exact amounts of energy to be attributed to masses separated from one another by a surface Avhere a discontinuity exists is touched on, but, in view of what has already been said above, this matter will probably be easily grasped by the reader, and in the immediately follow- ing pages the development follows that of the earlier parts of Gibbs' treatise, i.e., on pages 65 ei seq. Great care is required when we reach page 224 to observe just what Gibbs means by the energy and entropy of the dividing surface S, and the superficial densities of these and of the several components. The definitions and arguments are quite clear, and the figure SURFACES OF DISCONTINUITY 529 may help to visualize the situation; nevertheless it cannot be too strongly emphasized here in view of the references later to experimental work that e^, rj^, nii^, etc. do not refer to the actual quantities of energy, entropy, etc. in the discontinuous region, but to the excesses of these over those quantities which would be present under the arrangement postulated in the text with ref- erence to the surface S. The actual quantities present are of course precisely determined by the physical circumstances of the system; the quantities e^, rj^, mf, etc. are, however, partly determined by the position chosen for the surface S. (This is a point more fully elaborated later by Gibbs on page 234.) That being so, there is something arbitrary about their values unless we can select a position for S by means of some definite physical criterion. Such a criterion Gibbs suggests and deals with in pages 225-229. He calls this special position the surface of tension. 11. An Amplification of Gihhs' Treatment The criterion is based on the formal development of the fundamental differential equation for the dividing surface regarded as if it were a homogeneous phase of the whole system. As usual the energy e^ of the portion 5 of the surface is regarded as a function of the variables, rj^, mi^, m2^, etc. Among these variables must of course be included the area s of s; but in addition there exist two other geometric quantities; these measure the curvature of s (regarded as sufficiently small to be of uniform curvature throughout), viz., the principal curva- tures Ci and C2. It is a possibility that a variation of the curvature of s, which would obviously involve an alteration in form of the actual region of discontinuity, would cause a change in the value of e^ and in consequence we must regard e« as dependent to some extent on ci and C2. The partial differential coefficients de^/dci and de^/dCi are denoted by Ci and C2. Now we know that e^ is dependent in value on the position which we assign to s; also it appears that the values of the differential coefficients just mentioned depend to some extent on the posi- tion and form of s. Gibbs chooses that position of s, which 530 RICE ART. L makes dCi dC2 equal to zero, to be coincident with the surface of tension. The proof that such a position can be found and the reasons for choosing it are expounded at length. In view of the fact that Gibbs takes S to be composed of parts which are approximately- plane and which are supposed in the course of the proof to be deformed into spherical forms of small curvature, we may as well introduce that simplification into the argument at once and assume that Ci = c^ so that Ci = C2, and we have then to show that we can locate s in such a way that To-"' where c is the common value of Ci and C2. Let CDEF in Fig. 4 represent the portion of the region of dis- continuity, and suppose AB represents an arbitrarily assigned position of s so that EA = FB = x. We shall represent the thickness of the film EC by f . We now suppose that a deforma- tion to a spherical form indicated by the diagram with accented letters is produced. This means that c varies from zero to 1/R, where R is the radius of the sphere of which A'B' is a por- tion; i.e., 8c = 1/R. We also suppose that s does not vary in magnitude; i.e., that the area of the spherical cap indicated by A'B' is equal to the area of the plane portion indicated by AB; nor is there to be any variation of the other variables rj^, mi^, tUi^, etc. Hence, by [493], C 5e« = 2C8c = 2 ^• But the only possible reason for which e^ will vary under these circumstances is the fact that the volume of the element of film indicated by C'D'E'F' is different from that of the element CDEF. In short one must remember that a, though called a surface energy, is strictly an energy located in the film with a SURFACES OF DISCONTINUITY 531 volume density cr/f. Consequently de^ will be equal to the product of o-/f and the difference in the volumes of the elements just mentioned. On working this out we shall be able to obtain some information concerning the order of magnitude of C and justify the statements which Gibbs makes on this point in the paragraph beginning at the middle of page 227. It is true he begins the paragraph with the words: "Now we may easily convince ourselves by equation [493] ..." but the reader may well be pardoned if he doubts whether conviction is so readily obtained. Since the solid angle subtended by A 'B' at the centre of the sphere is s/W, it is proved by well-known propositions in solid geometry that the volume of the spherical film C'D'E'F' is 3 R -{{R-\-^ -xY- {R-xY], since R — x\q the radius of the sphere on which E'F' lies and R -\- ^ — X the radius of that on which CD' lies, R being the 532 RICE ART. L radius of A'B'. This volume is equal to 3 f, {sRHt -x) + 3R(r - xy + (r - xy + sr^x - srx^ + x^} = sf + ^ (f2 - 2^x) , neglecting the remaining terms which involve squares and products of ^/R and x/R. Hence the difference of the volume elements is ^ (f ^ - 2fa:) , and so the value for 8e^ calculated as suggested above is equal to = — (r - 2x). This is the same as 2C8c, i.e., 2C/R. Hence we find that C = ks(f - 2x). From this equation it is clear that C can have positive or negative values according as x is less or greater than f/2. C is zero if X = f/2, i.e., if the dividing surface is midway in the film. Also if C is the value of C when x = x\ and C" its value when X = x", these being in fact the values of C for two positions of the dividing surface separated by X, where \ ^ x' — x", we have 2(C" - C) = 2(Ts{x' - x") = 2as\. In this way we confirm the results obtained by Gibbs on page 227. These results show that we can choose in any general case a position for s which gets rid of the awkward terms Cibc\ + CibCi in [493]; our sole object in presenting an alternative method of derivation has been to show the physical basis for introducing these terms at all. It may also help the reader to a SURFACES OF DISCONTINUITY 533 further insight into the argument presented by Gibbs on page 226. Before leaving this topic, however, it may be as well to enjoin on the reader the necessity of keeping Gibbs' own caution in mind that in strict theory it is only for this specially chosen position of the dividing surface that the equation [500] is valid, and that only to it may the term surface of tension be correctly applied. VI. The Adsorption Equation IS. Linear Functional Relations in Volume Phases Let us revert for a moment to the substance of pages 85-87 of Gibbs, which leads to the equation [93]. Divested as far as possible of the mathematical dressing, the simple physical fact on which it rests is this. We are considering two homo- geneous masses identical in constitution and differing only in the volume which they occupy. If the volume of the first mass is r times that of the second, then the amount of a given constit- uent in the first is r times that of the same constituent in the second; also the energy and entropy of the first are respectively r times the energy and entropy of the second. Hence, when we express e as a function of the variables -q, v, mi, m^, ... w„, writing for example, e = <^(r?, V, mi, m2, ... m„), we know that (f){rr], rv, rmi, rm2, . . . rmn) = r4){y}, v, mi, W2, . . . w„). In other words, the function (/> is a homogeneous function of the first degree in its variables.* There is a well-known theorem of * It should be observed that this does not of necessity mean a linear function. Thus ax + by + cz is a linear function of the variables x, y, z; but ax^ + fcy^ + cz^ Ix + rny + nz is not. Yet both are homogeneous functions of the first degree; for if I, y, z are all altered in the same ratio, the values of these functions are also altered in the same ratio. 534 RICE ART. L the calculus due to Euler, which states that if ^{x, y, z, . . .) is a homogeneous function of the q^^ degree in its variables then d\p dyj/ d\J/ dx dy dz As a special case of this we see that 9<^ d(j) d d4> d 07] dv drrii 9w2 dnin But by the fundamental differential equation [86] which expresses the conditions of equilibrium 90 90 90 Hence e = tt] — pv + fxinii + M2W2 . . . + iJLnm„ , which is equation [93]. 13. Linear Functional Relations in Surface Phases Precisely similar arguments justify equation [502], since we assume as an obvious physical fact that if we consider two surfaces of discontinuity of exactly similar constitution then the entropy, energy, and amounts of the several components in each would be proportional to the superficial extent of each. Since e-^ is homogeneous of the first degree in the variables 7]^, s, nii^, W2'5, etc., it follows that the partial differential coeffi- cients of the function 4>{'r]S, s, mr^, m2^, . . .) of these variables, which is equal to e'^, with regard to the variables are individu- ally also homogeneous functions of the variables of degree zero, i.e., they are functions of the ratios of these variables. But by [497] 90 90 90 ^ = :97^' ^ = 7.' ^^ = ^' ^^'- (^2) Hence the n -\- 2 quantities t, a, \i\, 1JL2, ... are functions of SURFACES OF DISCONTINUITY 535 the n + 1 variables tjs = tVs> Ti = nii^/s, T2 = mz^/s, etc. By means of the n + 1 equations which express t, mi, M2, etc. as functions of the n + 1 quantities 77s, Ti, r2, etc., we can theoretically express 77s, Fi, r2, etc. as functions of t, mi, M2, etc. In consequence a, which is also, as we have just seen, a function of the former set of n + 1 quantities, can be expressed as a function of the second set, viz., t, /xi, 1x2, etc. This functional relation between a and the new variables t, ni, jU2, etc. is referred to by Gibbs as "& fundamental equation for the surface of dis- continuity." Now the values of the potentials jUi, 1x2, etc., are themselves determined by the constitution of the phases or homogeneous masses separated by the surface of discontinuity; so we see that o- is itself ultimately dependent on the constitu- tion of the adjacent phases and the temperature (unless any of the potentials relate to substances only to be found at the surface). Furthermore, as we know, the pressures p' and p" in these phases are also determined by the temperature and the potentials. Since by equation [500] pf _ p" Ci + C2 = , it follows that the curvature of the dividing surface is also dependent on the temperature and the constitution of the phases separated by it. 14- Derivation of Gibbs' Adsorption Equation Suppose the constitution of the phases suffers a change so that a new equilibrium is established at a temperature t + dt, with new values of the potentials in the phases equal to mi + dyn, H2 + dn2, etc. This will involve changes in the surface energy, entropy and masses to values e^ + de^, rj^ + drj^, mi^ + dnii^, rriz^ + dm2^, etc., and the surface tension will alter to o- + da. The equation [502] still holds for this neighboring state of equilibrium, so that e^ + de^ = {t-\- dt) (tjS + drjs) + {(T + da) (s + ds) -f- (jLii + c?jui) (mi^ + drui^) + etc. 536 RICE ART. L or, neglecting quantities of the second order, di.^ = tdt]^ -\- r]^dt + ads + sda + nidmi + Widiii + etc. But since e^ is equal to a function (^(tj'S, s, rrii^, m-f, ...) of ri^, s, rrii^, m^^, etc., d4) d(j> d4> 34) de^ = — dr]^ + — ris + - — : drui^ + - — : dnii^ + • • • 3?j* ds dmi^ dmf = tdrf + ads + /ii c^Wi'^ + jii dm%^ + . . . by equation (12) above. Hence by equating these two values of dt^ we obtain iq^dt + sda + TUi^dni + mo^dfXi + . . . = 0 , which is equation [503] of Gibbs. Equation [508] is just another way of writing it. We have already seen that a can be expressed as a function of the independent variables, t, jUi, H2, etc., and [508] shows that if this function were known so that a = fit, Hi, juo, • • •)> where/ is an ascertained functional form, then 9/ 9/ 9/ Vs = — — ' Ti = - —- , T2 = - — f etc. (13) 01 Ofil OfJ.2 Equation [508] is the "adsorption equation" and as we shall see presently the experimental verification of its validity is beset with difficulty and some doubt. One cause of this difficulty can be readily appreciated by considering the form of the equa- tions (13) which constitute another way of expressing the Gibbs law of adsorption. Considering the first component, we see that its excess concentration in the surface (estimated of course per unit area) is given by the negative rate of change of the surface tension with respect to the potential of the first component in the adjacent phases, provided the temperature and the remaining potentials are not varied. Now, quite apart from the trouble involved in measuring with sufficient precision the excess con- centration, it is impracticable to change the amounts of the components in the phases in such a manner that all but one of the potentials shall not vary. SURFACES OF DISCONTINUITY 537 15. Variations and Differentials The apparent formal similarity of equations [497] and [501] should not blind the reader to the different implications of the two, which the alternative method of writing the derivation of [508] may help to bring out. In equation [497] the functional dependence in the mathematical sense of e^ on the variables rj-s, s, mi-s, W2'5, etc., is kept in the background as it were; 8e^, dr}^, brrii^, 8m2^, etc., are any arbitrary infinitesimal variations of t^, etc., in other words, although t^ is some function of the quan- tities Tjs, s, rrix^, mi^, etc., presumably discoverable by experi- ment, €^ -{■ 8e^ is not necessarily equal to this same function of the quantities tjs + Srj^, s + 5s, nii^ + Snii^, rrbi^ -\- 8m2^, etc. ; i.e., the varied state is not of necessity one of equilibrium. Equation [497], while being the statement of the condition that the unvaried state is one of equilibrium, is from the mathematical point of view a way of writing down the n + 2 partial differential equations (12). But in [501] the quantities dr]^, ds, drtii^, dnii^, etc. are not arbitrary variations but differ- entials whose values must be chosen so that the varied state is one of equilibrium as well as the initial, i.e., so that t^ + de^ is the same function of ??« -f d-q^, s + ds, Wi^ + dmi^, nii^ + d?r.2^, etc., as e^ is of t?^, s, mi^, m2«, etc. If this is kept in mind it will be seen from the nature of the proof of [508] that, in passing from any state of the system for which [508] is assumed to be true to any other for which it is also true, we must pass through a series of equihbrium states; briefly all the changes involved must be reversible in the usual thermodynamic sense, not merely in the special sense in which Gibbs uses that word. More than one writer has pointed out that in some of the operations carried out in certain experiments made to test the validity of the adsorption equation this condition has apparently not been satisfied and irreversible steps have intervened. Further reference will be made to this presently, but it is this feature of the proof to which we have drawn attention that is involved. 16. Condition for Experimental Tests In many of the experiments made to test the truth of [508] the adsorption is measured at the surface of bubbles of a gas or 538 RICE ART. L liquid rising through another Hquid. Clearly such surfaces are not plane and yet in the argument it is generally implied that the conditions for a plane surface exist. Actually Gibbs has anticipated this point in his discussion on pages 231-233. The crucial point in this is reached on page 232 where he says "Now TiCci + C2) will generally be very small compared to 7/' — 71'." In general where adsorption is very marked Ti/f , which is the average volume concentration in the region of discontinuity, is greater than 7/ or 7 1 , the volume concentrations in the homo- geneous masses; but ri(ci + C2) is of the same order of mag- nitude as Vi/R, where 22 is a radius of curvature of any curve in which a normal plane cuts the surface, and so ri(c] + d) has the same order of magnitude as Fi/f multiplied by ^/R. If the thickness of the film is very small compared to R, the factor ^/R may easily be less than the factor by which one would multiply 7/ or 7/' to obtain Ti/f ; so that Ti (ci + C2) is negligible compared to 7/ or 7/' and therefore to their difference except in the rare cases where 71' and 7/' are extremely near to each other in value. Now even for small bubbles R must be much greater than f , and the conditions postulated would appear to be practically satisfied in the actual experiments. So that, although Gibbs says that "we cannot in general expect to determine the superficial density Ti from its value — {d(r/dfJLi)t.^ by measurements of superficial tensions," the conditions which render this feasible in particular circumstances seem to be satisfied in the usual experiments, and we must look in other directions for the source of the discrepancies which undoubtedly exist. Of course, the first sentence of the next paragraph on page 233 which refers to the practical impossibility of measuring such small quantities as Ti, r2, etc. has no application at present, as the skill of the experimenter has actually surmounted the difficulties. 17. Importance of the Functional Form of a in the Variables We have already pointed out that it is impracticable to obtain da/dfii directly by arranging to vary ni while keeping the other potentials constant. Hence has arisen the device, actually suggested by Gibbs himself, of altering the position of the SURFACES OF DISCONTINUITY 539 dividing surface from that which is termed the surface of tension to one determined so as to make a specified surface concentra- tion vanish. This is fully expounded in pages 233-237. In the case of plane surfaces the term CiSci -f C25C2, which necessi- tated the special choice of the surface of tension, disappears in any case, and although es, rjs, Ti, r2, etc. will change in value with a change in the location of the dividing surface, cr will not change in value. To be sure, the proof given by Gibbs of this statement is confined to plane surfaces, but it is easily seen to be practically true even for surfaces of bubbles of not too great curvature; for on using the equation p' — p" = a(ci -\- Ci) we see that the increment of a caused by a change of amount X in the position of the dividing surface, viz., X(ev" — ^v') — t\{r]v" — riv') — mACti" ~ 7i') — etc., is not actually zero, but equal to o-X(ci -f- Ci). As before, X, which is in all cases com- parable with the thickness of the discontinuous region, is so small that X(ci -f- C2) is an insignificant fraction, and so a is altered by a negligible fraction of itself. A difficulty, however, which might occur to an observant reader is the following. Since a- is a definite function of the variables t, ni, ju2, etc., (for so it has been stated), how comes it that da/ dm, da/dyLi, etc. will alter with the location of the dividing surface? We have just seen that cr does not alter, and certainly the variables t, m, M2, etc. are in no way dependent on where we place the surface; if (T is a definite function of t, m, H2, etc., so also are da/dni, da/dni, etc. definite functions of the same variables, and appar- ently they should no more change in value than a itself. The solution of this difficulty requires the reader to guard against confusing the value of a with the functional form of a. Actually, if after the alteration a remained a function of the variables t, Hi) M2, etc., the implied criticism would be valid; but a does not do so. It must be borne in mind, as indicated by Gibbs on page 235, that, with an alteration which makes Fi zero, a itself, although not changed in value, has to be regarded as an entirely different function, and moreover a function of the variables t, 1J.2, jU3, etc., jui being excluded. The equation V'(i, Ml, M2, ...) = P"(t, Ml, M2, • . •) 540 RICE ART. L enables us to express jui in terms of /, /X2, ms, etc. If this expres- sion for /xi is substituted in the original function expressing a, say f{t, Hi, /i2, . . . ) we obtain an entirely different function say x(^ M2, M3, . • .). No doubt but certainly a//aAi2 is not equal to 6x79^2, etc. The differential coefficients dx/dfii, 9x/9m3, etc., are the new values of the surface concentrations (with reversed sign); there is of course no dx/dfJ-i at all, in consequence of the fact that we have elim- inated Ti; it has no existence. To be still more explicit the equation p' = p" is by means of [93] equivalent to ev' — t-qv' — MiTi' — M2T2' — . . . = tv" - iw" - MiTi" - m2" - . . . (14) Hence ey' - ey" - tinv'- nv") - M2(72^- 72^0 - M3(73^- 73^0 - ■ . . Ml = —, -T, . 71 ~" 71 Inserting this value of mi in fit, ni, H2, . . . ) we obtain x{i, M2, M3, . . •). We can then derive dx/dfx^ by observing that dx df df dm dn2 dfii dni dn2 and obtaining 9mi/9m2 in this result from (14). Thus dx 9/ 9/ 72' - 72" 80 that dm dm dni ji — 7i" — ^2 — ii / _ „ dm 71 — 7i which is equation [515], obtained by Gibbs in another way. We observe in passing that if the dividing surface is considered to be moved a distance X toward the side to which the double SURFACES OF DISCONTINUITY 541 accent refers we increase the amount of the r*'^ component in the conceptual system, in which the two homogeneous phases are assumed to extend right up to the dividing surface, by ^(7r' — 7r") estimated per unit area of the surface, and so we diminish the value of Tr by this amount, so that the new Tr is equal to the old Tr — X(Tr' — y/'); if we choose X to be equal to Ti/iyi — 7/'), this obviously makes the new Ti zero, and the new Tr, i.e. Tra), equal to 71 ~ Ti which is the result [515] once more. VII. Other Adsorption Equations Having commented on the derivation and form of Gibbs' adsorption equation we will refer briefly to other equations, which have been suggested empirically or derived in other ways, concerning the concentration of components at a surface of dis- continuity. Some of these refer to adsorption at solid surfaces just as much as at liquid surfaces; indeed in their derivation the conditions at solid surfaces have been more in the minds of their originators when developing their views. In such cases the concept of surface tension hardly has any bearing on the matter; but of course surface energy is a wholly justifiable term to use, although in the nature of things it is only at liquid-vapor or liquid-liquid interfaces that measurements of change of interfacial energy are practicable. This, however, is a minor matter, as it happens that the surface tension does not enter into many of these laws, apart from the one derived by J. J. Thomson, and a few others. Nevertheless, in the discussions concerning the validity of the Gibbs relation it is hardly possible to avoid making some reference to a few of these other proposed forms of adsorption laws, and that must serve as an excuse for making a brief reference to two or three of the most important of them. For a very adequate account of the complete group of laws the reader is referred to a rev^iew of the literature by Swan and Urquhart in the Journal of Physical Chemistry, 31, 251-276 (1927). 542 RICE ART. L 18. The Exponential Adsorption Isotherm Historically, the oldest equation is one usually referred to as the "exponential adsorption isotherm." We have already mentioned that Gibbs does not use the term "adsorption," and the word itseh has been used somewhat loosely to cover effects complex in origin and due to the operation of more than one cause. It has been suggested that a rough criterion of adsorption proper is that it takes place very rapidly, whilst in many cases the effects produced by the presence of a porous substance such as charcoal immersed in a gas or gas-mixture or in a solution require considerable time to reach completion, McBain has suggested that the whole phenomenon should be called "sorption", and that portion of it which occurs rapidly should be termed adsorption proper. Rapidity of occurrence, however, can only be a rough guide at best. It is only in terms of the effect which Gibbs calls the "excess" (or defect in the case of negative adsorption or "desorption") of a component at a surface that a precise definition can be given. Actually adsorption is to some extent a phenomenon which recalls absorp- tion, i.e., the dissolution of a gas or solute throughout the entire space occupied by a phase. Adsorption, however, differs from absorption in certain fundamental respects. As is well known, absorption equilibrium in a heterogeneous system is governed thermodynamically by a relation which demands (in the simplest case) that the ratio of the concentrations (or more exactly the activities) of a gas or solute in the different phases present shall be independent of the absolute quantity of gas or solute in the system. However, no such constancy obtains in a system consisting of an aqueous solution in which finely divided material such as charcoal is immersed; the concentration term of the solute in the aqueous phase has to be raised to a power less than unity in order to obtain a relation which is capable of fitting with sufficient accuracy the observed values of the adsorption. It is this relation which is called the "exponential" adsorption equation and is written in the form re = A;c" , SURFACES OF DISCONTINUITY 543 where x is the mass of gas or solute adsorbed per unit mass of adsorbing material, c the concentration of the solution in the bulk or the partial pressure of the gas in a gaseous system, n an exponent which in general is less than unity. The exponent n and the constant k are in general functions of temperature. For substances feebly adsorbable n approaches unity. Ap- parently this type of equation appears to have been first applied to adsorption of gases by Saussure as early as 1814, and in 1859 Boedecker extended it to solutions. It has since been em- ployed by a large number of workers. The most complete examination of its applicability in relatively recent times has been made by Freundlich, whose name is now very generally associated with the relation itself. In his Colloid and Capil- lary Chemistry (English translation of the third German edition, p. 93 (1926)), he draws attention to the fact that some of the experimental results at liquid-liquid interfaces fit it fairly well ; for in them there appears a striking feature, corresponding to what is known to be true at solid boundaries, viz., a surprisingly large relative amount adsorbed at low concentrations, followed by a growth as the concentration rises which is not in proportion to the concentration but increases much less rapidly, ending up at high concentrations with a saturation which hardly changes. Actually the exact formula is only roughly valid numerically at high concentrations, but when the conditions are sufficiently removed from saturation it holds quite well. Although only one of many relations suggested, it is still regarded as one of the most convenient and reasonably exact modes of represent- ing existing data, especially for systems consisting of finely divided solids as adsorbing agents. For a discussion of the limi- tations of its applicability the reader is referred to Chapter V of An Introductio7i to Surface Chemistry by E. K. Rideal (1926). 19. Approximate Form of Gihhs' Equation and Thomson's Adsorption Equation Actually Gibbs' equation is the earhest theoretically derived relation; but in 1888, about ten years after its publication, J.J. Thomson obtained by an entirely different method a relation which resembles that of Gibbs. There is a rather prevalent 544 RICE ART. L impression that the two equations are the same, but that is not so; and both on grounds of priority and because of the wider scope of Gibbs' result, there is no justification for the use of the name "Gibbs-Thomson equation" which one sometimes meets in the hterature, although it is doubtless true that Thomson's work was independently carried out. In equations [217] and [218J Gibbs shows that, for a component the quantity of which is small, the value of the potential is given by an expression such as A log (Cm/v), or A log (m/v) + B , where m/v is of course the volume concentration of the compo- nent in question and A,C (or B) are functions of the pressure, temperature, and the ratios of the quantities of the other components. For a dilute solution regarded as "ideal" this result becomes M = Mo + -RHog c , where c is the concentration of the solute and /io is a function of pressure and temperature. This is proved in standard texts of physical chemistry. For non-ideal and concentrated solutions, the relation is given by fi = Ho + Rt log a , where a is the "activity," whose value in any case can be determined by well-known methods described in the standard works. As the concentration diminishes the activity approaches the concentration in value. On this account an approximate form of Gibbs' equation is frequently used for a binary mixture, where the dividing surface is so placed that the surface con- centration of one constituent (the solvent) is made zero. It is c da , . since b^x is put equal to Rt bc/c if temperature and pressure do not vary. Now in Thomson's derivation of his result he uses the methods of general dynamics. The reader may be aware that in that science a system is specified by the coordinates and SURFACES OF DISCONTINUITY 545 velocities or the coordinates and momenta of its discrete parts (the molecule, in the case of a physico-chemical system). The most usual method of attack on the problem of how its con- figuration will change in time is by the use of a group of differ- ential equations which involve an important function of the coordinates and momenta which is called the Hamiltonian function. There is another method, however, actually devel- oped by Lagrange before Hamilton's memoirs were written, which involves another group of differential equations asso- ciated with a function of the coordinates and velocities called the Lagrangian function. J. J. Thomson has made a brilliant application of this analysis to the discussion of the broad development of physico-chemical systems. Before the present- day methods of statistical mechanics had developed, he showed how to convert the actual Lagrangian function of a system into a "mean Lagrangian," expressed in terms of the physical properties of the system which are open to measurement, and by the aid of it to use the Lagrange equations so as to deduce macroscopic results. His work on this subject is summ.arized in his Applications of Dynamics to Physics and Chemistry (1888), a book that has never received the attention which it justly merited. By this method he deduced the following result for adsorption from a solution at its surface: P p = '''p{ii} ('«' In deducing it he assumes that we have a thin film whose area is s and surface tension cr connected with the bulk of the liquid by a capillary tube. The quantity ^ is the mass of the solute in the thin film itself, while p and p' are the densities of the solute in the film and in the liquid, respectively. R is the gas constant for unit mass of the solute, i.e., the gram-molecular gas constant divided by the molecular weight of the solute. Now on study- ing Thomson's work we realize that his mean Lagrangian function is formulated for dilute solutions in which ideal laws are satisfied. This limitation enables us to transform (16) into the approximate form of Gibbs' relation. Provided p'/p is 546 RICE ART. L not very different from unity the argument of the exponential function is sufficiently small to permit us to write 1 + (s/Rt) (da/dO for the right-hand side of (16), and so P — p' s da P ~ ~ Rt' d^ Now, if the dividing surface is placed at the boundary between the film and the vapor, then p — p' is the same as r/f, where ^ is the thickness of the surface film. Hence sf dcr ^ ^ ~ ^Rtd'^' But ^/(sf) is equal to p, and so P dcr , , which under the limitations assumed is practically the approxi- mate form of Gibbs' equation. The details of Thomson's work will be found in the Applications, Chapter XII. A critical inspection of the two formulae, Gibbs' and Thomson's, shows that they are not so similar as one imagines. We have already mentioned that the assumptions made concerning the dilute nature of the solution places a limitation on Thomson's result not ostensibly present in Gibbs'. Added to that, it is possible that the mathematical restrictions imposed by the neglect of higher powers in the expansion of the exponential function may place a further restriction on (17) which is more severe than that necessitated by the physical assumption concerning dilu- tion. Thomson actually makes no quantitative application of his formulae — indeed in those days there were no data available ; he draws from it just the same broad qualitative conclusions which can be inferred from Gibbs' result. If the presence of a solute lowers the value of the surface tension, so that da/dc or da/dp is negative, then T is positive by Gibbs' equation and p' < p by (16), which we can write in the form SURFACES OF DISCONTINUITY 547 p - ^^P \R^t dp, = '^P Km Tk. (18) where k is the surface density of the solute, not in Gibbs' sense of an excess, but of the actual amount in the film. If, on the other hand, the surface tension is increased by increasing con- centration of the solute, V is negative or p' > p, and the solution is less concentrated in the surface film than in the bulk of the phase; there is "desorption." Actually in the approximate form of Thomson's relation, viz. (17), a is differentiated with respect to p, the equivalent of the volume concentration in the surface; to make it the exact counterpart of the approximate form of Gibbs' equation it should be p;_da_ ^ ~ ~ Rtdp'' No doubt under the severe limitations imposed (which we have just referred to) this change is justified, but it is well to notice that in Thomson's actual result the concentration which is the variable on which a depends is the surface concentration. In Gibbs' adsorption law the variable is the chemical potential and it matters not at all whether we refer to the potential at the surface or in the bulk of the phase, since by the equations of equilibrium they are equal; when we approximate we naturally use the approximation for the potential in terms of the bulk concentration. This indeed will serve as a cue to raise a small point which, as the writer knows from experience, occasionally causes some perplexity. The surface tension is of course measured at the surface and we cannot help feeling that it should be directly dependent on the concentration there. When one sees the expression da/dc it is not altogether unpardonable to feel somehow that in this differential coefficient a is the surface tension at the surface of a hypothetical solution in which there is no concentration at the surface. Any such idea must be carefully avoided. Such a condition would of course be physi- cally unrealizable, and the conception is entirely valueless. To 548 RICE ART. L repeat it once more, cr is a function of t, m,, y.^, etc., quantities whose values in the bulk of the solution are meant, and any approximations make Rt dc where c is the ratio of solute molecules to solvent molecules and a is a factor obtained from the equation = log — 1 — ac J) P being the saturation pressure of an adsorbed gas or vapor and p its equilibrium pressure. In this the departure from the simple approximate Gibbs' formula is attributed to the forma- tion of loose compounds between the molecules of the solute and those of the solvent, which is termed solvation. This has the effect of altering the internal pressure of the solution and with it other properties such as surface tension and compressi- bility which depend upon the internal pressure. On account of the existence of this solvation Freundlich has criticized the approximate form of Gibbs' law even for dilute solutions, since this property certainly interferes with the application of the simple van't Hoff laws to them. Langmuir, however, has replied to this criticism by pointing out that there are deriva- tions of the law, e.g. Milner's, in which the gas laws are applied only to the interior of the solution. This, of course, does not invalidate in any case the complete form of Gibbs' law, although even this is almost certainly limited to true solutions and cannot be applied to colloidal solutions. This point has been empha- sized by Bancroft (J. Franklin Inst., 185, 218, (1918)); we have already drawn attention to the feature of the proof which im- plies thermodynamic reversibility of the adsorption process, and that is certainly in doubt in some instances where the equation has been applied. Undoubtedly in true solutions some equation of the form holds, where /(c, t) is some function which is positive; but this SURFACES OF DISCONTINUITY 551 cannot be formulated correctly until a general formula for potential in terms of concentration has been discovered. 20. The Empirical Laws of Milner and of Szyszkowski for ''") s V = g(l - e— ) , ^ (21) where g and a are constants. We see that this adsorption isotherm has the same feature as (19), viz., that tii/s the surface concentration of the solute approaches a hmiting value g as c increases. In fact, since g is ^/v, we see by the definitions of f and v that g is the surface concentration when the assumed unimolecular layer is quite full. By measurements of the surface and bulk concentrations at different states of dilution where the equation is valid we can eliminate g and measure the constant a. By repeating these measurements at another temperature we can determine the value of a at this other temperature, say a' at temperature t'. This gives us the ratio X'/X which is of course equal to a'/a. But X = exp(—u/kt); hence we obtain and knowing k, t and t' we can obtain u the energy of adsorption. VIII. Experimental Investigations to Test the Validity of Gibbs' Adsorption Equation S2. The Earlier Experiments to Test Gibbs' Equation The simplest conditions from a theoretical point of view for testing the Gibbs equation exist at the boundary separating a vapor from a liquid; however, this is not the easiest case to test by experiment, and measurements carried out at air-liquid or liquid-liquid interfaces make up the majority of the attempts in this direction. When we have a binary mixture, the equa- tion becomes (at constant temperature) da = —Tidjii — T2dijL2. As we have seen, this is only strictly valid for the surface of 558 RICE ART. L tension determined in the manner pointed out earlier. Practi- cally, however, any surface in the film will serve, provided that the values of Vi and r2 are adapted, as we have shown, to the chosen situation. It has been customary to choose the position of the surface so that the actual amount of one of the com- ponents in the discontinuous region is the same as if its density were uniform in each phase right up to the surface. This makes one of the excess concentrations (say Ti) zero, and the equation becomes da = — Fgd) dfi2 . Gibbs, himself, originally suggested this procedure and gives an example of its application in the footnote to page 235. In a number of the measurements, the simple formula for the chemical potential H = Hq -\- Rt log c has been used, and these on the whole indicate that a solute which lowers surface or interfacial tension is concentrated more at the surface than is deduced by the use of this formula. Measurements of the activity of solutes are not yet very numer- ous, but wherever the more accurate expression for the potential fi = Ho -{■ Rt log a can be used, the agreement is very much better, though there still appears to be a greater concentration than the equation would lead us to expect. However, in addition to direct tests of the vaUdity of the equation, it has been used to investigate the structure of the surface region, and the comparison of the results with the properties of films of insoluble substances at the surface of a liquid, obtained by Langmuir, Adam and others by different means, seems to lend considerable support to its va- lidity. There are a number of early investigations which show that a concentration of capillary-active solutes at the surface actually does take place. Plateau {Pogg. Ann., 141, 44, (1870)) showed that the viscosity of the surface layers of a saponin SURFACES OF DISCONTINUITY 559 solution in water was greater than in the interior. Zawidski (Zeit. physik. Chem., 35, 77, (1900) and 42, 612, (1903)) pre- pared saponin foams and showed by means of measurements of the refractive index that the saponin content in the foam was higher than in the original solution. Analogous qualitative information was obtained by Ramsden (Zeit. physik. Chem., 47, 336, (1904)) on the accumulation and consequent precipita- tion of protein at surfaces. C. Benson (J. Phys. Chem., 7, 532, (1903)) examined foams from aqueous solutions of amyl alcohol and also observed excess concentration of the alcohol in the foam. An important investigation was made by S. R. Milner (Phil. Mag., 13, 96, (1907)) on solutions of acetic acid and sodium oleate. He used the Gibbs equation in its simple form to calculate the surface excess in the first case and brought out the important fact that the surface excess for a normal solution of acetic acid is only about 15 per cent less than what it is for a solution eight times as concentrated. In the case of sodium oleate, its high capillary activity causes the surface tension to fall so rapidly that the ( 580 RICE AET. L where C and A are given at the top of page 268. Now 7/ is the density of the Hquid and 71" is the density of the Hquid's vapor in the gaseous phase, so that 71" is very much smaller than 7/; 72" is the density of the gas or vapor, whose adsorption is being considered, in the gaseous phase ; 72' its density in the liquid bulk phase, may be regarded as zero. Hence, practically, A = -7iV, C = ri72" + r2(7/ - 7/0 = ri72" + r27i' . Therefore c _ _ r3_ _ £2^ A ~ 7/ 72" ' Since Ti is zero by the choice of dividing surface, it follows that C _ _ £2 A " ~ 72" or da dp where 7 refers to the density of the adsorbed vapor in the gaseous phase.* Before passing on to consider the experi- mental results we may remind the reader of the mechanical explanation of gaseous adsorption given m the last paragraph of section IV of this article. The existence of a surface energy depends, as we saw, on a normal field of force existing in a molecular layer at the surface of the liquid and also extending a similar distance into the space above the liquid. Such a field would cause an increased concentration of gas close to the sur- face, just as the density of the atmosphere is greatest at the lowest level in the earth's gravitational field. Actually the outward attraction of this concentrated layer of gas would * Not of the liquid's vapor; 7/' is the density of that. SURFACES OF DISCONTINUITY 581 tend to weaken the field of force to which it is due and so produce a diminution in the surface energy. 30. The Experiments of Iredale We shall first briefly review the results obtained in Donnan's Laboratory by Iredale {Phil. Mag., 45, 1088 (1923); 48, 177 (1924); 49, 603 (1925)). He deals principally with the adsorp- tion of vapors of organic substances at the surface of mercury; these have the property of lowering the surface tension of mer- cury. The drop weight method of determining surface tension was used and its accuracy is carefully discussed. The vapors were generated by passing a very slow current of dry air at con- stant pressure through the organic liquids. The adsorption of the vapor at the surface of the drops appeared to be a fairly rapid process; for "the period of drop formation was never less than 3| minutes and with longer periods the weights of the drops were not found to decrease appreciably" thus indicating that a steady condition of surface tension had been reached. The re- sults with methyl acetate vapor showed a fall from 470 dyne per cm. to about 430 for a partial pressure of 40 mm. in the vapor; thereafter the fall was much slower, reaching a value about 412 dynes as saturation of the vapor at about 225 mm. was ap- proached. At this point there was a sudden fall of the surface tension to about 370 dynes which is the value of the surface tension of mercury in liquid methyl acetate. Taking the slope of the graph, which gives da/dp at 62 mm. pressure, where the conditions of maximum adsorption are being approached although the vapor pressure is still well away from saturation, and multiplying it by y for the vapor there, a value about 4.5 X 10~* gram of methyl acetate per sq. cm. is obtained. This corresponds to about 0.37 X lO^^ methyl acetate molecules per sq. cm. of mercury surface. This figure is near the values given by Langmuir {J. Am. Chem. Soc., 38, 2288, (1916)) for unimolecular layers of carbon dioxide, nitrogen, etc. "More- over the space taken up by each molecule (27 X 10^^^ sq. cm.) is near that required for molecules of esters and fatty acids on the surface of water, namely, 23 X 10"^ sq. cm., and it is possible that the same type of orientation obtains on the mercury surface. 582 RICE AKT. L There appears, however, to be a somewhat abrupt change from a simple adsorption process to a condensation." In later work Ire- dale examined more carefully the remarkable behavior exhibited at the saturation point of the vapor. Among the vapors studied was water vapor in the presence of air. In this case the slope of the {a, p) curve was practically uniform up to the saturation point, and so the adsorption increased uniformly with the den- sity and partial pressure of the vapor right up to the satu- ration point. Calculation of r at this point gives a value 1.8 X 10~^ gram per sq. cm. which is somewhat less than that required for a unimolecular film (3.8 X 10~^ gram per sq. cm. according to Langmuir). At the saturation point there is the same instability in the tension of the vapor-mercury interface, its value being entirely uncontrollable and lying anywhere between 447 and 368 dynes per cm. Iredale suggests that the primary phenomenon is the gradual formation of a uni- molecular layer, this being represented by the earlier portion of the curve. After the vapor reaches the saturation value a very thin film of liquid may be produced, the thickness of which "is not a determinate function of the pressure and temperature, though the most stable state corresponds to the formation of a film, which may, from the standpoint of intermolecular forces, be regarded as infinitely thick." Iredale also examined the adsorption of benzene vapor on a mercury surface. This showed one rather unexpected feature. He considered that near the saturation point the value of r attained a maximum and decreased slightly with a further small increase of pressure. He also found a similar tendency in methyl acetate, though not in water vapor. (This was criticized later by Micheli whose work we shall refer to presently.) The maximum value for benzene was such as agreed with an area 21 X 10~^^ sq. cm. for each molecule, very near to Adam's value (23.8 X 10~^^) for certain benzene derivatives on a water surface, and once more supported the view that the vapors adsorbed on the surface of mercury tend to form primary unimolecular films. Further measurements were made using the sessile drop method for measuring surface tension, and without admixture of air. These results were in fair agreement with the previous work and SURFACES OF DISCONTINUITY 583 gave much the same value for the area per molecule of adsorbed benzene on the mercury surface. Experiments were carried out with ethyl alcohol, propyl chloride, and ethyl bromide, showing that, as in the previous cases, the adsorption of these substances appears to be within certain limits a reversible phenomenon. Iredale expresses surprise that these substances, "which are more definitely polar than benzene and, especially in the case of the alkyl halides, possess an atom or group more likely to form a definite finking at the mercury surface, should have no more marked effect on the surface tension than benzene itself." SI . The Experiments of Micheli, Oliphant, and Cassel Subsequently Micheli at Donnan's suggestion {Phil. Mag., 3, 895 (1927)) took up the same problem. He examined the va- pors of benzene, hexane, heptane, pentane and octane, all in a high state of purity, at a water-vapor interface using the drop- weight method. It was found that if ' where on the left-hand side we suppose that Gibbs' "heat function," x, is expressed in terms of the variables t, p, s, r. Hence ,, X . da(t, p, r) dxjt, P, s, r) c{t,p,r)-t—^^—= ■' This will be found on careful examination to be equation 22 of Chapter XXI of Lewis and Randall's Thermodynamics. The equation [594] of Gibbs can be obtained by similar 592 RICE ART. L methods. Thus by the second equation of (22) dp and dUt, V, s, r) = v{t, p, s, r)*, = <^ii, V, r). ds Hence by cross-differentiation dv(t, p, s, r) _ dajt, p, r) ds dp The left-hand side is the quantity — F in Gibbs' text. This equation also appears in Lewis and Randall's book as equation 19 of Chapter XXI. 34. Empirical Relations Connecting a- and t. Degree of Molecular Association in Liquids We have referred above to the approximately linear relation between surface tension and temperature for many liquids. Also, since surface tension must vanish at or near the critical temperature of a liquid, the relation should then be (T = Co (■4). where o-q is a constant for the liquid and tc the critical tem- perature. Almost 50 years ago Eotvos from a not too rigorous argument suggested that the constant o-o should vary as the number of molecules in unit area of the liquid surface; since the number of molecules per unit volume varies inversely as MV, where M is the molecular weight of the liquid and V the specific volume of the liquid, ao would then vary inversely as (M7)* or directly as (D/M)^, where D is the density of the liquid. About ten years later Ramsay and Shields, in a series of well- Note that V is the volume of the whole system. SURFACES OF DISCONTINUITY 593 known researches, found considerable support for the law pro- vided M was taken to be the molecular weight of the liquid and not of the vapor. Indeed this work was used to calculate the degree of association in many liquids. Ramsay and Shields actually made another slight modification of Eotvos' law, writing it i OC02 / These two (condensed) terms of the original condition of equilibrium [600], viz. — SyWv + fabDs, are the two which offer the most trouble in being transformed into a convenient form. When we replace them in [600] by the expressions just obtained we can rewrite the condition [600] in the form ft Wyf + ft SDrj' + fW + gz.') 8Dm,'^ + /(mi" + gzx") 8Drm"^ + /(mi" + gzi') ^Dm,' + /(m/ + gz2') bDm-r + finz" + gz-n 8Drm"^ + finz' + gz^') dDm2^ dx' dy' fj, 8x" + „ , dx dy + /{k+S^^" + S-^^"+S^.'7o»' 602 RICE ART. L + [[[(v" - V') + ^(ci + C2)] hN + gV Sz^ r da da 1} ^ |_daJi 00)2 J) + fadTDl = 0. Now we introduce the usual conditions, viz., f8Dr,y + fdDrjs = 0 , fSDmi'y + f8Dmi"y + /5Dwi^ = 0 , fbDrrii'^ + fWm2"y + fWm^s = Q , and in addition to these the further conditions that Sx', by', 8z', 8x", 8y", 8z" are arbitrary, and that 8z^ = 8N cos 6 + ai5coi + a25w2 , where aiScoi + a25co2 is the tangential part of the displacement of a point on the surface, ai and a2 being functions of coi and wj and the angles between the vertical and the directions in the surface defined by 6coi and 80)2. It follows from the conditions of equilibrium and these addi- tional conditions that t = a, constant throughout the system, Ml' + 9^1 = Ml" + gzi" = Mi^ + gzi^, M2' + gz2 = M2" + 9Z2" = M2^ + gzi^K ^' _ ^' = n dx' dx" "' ^ _ ^' = n dy' dy" "' dp' [605] [617] dz' dz" = - gy = — gy ft [612] SURFACES OF DISCONTINUITY 603 p' - p" = a(ci + C2) + ^r cos 6. [613] Also gT (ai Soji + 02 5co2) = t 5coi + ~ 5w2. ocoi aw2 This means that for any arbitrary displacement of a point in the surface in a direction tangential to the surface the variation 8a in o- is equal to ^r multiplied by the vertical component of this displacement; for a reference to the expression for 8z^ above reveals that this is the meaning of ai5wi + a28u2. Hence we have 'i = sr. [6141 To summarize the matter we see that the potential of any component does not remain constant throughout a given phase; it decreases with altitude. What remains constant throughout the phase is /i + gz, and the constant value of this for a given component is the same in each homogeneous phase and on the surface of discontinuity. The pressures p' and p" and the surface tension a are functions of t and the constants Mi, M2, and are therefore functions of z, and their rates of change with respect to z are given in [612] and [614]. They are independent of X and y. We have omitted the last result faSTDl = 0. This has been written so far in too simple a form, in order to avoid causing trouble at the moment by an awkward digres- sion. We have been considering, it will be recalled, two homo- geneous phases and one surface of discontinuity. This would of course be realized if one phase were surrounded entirely by the other, but as in that case the dividing surface would have no perimeter at all the condition written would be meaningless. However, we are not necessarily confined to this case, but if we treat two phases in a fixed enclosure, then we must include the wall of the enclosure as a "surface of discontinuity" as well as the dividing film between the two phases. It is true that we assume 604 RICE ART. L that no physical or chemical changes take place in the wall, and no energy changes so caused are therefore involved, but the perimeter of the dividing surface may move along the wall (the creeping of the meniscus in a capillary tube up or down is a familiar example) and the condition above must then be written f(ai8Ti + CX28T2 + azbTz)Dl = 0 , where 8T1 is the tangential motion (normal to Dl) in the dividing surface, 8T2 the tangential motion in the surface between the single accent phase and the wall, dTs that in the surface between the double accent phase and the wall, and o-j, cr2, 0-3 are respec- tively the three free surface energies between the two phases, and between each phase and the wall. This means that at any point of the perimeter (T18T1 + 0-25^2 + (Ts8Ts = 0 , and this is the well-known condition ci cos a + 0-2 — o"3 = 0 , where a is the contact angle between the dividing surface and the wall. Actually, in the general case of several homogeneous phases and dividing surfaces, the condition is interpreted in a similar way for a number of surfaces of discontinuity (at least three) meeting in one line, as is shown at the bottom of page 281 of Gibbs' treatise. The constants Mi, M2 are the potentials at the level from which z is measured (positive if vertically upwards). It follows that p', p", 0-, r are functions of t, Mi, M2, z. If determined by experiment these functions enable us to turn [613] into a differ- ential equation for the surface of tension as shown in pages 282-283. Equation [620] is an approximate form of this differential equation. We refer the reader to the short note on curvature (this volume, p. 14) for an explanation of the left- hand side of it. SURFACES OF DISCONTINUITY 605 XIV. The Stability of Surfaces of Discontinuity 38. Conditions for the Stability of a Dynamical System When the stabiHty of a dynamical system is being investi- gated, the potential energy of the system is expressed as a function of the coordinates of the system. If the system were at rest in any configuration this function of the coordinates for this configuration would give the whole energy of the system. If this configuration is one of equilibrium then the partial differential coefficients of the function with respect to different coordinates are severally zero; for if /(xi, Xi, xs, . . .) represents the function, Xi, Xi, Xz, ... being the coordinates, we know that to the first order of magnitude f{x\^ Xi, xz, . . . ) must not vary in value when xi, x^, Xz, ... receive small arbitrary increments bxi, 8x2, dxz, . . . Thus 9/ 9/ 9/ — 8x1 + — 8x2 + — 8xz+ . . . =0, dxi dX2 dxz and since 8x1, 8x2, 8xz, . . . are arbitrary, it follows that 9/ 9/ df — = 0, r^ = 0, r" = 0, etc. dxi ' dx2 ' dxz We can express this simply by the condition 8f{xi, X2, xz, . . . ) =0. Now the equilibrium may be stable, unstable or neutral. If we wish to investigate the matter in more detail we must consider the value of A/(a:i, X2, xz, . . .). This is equal to the value of f(xi + 8x1, X2 + 8x2, xz + 8x3, . . . ) — f(xi, X2, xz, . . . ) when higher powers of 8x1, 8x2, 8xz, etc. than the first are re- tained in the expansion of f(xi + 8x1, X2 + 8x2, xz + 8xz, . . .). In many cases it is sufficient to retain the second powers and neglect those that are higher. For convenience we write ^1, ^2, ^3, ... for 8x1, 8x2, 8xz, . . . Then by Taylor's theorem 606 RICE ART. L A/(a:i, X2, X3, 9/ df df dxi dX2 dX3 + 1 + 2 ay aa;i2 ^' ^ aa;2' 32/ ^ 32/ + 32/ 3a:i 3x2 ^1 $2 + 2 32/ 3xi 3X3 ^1^3 + + 2 92/ 3rc2 3x3 ^2 $3 + ]■ The values of df/dxi, df/dXi, etc, are zero when xi, xi, xt, ... are the values of the coordinates for the configuration in question. For convenience let us represent the values of d^f/dxi^, d'^f/dx^^, . . . d^f/dxidXi, . . . for the same coordinates by the symbols flu, 022, . . . ai2, . . . The symbol 021 would represent d^f/dXidxi, but by the law of commutation for partial differentials this is the same as a^. Now if the configuration is one of stable equilibrium, the value of /(xi, X2, X3, . . .) is less at the equilibrium configuration than for any neighboring configuration. Hence if the equilibrium is stable the quadratic expression ail^l'* + ^22^2^ + «33^3^ + 2ai2^i$2 + 2ai3^i6 + 2a23?2?3 + . . . is positive for any arbitrary values of ^1, ^2, ^3, ... In short it is a "positive definite form."* The conditions which must be satisfied by the coefficients an, 022, . . . an, . . . for this to be the case are well-known and can be most readily expressed in terms of the determinant Oil ai2 ai3 . . ain an ^22 ^23 . . . azn 031 ^32 ^33 . . flan dnl am anz Or * See the note on The Method of Variations, this volume, p. 5. SURFACES OF DISCONTINUITY 607 and its minor determinants. Thus if the form is to be definitely positive, this determinant, the first minors obtained by erasing any row and a corresponding column, the second minors ob- tained by erasing any two rows and the corresponding columns, the third minors obtained in a similar way, and so on until we reach the individual constituents of the leading diagonal, must all be positive quantities. If this is not so the form will have negative values for some sets of values of ^i, ^2, ^3, ... and so the system will for some displacements not tend to return to, but will move further away from, the original equilibrium con- figuration. Indeed if the first minors, third minors, fifth minors and so on had one sign; the determinant, the second minors, the fourth minors and so on, the other; the system would be unstable for any displacement whatever. 39. Restricted Character of such Conditions as Applied to a Thermodynamical System In the investigation of the stability of a thermodynamic system a similar procedure can be followed, but it suffers from one limitation which Gibbs discusses. The energy of the system is regarded as a function of the thermodynamical variables, which in the present instance specify the condition of the homogeneous masses and of the film separating them. For equilibrium 6e must be zero for any arbitrary infinitesimal variations of these variables — ^at least, arbitrary apart from the familiar conditions such as [481].* For stable equilibrium Ae will be positive for all possible variations of the variables within the assigned limitations. If we then proceed to apply the method just outlined we must conceive e to be formulated as a function of the variables, (the entropy, masses of components, volume, area of film) and the first and higher differential coeffi- cients also so expressed and the tests applied. (See the proof for the thermodynamic system as given on pages 105-115, especially [173] et seq.) But this assumes that in any state, other than the initial one, whose energy content needs to be * This restriction in arbitrariness would render the analytical pro- cedure in such a case somewhat more complicated than that indicated above, but would not invalidate the general idea. 608 RICE ART. L considered, we are regarding the energy as expressible in the same functional form of the altered values of the variables, and this implies that such other states are states of equilibrium. In consequence, this method limits us to the consideration of the stability of the initial state with reference to the neighboring equilibrium states, but not with regard to all neighboring states, among which may be non-equilibrium states. In the purely dynamical problem, all states of the system, equihbrium or not, have their potential energy expressible in terms of the coordi- nates; but in the thermodynamical problem all the states of the system cannot have their energy expressed in terms of the variables. Indeed certain values of the variables inconsistent with equilibrium may "fail to determine with precision any state of the system." The question of instability would of course offer no difficulty in this case. If near the equilibrium state in question there exist one or more other equilibrium states which under the usual conditions possess less energy, the origi- nal state is certainly unstable; that requires no consideration of non-equilibrium states. However, although there may exist neighboring states of equilibrium which might prove, on investi- gation by the method outlined, to be states of greater energy, we cannot be so definite about the original state being one of stable equilibrium; for the method does not preclude the pos- sibility of the existence of non-equilibrium states of smaller energy. Having drawn the reader's attention to this matter, which we shall take up later, we proceed to a commentary on the subsection. 40. Stability of a Plane Portion of a Dividing Surface Which Does Not Move At the outset Gibbs deals with the problem of stability with the limitation that the dividing surface film is plane and uniform and is not supposed to move. He directs attention to the possibility of a small change taking place in the variables which specify a small portion of the fihn, and which are a small group of the entire collection of variables specifying the whole system. Denote the small part of the film by Ds; its variables are the temperature t, its entropy Dt]^', and the masses of the com- SURFACES OF DISCONTINUITY 609 ponents in it DrUa^', Drrib^', . . . Dnig^', Dnih^', ■ . ■ The change does not in the first instance involve an alteration in t, nor in the position or size of Ds; but Drria^' is changed to Dma^", etc., and Dri^' to Dri^"; in short, the single accent indicates the initial state, the double accent the state after change. Of course the changes of mass in this small portion of the film must be drawn from (or passed into) the remaining portion of the system, i.e. , the rest of the film and the homogeneous masses. Similarly as the total entropy must remain constant the rest of the system must experience a change of entropy equal to Drj^'—Drj^". The homogeneous masses are assumed to be relatively so great that these small changes in them do not practically affect the values of the potentials Ha, f^b, ... of the components a, h, . . . which are both in the volume phases and the surface phase, so that no accenting is required in writing them. A similar remark applies to the large remaining portion of the film. However, as regards the g, h, . . . components which only occur at the surface, the value of the potentials will alter in Ds from /Xg', Hh, ... to Hg", fjLh", . . . , but for the rest of the film they will remain at their original values fig', nh, . . . for the reason already specified, viz., that the changes of masses and entropy in this part of the film are relatively too insignificant to effect a change in the potentials. It is very important to keep in mind the fact that it is assumed that there are components in the surface which are not in the homogeneous masses; otherwise the discus- sion of this particular special case would be pointless. The new condition of the portion Ds of the film is supposed to be one which is still consistent with equilibrium between it and the neighboring homogeneous masses. (This of course places the limitation mentioned above on the generality of the investiga- tion. It will be quite definite in its answer concerning instabil- ity, but leaves a possibility of failure to lead to a definite conclu- sion concerning stability.) In consequence, the energy of the small portion, Ds, of the film will be De^", where D^" is the same function of the variables t, D-q^", Dnia^", etc., and Ds, that Z)es' is of t, Drjs', DiUa^', etc., and Ds. The energy of Ds is therefore increased by Dt^" — D^'. The energy of the rest of the system is increased by an amount which is equal to 610 RICE ART. L t8r]' 4- tia'^rria + Hh'^rrih . . . + ixg'hvfhg' + lihhmh + . • . where 677', 5ma', etc., are the increases of entropy and of the masses of the various components in the rest of the system. But we have seen that these increases are Dtf' — Drj^", Dma^' — Dnia^", etc. Hence the increase in the energy of the rest of the system is tiDri^' - Dr,s") + tiJ{Dm.^' - Drua^") . . . + tio'il^m/ - Dm/') + . . . (24) The increase in energy of the whole system is therefore D^" - D^' + tiDri^' - Dr}S") + fiaiDma^' - Dnia^") . . . + IX,' {Dm/ - Dm/') + ... where we have dropped as unnecessary the accents over Mo, Hb, . . . , the potentials which do not alter between the first and second state. Now by [502] applied to the small portion of the film, which it will be remembered is in an equilibrium condition in both states De^' = t D/ + 0-' Z)s + yiaDma^' ... + fx/ Dm,^' + . . . , De^" = t D/' + (t"Ds + tioDm/' ... + iiJ'Dm/' + . . . , where a' and a" are the values of the surface tension in the small portion in the two states. Hence we easily see that the increase in energy of the whole system is equal to ia" - a'. There appears to be a contradic- tion here; we have seen that o- is a function of t and the potentials Mo^, Mb^> • ■ • M(7^, fJ'h^, • ■ ■ and it appears absurd to assume that • • • )"»", M^", • • • But this is to over- look the possibility of a being a double-valued or multi-valued function of the temperature and potentials, so that if the variables ^a, M6> • • • M^j y-n, . ■ • experience a change of values corresponding to changes in the masses of the components, and presently retake the same values, the surface tension may not retake its original value. (We have already made use of this result in an earlier part of this commentary to show that if there are, say, a "gaseous" and a "liquid" phase in the surface of discontinuity, they must, if stable, have the same value of a.) The second conclusion drawn concerns the sign of a. In the argument so far there has been no displacement or def- ormation of Ds. It is implied also that s is practically plane. If Ds being plane is deformed, its area must increase. This will necessitate the withdrawal of small amounts of the com- ponents from the homogeneous masses or from the rest of the film in order to maintain the nature of the film in Ds unchanged. These amounts, as before, will be infinitesimal for the rest of the system. The amounts will have gone from a place where the potentials have been at certain values to a place where they are at the same values. This will cause no change in the energy of the system; the term of the energy expression which will have altered will be aDs which will become a{Ds + 8Ds). SURFACES OF DISCONTINUITY 613 The energy change will be adDs. For stability this must be positive, and as 8Ds is positive, a must be positive. The paragraphs on pages 240, 241 elaborate this. The third conclusion occurs in the paragraph beginning towards the bottom of Gibbs, I, 241. It is very elusive indeed and the final sentences of the paragraph are not very happily chosen for a reader not expert in mathematical technique. First of all the reader must realize that there may be a whole con- tinuous series of states of the system differing in the nature of the film, which will be states of stable equilibrium. A change from any one of them to any state infinitesimally near it, whether a non-equilibrium state or one of its equilibrium neighbors, will involve an increase of energy. Let the single and double accents refer to two neighboring infinitesimally different states of stable equilibrium. We have seen then that (a" - a')s + W - m/)w/" + W - Hk')mH'" + • • • must be positive. But exactly the same reasoning will show that {a' - a")s + (m/ - lij')^/ + U' - tJ^h")m,^' + • • . must also be positive. Now write fXg for /x/', Hh for ixh^', . . . Hg -{■ Afig for fXgS", fx h ^ A)U/, for nh^", etc.; o- for a', (t -\- Acr for a", m/ for Mg^', Mh^ for Mh^' , ... m^ + Anig for m/", rrih + Anih for w^-s", etc. From the expression given four lines above we obtain the result s(-Ao-) + m/(- Arrig) + mh^{— Amn) + ... > 0, which is just the equation preceding [521]. Considering [521] we may write it, remembering that Hg, Hh, . . . are the only quantities which are varying, d 0 the complete system is stable as regards the movement of the surface. Since the total entropy and masses are constant we can state that if aAs - p'Av' - p"Av" > 0 * Finite, that is, with reference to the system; they are small com- pared to the external mass. SURFACES OF DISCONTINUITY 617 the complete system is stable. Now if the complete system is stable, the original system (without communication with external mass) is certainly stable. For blocking up the tubes and isolating the original system is equivalent to imposing a mechanical constraint on the complete system; and it is well known in mechanics that if a dynamical system is in a stable state of equilibrium, the imposition of a constraint does not upset that condition. Indeed this fact is intuitively obvious. The inequahty [549] is simply the same result extended to a wider system. But, of course, the condition may not be necessary for stability of equilibrium as regards movement of the surfaces; in short it insures stability for the system under wider conditions than are actually envisaged at the outset and so under more restricted conditions than these the system might be stable without [549] being satisfied, 43. Gibhs' General Argument Concerning Stability in Which the Difficulty Referred to in Subsection {39) Is Surmounted The general argument of Gibbs on the conditions of stability or instability will be found on pages 246-249, (On pages 242-246 he discusses the problem by a more specialized method which can be passed by for the moment.) At the outset of the argument he raises the point which we have already noted, that if we use an anal3rtical method, analogous to that employed in dynamics, we are virtually excluding from consideration those states of the system which are not in equilibrium and for which the fundamental equations are not valid and the usual func- tional forms for energy, etc. have no meaning, since in these states the systems cannot be specified with precision by values of the usual variables. That is dealt with on page 247. He proposes then to surmount this obstacle by introducing the consideration of an "imaginary system" which is fully de- scribed at the top of page 248. This system agrees with the actual system in all particulars in the initial state, which is one of equilibrium for both systems, though whether it is stable or not for the actual system is the point under consideration. His argument, however, may be framed so as to exclude any express consideration of his imaginary system and may appear simpler 618 RICE AET. L on that account. We may,for simplicity of statement, consider a system of two homogeneous masses with one dividing surface; the statement can easily be extended to cover wider cases. Let us suppose the system is varied to a state in which the condi- tions in the phases and dividing surface are not conditions of equilibrium as regards temperature and potentials, and the dividing surface is changed in position ; also let it be found that this is a state of smaller energy than the unvaried state, the total entropy and total masses however being the same as originally. Now imagine that the dividing surface is "frozen," as it were, in the varied position. (This is equivalent to the postulate of Gibbs as to constraining the surface by certain fixed lines.) If left alone, the system in this "frozen varied" state would tend to a new state of equilibrium; we are conceiv- ing that its total energy is not altered from the varied value, nor, of course, the individual volumes of each phase; the total masses are not to vary either, but there may still be passage of components through and into or out of the dividing surface (its rigid condition is not to interfere with that). In this third state (second varied state) the entropy will of course have increased above that of the first varied state and so above that of the original state of equilibrium. Now by the withdrawal of heat (the rigidity of the system being still preserved) we can arrive at a third varied state, which is also one of equilibrium, in which the total entropy, etc., will be as originally, but the energy less than that of the second varied state and therefore less than that of the original state. Of course, on imagining the surface now to be "thawed out," that is, the constraint on it removed, we cannot be sure that the varied pressures established in the phases and the varied tension in the surface will be con- sistent with the curvature of the dividing surface, which must of course remain in the same varied position all the time (for if it moves from this the volumes and therefore the potentials will change from the values arrived at in the last state and might not be in equilibrium in the two phases in the final state). The point, however, is that if there is a non-equilibrium state infinitesimally near the original state which is one of less energy, there is also a quasi-equilibrium state infinitesimally near which SURFACES OF DISCONTINUITY 619 is also one of less energy — using the word "quasi-equilibrium" to designate a state in which the equilibrium conditions for the temperature and potentials are satisfied, but not the mechanical condition which connects the difference of pressures in the two phases with the tension and curvature. More than that, if there is no quasi-equilibrium varied state which has less energy than the unvaried state there is no non-equilibrium varied state which has less energy; for as we have just seen if there were one such non-equilibrium state there must be at least one such quasi-equilibrium state. Thus if there is no equilibrium state, or quasi-equilibrium state, infinitesimally near to the given state which has a less energy than that state, it is one of stable equilibrium. Now all such states, equilibrium or quasi- equilibrium, are states for which e is given by the fundamental expression in terms of the variables 77', 77", 77^, v', v", s, w/, rrii', . . . , and so we can apply the analytical method of maxima and mimima outlined above to the solution of the problem of the stability of a given state, without concerning ourselves about the mechanical equilibrium of the dividing surface in any adjacent state. 44- Illustration of Gibbs' Method by a Special Problem The problem with which Gibbs illustrates this method on pages 249, 250 concerns the system which we have used, for simplicity, to expound the method, with the limitation that the edge of the surface of discontinuity is constrained not to move, so that the two fluid phases are, as it were, separated by an orifice to the edge of which the film adheres. The whole is enclosed in a rigid, non-conducting envelop. Suppose a small variation takes place from this condition of equilibrium, so that the volumes change from v' and v" to u' + 8v' and v" + 8v''' where, of course, 8v' + 8v" = 0. This will entail a change in the position and size of the surface, its area becoming s + 8s. The total quantity of any component remains unchanged, but the potentials in the masses and at the surface change. Since the first component has a given amount for the whole system liv' + 7i"v" + TiS = constant, 620 RICE ART. L and therefore + U'7^ + e^"^ +s— 5M2 + etc. = 0. \ dfX2 dfJL2 dH2/ (This is the equation [546] on page 251, generahzed to deal with the variation of several potentials and not merely of one.) There are several points about this equation which require careful consideration before we proceed, for they reveal the nature of the assumptions implied. First, it is clearly assumed that in the varied state the potentials of any component are still equal in the two masses, and also equal to the varied potential of that component at the surface; for example, the first com- ponent has the potential /xi + 8ni everywhere. Thus we are assuming that the varied state is one which does "not violate the conditions of equilibrium relating to temperature and potentials." Second, since the equation is meaningless unless dji'/dni, dji'/dni, 9ri/a;Lti . . . have definite values, we are assuming that 7/ = dv'/dni, 7/' = dv"/diJLi, Ti = —da/dni and so on, and that dji/dfjLi, etc., are obtained from these by further differentiations. So it is implied that the fundamental equations are valid. The equation is not quite in the form of [546]; to make it so we should have to write the first three terms in the form (T;-7'; + r:|,)a.'. But this implies that s is a function of v'; otherwise ds/dv' has no meaning. This, however, is taken care of by the necessary condition of stable equilibrium that the surface of tension has the minimum area for given values of the volumes v' and v" separated by it. This minimum-area condition is not sufficient for stable equilibrium, but it is necessary, and therefore in discussing the stability of a state of equilibrium there would be no necessity to proceed further if we knew that it was not satis- fied. This condition therefore gives a unique value to s for a SURFACES OF DISCONTINUITY 621 given value of v' (or v"; v' + v" is constant). So s is a single- valued function of v', and ds/dv' has a definite meaning. We can obtain n — 1 similar equations (.'-." + r,^).' + (/£ + ."^' araX + s — 5mi + etc. = 0, etc. These n equations give us the theoretical means to calculate the n quantities d\i\ldv\ dyti/dv' , ... in terms of the state of the system. In this way we see, as is stated at the top of page 250, that all the quantities relating to the system may be regarded as functions of v'. Thus we can obtain d-p' /dv'; for it is equal to dux dv' ^ dti.dv' '^ • • • " ^' dv' "^ ^'' dv' Similarly dy" „djii . „dji2 dv' - ''' dv' + ^^ dv'^ ■- and da djii dyii d^' ^ ~ ^'d? ~ ^'d^' ~ ••• In the initial state we assume that p' — p" = o-(ci + C2); in the varied state the pressures and surface tension p' + 8p', p" + bp", (J -{- b(T are of course the same functions of t, Ml + ^Mi, ... as p', p", a are of t, ni, ... But nowhere do we have to assume that (p' + bp') - ip" + Sp") = ( 2(r/r' if r' > r, and so the internal sphere ex- pands encroaching on the outer phase ; whereas p' — p" < 2a I r' ii r' < r and the internal sphere gradually disappears as the outer phase encroaches on it. The treatment of stability on pages 285-287 will now be easily followed. Certain obvious generalizations to be intro- duced when gravity is taken into account are given there, the result in [625] being, for instance, a wider statement of the result [549] on page 252. XV. The Formation of a Dififerent Phase within a Homogeneous Fluid or between Two Homogeneous Fluids 4-6. A Study of the Conditions in a Surface of Discontinuity Somewhat Qualifies an Earlier Conclusion of Gibbs Con- cerning the Stable Coexistence of Different Phases The possibility of the stable coexistence of different phases has been treated earlier in Gibbs' treatise without reference to the special nature of the surfaces of discontinuity separating them. (See pages 100-115 of Gibbs.) There it is shown that if the pressure of a fluid is greater than that of any other phase of its independently variable components which has the same tem- perature and potentials, the fluid is stable with respect to the formation of any other phase of these components; but if the pressure is not as great as that of some such phase, it will be practically unstable. ''The study of surfaces of discontinuity throws considerable light upon the subject of the stability of such homogeneous fluid masses as have a less pressure than others formed of the same components . . . and having the same temperature and the same potentials. ..." Suppose for in- 626 RICE ART. L stance we have two phases of the same components whose pres- sures are the functions p'(t, mi, M2, . . .) and p"(t, ni, m, . . .) of temperature and potentials (written p'(t, ju) and p"(t, ju) for brevity). A surface of discontinuity between two such phases would have a surface tension which is the function a{t, mi, M2 . ■ . )> or (T{t, ju), of the same temperature and potentials. For the purposes of the argument we are assuming that these functional forms are known. Now if the surface were plane, the condition would not be one of equilibrium; the phase for which the pressure function has the larger value at given values of t, Hi, H2, ... would grow at the expense of the other. Actu- ally, if the phase of greater pressure, say the single-accent phase, were confined in a sphere whose radius is equal to 2 (Tjt, m) p'(t, m) - p"it, /x) there would be equilibrium when surrounded by the phase of smaller pressure. However, as we know, if the second mass is indefinitely extended the equilibrium is unstable (provided there are no components in the internal phase which are not in the external), and the first mass if just a little larger will tend to increase indefinitely; while one a little smaller would tend to decrease, leaving the field to the second mass. So under cer- tain circumstances the mass of smaller pressure, if indefinitely extended around the mass of larger pressure would be the one to grow, thus somewhat qualifying the conclusion from the earlier part of Gibbs' discussion. However, since the possibility of this qualification depends on the smallness of the internal mass of the higher pressure phase, it becomes necessary to take into account the case where this mass "may be so small that no part of it will be homogeneous, and that even at its center the matter cannot be regarded as having any phase of matter in mass." Pages 253-257 of Gibbs treat this problem. The reader is to keep in mind that the phase which might be conceived to grow out of this non-homogeneous nucleus under favorable circum- stances is supposed to be known, with its fundamental equa- tions, as well as, of course, the second phase inside which it may grow; i.e., p'(t, /x), p"{t, /x) and ait, m) are to be regarded as SURFACES OF DISCONTINUITY 627 known functions. Let E represent the energy of the system if the space were entirely filled with the second phase; then E -\- [e], by the definition of [e] in the text, is the energy of the system with the non-homogeneous nucleus formed inside. But of course [e] is not the e^ (nor are [77], [mi], . . . the same as rj^, mi«, . . . ) by means of which a is defined. As usual, we postu- late a definite position for the dividing surface, a sphere of radius r. For the purpose of defining e^ this is supposed to be filled with the homogeneous phase of the first kind right up to the dividing surface, the second phase occupying the space beyond ; the energy then would be E+v' (e/ - 6/0, 4 where v' = i^rr^, and so o es = E + [e]- {E + v'(ey' - e/')} = [e] - v'iey' - e/O, with similar definitions for rj^, mi^, ... as in the text. 47. The PossihiliUj of the Growth of a Homogeneous Mass of One Phase from a Heterogeneous Globule Formed in the Midst of a Homogeneous Mass of Another Phase Imagine the heterogeneous globule to be formed in the midst of the originally homogeneous mass of the second phase, the formation being achieved by a reversible process and the globule being in equihbrium. The additional entropy and masses, Iv], [wi], [mi], ... in the space where the globule is situated are supposed to be drawn from the rest of the system, which may be conceived to be so large that these withdrawals do not appreciably affect the temperature and potentials in the exterior parts. The change of energy in the exterior will be a decrease of amount t[v] + MiNi] + M2N2] + . . . The increase of energy in the space occupied by the globule is [c]. Hence the increment of energy in the whole system, above 628 RICE ART. L that of a system in which the second phase occupies the whole space, is [e] - t[r]] - ni[mi] - )U2[W2] - . . . , which is denoted by W (Equation [552]). This is a function of the temperature and potentials and is independent of any selected situation for the dividing surface; so we write it W{t, ju). Now, as Gibbs himself notes at the outset of this subsection, the method of selecting the surface of tension in former cases is hardly applicable here, and it is not at all clear just how he proposes to select it since his remarks concerning the Ci8ci + C25C2 terms do not appear very convincing. As he says, the |(Ci — C2) 5(ci — C2) term does not concern us for spheri- cal surfaces. But what of the ^(Ci + C2) 5(ci + C2) term? However, on closer investigation it becomes clear what he does. In the earlier parts he showed that the special choice which got rid of the Ci8ci + €2602 terms placed the dividing surface so that it satisfied the condition p' - v" = o-(ci + C2), so here he takes the dividing spherical surface to have a radius given by 2 a{t, ix) r = v'{t, n) - p"{t, m) This is tantamount to assuming that the ideal system which replaces the heterogenous globule and exterior mass, supposed to be in equilibrium, is a homogeneous sphere of the first phase, an ideal surface with the tension ait, ju) and the exterior mass of the second phase, which is in equilibrium mechanically, as well as with regard to temperature and potentials. The radius of this surface then becomes a definite function of the temperature and potentials; for as is shown on page 254 as = e^ — tt]^ — nirrii^ — ^2^2^ — . . . = TF + v'{v' - V"), SURFACES OF DISCONTINUITY 629 and since r(p' - v") = 2cr, and 47rr^ s = 47rr2, v' = -y-, it follows easily that W{t, m) = \ Sa(t, m) = hv'lp'it, m) - P"a, m)}, and so 3 W(t, m) ^ p W{t, m)T 1_ 47r(r(f, m) J [556] The reader can now follow the course of the reasoning on pages 256-257. If, for given values of temperature and poten- tials, there are two phases possible with different pressures such that equilibrium is possible with an inner /iowogre/ieows sphere of the higher pressure phase, an exterior phase of lower pressure and a surface of discontinuity, we see that since r in [556] is then a real positive quantity and p' — p" is positive, W{t, n) is positive for these values of t, mi, M2, • • • In other words, this system has actually greater energy than the system made up of the lower pressure phase alone, and so there would be no tendency for the latter system to transform naturally into the first. If however, by any external agency, the spherical mass of this size and constitution were formed, then it would be unstable, as we have seen, at least if the external mass is indefinitely extended, which means in practice that if any disturbance caused a small increase in the size of the sphere, it would tend to increase still further up to a limit set by the extent of the exterior phase. Now if, by alteration of the tem- perature and potentials of the system, we find values ^o, Mio, JL120, ... for which p'(to, juo) = p"(fo, Mo), 630 RICE ART. L then W{tQ, fxo) is infinite for these values. It is to be noted that near the top of page 255 Gibbs says that W can only become infinite when p' = p", which is true enough in view of [555] or [556]; for since at such values of the potentials equilibrium between the two phases could only occur at a plane surface, r must be infinite, and so W might be infinite, but not necessarily infinite on account of [556], since by that equation r could be infinite when p' = p" even if W were finite. But in any case W could not be infinite under other conditions. However, on page 256, Gibbs says quite definitely that when p' = p" the value of W is infinite, thus invoking implicitly some other reason than the purely mathematical, but not perfectly cogent, argument just cited. Apparently it is the physical fact that an infinitely extended sphere of the first phase will have an excess of energy of infinite amount over the same sphere of the second phase, since v'{iY' — c/') tends to infinity with v' if €y' — ty" remains positive and finite, which must be assumed to be true or otherwise the discussion would be pointless. Returning therefore to the state indicated by the values to, yuio, M20, • . . let the temperature and potentials change gradually from these so as to make p'{t, n) increasingly greater than p"(t, n) ; W{t, n) will gradually decrease. It may ultimately reach the value zero, but if it does so then r and a will also vanish for the values of t, Hi, H2, ... which make W vanish, the difference p' — p" still being finite. For any values of temperature and potentials in the range up to this stage the conditions of stability remain as stated ; the second phase is stable, there would be no tendency for a "fault" to form in it. At this stage the matter is in doubt. The argument in the last few lines of page 256 is very subtle indeed. The quantity r may be zero, but this does not imply that a heterogeneous globule might not exist in equilibrium since r is not the radius of the globule. If, however, the globule dimension vanishes when r is zero, Gibbs says that the second phase would be unstable at the corresponding value of temperature and potentials. To see this we must remember that if, at any values of temperature and potentials, we created by any physical means the internal mass corresponding to the finite r for these values of t, ni, H2, . . . , then the slightest dis- SURFACES OF DISCONTINUITY 631 turbance causing a slight growth in its size would cause the first phase to encroach on the second; but, of course, finite energy- would be required for the initial creation of the sphere before the infinitesimal disturbance in the right direction is applied. But if conditions were such that "zero globule" corresponded exactly to "zero r," no finite energy would be required to create the globule ; any infinitesimal impulse in the right direction pro- ducing any globule however small would produce one larger than the "critical globule," which in this case is "zero globule," and at once the encroachment of the first phase on the second phase would begin. This argument does not apply if the globule does not vanish when r reaches zero, and the second phase is not unstable in the strict sense. Gibbs clearly regards the second case as the most general in nature. Doubtless he had in mind the example of the formation of water drops in saturated vapor. This instance is a good illustration of the application of the abstract reasoning of these pages. When a drop of water is in equilibrium with its vapor in a large enclosure, the vapor, over its convex surface, is supersaturated as compared with vapor over a plane surface; there is a tendency, on the slightest dis- turbance in the right direction, for the drop to grow in size (as we have frequently pointed out); as it does so its surface flattens and the equilibrium vapor around it decreases in pres- sure and density, as it naturally would do if it were being in part condensed. Nevertheless, it is a commonplace physical fact that it is next to impossible to start condensation in a mass of saturated vapor quite free from dust particles or ions. 48. The Possibility of the Formation of a Homogeneous Mass between Two Homogeneous Masses We now pass on to the possibility of the formation of a fluid mass between two other fluid masses. The latter are denoted by the letters A and B. In the discussion on pages 258-261 they are supposed to be capable of being in equilibrium with one another when meeting at a plane surface, so that the func- tions p^it, n) and psit, ij) are to be equal to each other for all values of t, ni, /X2, • • • On page 262 the problem is generalized, but in the meantime this condition is to be kept well in mind. 632 RICE ART. L Now a third fluid mass C is conceived to exist, made up entirely of components which belong to A or B; i.e. C, having no com- ponents other than those in A and B, might conceivably form at the surface dividing A and B, and we are once more supposed to know the fundamental equations of this fluid C so that Pc(t, m) is a known function whose numerical value can therefore be calculated for given values of t, ni, /X2, • • • In addition, (TABit, m), (^Ac{t, m)> <^Bcit, fJi) are also known functions. For the problem to be not merely trivial it is essential that (XAsit, /x) should not be greater than (7Ac{i, m) + o-Bc{t, n). To see this conceive a very thin layer of C to be situated between A and B. This is equivalent to a dividing surface between A and B whose surface tension is o-^ c + ctb c- Referring to the previous subsection on conditions of stability (Gibbs, I, 240), we see that if aAB > ctac + o'sc this is a more stable state than if A and B exist with the ordinary surface of discontinuity between them having the surface tension (Tab, which is presum- ably greater than (Tac + (^b c- Thus for such a condition the problem is settled offhand— the layer of C would certainly form on the slightest disturbance. The problem is really worth considering if (Tab ^ ??A(i', m')> (PsCi'jM') still remaining equal to Pa (f, n') as postulated originally) , then equilibrium could not be maintained unless the surfaces separating A and B from C became concave towards the latter phase, tending towards a lens form. This would upset the balance of the surface ten- sions at the edge where the surface A-B meets the surfaces A-C and B-C, The conditions of this equilibrium can, for purely mathematical purposes, be regarded as equivalent to the equilibrium of three forces. Now the directions of the forces equivalent to cac and cbc are no longer opposite to that equiv- alent to (Jab- The force equivalent to (Tab is greater than the resultant of the inclined forces equivalent to Pa, then presumably a lentiform mass might be in equilibrium both as regards pressures and also surface tensions, since the resultant of the force equivalent to psit, m), all the preceding Hne of reasoning can easily be adapted to the wider condition. This is done on pages 262-264. As before, the condition (Xab > (Tac + (Tbc is set aside. If we would gradually alter the temperature and potentials in such a way as to make pc(t, n) grow larger than the value of the corresponding expression on the right-hand side when (t, fj.) is substituted for (^0, Mo). Notice that this would probably involve a gradual change in the curvature of that portion of the surface not embraced by the lens of C, as pA(t, m) — Pait, m) and (TAsit, m) would probably change in value as t, mi, M2, • • • change in value. The process would end up in the condition and size indicated in the figure. Now to judge if this would happen naturally we need not consider so complicated a change. We have only to conceive any reversible process in which the system begins as imagined with the lens of C formed, and ends up in a final state in which A and B are separated by a surface having the same curvature, but with no lens there. That is, in the final state the temperature and potentials would be the same as they are at the end of the process which is supposed to have formed the lens originally. This is the process conceived by Gibbs, and what we have to do is to determine the sign of the energy change in this conceived process. During it the pressure in A and in B, as well as the surface tension between A and B, will remain at one set of values ; i.e. , Pa, Pb, (Tab will be constant during the process. We are also to conceive that between A and C and between B and C are membranes which gradually contract, keep- ing at constant tensions which are equal to the values of (Tac and aBc in the initial state of this process, i.e., when the lens of C exists in its fully formed state. These membranes are not SURFACES OF DISCONTINUITY 637 to be permeable. The necessary amount of the fluids A and B can be fed in from large reservoirs through narrow tubes let in through the exterior envelop of the whole system, and the liquid C can be passed out through a similar tube into a reservoir of C in which the potentials and pressure can be adjusted; for throughout this process the one variable is the pressure of the fluid C in the gradually contracting lens. It is very necessary to observe that for equilibrium at each stage of the process this pressure increases with contraction of the lens, as can be readily seen by considering the simple case of a spherical membrane contracting with a constant external pressure on it and a con- stant tension in it. This conceptual process may help the reader to realize that the sentence near the bottom of page 263, beginning: "It is not necessary that this should be physically possible . . . ," is not an entirely arbitrary statement support- ing a doubtful line of reasoning. Now let x stand for this internal pressure which increases from a value p c which exists in the fully formed lens and ends up at a larger value p c" when the lens just disappears. During the process the values of the surface areas between A and C, and between 5 and C will change, and we will represent them as functions of x^ viz. Si{x) and S'iix), respectively; the initial values of these functions are S>ac, Sbc and the final values zero. The value of the part of the surface which would lie between A and B extended into the lens, and which decreases as the lens contracts, we will represent by S3 (a:) ; its initial value is Sab and final value is zero. Similarly Vi(x) and V2{x) will respectively represent the volumes between the surface A-C and the surface A-B extended into the lens, and between the surface B-C and the surface A-B so extended, while V3{x) will represent their sum, the volume of the whole lens at the stage when the internal pressure is x. The initial values of Vi(x), V2(x) and ^3(2;) are Va, Vb and Vc respectively; their final values are zero. Now consider the function of x, f{x), defined by fix) = (TAcSiix) + (TBcSiix) — (Tab Si(x) + Pa Vi{x) + Pb Viix) — xvi{x). 638 RICE ART. L The initial value of this function is the quantity W defined in equation [573]. Its final value is zero. If we differentiate it with respect to x we find that df{x) = [(Tag dSi{x) + (Tbc dSiix) — (Tab dsaix) + Pa dvi{x) + Pb dviix) — X dv3{x)] — V3(x)dx, and by the fact that there is equilibrium at every stage of this process, which is conceived to take place reversibly, the expres- sion inside the square brackets on the right-hand side is zero. Hence df{x) = —Vi{x)dx. Integrating we obtain f(pc") - Kpc') = - r^" v,(x) dx. J PC Since the upper limit pc" is larger than pc, as we have men- tioned above, and since V3{x) is a positive quantity throughout, the integral on the right-hand side must be positive also. Therefore the expression on the right-hand side is negative. Hence SiPc') >Kpc"). But/(pc") is zero, since at the final stage Si (a:), S2 (a;), . . . and V2,{x) are all zero. Hence /(pc')> or W, is positive. Now W is the energy excess in the initial state of the system over the final state. Since it is positive, the initial state of the system has really more energy than the final state, and moreover it is free energy, as the expression [573] shows. Thus the initial state would be unstable and so would not tend to form. The treatment of stability given by Gibbs in this subsection and the one preceding must form an important part of any body of principle from which one may hope to obtain in time a satisfying explanation of the colloidal state. Looking back to SURFACES OF DISCONTINUITY 639 page 241 of Gibbs, the reader will see that he comes to the con- clusion that "the system consisting of two homogeneous masses and the surface of discontinuity with the negative tension is ... at least practically unstable, if the surface of discontinuity is very large, so that it can afford the requisite material without sensible alteration of the values of the potentials." In conse- quence Gibbs excludes from the discussion of stability surfaces with negative tensions. Nevertheless the proviso about the size of the surface is important; for if it is not satisfied the con- clusion may not be entirely valid, and so stability might be insured in cases where the interfacial surface is very small. Another instance where the conclusion might not be justified would arise if one of the masses took the form of a stratum so thin that it no longer had the properties of a similar body in a less laminated shape. (See the remark at the bottom of Gibbs I, page 240.) The reader's attention is drawn to these points because in the treatment of the colloid state negative interfacial tensions must come into consideration. A large drop within another medium will only break up "spontaneously" into two or more drops if the free energy of the latter system is less than that of the single drop. As the sum of the surfaces of the separate drops is certainly greater than the surface of the parent drop, this is impossible with a positive interfacial tension; but a de- creased free energy becomes a possible result if the tension is negative. In a paper published in the Z. physik. Chem., 46, 197 (1903) Donnan showed that from the point of view of the Laplace-Gauss theory of capillary forces (briefly outlined in the introductory sections of this article) it was possible to introduce negative interfacial tensions and draw the conclusion that "in certain cases the theory leads us to predict the spontaneous production of extremely fine-grained heterogeneous mixtures, in which one phase is distributed throughout another in a state of very fine division." Of course the difficulty of the problem is not in simply applying the notion of a negative tension, but in demonstrating that at a certain critical thickness the free energy of a film which is thinning out reaches a minimum and thereafter increases if further thinning is continued, or that at 640 RICE ART. L a definite size a drop reaches a similar critical state as regards its free energy. Considerations of space prevent us from anything more than a passing reference to this very important theoretical problem ; but the interested reader will find further discussions, which bring in thermodynamical principles and the effects of surface electric charges, in papers by R. C. Tolman (J. Am. Chem. Soc, 35, 307, 317 (1913)) and N. von Raschevsky (Z. /. Physik, 46, 568 (1928); 48, 513 (1928); 51, 571 (1928)). In particular, Raschevsky's papers emphasize the fact that in addition to the purely surface phenomena a further important factor consists in the rate at which differences of concentration arising from a fast enough velocity of diffusion may give rise to inhomo- geneities in the drop. XVI. The Formation of New Phases at Lines and Points of Discontinuity 49. The Possible Growth of a Fifth Surface at a Line of Dis- continuity Common to Four Surfaces of Discontinuity Separating Four Homogeneous Masses Pages 287-300 deal with fresh possibilities in the way of new formations in addition to the natural processes studied in pages 252-264. It might be possible under certain circumstances for a new surface phase to develop in a system consisting of more than three homogeneous masses. If there were three homo- geneous masses a surface of discontinuity would already exist between any pair, but if four masses were in existence and four surfaces of discontinuity had one line in common, there would be no surface between two pairs of the masses, and the problem arises as to the possibility of the growth of a fifth surface be- tween such a pair. This problem is treated in pages 287-289. The condition of equilibrium used is stated in equation [615]. In Figure 11 on page 287 of Gibbs, the common line is supposed to run perpendicular to the plane of the paper. We consider ci, 0-2, o's, 0-4 to be the four tensions in the surfaces A-B, B-C, C-D, D-A of which the lines in the figure are supposed to be sections by the plane of the paper. Conceive any virtual dis- SURFACES OF DISCONTINUITY 641 placement of the line of discontinuity to an adjacent position which is cut by the plane of the paper in a point 0'. (Not as represented in Figure 12, however, but with four displaced Imes all branchmg from 0'.) If the resolved components of the displacement, perpendicular to the line of discontinuity and lying individually in the surfaces, are 6Ti, 8T2, 8T3, 8Ti, then the system of surfaces is in equilibrium if cid + obd no formation of D would take place naturally; the problem of stability as regards formation of Z> is settled at once. Thus for a problem to exist at all we must postulate CfiC ^ o'bd "T O'er), <^CA = <^CD + ^AD, CTaB = O-AD ~\~ CFbD- If now it happened to be true that cab = o-ad -\- o-bd we might have the formation of Z) as a film between A and B, as in Figure 6. This would resemble the similar cases dealt with on pages 259-264 of Gibbs; the film would form if po were greater than a certain critical pressure (TadPa + (TbdPb (Tad -j- (Tbd If (Tab < (Tad -h (Tbd we would not have formation of D in this way even in a lentiform mass, the argument being once more that of pages 259-264. But taking the tension conditions to be (Tbc ^ (Tbd "r (Tcd, (T CA <^ (Tcd "I (Tad, (Tab <^ (Tad ~r (Tbd, we may consider the possibility of the mass D forming as a fil- ament of triangular section stretching along the direction of the original line of discontinuity. If the three pressures Pa, Pb, Pc were equal, the sections of the surfaces B-C, C-A, A-B by the plane of the paper would be straight lines, as in Figure 14 of Gibbs, da, db, dc being the continuations of these lines. If the pressure po happened also to be equal to Pa (or Pb or p c) the sections of the surfaces A-D, B-D, C-D by the paper, i.e, the lines be, ca, ab would also be straight; but if po 9^ Pa the surfaces A-D, B~D, C-D will be cylindrical with their generating lines per- pendicular to the plane of the paper (Fig. 7) . Thus the lines be, 646 RICE ART. L ca, ah will be circular with their convexity outward if po > y^, but with their convexity inward if po < Pa. In general however Pa, Pb, Pc would not be equal, and in that case the lines da, dh, dc with their continuations would be curved also, and the convexity or concavity of any of the lines be, ca, ab would be determined by the conditions as to whether Pd > Pa or po < Pa, etc. If Pd = Pa; of course be is straight. (To avoid awkward digression later we deal with a few geometrical facts Fig. 7 now. The total eurvature of a limited curved line is the exterior angle between the tangents at its extreme points and is equal to the sum of the two angles between the chord joining these points and the tangents. The angles of a curvilinear triangle are the angles between the pairs of tangents drawn to pairs of adjacent sides where they meet. It will be easily seen that the excess of the sum of the angles of a curvilinear triangle over two right angles is equal to the algebraie sum of the total curvatures SURFACES OF DISCONTINUITY 647 of its sides, the curvature being reckoned positive for a side if it is convex outwards, negative if concave. On account of this convention of signs it will be seen that the excess may be posi- tive, negative or zero, showing that it is possible for a curvilinear triangle to be like a rectilinear in having the sum of its angles equal to two right angles.) If now a mass of the phase D can exist in equilibrium there is an equilibrium for each of the three triads of tensions at each of the new lines of discontinuity; there is also an equilibrium for the triad of tensions at the original line of discontinuity whose section by the paper is d. We construct a rectilinear triangle whose sides represent the mag- nitudes asc, (TcA, (Tab. Its angles must then be the supplements of the angles between the tangents (or normals) at c^; so we can Fig. 8 set it in such an orientation that its sides are parallel to the normals at d. This is the triangle ajSy of Figures 15 and 16 of Gibbs. On ^y we can construct a triangle ^y8' whose sides represent the magnitudes 0, which means that for given values of the tensions and pressures the quantity 'Zas — Spy is a minimum for a stable configuration of the surfaces and volumes. (For convenience we denote the points where the SURFACES OF DISCONTINUITY 651 lines in which the section by the paper cuts the exterior envelop of the whole system by the letters e,f, g.) Then So-s = CAD-hc + (TBD-ca -\- (TcD-oh + oTBc-ae + (TcA-hJ + (TAB-cg, since the lengths of the curvilinear lines be, ca, ah, ae, bf, eg, are equal to the areas of the respective cylindrical dividing surfaces for that part of the system which lies between two sections unit distance apart. Also Xpv = Pa -fbeg + Pb ■ geae + p c • eabf + po • abc. Now let us subtract from Zas — Xpv the quantity (TBc-de + (TcA-df + (TAB-dg — pA-fdg — PB-gde — pc-edf which is unchanged in value by any variation of the surfaces A-D, B-D, C-D. The result of this subtraction is (TAD'be + (TsD-ea + crcD-ab — aBc-dd — ccA-bd — CAs-cd — (pD-abe — pA-bcd — pa-ead — pc-ahd). This is the quantity Ws — Wv of page 292, and since it differs from Xcrs — 2py by a quantity which is unaltered by any variation of the surfaces A-D, B-D, C-D, it is also a minimum for a stable configuration provided the tensions and pressure are given. This leads directly to Gibbs' equation [629]. In order to grasp what Gibbs is doing in the subsequent portion of page 292, let us consider what would happen to the equilib- rium configuration which involves a mass of the phase D were the six functions (TBc(.t,iJ.), . . . Tad' Pb - Pc (Tbc Tbc' etc.. it appears that TAi 0 • '"XD = (Tad • (Tad, Tb t7 • ^BC = ^ 1 . (Tbc • (Tbc, etc. SURFACES OF DISCONTINUITY 653 Thus the figure representing the configuration would shrink so that the lengths of the lines in the figure would be proportional to the changing values of the tensions; therefore (Tad • o'xo = {Sad "T CISad) • Sad or ((Tad + da-Ao) : (Tad = (Sad + dsAo) : Sad, and so Sad d(TAD ^= (Tad uSad- Hence d{cFAD Sad) = 2sad ddAD, etc. Thus it appears that d{Ws — Wy) = i d{(TAD SaD+ . . . -(TbC SbC— . . ■) = h dWs. Since Ws = 0 when Wv = 0, it follows that Ws -Wy = l Ws or Wa = 2Wy. This disposes of the details in the first step. Turning to the second we again consider a variation of the type just considered from the equilibrium configuration, i.e., such that the new figure a'h'c'd remains similar to abed. This varied configura- tion is of course not one of equilibrium for the actual tensions and pressures, but this is of no importance as regards the conditions of equilibrium and stability of the unvaried con- figuration; Ws and Wy' can be reckoned for this varied con- figuration, but of course Ws is not equal to 2Wy since this 654 RICE ART. L configuration is not one of equilibrium; actually Ws' in- volves the same Q if Wv is negative, since {AWs/Ws)"^ is positive for any sign of AWs- For stable equilibrium A{Ws — Wv) must be positive for all variations; thus a necessary condition of stability is that Wv should have a negative value in the equilibrium configuration. This is the result obtained in the second step. The reader can now probably manage the remaining points on pages 294, 295. Note that on page 294 a well known theorem in the mensuration SURFACES OF DISCONTINUITY 655 of triangles is employed, viz., that the area of a triangle whose sides are a, h, c in length is l[(a + 6 + c) (6 + c - a) (c + a - 6) (a + 6 - c)]K 54' Consideration of the Case When the New Homogeneous Mass is Bounded by Spherical Lunes To follow the reasoning in the last two paragraphs of this sub- section (pp. 296, 297) one must visualize somehow the form of D in Fig. 9 this case. First imagine (Fig. 9) a thread stretched between two points I and m; mark two points between I and m on the thread and call them di and c?2. The thread represents the original line of discontinuity, and three surfaces B-C, C-A , A— Ball con- taining the thread divide the space round the thread into three portions, each of which contains one of the fluids^, B, C which are supposed to be in equilibrium at these surfaces. Now consider a plane drawn at right angles to the thread with di and c?2 lying on opposite sides of it. Let the thread cut the 656 RICE ART. L plane in d, and let de, df, dg be the line sections of the plane by the three surfaces. If a, h, c are three points on de, df, dg, we can conceive an arc of a circle drawn through diadi and similarly arcs also drawn through dihd2, dicd^. Further, we can conceive a portion of a sphere (a "spherical lune") drawn so as to connect the arc ^16^2 with dicdi, etc. The mass D, if formed, is supposed to be inside the space bounded externally by three such lunes, and the lune joining dihd^ with dicd^ is the surface D~A, and so on. We now name various portions of surface as follows. The lune dibd^cdi is named Sad, and so on. The portion of the surface B-C which is marked off between the arc diadi and the line diddi is named Sbc- It is in fact the portion of the surface B-C which is, as it were, destroyed by the formation of the phase D. Similar definitions are given to Sca and Sab- Simi- larly Vd stands for the volume occupied by the phase D and Va, vb, Vc for the volumes of the three portions of it originally occupied by the phases A, B, C before the phase D was formed. The discussion of the stability follows the same course as before. Representing the expression Cad' Sad + . . . — (^bcSbc — • • • by Ws, and the expression Pd Vd — PaVa — PbVb — Pc Vc by Wr, we have to investigate when Ws — Wv is a minimum or maximum in the assumed state of equilibrium. (Its variation is zero when we neglect higher powers than the first of the variations of the variables.) We can find the ratio of Ws to Wr in an equilibrium state by the same method as before. The only difference in the result is that although, in the changes of size which keep the figure similar to itself, cxad, (Tbc, etc. all vary as the linear dimensions of the figure (since, for instance, ^cjadItad is to be maintained constant and equal to pt> — Pa), the surfaces Sab, etc. vary now as the squares of the linear dimensions. From this it follows that d{(TAD Sad) = 3cr^o dsAo SURFACES OF DISCONTINUITY 657 so that the analogous result to [632] is d(Ws — Wr) = i d{(rAD Sad+ • . . — (Tbc Sbc— ' . .) = ^dW. and it follows that Hence 3) Wa= IWy Ws- Wy == i Wy. In the subsequent steps one need only consider conditions of temperature and potentials for which pD{t, m) is greater than the other pressures. Clearly the figure would not be possible otherwise. 55. The Stahility of a New Homogeneous Mass Formed at the Point of Concurrence of Four Lines of Discontinuity In the last subsection on stability we have to return to the equilibrium considered in the last paragraph on page 289 and to the commentary thereon. Exactly the same principles are applicable as before, and there will be no difficulty experienced in following the argument, once the figure has been visualized. The modification in the thread diagram used in commenting on page 289 can easily be indicated. Above the drawing board used there we place a wire frame in the shape of a tetrahedron abed, with the vertex d uppermost and the base ahc nearest the drawing board. Tie aioX,h to Y, cto Z and d to U, which is above the frame, by tight threads. We now conceive the phase D to be in the space in the truncated tetrahedron abcXYZ between the surface ahc and the exterior envelop of the whole system, and so on. The phase E is supposed to form inside the tetrahedron. We are not to suppose that the surfaces abc, etc., i.e. E-D, etc., are necessarily plane, nor for that matter the surfaces D-A, etc. There are ten of these surfaces now. 658 RICE ART. L viz. E-A, E-B, E-C, E-D, D-A, D-B, D-C, C-B, C-A, B-A, and when we construct all the triangle-of-force diagrams for the various triads of equilibrating tensions we can fit them together as follows. The original system of A, B, C, D being in equilibrium round a point we can construct a tetrahedron of forces for this equilibrium, as pointed out earlier, and call it a^yd. (It is of course rectilinear.) Now in the new system we have, for instance, at the point a of the system a similar equilib- rium existing for the surfaces E-B, E-C, E-D, B-C, B-D, C-D. Hence we can construct a rectilinear tetrahedron of forces for it, and we can arrange three sides of it to coincide with ^yS, with the fourth vertex at a point e'. Similarly a tetrahedron €"y8a can be constructed to represent the tensions of the surfaces E-C, E-D, E-A, C-D, C-A, D-A, and one t"'ba^ to represent the tensions of the surfaces E-D, E-A, E-B, etc., and finally e""a^y to represent the tensions of E-A, E-B, E-C, etc. In the special case when all the surfaces in the system are plane, the four points e', e", t'" , t"" coincide at one point c inside a^yb, and the tetrahedron a^yb can be oriented into a position in which its six edges and the four lines ea, e)3, ty, c5 are normal to the surfaces in the system. As before, we construct an expression Zo-pSp — S(r„ Sn, where Sp stands for a new surface which has been formed in developing the system with the phase E from the original system without E, and s„ stands for a portion of one of the original surfaces which has disappeared. We call this expression Ws- As before, Wr = PbVb - VaVa — VbVb — VcVc ~ PdVd, where Vb is the volume of the phase E, and Va, etc. the volumes of the parts of it originally occupied by the phases A, etc. We can now prove that Ws = I Wv\ for in this case the preservation of similarity of shape in a conceptually growing phase E would require the tensions to vary with linear dimensions of the figure E (the pressures not changing) while the surfaces Sp, Sn vary as the square of the linear dimensions. The argument proceeds in the now familiar way. If we are considering the stability of the system without the phase E, we need only consider the conditions relating to the system when the amount SURFACES OF DISCONTINUITY 659 of phase E formed is very small. In that case, for purely geometrical calculations, we can regard the faces of tetra- hedron abed and also the portions of the surface D-A etc. within it as plane. This means that the tetrahedron a^yS is similar to ahcd and the point e is situated within it just as is the point c within abed (e is the point which we originally named 0). This justifies the various steps in the geometrical argument leading to [641]. XVII. Liquid Films [Gibbs, I, pp. S00-S14] 56. Some Elementary Properties of Liquid Films. The Elasticity of a Film Since soap solutions are generally used for experimental illustration of the properties of liquid films between two gaseous phases, it may be of advantage to mention briefly some of the most striking facts concerning such solutions. In the first place it is remarkable how great a reduction is produced in the surface tension of water by quite small concentrations of soap. This is, of course, due to the excess concentration of the capillary active soap in the surface layer. Actually, when the bulk con- centration of a sodium oleate solution attains 0.25 per cent the surface tension has decreased from about 80 dynes per centimeter to about 30, a figure at which it remains during fur- ther increases in concentration. However, it is known that these values are only attained some time after the formation of the surface layer. If the surfaces are continuously renewed nothing like such a lowering of surface tension is observed. Thus Lord Rayleigh obtained for a 0.25 per cent concentration a "dynamic" surface tension equal to that of pure water, as distinct from the "static" value given above. Even a 2.5 per cent solution with a continuously renewed surface recorded 56 dynes per centimeter, or about twice the "static" value. This can only mean that the specific surface layer with the very low surface tension takes some time to form. Some work by du Nouy (Phil. Mag., 48, pp. 264, 664, (1924)) on extremely dilute solutions shows that concentrations as low as 10~^ hardly affect 660 RICE ART. L the surface tension initially, but after two hours produce a drop of about one-third in value. This fact should be borne in mind in considering the variations in the tension of soap films which are instanced by Gibbs, and of which many illustrations can be found in A. S. C. Lawrence's book on Soap films: A Study in Molecular Individuality (London, 1929). Of course the thin film between two gaseous phases is not to be regarded merely as a very thin layer. As Gibbs clearly states at the top of page 301, it is in general a hulk phase with two surfaces of discontinuity each with its appropriate dividing surface and superficial energy or tension. One point must however be noted; owing to its thinness any extension of its area finds no large source of the capillary active substance to draw on so as to maintain the surface layers in the same condi- tion, and the resulting reduction in excess surface concentration produces an increase in the surface tensions and therefore in the combined tensions or "tension of the film." This gives rise to the conception of an elasticity of the film, analogous to that of a stretched string or membrane. This will of course have different values according to the conditions imposed, just as occurs in the case of deformable solids. A formula for the value under the conditions prescribed at the bottom of page 301 is worked out by Gibbs on pages 302, 303. In the case of solids or fluids, what is called the "bulk modulus of elasticity" is defined by the quotient of an increase of external uniform pressure on the surface by the resulting decrease in unit volume, i.e., by — 8p/{8v/v) . The definition of E in [643] is analogous to this. 2cr being regarded as the tension of the film. If Gi and G^ are the total quantities of Si and S2 per unit area, as defined in [652] and [653], then under the conditions prescribed GiS and G^s are constant, so that Gids + sdGi = 0, Gids -\- sdGi = 0. These yield [644]. The rest of the analysis on pp. 302, 303 is of a simple mathematical character and can be easily followed. It will be noted that the statement after [655], that E will be \ SURFACES OF DISCONTINUITY 661 generally positive, is based on the assumption that /i2 in general increases in value with G2. It is clear that the elasticity is not simply dependent on the thickness of film. The extension must produce some change in the concentration of the com- ponents in the actual surfaces of the films, so that in a film held vertically, for instance, the conditions of distribution of the components in successive elements of the film must be different as we move up and down. Draining away of the liquid from the interior of the film does not of necessity cause a change in tension even although the thickness diminishes. The statement in parenthesis at the very bottom of page 303 may be justified as follows. All the other potentials except those of Si and S2 remaining constant, a change in composition with respect to these components produces a change in a given by da = —Tidni — T2dfi2. In the argument just preceding we have chosen the dividing surface so that Fi is zero. Then r2(i) is positive on the assump- tion that *S2 exists in greater proportion at the surface, as compared with the interior, than Si. Suppose, however, that we choose the dividing surface so that r2 is zero. This makes ri(2) negative, and we have of course d(T = — Ti(2) d/JLi. But a reduction of ^Si by evaporation, S2 remaining constant, makes the potential of Si diminish so that dfxi is negative in value. In consequence Ti(2)dpLi is positive and therefore da is negative. Pursuing the commentary for the moment, before reference to more recent experimental evidence on these matters than that offered in Gibbs' treatise, we find that on page 305 we meet some remarks on films gradually approaching the tenuity attained by the films which show interference colors by reflected light. The elasticity of a thin film is greater than a thick one as we can see from the equation [650] ; for E increases as X diminishes so long as the interior retains the properties of the matter in bulk, and so the quantities 71, dr/dn2, dT2a)/dii2 are not different in value 662 RICE ART. L for the thick and thin films. This is held by Gibbs to justify his statement near the top of page 305 that, just as the film reaches the limit where the nature of the interior begins to alter, the elasticity cannot vanish and the film is not then unstable with respect to extension and contraction, a statement which has proved to be a remarkably acute prevision of the true state of affairs despite the qualifications of the following paragraph; for quite recent investigation has shown that the thinnest possible film, that showing black by interference, is remarkably stable under proper conditions, and the old idea that thinning necessarily leads to rupture has been disproved. 57. The Equilibrium of a Film Returning to the thick film, Gibbs shows on page 306 how the mechanical conditions for its equilibrium can be approximately satisfied by regarding it simply as a membrane of evanescent thickness, its plane being placed between the two dividing surfaces of the film according to the rule which connects the line of action of the resultant of two parallel forces with the lines of action of the forces. But the following paragraph shows that such a method of dealing with these conditions of equilibrium is really inadequate, and that the film is not really in equilibrium when it apparently is at rest and the conditions called for by this restricted point of view presumably satisfied. The argument reverts to the equations developed on pages 276-282, and resembles in some particulars the line of reasoning on page 284. Thus according to [612] since the pressure in the film satisfies "^ = — gill + 72 + . . .) it should decrease rapidly with height in a vertical film, yet by [613] if we suppose p' to be the pressure at an interior point and p" the pressure in one of the contiguous gaseous masses the value of p' anywhere in the film must be between the pressures of the gaseous masses for a film in any orientation, since p' - Pa' = o-a(ci + C2) + ^(Sr) cos Ba, Ph" - p' = (Tb{ci 4- C2) + £7(2r) cos dh, SURFACES OF DISCONTINUITY 663 where the suffixes a and b refer to the two faces of the film. This means that in a vertical film both these conditions cannot be established, and in the thick film apparently in equilibrium the liquid is in reality draining away between the faces towards the bottom.. As was noted in somewhat similar circumstances on pages 283, 284, there will also be considerable doubt as to the adjustment of the various potentials to equation [617]. If this adjustment took place, then by [98] dp = yid/ii + y^dni = - g(yi + 72 + ...)dz since Hr + gz would be constant in the film if the condition [617] were true for the r"* component. But this is equation [612] which we have just seen cannot hold; so the assumption that [617] is true for all the components leads to a contradiction. Thus there must be at least one component for which the con- dition [617] is not true. It might appear that this requirement could be met if this one component were a component not actually present in the contiguous masses, since then iir + gz in the film for such a component cannot exceed a certain constant Mr, viz., the value of the potential in the gas at the level, 2 = 0, but is not necessarily equal to it. However, as Gibbs points out, one such component is not enough, the situation being similar to one already discussed on page 286. If there were only one such component, it must satisfy equation [617] or else the condition [614] will not be obeyed. For by [508] dar = — Tidni — Vidii^ ... — T^ dyir, where the suffix r refers to this special component not found in the gaseous masses. Hence da = g{Vi + Tj . . . + Vr-i)dz - T, d^r. But by [615] (which, unlike [612], must be satisfied even for apparent equilibrium) d(T = g{Vi + V2 ... + r,-i + Vr)dz, 664 RICE ART. L and so dur = — g dz, or Hr -\- gz = constant throughout the film. However, if there are two such components, r and s, a similar line of reason- ing will show that Trdur + Tsdfx, = - g{Tr + T,)dz, which only necessitates that Trinr + gz) + Tsins + gz) = constant, but not two such independent conditions. In following up the arguments on pages 307-309 the reader may possibly be familiar with Poiseuille's formula for the efflux of liquid from a narrow tube, in viscous flow and under a pres- sure gradient which is small enough to permit the motion to be zero at the wall of the tube and not to cause turbulent motion. It is Trpr^ dp^ '^^ ~~^ ~dl where m is the mass crossing any section in unit time, p the density, t? the coefficient of viscosity, and d-p/dl the pressure gradient along the length I of the tube. This makes the volume of flow per unit time, i.e., 7n/p, proportional to the fourth power of the radius, other things being equal, and this would require a mean velocity across a section equal to pD^ dp 32»7 dl (where D is the diameter), and so proportional to the square of the diameter. The formula for the mean velocity of flow between parallel plates at a distance apart equal to D (again for non-turbulent slow motion) is also known to be pD^ dp, 12r, dl SURFACES OF DISCONTINUITY 665 or 8/3 times the corresponding Poiseuille value for equal values of D. It is this fact which enables Gibbs to convert Poiseuille's experimental result for tubes into the result [657], somewhat greater than [656], but of the same order of magnitude and sufficiently approximate for the purpose in hand. Towards the end of the succeeding paragraph there occurs one of those almost casual statements, so common in Gibbs' writings, which have the appearance of extreme simplicity but are not so easy to justify as one might imagine. Somewhat earlier we have shown how the evaporation of Si, would diminish the tension of the film. (This volume, p. 661, referring to Gibbs, I, 303.) This implies that if we have two elements such that the ratio of the quantity of S2 to the quantity of Si in the first is greater than the corresponding ratio in the second, then the tension in the first element would be smaller than in the second. Suppose the second element to be in equilibrium at the level which it occupies, and that the first element should happen to be situated at the same level. Clearly a small strip of the film lying between this first element and the part of the film immediately above this level would not be in equilibrium. The pull upwards on this strip, which would be balanced by the pull downwards on it if the second element were below it, is greater than the pull downwards on it due to the first element ; thus the first element would tend to rise and of course to ex- perience a stretching and have its tension increased. In the final paragraph of page 309 the observation referred to is now generally known by the name, the "Gibbs ring," and we shall comment on it presently when giving a few details concerning experimental work on films. Passing on to the middle paragraph of page 310, the writer supposes that the reasoning by which the stated conclusion "may easily be shown" is as follows. We have already seen that a vertical film is not an example of true equilibrium, and although the variation of a with the height z necessitates varia- tion of some at least of the potentials with z, since equation [508] must be satisfied, the law of variation is not necessarily the genuine equilibrium law [617]. For, if that were valid for all the potentials, p would have to vary with z according to the 666 RICE ART. L equation [612], whereas, owing to [613], pis practically constant throughout the interior of the film. The law of variation to which the behavior of the potentials will actually approximate may be worked out in the simple case dealt with in this para- graph. Let *Si be the water and S2 the soap, which exists in excess at the surface, so that r2 > Ti; we may take it that in the interior 71 > 72. Since and da = — Ti dn\ — V2 d/jL2 da , . it follows that TiMi + r2ju2 + (Fi + V2)gz = constant. Moreover, since the pressure is practically uniform through- out the interior dz ' and so by [98] or dm dn2 TiMi + 72M2 = constant. From these two equations in /xi and H2 we can eliminate fii and obtain (ri72 - r27i)Mi + 72(ri + T2)gz = constant. Since by our assumptions the coefficient of ni in this is essentially negative while that of z is positive, it follows that fii, the potential of the water in the film, increases as we rise. On the other hand in the atmosphere the potential of the water SURFACES OF DISCONTINUITY 667 will fall according to the usual equilibrium rule [617]. As they are supposed to be equal at the midway level it follows that above that level the potential of the water in the film is greater than that in the atmosphere and there the water will escape into the atmosphere from the film, with the reverse process occurring below. Following a similar line of argument the reader will now find that the subsequent statements on page 310 are not difficult to verify. - The material in pages 312, 313 will be referred to in the brief account of experimental work on soap films which follows. 58. Foams. The Draining of a Film. The "Gibbs Ring" Apart from the blowing of soap bubbles the most common illustration of the existence of liquid films is to be found in foam, which is really a collection of bubbles of various sizes which coalesce according to the following simple rule: when three films meet they intersect in a line and their planes are equally inclined, i.e., at an angle of 120°. Six such films can meet at one point with the four common edges also passing through this point in a manner which we have already discussed at an earlier stage of the commentary. Thus in the interior of the foam each bubble is bounded by hexagonal plane faces (in general irregular hexagons). The pressure of the confined gas is everywhere the same. Only the outer faces between the foam and the atmosphere are curved to any extent, and only at these faces is there any difference of pressure on the two sides. The whole mass quickly drains to the "black stage" by the inter- connected liquid channels. The existence of foam indicates the presence in the liquid of capillary active substances such as saponin. Such substances are to be found in many plants, and the occurrence of stable foams is very marked on that account in tropical rivers. Actually the line of intersection of three films is not a "line" but a channel of finite cross-section which is in the form of a curvilinear triangle as in Figure 10, where A, B, C, represent three adjacent bubbles, D being the channel of liquid. On account of the curvature the pressure of the gas in A, B or C is greater than the internal pressure of the liquid in D, while 668 RICE ART. L the liquid pressure in the films between A and B, etc. is practi- cally equal to that in the gas. This state of affairs causes the "suction" referred to by Gibbs on page 309, and the liquid is forced by this excess of pressure from the films into the channels, thus assisting other influences such as gravity in the draining of the films. When a film of soap solution is drawn up from a mass of such solution at the mouth of a cup, we have a ring shaped channel of this kind where the film meets the horizontal surface of the general mass and into this "Gibbs ring" there is a considerable draining of the film by this suction and gravity. 59. The Black Stage of a Soap Film In general a newly formed soap film passes through a regular succession of changes. Recently, much more light has been thrown on the nature of the succession by improvement in the methods for preventing mechanical shock, sudden large changes of temperature and, more especially, contamination of the solu- tion. In this way it has been shown that the fundamental change is the thinning down to the black stage, so that the black stage is the only film in true equilibrium. It is true that it can hardly be called a stable equilibrium in the accepted sense of stability since the black stage is extremely susceptible to me- SURFACES OF DISCONTINUITY 669 chanical shock, being much less resistant to this than the thicker, colored films. Nevertheless, with extraordinary pre^ cautions soap films have been kept "alive" for many days, and in one case certainly for a year. For further information on the preparation of the solutions and on the experimental technique, the reader can consult Lawrence's book already mentioned. In a vertical film the black stage appears at the top and gradually spreads downwards, the boundary between it and the thicker film immediately below being quite a sharp horizontal line. In the lower part of the film illumination by mono- chromatic light shows, by the appearance of horizontal bands of color across the film, that stages of different thickness succeed one another, the whole mass draining all the time and the banded appearance going through characteristic changes accord- ingly. In a horizontal film the black appears as a small circular disc. The sharp boundary between the black and the adjacent part indicates a change in thickness with a very steep gradient, involving changes occasionally as much as several hundred to one between black and adjacent parts, and never less than ten to one. As stated on p. 662 of this volume, it used to be believed that the appearance of black necessarily led to early rupture of the film, but this is not a fact provided shock and contamination are avoided. The thinning of a horizontal film in this way is of course not due to gravity; actually the Gibbs ring formed where the film meets the solid boundary to which it is attached is responsible for this draining. We have referred briefly to the normal thinning of a film, under, of course, careful conditions, but certain abnormal developments occur at times, and Gibbs himself knew of these as we see on reading pages 312 and 313. Sir James Dewar made many experiments on vertical films in which he observed that instead of the black spreading steadily over the film, black spots appeared in many places, especially at the thicker parts. These spots rise to the top of the film and there coalesce to produce an apparently normal black film, and the film settles down thereafter to the usual course of development. This so called "critical" behavior of the film seems to require some definite stimulation from external sources to bring the film to the state in which the "critical black fall" begins. 670 RICE AKT. L Space permits us to mention only one more point, first clearly established by Perrin, viz., that soap films can be "stratified," the layers of a stratified film being formed by the superposition of identical elementary leaflets in suitable numbers. The thickness of each layer is an integral multiple of an elementary thickness which is of the order of 5 to 6 millimicrons. Actually it is known also that under certain circumstances more than one thickness of black film can be formed ; but the thicker blacks do not last long and quickly give place to the thinnest. With this extreme tenuity of the ultimate black film, it becomes porous and the air inside a bubble which has reached the black stage is gradually forced out by the excess of internal pressure, thus leading to the collapse of the bubble. The reader will find a wealth of interesting material in Lawrence's book, with abun- dant references to original papers on the subject. XVIII. Surfaces of Solids [Gihhs, I, pp. 314-831] 60. The Surface Energy and Surface Tension of the Surface of a Solid In the first portion of this subsection Gibbs returns to the treatment of a problem which he has previously considered in pages 193 et seq. of the section on the conditions of equilibrium for solids in contact with fluids, viz., the expression of the con- dition which relates to the dissolving of a solid or its growth without discontinuity. The problem is now studied with regard to the effect of the existence of surface energy on the course of events, a point not raised in the earlier discussion. He defines his terms for surfaces between a solid and a fluid in a manner similar to that employed for fluid interfaces, and it is to be observed that his symbol a is now definitely associated with surface energy and not surface tension. We have already referred to common misconceptions in this connection in the case of fluids, where, however, the concept of a surface tension may prove serviceable at times as a fiction whose use can be justified by mathematical convenience. But here the various states of strain in a solid can perhaps justify us in the conception SURFACES OF DISCONTINUITY 671 of a tension depending on a stretching of the surface arising from a deformation of the soHd itself, but this is entirely- different from the surface energy. In the case of a fluid the quantity o-, whatever name we give it, is not the measure of the work of a force stretching the fluid surface by unit amount but of the increased energy acquired by molecules which have come from the interior of the fluid to form a new unit of surface, the surface itself being otherwise in the same physical condition as before. It may be, as Gibbs remarks, that in certain cases the actual numerical values for the two quantities in the case of a solid approximate to each other, and so, for example, equation [661] can receive an interpretation, as explained in the last paragraph of page 317, which makes its content identical with that of equation [387]. However, the writer has some reserva- tions to make on this matter which will be given presently. A reminder to the reader may not be out of place when he begins to read this subsection. The words isotropic and anisotropic can be applied to states of stress in solids, as well as to the solids themselves. This matter has been already dealt with in the commentary on "The Thermodynamics of Strained Elastic Solids" (Article K) which may well be referred to in this connection. On pages 316-320 the equation equivalent to [387], viz. [661], is deduced for isotropic solids. On pages 320-325 crystalline solids are considered. The proof of [661] will offer no difficulty, as the reader will now be familiar with the type of argument employed. One special point alone calls for comment. If a closed curved surface is displaced by an amount ^A'" along its normals so as to take up a new position "parallel" to its original form, each element of its surface, Ds changes in area by an amount (ci + C2)8NDs where Ci and C2 are the principal curva- tures of the element. This fact, the proof of which will be found in the section on curvature in Article B of this volume, is used in the expression for the increment of energy with which the argument starts and in the subsequent expressions for incre- ment of entropy, etc. Just after equation [661] there occurs a statement concerning the expression p" -{- (ci + C2)a. This is dependent on the same considerations as were used in our dis- 672 RICE ART. L cussion on p. 521 of the connection between the external pres- sure on the spherical surface of a liquid and its internal pressure at the surface, the quantity Ci + Ca here replacing the quantity 2/R there, R being the radius of the sphere. It is in fact equivalent to the use of equation [500]. The writer, however, feels that the qualification in the text concerning o- being the "true tension of the surface" is uncalled for. If a is the free surface energy per unit area, the same form of proof will hold as before for the statement, and will lead to the same conclu- sion, viz., equation [500]. It is true that in the case of the solid the causes giving rise to free surface energy will include changes in the relative configuration of molecules in the surface arising from surface stretching, as well as the already familiar inward attractions of underlying molecules ; but whatever be the causes, o- has the same meaning in these formulae as before, and p" + (ci + ^2)0- is the internal pressure under all circum- stances. On the same grounds the writer is somewhat critical concerning the remarks at the end of the first paragraph on page 318. He feels that the conclusion there drawn is based on a mistaken view that the surface phenomena resemble in this respect those in a stretched membrane separating two bodies of fluid, and he cannot persuade himself that one should adopt any other view concerning a than those already indicated ; if he is right in this contention and if one introduces the con- ception of an isotropic internal pressure, he fails to see how the familiar proof from energy considerations already used on pages 228-229 of Gibbs' work is not as valid as before. In short he cannot satisfy himself that there is any need in these arguments to separate artificially a certain portion of the free surface energy, viz., that arising from stretching apart of the surface molecules, from the whole amount of it, and to introduce it as the sole determining factor in the difference between internal and external pressure. In order to convince himself of the truth of the statements made in the second paragraph on page 318, the reader should refer back to the conclusions drawn in Gibbs' discussion of strained solids at the bottom of page 196, which might other- wise not be recalled. The additional argument when gravity is taken into account needs no comment. SURFACES OF DISCONTINUITY 673 The gist of the long footnote on page 320 is that since two pieces of ice, for example, do not freeze together spontaneously but only under pressure, the free energy of the discontinuous region formed between the two pieces on freezing, denoted by (T// is not less than, and is most probably greater than, the sum of the free energies of the two surfaces in existence before the regelation, denoted by 2(tjw. The argument concerning crystalline solids follows the same course. To enable the reader to grasp the reason for the second part of the expression on page 320, Figure 11 is supplied. It represents a section of the crystal at the edge V which is sup- posed to extend at right angles to the plane of the paper; BE is part of the section of the surface s by the paper, AB a. part of Fig. 11 the section of s'; CF is a part of the section of the surface s after growth of the crystal, so that the angle EBC is w', and CD is equal to bN. The face s' has, as far as the phenomena around the edge at D are concerned, increased by an area I'BC, i.e. V • CD cosec co' or V • cosec co' 8N; the face s has decreased by an area I' ■ BD or V cot w' 8N. Of course if co' is greater than a right angle, at any edge, the term involving cot co' in the correspond- ing portion of the summed expression will be essentially nega- tive and the term will be virtually an addition term, as is clear from the fact that at such an edge s increases in area. The argument on page 322 concerning stability follows precisely the same course as those employed earlier in the case of fluids, on which we have already commented fully. It should offer no difficulty. Nor is there anything in the three following 674 RICE ART. L pages requiring any special explanation, except perhaps the remark in the footnote on page 325, that the value of the poten- tial in the liquid which is necessary for the growth of the crystal will generally be greatest for the growth at that face for which a is least. The reader will note that if formation of solid material is taking place on this face, it is the faces with larger values of a which are increasing in size, and therefore the crystal is receiving greater increments of energy per unit increase of area than would be the case if growth took place on one of the sides of low a. It should be mentioned that attempts have been made, especially in recent years, to measure the free surface energy and total surface energy of solids, but with very doubtful success owing to the inherent difficulties of the situation. Owing to the absence of mobility the usual methods applicable to liquids fail. However, one can resort to a method which treats the solubility of small particles as varying with size in the same way as the vapor pressure of small drops of liquid. The method is theoretically sound but there are unavoidable errors in its application. It is known that the vapor pressure, p, of a liquid above a plane surface and p', the vapor pressure in equilibrium with a spherical drop of radius r, are connected by the relation Rt v' 2(r — log — = — ' M p rp where M is the molecular weight of the vapor and p the density of the liquid. The solubilities of a solid in large bulk, and in the form of small spherical particles, are related in a similar manner. However, there are considerable difficulties in grind- ing suitable particles, or in preparing them by rapid condensa- tion from vapor or by deposition from solution. It is not prob- able that the surface atoms in such small portions will have the same regular arrangement as in a plane surface. The reader should consult the following papers for details: Ostwald: Z. physik. Chem., 34, 495 (1900). Hulett: Z. physik. Chem., 37, 385 (1901). SURFACES OF DISCONTINUITY 675 Hulett: Z. physik. Chem., 47, 357 (1904). Dundon and Mack, and Dundon: /. Am. Chem. Soc, 45, 2479, 2658 (1923). Thompson: Trans. Faraday Soc, 17, 391 (1922). Attempts have also been made to measure the change in total surface energy owing to smallness of particle by determin- ing the heats of solution for small and large particles. See papers by Lipsett, Johnson and Maass in the /. Am. Chem. Soc, 49, 925, 1940 (1927); 50, 2701 (1928). 61. Contact Angles. The Adhesion of a Liquid to a Solid. Heat of Wetting Pages 326, 327 of Gibbs' treatment deal with the derivation B Fig. 12 from the very general method, used earlier on page 280, of the well-known contact-angle relation [672]. The double relation [673] is necessary for an edge. Thus if the line of meeting receives a virtual displacement from the edge of the solid along the face of s in contact with A (Fig. 12) so as to allow the liquid B to come into contact with unit of area of this face, the inter- face between A and B is reduced by an area of amount cos a, where a is the angle YXP. (This is' in general actually an increase since a is usually obtuse.) Thus there would be a 676 RICE ART. L change of free surface energy of amount (Tbs — (Tas — <^ab cos a. For equilibrium this must be positive or zero, and so (Tbs — (Tas '^ Oab COS a. Similarly (Tab — (Tbs "^ ctab COS /3, where (8 is the angle QXP. If A and B are in contact with a single face, a and jS are supplementary angles, and the signs of inequality must be removed since the two statements would be contradictory in that case; thus we obtain [672]. A very good account of the measurement of contact angles is given in Adam's book on the Physics and Chemistry of Surfaces, Chap- ter VI, where, in addition to the well-known troubles due to contamination, the effect produced by a movement of the liquid along the surface of the solid is discussed, an effect which is not sufficiently recognized in much of the literature. The contact angle gives a very good idea of the relative mag- nitudes of the adhesions of different liquids to a given solid. The measure of such an adhesion is the energy per unit area re- quired to separate the solid and liquid from contact. Thus if (tla is the surface tension of the liquid in contact with air, csA that of the solid in contact with air and (Tls that of the interface between solid and liquid, this "work of adhesion" is equal to (Tla + (Tsa — (tls- If now a is the contact angle at which the liquid-air interface meets a wall of the solid (measured in the liquid) we have from [672] (Tla cos a = (Xsa — (Tls- Therefore the work of adhesion, being measured as above, is equal to (tlaO- + cos a). If the contact angle is zero the work of adhesion is equal to 2(rLA, which is the energy required to separate the liquid from itself (since such a separation produces two surfaces in contact with air, where there were none previously), and so if SURFACES OF DISCONTINUITY 677 the liquid attracts the surface as strongly as (or indeed more strongly than) itself, the contact angle is zero. On the other hand, an obtuse angle of contact, such as in the case of mercury and gl ass, indicates relatively small adhesion or absence of wet- ting. Reference should also be made to the "heat of wetting" in this connection. Heat generally results from the making of a contact between the surfaces of a liquid and a solid. This heat is the total energy of the wetting of the solid by the liquid, and is connected with the adhesion or free energy of wetting by the same relation as exists between the total and free energies of a surface, as can be easily shown by combining the three equations derived thus for the three interfaces, solid-air, liquid- air, solid-liquid, with the definition of adhesion given above. In fact if WsL is the work of adhesion, the expression for the heat of wetting per unit area is dWsL However, there seems to be considerable difficulty involved in calorimetric determinations of the heat of wetting, as widely divergent results are obtained by different experimenters, although the existence of the phenomenon has been known for over a hundred years. In consequence, the result just quoted has not been verified, since it would require, in addition to a knowledge of the changes of aLA and a with temperature (which could be obtained with sufficient precision), reliable values of the heat of wetting, which appear to be wanting. The reader should consult Adam's Physics and Chemistry of Surfaces and Rideal's Introduction to Surface Chemistry, Chapter V, for fur- ther information and references. The matters just dealt with are also closely connected with the question of the conditions under which a liquid will spread as a film over a solid, or remain in compact form as a drop. For an adequate treatment of this important point and its bearing on lubrication reference can be made to Chapter VII of Adam's book, as space is not available for more than a passing remark here. In the same volume a brief account is given of the connection between contact angles 678 RICE ART. L and the separation of minerals from a mixture by the "flotation" process. There is of course an "adsorption equation" for a soHd- fluid interface; it is [675] of Gibbs, or its equivalent, [678]. Reference to adsorption at a solid surface has already been made earlier in this commentary, where an account is given of Langmuir's deduction of his adsorption equation from statistical considerations. Here the experimental results are once more so difficult to interpret that the situation is far from satisfactory as regards proving or disproving any theory. The reader is once more referred to Adam, Chapter VIII, for an adequate account with references. XIX. Discontinuity of Electric Potential at a Surface. Electrocapillarity [Gibbs, I, pp. 331-337] 62. Volta's Contact Potential between Two Metals and Its Con- nection with Thermoelectric and Photoelectric Phenomena The brevity and caution with which Gibbs refers to these matters is natural when one remembers the date of publication of this memoir. In this connection a letter written to W. D. Bancroft, printed at the end of the volume (Gibbs, I, pp. 425- 434) , will prove of interest, especially the paragraph at the top of page 429. The situation has been, of course, radically al- tered since those days, experiment having in the meantime clarified obscurities and removed doubts inherent in any treat- ment undertaken at that time. Historically, the question of electrode potentials dates back to Volta's early researches on contact potentials between metals. The discredit into which that hypothesis fell during the nine- teenth century was due, of course, to the extreme insistence by the physical chemists and some physicists on the source of the energy transformations in the cell. This led them to look for the source of the E. M. F. of the cell entirely at the metal-electrolyte interfaces, though it must be remembered that Volta's theory was ably defended by many physicists, among whom must be reckoned Lord Kelvin and Helmholtz. An account of the SURFACES OF DISCONTINUITY 679 famous controversy will be found in Ostwald's Elektrochemie, Ihre Geschichte und Lehre, or in briefer guise in the first few pages of a paper by Langmuir, " The Relation between Con- tact Potentials and Electrochemical Action" (Trans. Am. Eledro- chem. Soc, 29, 125 (1916)). The great temporary success of Nernst's "solution pressure" hypothesis still further intensified the neglect of Volta's ideas. It was the essence of Volta's theory that the contact P.D. between two metals is the differ- ence between two quantities, each one being a characteristic of one metal only, and Volta recognized that such an assump- tion fitted very simply with the fact that in a closed chain of different metals in series no current flows. It must be admitted that the great discrepancies between the different experimental attempts to measure Volta potentials militated against the success of the theory as a working hypothesis, and led people generally to believe that such potentials, if they existed, were the result of chemical actions at the surfaces of metals and not characteristic of the metals purely and simply. But today investigation of thermionic and photoelectric phenomena has greatly altered the status of Volta's ideas just when the validity of Nernst's hypothesis is being seriously ques- tioned by the physical chemists themselves. The work initiated by Richardson on thermionic emission, and the great power which experimentalists possess in producing high vacua and maintaining scrupulously the cleanliness and freedom from con- tamination of metal surfaces, has demonstrated beyond question that electron emission from metals is an intrinsic property of pure metals, and that for each metal there is a characteristic quantity, viz., the energy absorbed when an electron escapes from the metal across the surface. If this be postulated it follows as a logical result that when two metals are in electric equilibrium there must be a P.D. between them if their "electron affinities" are different. (The electron affinity is defined as the quantity cf), where e4> is the characteristic energy of escape referred to, e being the numerical value of the electron charge.) Further, the experimental work of Langmuir, Millikan and others has placed the existence of this P.D. beyond the pale of doubt. To demonstrate the logical dependence of contact 680 RICE ART. L potentials and electron affinities is not a difficult matter, but it requires the reader to be very clear on certain elementary points in the theory of electricity. Thus the definition of elec- tric potential at a point is given in the words "the work required to bring unit positive change from infinity to the point," but it is not always borne in mind that the transference of the charge is assumed not to disturb the existing distribution of electric charge in space. The neglect to take account of this proviso will lead to paradox and perplexity in some cases. Thus suppose we have an uncharged conductor far away from all other conductors so that it is at zero potential. Now imagine the test positive charge to approach the conductor from infinity; as it gets near, a negative charge is iijduced on the proximate face of the conductor and a positive on the re- mote; an attraction is exerted on the test charge, which means that work has been done on the charge in coming from infinity to the conductor. Or, if a test charge be taken away from the conductor, the disturbance of the distribution of charge which existed in the conductor before the test charge was placed near it will produce an attraction on the charge, and the unwary might therefore infer that the uncharged conductor is at a negative potential, the potential at infinity being taken to be zero as usual; but of course that is an erroneous conclusion and due to neglect of an essential feature of the definition of potential. Another point to be borne in mind (but often overlooked) is that there is no discontinuity of potential between a point in a charged conductor and a point just outside it. The quantity which is discontinuous is the intensity of electric force (which is zero inside a statically charged conductor and equal to 4tk just outside, where k is surface density of charge), and this intensity is the gradient of the potential. A geometrical illus- tration can be observed at a point on a graph where there is a sharp break in the slope. There is no discontinuity in the ordinate y, but one in the slope, i.e., ui the gradient of y, viz. dy/dx. If there is a discontinuity in the potential at the sur- face of a conductor, or at an interface between two conductors, it can only arise owing to a "double layer" of opposite charges, say a positive surface charge and, at a physically small distance SURFACES OF DISCONTINUITY - 681 further out, a negative charge (either in the form of a surface charge or in a more or less diffuse layer) not actually coincident with the positive charge. We can now give the theoretical connection between electron affinities and contact potentials quite simply if the reader will recall the few remarks on statistical conditions in subsection (9) of this article. Conceive a metal body to be in a vacuum in an enclosure. Electrons escape from it and gradually the metal will become positively charged. (At room temperatures this process would be very slow, but this does not affect the validity of the calculations which are concerned with the ultimate state of equilibrium, attainable of course at much greater speed at high temperatures.) A state of equilibrium is reached (anal- ogous to that of an evaporating liquid in an enclosed space) when as many electrons return to the metal body as leave it in unit time. There is no difference of potential between the metal and a point just outside, but there does exist a difference between the metal and a distant point, since the metal is charged. Let the electron concentration in the metal be n and that in the space adjacent to the metal surface n'; then we have by a well-known statistical relation n = exp i-t) 11 or kt(\og n — log n') = e. If an electron travels from a point near the surface to a point P in the "space charge" where the potential is V p, the electron loses kinetic energy of an amount e{V — V p) where V is the potential of the metal body and also the potential at a point just outside it. (It would gain that amount if the electron were charged positively.*) This follows from the strict definition of potential; for it is assumed that by the time the electron has travelled a physically small distance from the surface the * Observe that e is treated here as a number without sign; the numeri- cal value 4.8 X 10""" of the electron charge. 682 RICE ART. L effect of its "induced charge" (i.e., the corresponding positive charge left unneutraUzed by its exit) on it has vanished and no further work is done against its motion on that account; that has already been reckoned in e^ and the movement from the surface to P produces no further disturbance of the surface charge and no practical change in the "electron atmosphere" or "space charge" in the enclosure, which has a very low con- centration. Hence by the same statistical rule np ( e\V -Yp\ = exp (e\Y-YA\ n or A;/ (log n' - log np) = e{V - Vp). Let us now consider two metal bodies not in contact with one another but inside the same enclosure. When in equilibrium the bodies will be at potentials Vi and V2. We then have the following relations kt(\og rii — log n/) = e<^i, ktilog n/ - log np) = e(Vi - Vp), and two similar relations for the other metal. It follows easily that ktlogui - 601 - e{Vi — Vp) = U log np = kt log n2 — €(f)2 — e(V2 — Vp), and therefore kt Ti — T2 = "~ (log Wi — log n2) + <^2 — 01. B This relation is not disturbed by bringing the metals into con- tact; it holds for any relative position of the bodies; when they come into contact the electron concentrations on their contiguous parts adjust themselves to produce a double layer consistent with the discontinuity of potential Vi — V2 across the interfacial boundary. The body with the smaller electron affinity has its SURFACES OF DISCONTINUITY 683 normal concentration reduced at the interface thus producing the positive side of the layer there, while the excess electrons go to increase the local concentration in the other body, produc- ing the negative side of the layer. It will be seen that this contact potential Vc = Vi — Vz depends on temperature. Now long ago Lord Kelvin and Helmholtz in combating the view that Volta potentials could be identified with the Peltier effect, showed that the latter is really dVc/dt being thus simply the temperature coefficient of the Volta effect. (See for exam- ple Lord Kelvin's paper, Phil. Mag., 46, 82 (1898).) If this is so we see that the Peltier effect, i.e., the "thermoelectric power" of two metals is (k/e) (log Wi — log W2). But we know that this is very feeble compared to Vc, and there is also evi- dence from the values of electric conductivities and from recent work on the electron theory of metals that the electron concen- trations in different metals are of the same order of magnitude, so that the term (kt/e) (log ni — log 712) is negligible. Thus, practically, Vc = 2 — <^l. This is the modern formulation of Volta's theory, expressing the contact potential as the difference of two electron affinities, each one a characteristic of its metal. As regards the production of current, suppose the metals to be in contact at a pair of faces, and bent so as to face each other across a relatively wide gap at another pair. If an ionizing agent were placed near the air gap, ions would be created in the gap and be driven one way or the other by the electric field between the two faces at differing potentials, thus tending to annul the field. If the ionization ceases, the P.D. is restored in the air gap ; fresh ionization will create fresh current and so on. It will be observed that the energy of the currents is not obtained from the surface of contact of the metals but from the ionizing agent. This vitiates at once one of the implicit assumptions of earlier generations of workers, viz., that one must look for the source of the E. M. F. at the same place as one finds the source of the energy changes. The function of the electrolyte, as Lord Kelvin always emphasized, 684 RICE ART. L is to discharge the charged surface of the plates. It does so by means of the ions arising naturally from its own dissociation. Indeed Volta had vague notions of the same kind, although naturally he could have no prevision, in his time, of modern ideas of dissociation and energy. Of course this changed attitude towards the Volta effect does not carry with it a denial of the existence of a P.D. at a metal- electrolyte interface; it merely asserts that the metal-electrolyle discontinuities in potential do not account for the whole of the E.M.F. of a cell. 63. Discontinuity of Potential between a Metal and an Electrolyte As is well known, the hypothesis of Nernst concerning the origin and magnitude of the potential discontinuity at a metal- electrolyte interface has been accepted until recently by most physical chemists as an adequate formulation. Nernst's proof of his formula is thermodynamical, and he deduces the result M Ve = — (log p, - log Pa) , where po is the osmotic pressure of the ion which is the common component of electrolyte and electrode, ps its "solution pres- sure" in the metal, v the valency of the ion, and Ve the excess of the potential of the electrode above that of the electrolyte. The "solution pressure" in the metal cannot be intuitively apprehended like the pressure in a gas, or even like an osmotic pressure, which at all events is open to observation by means independent of all considerations of electrode potentials. It is merely brought into the proof to provide a work term in a usual isothermal cycle when electrons occupying volume v in the metal pass into a volume v' in the solution, The proof is well known and can be found in standard texts (e.g., F. H. Newman's Electrolytic Conduction, London, 1930, pp. 184-185). The great objection to the hypothesis is the perfectly monstrous values of solution pressure which must be postulated to make the formula fit the facts. Thus for zinc Ps is almost 10"^^ atmos- pheres, while for palladium it is about 10~^^ atmospheres; in SURFACES OF DISCONTINUITY 685 the latter case the solution would have to be so dilute round the electrode that a quantity of it as large as the earth would contain two palladium ions at most! With such a huge solu- tion pressure zinc would have to part with over one gram of ions per sq. cm. in order to attain equilibrium when placed in an ordinary solution of a zmc salt; to avoid such an obviously impossible result one has to make ad hoc hypotheses concerning the extreme slowness with which equilibrium is reached. It is true that, by abandoning the assumption that ionic atmospheres obey the gas laws, Porter and others have shown that more moderate values for p^ can be obtained; but investigators have of late considered other possible explanations of metal-solution pressure. References to these will be found in Newman's book Chapter VI and Rideal's Surface Chemistry. A feature of Nernst's formula is its logarithmic form, in which it resembles the contact potential formula obtained above — indeed Nernst's formula could be obtained by somewhat similar statistical argu- ments provided the physical environm.ent of the metal were as simple as in the case of contact potentials. Now Rideal (Trans. Faraday Soc, 19, 667 (1924)) has observed that the order of different metals as regards electron affinities is much the same as the ordinary electromotive order. Nevertheless, the fact that an electrode P.D. depends upon the concentration of the electrolyte shows that it is impossible to interpret such a P.D. entirely in terms of a quantity such as is adequate to account for contact potentials. However, Rideal has derived a formula in which the difference between the electrode potential and the electron affinity of the metal is dependent on its atomic volume. Its form is kt F. - * = -f(A), where A is the atomic volume of the metal. Schofield (Phil. Mag., [7], 1, 641 (1926)), by an argument based on Gibbs' chemical potential of an ion, derives a formula J. _ kt(\og c — {km — ke}) Ve — - ) ve 686 RICE ART. L where c is the concentration of the ion in the solution, km. a quantity "representing the concentration and environment in the metal" and ke "represents the environment in the electro- lyte". The solution is supposed to be dilute; in stronger solutions log c would be replaced by the logarithm of the activity. This is formally somewhat like Nernst's formula, km — ke replac- ing the term containing the logarithm of the solution pressure. Butler has derived from a statistical argument the result y. _ u + kt{\og r + log g) ve where u is the energy change for the transference of one ion from metal to solution, a the activity of the ion in solution and r a small constant characteristic of the metal and depending on the number of metal ions per sq. cm. of the metal surface. (See Trans. Faraday Soc, 19, 729 (1924)). All these formulae for electrode potentials exhibit one common feature. They attempt to express the P.D. as the difference of two quantities, one related to the metal and one to the electrolyte, and in that respect they resemble the theoretical result obtained above for a contact potential between metals; but the quantity related to the metal can scarcely be said to be "characteristic" of the metal in the sense that it depends only on the metal. Thus consider the formula of Butler; it appears in the proof that uisw2 — wi, where Wi is a loss of energy by the ion in travelling from the surface to a certain point in the liquid against the ordinary attractive forces of the solid and adjacent liquid, and w^ is a similar quantity for a movement from the interior of the Hquid to the point. A careful examination of the proof shows, however, that the position of this point would alter with the concentration of the electrolyte, so that Wi would change with this concentration; and so the quantity related to the metal depends as regards its value on the nature of the electrolyte. But, of course, the simpler state of affairs which holds for metals in a chain could not be true for metals and electrolytes; for if it were, no current would flow in any complete circuit made up of metals and electrolytes, as is true in the case of a complete chain of metals. SURFACES OF DISCONTINUITY 687 64. Gibbs' Comments on Electrode Potentials Leaving these matters, and turning to a few brief comments on Gibbs' own pages, we meet a statement in a footnote to page 333 to the effect that for a cell with electrodes consisting of zinc dissolved in mercury in different proportions equilibrium would be impossible. For, considering a certain solution, if we slightly alter the relative masses for two constituents but maintain the pressure constant, then dp is zero and so (mi/v)dni + {m2/v)dn2 is also zero ; so that if d/xi is positive, dn2 must be negative, or an increase in ^i involves a decrease in nz. Hence if Hm' > y-J' then /i/ < Hz" . Thus it would be impossible for the conditions of equilibrium ■m } V + a„Mm' = V" + a„M to be true simultaneously. With regard to paragraph (II), p. 334, a discharged ion going into solution would no longer be related to other components by equation [683] ; it would be an independent component with in general an entirely different chemical potential from the charged ion. If there were current flowing, a charged ion would appear to have no definite chemical potential since it would not be in equilibrium, but we would infer by [687] that for small currents its chemical potential, if it were a cation, would increase as it travelled towards the cathode, (if an anion, towards the anode) on account of changing electric potential in the solution. The discharged ion would not be affected by the electric field. How- ever, the paragraph indicates the case of minor interest where the chemical potential might remain unchanged by the dis- charge. Paragraph (III) introduces the possibility of an equilibrium being effected by absorption of an ion by the elec- trodes, as in the case of the well known polarizing effect of hydrogen bubbles in a simple copper-zinc cell. The phe- nomena of polarization and of overvoltage can be studied in standard texts. (See for example Chapter VIII of Newman's book, cited above. Chapter VI of the same work gives a good account of the experimental methods used to measure electrode potentials.) 688 RICE ART. L 65. Lippmann's Work on Electrocapillarity and Its Connection with Gibbs' Equation [690] The paragraph marked (IV) makes a brief reference to electrocapillarity, and in it Gibbs derives equation [689] which, under the conditions that govern the use of the capillary electrom- eter, reduces to a simpler form without the second term on the right-hand side, and this is shown to be equivalent to [690] which is the well-known equation due to Lippmann. The fact that the tension in an interface between mercury and acidulated water is dependent on the electric conditions was first discovered by Varley (Phil. Trans., 161, 129 (1871)). Two or three years later Lippmann began a fuller investigation of the phenomenon. He derived the equation which goes by his name, and designed the capillary electrometer to test his conclusions.* The essence of his experiment is the use of an electrolytic cell consisting of sulphuric acid solution and mercury electrodes; the anode has a large surface exposed to the solution, the cathode a very small surface (actually the section of a capillary tube). A current is passed, and if it is not too large the density of the current per unit area of the anode is very small, while the current density at the cathode is so great that the cathode surface becomes highly polarized while little or no effect is produced at the anode surface, and the current is stopped by the reverse E.M.F. set up. A new state of equilibrium is produced which varies as the applied E.M.F. from the external source is increased up to a limit beyond which the current cannot be stopped and equi- librium becomes impossible. The theory which he gave for his results is essentially the theory of a charged surface — purely electrical with no hypothesis as to the physical occurrences at a mercury electrode. A charged conductor like a body of mer- cury has its charge on the surface. Looking at the surface ten- sion as if it were due to tangential attractions in the surface, the conclusion that a surface charge should reduce the surface ten- sion by reason of the mutual repulsions of its parts is very ♦ Comptes Rendus, 76, 1407 (1873); Phil. Mag., 47, 281 (1874); Ann. chim. phys., 6, 494 (1875) and 12, 265 (1877); Comptes Rendus, 95, 686 (1882). SURFACES OF DISCONTINUITY 689 plausible; but there is no need to resort to this fallacious view of the nature of surface energy. Actually there is at the surface an amount of energy Ag + HCl (w), which will take place from left to right. To measure the reversible electromotive force, E, and the reversible electrical work, NEF, corresponding to the equation of the reaction, the electromotive force of the cell is exactly balanced against an outside electromotive force just sufficient to prevent its dis- charge and not sufficient to charge it. This is the electromotive force of the cell when no current is passing through the cell, or when the entire system is in equilibrium. If we imagine the cell to discharge against this electromotive force until the quan- tities specified in the equation have reacted, the cell process will have taken place reversibly. The electrical work, NEF, will then be the maximum, and will be denoted the reversible electrical work. We shall now follow Gibbs in determining the total energy increase of the cell. Four kinds of changes are possible (Gibbs, 1,338): "(1) The supply of electricity at one electrode and the withdrawal of the same quantity at the other. (2) The supply or withdrawal of a certain amount of heat. (3) The action of gravity. (4) The motion of the surfaces enclosing the apparatus, as when the volume is increased in the liberation of gases." In the cell just described, there will be a contraction in volume due to the disappearance of one-half mol of hydrogen at a con- stant pressure of one atmosphere. These changes are neces- sary and sufficient for the evaluation of the energy change accompanying cell action. Indeed, the third is usually negli- gible. Since, according to the first law, the increase in energy is equal to the algebraic sum of the work and heat effects received 712 EARNED ART. M by the system, we obtain de = (V - V")de + c?Q + dWa + dWp, (1) [691] in which de is the increment in internal energy of the cell, de is the quantity of electricity which passed through the cell, and V and V" the electrical potentials of leads of the same kind of metal attached to the electrodes. Therefore, {V — V")de is the electrical work necessary to charge the cell reversibly, dQ is the heat absorbed from external bodies, dW a is the work done by gravity upon the cell, and dWp, the work done upon the cell when the volume changes. Since no current is flowing, {V" — V) equals the electromotive force, ±^, of the cell.* Since all changes are to be reversible, dQ will be transferred to or from the cell under conditions of thermal reversibility, that is to say, the cell at every instant must be at the same tem- perature as the external source from which it receives the heat or by which the heat is withdrawn. This is the only source of change of entropy, and since the above condition of reversibility prevails, the increment in entropy at constant temperature will be dv = y • (2) [692] The first and second laws, therefore, lead to the equation for the energy increment of the cell, de = (F' - V")de + tdtf + dWo + dWp, (3) [693] or the equation for the electromotive force, , „ „ de td-q dWo dWp , , , , * Two conventions regarding the sign of electromotive force are in use. For a given direction of the current through the cell its elec- tromotive force is V" — V or V — V" according to the convention which we adopt. Since this is largely a matter of personal preference, the adoption of one convention or the other will add nothing to the pres- ent general development. Therefore, we shall write ±E for the electro- motive force. ELECTROCHEMICAL THERMODYNAMICS 713 If the cell actually discharges at a finite rate, the conditions of reversibility no longer prevail, and the cell is no longer a thermodynamically useful "perfect electrochemical apparatus." On the other hand, if the cell is maintained at constant tem- perature, we have, in general, dO dv^-J (5) [695] and, therefore, for the electrical work done by the cell, (7" - V')de ^ -de + tdr, + dWo + dWp. (6) [696] Before proceeding to further discussion of these equations, we shall consider the relation of the reversible electrical work to the work content function \p and the thermodynamic poten- tial f (Gibbs, I, 349). The definition of \p is given by the equation yP = e-tn, (7) [87] and, therefore, at constant temperature, dyp = de - tdr]. (8) If this value of {de — tdr]) be substituted in equations (4) and (6), we obtain , „ ,s # dWo dWp , , , , for the electromotive force of a reversible cell and (V" - V')de ^- d^p + dWa + dWp (10) [698] for the electrical work of any cell at constant temperature. The value of the term due to gravity is extremely small, and negligible in ordinary cells. Further, dWp is the reversible work done on the cell corresponding to the volume contraction or expansion against a pressure p, and is equal to — 'pdv. Hence, for the reversible cell at constant temperature, (J" - V')de = -d^p - pdv, (11) 714 HARMED ART. M which, at constant volume and temperature, becomes simply (7" - V')de = -#. (12) Thus, if the cell is maintained at constant volume and tem- perature, the reversible electrical work done by cell discharge equals the decrease in work content. In actual experimental studies, we are more likely to be con- cerned with processes at constant pressure and temperature, and for this reason Gibbs' thermodynamic potential f is of extra- ordinary usefulness. This function is defined by ^ = e-tv + pv (13) [91] and, consequently, at constant pressure and temperature, an increment in ^ is given by d^ = de - tdrj + pdv. (14) Since equation (4) [694] may be written — dt -\- tdrj — pdv ,^ . Y" - y = ^^-^ ^ (15) de if we neglect dW a, we immediately obtain for the reversible cell, (F" - Y')de dr, (16) [699] and for any cell, (7" - Y')de ^ -dr. (17) [700] The reversible electrical work at constant pressure and tem- perature is equal to the decrease in thermodynamic potential due to the chemical reaction taking place in the cell. This equation is of great importance since it affords a method of evaluating directly the changes of thermodynamic potential in many chemical reactions which otherwise could not readily be obtained. These few considerations, deductions, and equations represent Gibbs' explicit contribution to the thermodynamic theory of the galvanic cell as contained in the "Equilibrium of Hetero- ELECTROCHEMICAL THERMODYNAMICS 715 geneous Substances." The directness and simplicity of his method are strikingly manifest. Let us consider for the moment equation (15), which, allow- ing for an irreversible process, is (7" - V')de ^ -de + tdr, - pdv. (15a) If the cell is maintained at constant volume, the last term vanishes, and if no heat is absorbed or evolved by the cell, the term tdr] vanishes, and the electrical work is equal to or less than the diminution of energy. Owing to the lack of very accurate experimental results as well as a confusion regarding the fundamental concepts involved, and to the fact that, in some cases of familiar cells, the term td-q is small compared to de, many investigators of the last century were of the opinion that the electrical work is entirely accounted for by the diminu- tion of energy. Since cells are measured at constant tem- perature and not at constant entropy, there is no reason why the term td-n should vanish. Gibbs, therefore, takes great care in the subsequent discussion (Gibbs, I, 340-347) to place this matter in the correct light. We shall postpone the consideration of this matter and consider the alternative deduction of the general law (equation [6]) given in the second letter to the Secretary of the Electrolysis Committee of the British Association for the Advancement of Science (Gibbs, I, 408-112). Gibbs wrote this letter in order to explain more fully his position, and its contents constitute the only other explicit statement of his thermodynamics of the galvanic cell. Consider a reversible cycle in which a cell discharges at a constant temperature t', producing electrical work, mechani- cal work and possibly heat effects. Chemical changes will take place. Then, by reversible processes which do not involve the passage of electricity, bring the system back to its original state by supplying or withdrawing the necessary work and heat. Let W and Q equal the work done and the heat absorbed by the system during the discharge of the cell, and [W] and [Q] equal the corresponding work and heat changes during the reversible processes employed to bring the cell back to its 716 HARMED ART. M original state. Since by the first law of thermodynamics the algebraic sum of the work and heat effects in a cycle is zero, W + Q + [W] + [Q] = 0. (18) ([1] p. 408) By the second law the algebraic sum of the entropy changes throughout such a cycle is zero. Hence, we obtain P + I 7 = 0, (19) ([2] p. 408) where t' is the temperature at which the cell charges or dis- charges. In the reverse process, the heat is supplied or with- drawn throughout a range of temperatures. If we neglect the term due to gravity, the reversible work during cell discharge involving the passage of one unit of elec- tricity is W = (V - V") + Wp. (20) ([3] p. 409) From equations (18), (19), and (20) we readily obtain 7" -v' = Wp+ [W] + [Q] - ^' / 7 • (21) ([4] p. 409) [W] + [Q] is the increase in energy Ac, supplied in bringing the cell back to its original condition, and this by the first law is equal numerically, but opposite in sign to the decrease in f dQ . ^ energy, — Ae, during cell discharge. Further, / — is the entropy change during the reverse process, and is equal, but opposite in sign, to the entropy change At/ during discharge. Therefore, V" -V = -Ae + t'Ar, + Wp. (22) ([5] p. 409) Since the variables of equation (15) are all extensive, it may be integrated term by term to give equation (22). Let us now define a temperature t", such that [Q] t' P = J ^' (23) ([7] p. 410) ELECTROCHEMICAL THERMODYNAMICS 717 which shows how, by means of a reversible process, the heat [Q] absorbed at constant temperature t" may replace that ab- sorbed at a series of temperatures denoted by i. The tempera- ture ^" is the highest at which all the heat may be supplied to f dQ the system. Eliminating / — from equation (21) by means of equation (23), we obtain V" -r = ^—^ [Q] + [W] + Wp. (24) ([6] p. 410) This equation can be derived from the usual form of reversible cycle in which the cell is discharged isothermally at t', heated to t", then the changes produced reversed isothermally at t" without the flow of electricity, and finally cooled to t'. The above equation would be true for such a process if the heat absorbed during the heating from t' to t" cancelled that evolved during the cooling from t" to t'. This may not be true for a specific case, but if we define t" by equation (23), then equation (24) is strictly valid. We shall find later that this definition considerably simplifies theoretical discussion. The remainder of the letter which we have been discussing is devoted to showing that the equations developed are in accord with those derived by Helmholtz. Gibbs proceeds to deduce the equation of Helmholtz, Yt = -~t (25) ([11] p. 411) by simple transformations of equation (22), and thus shows that his methods lead to the same conclusions as those of this investigator. II. On the Question of the Absorption or Evolution of Heat during Galvanic Cell Processes As we have shown by consideration of equation (15), there is every reason to beHeve that during charging or discharging of a galvanic cell at constant temperature, heat may be absorbed or evolved. Gibbs uses three lines of argument to show the 718 HARMED ART. M error made in neglecting these heat changes. The first depends upon the conception of a cell at constant volume, or "in a rigid envelop," which, during charge or discharge, does not change in intrinsic energy. In this case, the reversible electrical work performed by the cell is equal to the heat absorbed. The second argument depends on the theoretical conclusion that unless a reaction can produce all its heat at an infinitely high temperature the reversible electrical work cannot equal the decrease in energy. The third argument is empirical. Gibbs computes, from the best data obtainable at that time, the values of the electrical work, change of energy, and heat absorbed, and shows that the heat term tdrj always exists and is some- times very considerable. We shall consider these arguments in turn. That it is possible to construct a cell such that (V" - V')de ^ tdr, (26) is easily shown. Consider two hydrogen electrodes in two limbs of a U-tube. Let the pressure on a large constant volume of hydrogen on the left side be two atmospheres and the pres- sure on a large constant volume of hydrogen on the right side be one atmosphere. This difference in pressure is compensated for by the difference in heights between the columns of hydro- chloric acid in the two limbs. If we neglect the small effect of gravity, the net effect of the cell reaction will be H2 (2 atm.) -> Ho (1 atm.) at constant volume and temperature. Since there is no increase or decrease in energy in the above process provided that hydro- gen is a perfect gas, and since the term pdv vanishes, the reversible electrical work will equal tdrj. This may be more concisely stated by equation (12) whereby (7" - V')de = -#]„,« = -de-}- tdrj = tdtj, since there is no energy change. Gibbs now proceeds to show that the absorption or evolution of heat is a usual phenomenon accompanying galvanic cell ELECTROCHEMICAL THERMODYNAMICS 719 action at constant temperature. He asks us to consider a change in which two molecules, A and B, combine to form a third, AB, with the evolution of heat Q. Now imagine them to react in a galvanic cell at a temperature t', and then complete a cycle by bringing the system back to its initial state by a series of reversible processes which involve the supplying of heat, but which for the sake of simplicity involve no work. This cycle can be represented by A+B-^AB-^W + Q (t = t') A+B^AB + [Q] {t = t") in which the intrinsic energy changes are Ae = [Q] at t", and — Ae = W -\- QbXI', respectively. According to equation (19), we have the well known relation Q [Q] p + ^ = 0, (27) where t" is defined by equation (23), and equals the highest tem- perature at which all the heat may be obtained. Obviously, if [Q] exists and possesses a finite value at a finite temperature, Q must exist at a temperature, t'. Since a change in a finite quan- tity of substance will be accompanied by a finite change in internal energy, [Q], the only condition which will cause Q to vanish will be that under which all the heat may be obtained at an infinite temperature. Gibbs does not deny this possibility, but simply states that this certainly does not represent the usual case. t' Further, the magnitude of Q is given by -r, [Q], and the work t" - t' performed by the cell, W, is given by — -f, — [Q]. These con- siderations form the basis of the discussion on pp. 342-344 of the "Equilibrium of Heterogeneous Substances," and in the first letter (Gibbs, I, 406) to the Secretary of the British Asso- ciation for the Advancement of Science. The remainder of the discussion of this subject on pp. 344-348 of the "Equilibrium of Heterogeneous Substances" has simply 720 HARMED ABT. M to do with proving that the data which existed at the time of writing, and which were obtained chiefly by Favre, substantiated the existence of heat changes during cell action. Since a great many accurate observations obtained in recent years completely confirm the contentions of Gibbs, and since the illustrations employed by him are far less accurate, it seems unnecessary to discuss this matter further. III. The Extension of the Theory of Galvanic Cells Not Explicitly Developed, but Contained Implicitly in the Thermodynamics of Gibbs Equation (17) [700] has proved to be of the greatest impor- tance to chemistry, and since the f function is peculiar to Gibbs it is to this extent unique in the history of the subject. This equation states that the reversible electrical work obtainable from a cell at constant temperature and pressure is equal to the decrease — d'f, in thermodynamic potential, corresponding to the cell processes. Since it is far more convenient to measure a cell at constant pressure and temperature than at constant volume and temperature, d^ is more easily obtainable than d\j/. If then a reversible cell can be constructed in such a way that the net effect of all the changes in the cell during the flow of current corresponds to a chemical reaction, the change in thermodynamic potential may be computed. This affords a very powerful experimental method for investigating the increase or decrease of thermodynamic potential correspond- ing to reactions which occur between solids, between solids and liquids, or between solids, liquids, and gases. In fact, in recent years cells have been constructed by means of which the changes in thermodynamic potential of all types of chemical reactions have been studied.* Early in the "Equihbrium of Heterogeneous Substances," Gibbs has shown that the differential of the thermodynamic * Recent surveys and discussion of this subject may be found in Taylor, Treatise on Physical Chemistry, 2nd Ed., Vol. I, pp. 731-745, D. Van Nostrand Company, New York (1924). See also International Critical Tables, Vol. VI, pp. 312-340, McGraw-Hill Book Co. (1930). ELECTROCHEMICAL THERMODYNAMICS 721 potential, rff, of a phase of variable composition is given by d^ = — r]dt + vdp + nidni + H2dn2 . . . + Undun, (28) an equation which is equivalent to equation [92] (Gibbs, I, 87) if ni, n2, etc., are the numbers of mols of the components, respectively, and m, ^2, etc., are the partial derivatives of ^ with respect to ni, n2, etc. From this we immediately find that, at constant composition, 11 = - - (-> and '^l = .. (30) dp Further, from the fundamental equation relating f to Xt the heat content function, we obtain ( = x-tv = x + tf\. (31) From equation (17) we obtain for a reversible cell at constant temperature and pressure the equation d^ = ±Ede. (32) As long as the various phases of the cell are sufficiently large so that their compositions will not be appreciably altered by the flow of a finite quantity of electricity e, then E will remain independent of e, and equation (32) may be integrated. Let us choose the path of integration to correspond with a chemical equation involving a flow of N faradays. Let us denote the faraday by F and employ the subscripts 1 and 2 to refer to the states of the system before and after the process represented by the given chemical equation. Further, let the symbol A denote the increase in the value of a function during the given finite process. We obtain Ar = r2 - n = r ^f = ± j^^' Ede = ± nef (33) 722 EARNED ART. M Therefore Af for the chemical reaction involving quantities of reactants and resultants corresponding to the passage of 96,500 coulombs or any multiple thereof may be measured at constant pressure and temperature. If E is expressed in volts, Af is in joules. Substituting this value of A^ in equations (29), (30), and (31), we obtain where At; and Ay denote the finite changes of entropy and volume respectively in the cell reaction, and ±iViJF = AX±(<*^)1. (36) Thus, not only do we obtain the pressure and temperature coefficients of electromotive force, but also the important equations by means of which the changes of entropy and heat content of chemical reactions can be obtained from measure- ments of E. Equation (34) is equivalent to equation (25). This method of measuring the entropy change in a reaction has proved to be of great importance in obtaining the data necessary for the verification of the so-called "third law of thermo- dynamics."* Let us now consider two cells which are to be measured at constant pressure and temperature: Pt I Ha (1 atm.) | HCl(wi) 1 AgCl 1 Ag; ±^i, and Pt I H2 (1 atm.) I HC1(W2) | AgCl | Ag; zt^2, and their corresponding reactions, ^Ha (1 atm.) + AgCl -> Ag + HCl(w:), * Lewis and Randall, Thermodynamics and the Free Energy of Chem- ical Substances, Chapter XXXI, McGraw-Hill Book Co., New York (1923). ELECTROCHEMICAL THERMODYNAMICS 723 and iHo (1 atm.) + AgCl ^ Ag + HCl(w2). By combining these cells we obtain the very important con- centration cell without liquid junction, Ag I AgCl 1 HCIK) 1 H2 I Pt I H2 I HCl(wO 1 AgCl | Ag; to which will correspond the cell process HCIK) ->HCl(wi). This means that the sum of all the changes occurring in this cell during the passage of the current is the transfer of hydro- chloric acid from a solution at a concentration wa to one at a concentration rtii. In other words, the process may be regarded as the reversible removal of one mol of hydrochloric acid from an infinite quantity of solution at a concentration W2, and its addition to an infinite quantity of solution at a concentration mi. The reversible electrical work will be ±(£"1 — E2)F. According to equation [104] (Gibbs, I, 89), the chemical po- tentials of the components of a phase are (37) [104] ar 1 9f 1 '"I = IIT ' ^2 = -7— , etc. OUi J p, «, nj, . . . Tin "'^2 Jp, t, ni, n„ ... nn This formula refers to the change in f for an infinitesimal change of composition in a finite phase. Correspondingly we have for a finite change of composition in an infinite phase iui=^l »M2 = ^^1 ,etc. (38), ZiTil Jp, t, nj, • • • n„ AW2 Jp. t, n,, nj, • • • nn where the operator A refers to the change in value of a function or a variable in a finite process. Thus, if we add one gram of component 1 to a very large quantity of the solution under the conditions specified by the subscripts, mi will equal the in- crease in f of the phase. If the unit of mass is the mol, ni will equal the corresponding increase in total thermodynamic poten- tial upon the addition of one mol. 724 EARNED ART. M With this fundamental consideration in view, it immediately becomes clear that the reversible electrical work of the cell without liquid junction just described measures the change in thermodynamic potential when one mol of hydrochloric acid at a concentration m2 is removed from one solution and then added to the solution at a concentration mi. Therefore, for the transfer of one mol of acid, we obtain by (38) /i/ - Ml" = Af = ±F(E, - E,). (39) These considerations show that the measurements of electro- motive forces of reversible cells containing various electrolytes of the same or different valence types afford direct measurements of the changes in chemical potentials of ionized components with their concentrations. Further, by measurements of the tem- perature coefficients of electromotive forces of cells of this type, and by employing the fundamental equations (34) and (36), the corresponding changes Ax of heat content, as well as of entropy may be determined. Further, by equation [97] (Gibbs, I, 88) the chemical potential of one component, the solvent for example, may be computed from that of the solute, or vice versa. Therefore, since we may measure the chemical potential of the solute from cell measurements, we may compute that of the solvent. In this way we may relate the electromotive force of a cell with the lowering of the vapor pressure, the lowering of the freezing point, and the osmotic pressure of the solution. Since the development of both the experimental side and the theory of the physical chemistry of solutions has depended to a considerable extent upon the evaluation of the chemical poten- tials, the value of this powerful and direct method of measure- ment of these quantities cannot be overestimated.* IV. Developments of Importance to the Theory of the Physical Chemistry of Solutions since Gibbs The general thermodynamics of Gibbs is complete and affords a basis for the exact treatment of the problems * A more detailed and systematic presentation of recent work on this subject is given by Harned in Taylor's Treatise on Physical Chemistry, Chap. XII. ELECTROCHEMICAL THERMODYNAMICS 725 which have arisen. Consequently, any further advance must rest upon some extra-thermodynamical discovery, for example, some empirical law. We have found that by a suitable mech- anism, we may obtain the change in chemical potential of an ionizing component from the study of a process represented by HCl(m2) -^HCl(wi). If we let niz vary and keep mi constant, at unit value, or at an arbitrary standard value, then we can measure the change in the quantity, ni' — ni", with the concentration. If this is done, we find that as m2 approaches zero, ni" changes with the con- centration at constant temperature according to the law m' - Ml" = 2Rt log — ' m2 or, since both /xi' are 2 Rt log mi are fixed, Ml" = 2Rt log W2 + /, (40) where 7 is a function of t and p only. Since the electrical process involves the transfer of both hydrogen and chloride ions, the factor 2 occurs in the expression on the right. This is the form of the expression derived from the perfect gas laws. It is, therefore, the equivalent of van't Hoff's law for dilute electro- lytes. This experimental discovery of van't Hoff, coupled with the ionic theory of Arrhenius, marked the beginning of a very extended experimental investigation of solutions of electrolytes. As a result, it was soon found that, in the cases of solutions of strong electrolytes, wide departures from this law occur. Without any addition to the fundamental thermodynamic theory, we may numerically overcome this difficulty by insert- ing a term which serves to measure the deviation from van't Hoff's law. Thus, m" = 2Rt log Ui, -{■ I = Rt log a^aci + I, or n" = 2Rt log ma + 2Rt log y + I, (41) 726 HARMED ART. M where anaci is the activity product of the ions as defined by Lewis,* and 7, or -^, is the activity coefficient. Hydrochloric m acid is a uni-univalent electrolyte and, consequently, the reaction of this cell represents the transfer of one gram ion of hydrogen ion and one gram ion of chloride ion. The modifications necessary for the general treatment of electrolytes of different valence types can easily be made. Consider any strong electro- lyte at a molal concentration, m, which dissociates according to the scheme C,+Ay_ = v+C + v-A, and let a2 = a+''+ aJ'-, where a+ and a_ are the activities of the cation and anion, respectively, and az, defined by the above equation, may be regarded as the activity of the electrolyte, and a± = (a+''+ aJ'-)'. Then equation (41) may be written in general n = Rt log a2 + I = vRt log a± + J, (42) which serves to define the activity. 7 is a function of the pressure and temperature, but not of the concentrations of the solute epecies. Further, we define the activity coefficient of any elec- trolyte by '^ = 7~7^ Zv. ' (43) and always measure it in reference to a value of unity when m equals zero. By means of cell measurements we obtain y. in reference to an * Lewis, Troc. Am. Acad., 37, 45 (1901); 43, 259 (1907). ELECTROCHEMICAL THERMODYNAMICS 727 arbitrary standard state, and, therefore, a^ may also be obtained. Now 7 may be computed if we let m be the molal concentration of the electrolyte. This is purely arbitrary since the molal concentration of the electrolyte tells us nothing regarding the real concentrations of the ions in the solution. The activity coefficient 7, however, acquires an important physical significance if the real ionic concentrations are known. According to the classical theory of Arrhenius, 7 was thought to measure the actual degree of dissociation of an electrolyte. Later, it was called by Lewis "the thermodynamic degree of dissociation". If this quantity measures the degree of disso- ciation, then the law of mass action in its classic form should be applicable to all classes of electrolytes. In the case of strong electrolytes, this conclusion was found to be erroneous, and therefore the first suppositions regarding 7 were entirely incorrect. The difficulty resides in the failure of these early theories to take into account the effects of the attractive and repulsive forces between the ions, which for charged particles vary inversely as the square of the distance. The careful con- sideration of these effects constitutes the departure of the recent developments of the theory of solutions from the classical theory. The most fruitful advance has come from the assumption that, in moderate concentrations in a solvent of high dielectric constant, the strongest electrolytes are completely dissociated into ions. Thus m in the cases of hydrochloric acid solutions, sodium chloride solutions, etc., is the true ionic concentration. If this is true, 7 acquires a definite physical significance. Fur- ther, if the assumption of complete dissociation is correct, then 7 must be calculable from fundamental considerations regarding the forces of attraction and repulsion between the ions. The various attempts to solve this problem have culminated in the theory of Debye and Hiickel* By the skillful application of Poisson's equation to a system of charged particles in thermal motion, they have succeeded in proving that in moder- ately dilute solutions 7 is a function of the electrostatic forces. ♦ Debye and Huckel, Physik. Z., 24, 305 (1923). 728 HARMED ART. M Since their calculation of 7 is numerically a very close approxi- mation, it justifies their initial assumption of complete disso- ciation of strong electrolytes. Even a conservative estimate of this theory will convince us that by far the larger part of the deviation factor, 7, is due to interionic forces in the case of strong electrolytes in media of high dielectric constant, such as water. It would be far beyond the purpose of the present dis- cussion to develop this theory and its many ramifications, but the knowledge that m is an ionic concentration or very nearly so in the case of strong electrolytes permits us to develop the possibilities of the study of reversible cells to a considerable extent without any sacrifice in accuracy. We shall now sketch briefly some developments which illustrate the more recent means of obtaining valuable data regarding strong electrolytes, weak electrolytes, and ampholytes from reversible cell measurements. To assure exactness, we shall omit measurements of all cells with liquid junctions since these all involve an undefinable and physically meaningless hquid junction potential.* (1) The Activity Coefficients of Strong Electrolytes We have already shown how the change in chemical potential of hydrochloric acid in passing from a solution at one concen- tration to a solution at another concentration may be measured by a cell without Uquid junction. For the change CA(m2) ^CA{mi), we have, according to equation (42), - Ar = (m' - m") = Rt log ^—^Tr (44) etc dA If we adopt the convention that a positive electromotive force accompanies a decrease in thermodynamic potential, we obtain from equation (39) *Harned, J. Physical Chem., 30, 433 (1926). Taylor, /. Physical Chem., 31, 1478 (1927). Guggenheim, /. Phtjsical Chem., 33, 842 (1929); 34, 1540 (1930). ELECTROCHEMICAL THERMODYNAMICS 729 NEF = Rt log ac'a/ ac'W E = 2Rt NF log y'mi y"m2 or 27?/ -v'm. (45) Thus, ifwe know y at one concentration, we may compute it at another. The activity coefficient, however, is always computed in reference to unity at infinite dilution. If we let Eq equal the electromotive force of the cell when y[ini equals unity, and refer all values of E and y"m2 to this standard value, we obtain r. ^ 2i2i , „ 2Rt , , E -Eo= - ]^log7" - -^ \0gm2 (46) or 2Rt 2Rt E -\- — \ogm2 = E,-— log y". (47) Since y" is taken to be unity as m2 equals zero, the left-hand mem- ber of the equation (at zero concentration) equals the normal electrode potential, Eo. By plotting the left-hand member against a convenient function of the concentration, Ea may be evaluated, and subsequently 7 may be calculated by equation (47) at any concentration, nii, at which E is known. Such a method permits the determination of 7 at a constant tempera- ture from electromotive force data only. In recent years the activity coefficients of many electrolytes have been determined by measurements of cells of this type. If we replace the hydrochloric acid by a halide of an alkali metal and the hydrogen electrode by a dilute alkali metal amalgam, the cell, Ag I AgZ 1 MX{m2) I ilfxHg 1 MX{m,) \ AgX | Ag, is formed. The electromotive force of this cell measures the change of thermodynamic potential corresponding to the reaction MX{m2) -^ MX{mi), whence n" and n' may be determined.* *MacInnes and Parker, J. Am. Chem. Soc, 37, 1445 (1915). Mac- Innes and Beattie, J. Am. Chem. Soc, 42, 1117 (1920). Harned and Douglas, J. Am. Chem. Soc, 48, 3095 (1926). Harned, /. Am. Chem. Soc, 51, 416 (1929). 730 HARMED ART. M Further, we mention the cell, H2 1 M0H(W2) 1 MxHg I MOH(wi) 1 Ha, which measures the transfer corresponding to M0H(w2) + H20(mi) -> MOH(wi) + HzOK), whence the activity coefficients of alkali metal hydroxides may be measured. By other cells of the same types, alkali metal sulphates and alkaline earth chlorides have been studied. All these data have an important bearing on the theory of electroly- tic solutions.* Not only may we obtain these changes in chemical potentials for single electrolytes by these measurements, but also the chemical potentials of one electrolyte in a solution containing another electrolyte may be computed. From the cell, Ag I AgX 1 HX{mO, MXim^) \ H2 1 HX(m) \ AgZ 1 Ag, we may measure the change of thermodynamic potential of a halide acid from the solution containing the chloride to the pure acid solution, which we represent by HX(mi) [MXim^)] -^ HX(m). Thus, we may obtain the activity coefficient of the acid at a concentration (wi) in a salt solution of a concentration (wz). Suffice it to say that by similar cells we now know the value of this important quantity for hydrochloric acid, sulphuric acid, and hydrobromic acid in many salt solutions, f Further, cells of the type, H2 I MOH(wi), MZ(m2) | MxHg | MOH(w) | H2, permit the calculation of the activity coefficients of hydroxides in salt solutions. I * Knobel, /. Am. Chem. Soc, 45, 70 (1923). Harned, /. Am. Chem. Soc, 47, 676 (1925). Harned and Swindells, J. Am. Chem. Soc, 48, 126 (1926). t Harned, /. Am. Chem. Soc, 38, 1986 (1916); 42, 1808 (1920). Harned and Akerlof, Physik. Z., 27, 411 (1926). t Harned, /. Am. Chem. Soc, 47, 684 (1925). ELECTROCHEMICAL THERMODYNAMICS 731 (2) The Activity Coefficients of Weak Electrolytes in Salt Solutions (a) The Ionic Activity Coefficient of Water in Salt Solutions. We have described a cell by means of which the activity coeffi- cient of hydrochloric acid may be obtained in a chloride solution. Suppose we maintain (mi + ^22) constant and measure 7 in the solutions of varying acid and salt concentration. It is found that 7 varies with the acid concentration according to the law* log 7 = ami + log 70. (48) Thus at constant total molality 7 extrapolates to 70 at zero con- centration of acid, whence we know 7hTci in the salt solution which is free from acid. In a similar manner from measure- ments of the cells containing sodium hydroxide in the sodium chloride solutions, we may obtain ^^ ^^ in the hydroxide-free salt solution. Also, from measurements of the cells containing sodium chloride, we know 7Na7ci ^-t the concentration (wi + nh). Therefore, if we multiply 7h7ci by '^^^^^^ and divide by TNa7ci> we obtain the ionic activity coefficient product of water, ThToh^ at this concentration of salt. Obviously, by this method, may be obtained at other salt concentrations. flHiO 7hToh ajiiO The primary dissociation of water is represented by H2O ;=± H+ -f OH- and the thermodynamic dissociation constant, K, is given exactly by ^ ^ OhOoh ^ 7H70H ^^^^^ (49) OHjO CLRiO Since we may determine in the salt solutions, the classical CtHiO • Earned, /. Am. Chem. Soc, 48, 326 (1926). Guntelberg, Z. physik. Chem., 123, 199 (1926). 732 EARNED art. m dissociation product, mnWoH) may be determined if we know K, and in this way we may study the effects of electrolytes on the dissociation of the solvent.* We have still to determine K from the electromotive forces of cells without liquid junction. Consider the cell, H2 1 MOB.{mi), MC1(W2) | AgCl | Ag.f Its electromotive force at 25° is given by E = Eq - 0.05915 logio mnwci - 0.05915 logio ThTci, (50) where Eq may be obtained from the cell containing hydrochloric acid. If we substitute the value of m^ obtained from equation (49), we obtain E = Eo- 0.05915 logio ^^^^'^^^ - 0.05915 logio thTci ThTohWoh = Eo- 0.05915 logio K - 0.05915 logio '^^^^^^^'" 7H70H -0.05915 logio ^^. (51) moB. Eo is known. In dilute solutions the third term on the right is very close to unity since it contains the ratio of activity coeffi- cient products. Therefore, E + 0.05915 logio ^^^ moH in very dilute solutions has very nearly a constant value. Thus, the extrapolation of this quantity to zero ionic concentration is a simple matter, and its value at infinite dilution is equal to [£'0 — 0.05915 logio K]. We have, therefore, an independent measure of K. (b) The Ionic Activity Coefficients and Dissociation of Weak Acids and Bases in Salt Solutions. By the application of the * Harned, /. Am. Chem. Soc, 47, 930 (1925). t Roberts, J. Am. Chem. Soc, 62, 3877 (1930). ELECTROCHEMICAL THERMODYNAMICS 733 principles just discussed, very important information concern- ing weak acids and bases in solvents containing salt solutions may be obtained. We shall consider the acid case only, since the bases may be investigated in exactly the same manner. Let us construct the cell, Ag I AgCl I HCl(wi), MCl(m2) | H2 | HAc(m), MC\{mz) \ AgCl I Ag, in which HAc is a weak acid, mi is 0.01 molal or less, and the concentrations are such that the total ionic concentration on the two sides is the same or very nearly so, so that Wi 4- W2 = Wh + W3, where m^ is the hydrogen ion concen- tration in the solution of the weak acid. The electromotive force of this cell at 25° is given by E = 0.05915 logic ^^5!^^' + 0.05915 logio ^^^^^ , (52) where the double accent refers to the hydrochloric acid solution and the single accent to the weak acid solution. Since Wi, W2, and ms are known mn may be evaluated if the first term on the right of this equation is known. Two secondary effects influ- ence this term, which can be completely taken into account if sufficient care is exercised. The first and most important is the effect of the presence of the undissociated molecule of the weak acid which causes th'tci' to differ from its value in pure water even though the concentrations of the ions in the two cell compartments are the same. The second effect is much simpler and merely requires a knowledge of the activity co- efficient of hydrochloric acid in the salt solution. This situation has been investigated very thoroughly by Harned and Robinson, and Harned and Owen, who show that both 7h"tci" and th'tci' as well as mn can be determined without the intro- duction of any inexact considerations. The dissociation of the acid is represented by HAc ^ H+ + Ac-, 734 HARMED ART. M and the ionization constant by K = ''-^^^ "^'^^^ = y.' ^^^ = 7x^ K., (53) THAc whac w — mn where m is the original concentration of the weak acid, and 7^ its activity coefficient in the salt solution. Since we determine /wh, Kc becomes known at various salt concentrations. We have yet to find its value at infinite dilution or when 7^ equals unity. This can be done very simply by the use of a function which gives the variation of 7 with the total ionic concentration, li, in dilute solutions; namely, logio 7^^ = - Vm + a/^, (54) where a is an empirical constant. If we take the logarithm of equation (53) , we obtain logio K = logio Kc + logio 7x^ (55) Substituting for logio 7x^ and rearranging terms, we find that logio Kc — \/ n = logio K — an. (56) Therefore, if we plot [logic Kc — \/ m]) which has been determined against /j., we obtain a straight line in dilute solutions, and the value of the function on the left is equal to logio K when /x equals zero. By this means we have an independent measure of the dissociation constant, the ionic activity coefficient, and dissocia- tion of a weak acid in a salt solution. The same or very similar methods will also afford very valuable evidence concerning similar properties of weak bases, and ampholytes.* These considerations, although very brief, serve to show the extent and power of the method of cell measurements when applied to the study of all kinds of electrolytes. It would be far beyond the scope of this discussion to treat the various * A thorough discussion of this subject is to be found in the contribu- tions of: Harned and Robinson, /. Am. Chem. Soc, 50, 3157 (1928); Harned and Owen, ibid., 52, 5079 (1930); 52, 5091 (1930); Owen, ibid., 64, 1758 (1932); Harned and Ehlers, ibid., 54, 1350 (1932). ELECTROCHEMICAL THERMODYNAMICS 735 ramifications which would develop upon considerations of the variations of these quantities with temperature and pressure. Suffice it to say that everything comes back to the experimental evaluation of the chemical potentials of electrolytes, which would have been impossible without the fundamental contribu- tion of Gibbs. Retrospect and Prospect We have emphasized the completeness and exactness of Gibbs' treatment of the perfect electrochemical apparatus. If we work in the spirit of the original method, then we must eliminate uncertainties inherent in the use of cells such as those containing liquid junction potentials. The invention and use of the concentration cell without liquid junction is an excellent illustration of an exact method of study. However, the power of this experimental method only becomes apparent when we introduce the chemical potentials and develop the general thermodynamics of Gibbs in its relation to such cells. But even this has not been enough. Extra-thermodynamical con- siderations which must be experimentally verified and finally proved by fundamental electrostatic theory have been required, and will continue to be necessary before the intricate subject of the nature of the ionic state in solutions will be unravelled and explained. But there will be nothing in these modifications to detract from the value of the contribution of the first master of this subject. AUTHOR INDEX Adam, 554, 556, 562, 567 ff., 575, 576, 582-584, 597, 676-678 Akerlof, 730 Alkemade, 324 Allen, 249 Amagat, 569, 571 Arrhenius, 725, 727 Avogadro, 27, 337 Bachman, 561 Bancelin, 561 Bancroft, 187, 550, 632, 678 Barker, 560 Beattie, 729 Bennett, 594 Benson, 559 Berkeley, 139, 140 Beudant, 329 Bircumshaw, 572, 586 Bjerrum, 211 Bocher, 10 Boedecker, 543 Boltzmann, 327 Bowen, 269 Boyle, 25, 337 Bradley, 574 Bredig, 331 Bruyn, de, 331 Bumstead, 19 Burton, 140 Butler, 211, 686, 693, 697, 701 Byk, 236 Calcar, van, 331 Carnot, 20, 64, 66, 67 Cassel, 586 Chaperon, 329 Chapman, 693 Charles, 25 Clapeyron, 109, 237, 349, 350, 595 Clausius, 20, 21, 61, 65, 67, 68, 109, 237, 339 Dalton, 339, 355 flf. Daniels, 388, 391 Davies, 562 ff. Davy, 61 Day, 249 Debye, 375, 727 Devaux, 567 Dewar, 669 Donnan, 211, 559, 560, 581, 583, 639 Douglas, 729 Downes, 135, 141 Duhem, 123, 134 Dundon, 675 Ehlers, 734 Einstein, 329 Eotvos, 592, 593 Euler, 89, 322, 534 Fihraeus, 332 Favre, 720 Tenner, 269 Frazer, 137, 139 Frenkel, 554 Freundlich, 520, 543, 550 Frumkin, 561, 693 Galileo, 327, 329 Gauss, 513, 639 Gay-Lussac, 25, 329, 337 Geiger, 455 Gerry, 367 Gillespie, 351, 355, 356, 367 Goard, 576 Goranson, 433, 491 Gouy, 329, 693, 708 Green, 461 Guggenheim, 211, 699, 728 Guntelberg, 731 Gyemant, 513 Hamilton, 545 Harkins, 562, 575, 576 Earned, 724, 728-734 Hartley, 139, 140 737 738 AUTHOR INDEX Helmholtz, 61, 85, 91, 234, 346, 678, 683, 691 ff., 708, 717 Henry, 123, 194, 363, 371 Herzfeld, 693 Hewes, 19 Huckel, 375, 727 Hulett, 674, 675 Humphreys, 578 Iredale, 581-585 Johnson, 675 Joule, 21, 61, 338 Katayama, 593 Kelvin, Lord (W. Thomson), 21, 61,66,109,338,678,683 Knobel, 730 Konig, 692 Konowalow, 113, 177 Kracek, 243, 269 Kundt, 25 Lagrange, 459, 545 Langevin, 334 Langmuir, 549, 550 ff., 567 ff., 576, 581, 582, 678, 679, 720 Laplace, 329, 510, 513, 517, 520, 639 Lawrence, 660, 669, 670 Le Chatelier, 233 Lerberghe, 375 Lewis, G. N., 85, 128, 130, 131, 137, 139, 211, 234, 344, 356, 371, 375, 591, 592, 726, 727 Lewis, W. C. M., 559, 560 Liebig, 382 Lippmann, 688 ff., 697, 702 Lipsett, 675 Lovelace, 137 Lurie, 355 Maass, 675 McBain, 542, 562 ff., 575, 578 Mack, 675 Maclnnes, 729 McLeod, 594 Mariotte, 25, 337 Markley, 263 Massieu, 56, 85 Maxwell, 20, 27, 50, 85 Mayer, 61 Meunier, 11 Micheli, 582-584, 595 Millikan, 679 Milne, 211 Milner, 550 ff., 559 Mitchell, 594 Morey, 243, 249, 252, 269, 287 Morgan, 575 Morse, 141 Nernst, 679, 684, 685 Newman, 684, 685, 693 Nouy, du, 659 Oliphant, 585 Onnes, 234 Ostwald, 674, 679 Owen, 733, 734 Parker, 729 Pascal, 511 Patrick, 560, 561 Pedersen, 333 Peltier, 683 Perier, 329 Perman, 135, 139-141 Per r in, 329, 670 Planck, 375, 692 Plateau, 558 Pockels, 566 Poiseuille, 664 Poisson, 705, 727 Pollard, 564 Porter, 139, 549, 685 Poynting, 355, 454 Quincke, 584 Ramsay, 592, 593 Ramsden, 559 Randall, 85, 128, 130, 137, 211, 344, 356, 375, 591, 592, 722 Raoult, 128, 194, 372 Raschevsky, von, 640 Rayleigh, Lord, 363, 566, 567, 659 Regnault, 338 Rhodes, 263 Rice, O. K., 693 Richardson, 679 Rideal, 543, 554, 556, 562, 570-573, 576, 578, 584, 677, 685 AUTHOR INDEX 739 Roberts, 732 Robinson, 733, 734 Rogers, 137 Roozeboom, 249, 256 Riidorff, 118 Rumford, 21 Saussure, 543 Scheel, 455 Schofield, 561, 571-573, 584, 585, 685 Schreinemakers, 274, 287 Shields, 592, 593 Smits, 259, 287 Stern, 693, 705 ff. Svedberg, 331, 332 Swan, 541, 549 Swindells, 730 Szyszkowski, 551, 555, 569 Tait, 21 Taylor, 720, 728 Thompson, 675 Thomson, James, 477 Thomson, J. J., 541, 543, 545 Thomson, W., vide Kelvin Tolman, 334, 640 Traube, 551, 569 Urquhart, 541, 549 Urry, 139, 140 van der Waals, 259, 342, 512, 569, 593 van't Hoff, 124, 197, 550, 725 Varley, 688 Verhoek, 388, 391 Volta, 678, 679, 683, 691 Warburg, 549 Washburn, 211 Wegscheider, 236 Westgren, 330 Williamson, 252, 269, 287 Wiillner, 118 Wynne-Jones, 564 Zawidski, 134, 559 SUBJECT INDEX Acetic acid, concentration at interface, 559 Acetone, activity coefficient in chloroform, 134 Activity, 131 ff., 726 Activity coefficient, 133, 190 ff ., 203, 726 ff. Adjacent phases, stability, 153 Adsorption, 542, 579 ff. Adsorption equation, Gibbs', 535 Adsorption isotherm, 542 Ampholytes, in voltaic cells, 734 Amyl alcohol, concentration at interface, 559, 575 Anticlastic, 14 Atmosphere, pressure gradient in, 329 Barometric formula, 329 Benzene-Alcohol system, vapor pressure of, 113 Black stage of soap films, 668 ff. Bromobenzene-iodobenzene sys- tem, vapor pressure of, 114 Cane sugar, activity coefficient, 135 Cane sugar, osmotic pressure, 140 Calcium chloride-water system, 256 ff. Catalysis, 178, 179 Catalyst, poisoning of, 554 Catenary, 15 Centrifugal force, equilibrium under, 330 ff. Chemical constant, 345 Chemico-motive force, 207 Chemo-electrical equivalent, 698 Chloroform, activity coefficient in acetone, 134 Coefficient, activity, 133, 190 ff., 203, 726 ff. , osmotic, 197 , strain, 402 Coexistent phases 235 Cohesion, 512, 517 ff. Colloidal solutions, 329 Component, actual, 93 , convertible, 382 , independent, 185 , possible, 93 Contact angles, 675 ff. Contact equilibrium, electrical, 206 ff. Convertible components, in gas mixtures, 382 Critical liquid, 313 Critical phases, 163 Cryohydrate, 242 ff. Curvature, of surfaces, 10 , total, of surfaces of discon- tinuity, 646, 647 Cycle, Carnot's, 20, 66 ff. Desorption, 547, 575 ff., 595 Dilatation, 489 Dipole gases, 342, 343 Dissipated energy, 178, 378 Dissociation of electrolytes, 727 Double layer, Helmholtz, 691 ff. Dyestuffs, adsorption of, 561 Efficienyc of heat engine, 64 Efflux of liquids, 664 Elastic constants, 430 ff. Elastic moduli, 431 Electrical work, reversible, 711 Electrocapillarity, 688 ff. Electrochemical apparatus, per- fect, 710 ff. Electrochemical potential, 199, 699 Electrode potentials, 678 Electromotive force, 209, 709 ff. Electron affinity, 679, 683 Electron atmosphere, 682 Elongation ellipsoid, 483 Enantiotropic forms, 254 740 Enkaumy, 234 Enthalpy, 234 Entropy, 23, 68 Equilibrium, thermodynamic, 72 Ethyl alcohol, surface excess, 572, 573 Eutectic composition, 250 Eutectic temperature, 304 Extruding of metals, 368 Ferric chloride-water system, 114 Films, draining of, 667 , impermeable, 566 fif. , liquid, 659 flf. , oil, 566, 567 , soap solutions, 659 ff. Flotation, 678 Foams, 667 Free energy function, 216 ff., 227 ff., 295 ff. Freezing point lowering, 125 Fugacity, 367, 371 Galvanic cells, 709 ff. Gibbs ring, 665 ff. Gravity, 327 ff. Heat function, 214, 220, 224 Heat of adsorption, 594 ff. Heat of wetting, 596, 677 Hemoglobin, molecular weight, 332 ff. Hydrochloric acid, in voltaic cells, 710, 722 Hydrostatic stress, 475 Hydroxides, in voltaic cells, 730, 732 Hypsometric formula, 329 Ideal gases, 337 ff. Ideal solutions, 188 Impermeable films, 566 ff. Independent components, 185 Interionic forces, 728 Internal pressure, 512, 520 ff. Intrinsic potential, 328 Intrinsic pressure, 512, 520, 521 Invariant point, 236 lodobenzene {see bromobenzene) Isothermal curves, 30 Isotropy, 482 ff., 490 SUBJECT INDEX Liquid films, 659 ff. 741 Mannite solutions, freezing point and vapor pressure, 138 Melting point, minimum, 257 Membrane equilibria, 181 ff. Molecules, cross-sectional area, 568 ff. Mol fraction, 187, 188 Negative adsorption, 547, 575 ff. Oil films, 566 ff. Osmotic coefficient, 197 Osmotic equilibrium, 192 Osmotic pressure, 124, 138, 330, 684 Overvoltage, 687 Partial pressure, 358 Peptisation, 145 Phase rule, 106, 233 ff. Phenol-water system, 164, 263 ff. Poisoning of catalysts, 554 Polarization, electrode, 687 Polymerization, reversible, of gases, 383 Potassium nitrate-water system, 241 ff. Potassium silicate (see silica) Potential, chemical, 95, 234 , electrochemical, 199, 699 , electrode, 678 Pressure, gradient in atmosphere, 329 , hypsometric formula, 329 , internal, 512, 520 ff. , intrinsic, 512, 520, 521 , lowering of vapor, 127 — , osmotic, 124, 138, 330, 684 , partial, 358 , surface, 567 ff. , vapor, 349 ff. Principal axis, of strain, 406 , of stress, 429 Protein, precipitation at interface, 559 Pyridine, surface excess, 572, 573 Quadric surface, 15 ff. 742 SUBJECT INDEX Reversibility, 68 Rigidity, modulus of, 431, 433 Silica-potassium silicate - water system, 269 ff . Shear, 400 Shearing tractions, 420 Soap solutions, 659 S. Sodium oleate, cross sectional area of molecule, 575 , surface tension of solutions, 559, 659 Solution pressure, 679, 684 Sorption, 542 Space charge, 682 Specific heat, 24, 341 Strain, 395 ff. Strain coefficient, 402 Strain-energy function, 437 Stress, 417 flf. Surfaces, curvature of, 10 ff. Surface energy, 515 Surface pressure, 567 ff. Surface of tension, 529 Surface tension, oil on water, 567 Synclastic, 14 Thermionic emission, 679 Thermoelectric power, 683 Two-dimensional systems, 567 ff. Ultra-centrifuge, 331 ff. Unimolecular films, 567 ff. Vapor pressure, 349 ff. , lowering of, 127 Variations, method of, 5 Vector function, 419 Volcanism, 249 Water, entropy of, 238 ff. Weak acids, in galvanic cells, 733 Wollastonite, 255 Work function, 214 ff., 226 Zeta function, 216 ff., 227 ff., 295 ff.