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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

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

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 <r 507

II. Surface Tension 509

3. Intrinsic Pressure and Cohesion in a Liquid 509

4. Molecular Potential Energy in a Liquid 513

5. An Alternative Method of Treatment 516

III. The Quasi-Tensional Effects at a Curved Surface 518

6. Modification of the Previous Analysis 518

7. Interpretation of o- as a Tension 520

IV. Statistical Considerations 523

8. The Finite Size of Molecules 523

9. Distribution of Molecules in Two Contiguous

Phases 525

V. The Dividing Surface 527

10. Criterion for Locating the Surface of Tension .... 527

11. An Amplification of Gibbs' Treatment 529

VI. The Adsorption Equation 533

12. Linear Functional Relations in Volume Phases. . . 533

13. Linear Functional Relations in Surface Phases . . . 534

14. Derivation of Gibbs' Adsorption Equation 535

15. Variations and Differentials 537

16. Condition for Experimental Tests 537

17. Importance of the Functional Form of a in the

Variables 538

VII. Other Adsorption Equations 541

18. The Exponential Adsorption Isotherm 542

19. Approximate Form of Gibbs' Equation and Thom-

son's Adsorption Isotherm 543

20. The Empirical Laws of Milner and of Szyszkowski

for <T and c. Langmuir's Adsorption Equation.

Frenkel's Equation 551

21. Energy of Adsorption 554

VIII. Experimental Investigations to Test the Validity of

Gibbs' Adsorption Equation 557

22. The Earlier Experiments to Test Gibbs' Equation. 557

CONTENTS xxi

23. The Experiments of McBain and His Collabora-

tors 561

IX. Gibbs' Equation and the Structure of Adsorbed Films . . 566

24. Impermeable or Insoluble Films 566

25. The Work of Langmuir and Adam. The Concept

of "Surface Pressure." Equations of Condition

for Surface Phases 567

26. Unimolecular Layers and the Dividing Surface . . . 572
X. Desorption 575

27. Unimolecular Layers and Negative Adsorption. . . 575

28. The Recent Experiments of McBain and Humph-

reys on Slicing Off a Thin Layer at a Surface .... 578
XI. Adsorption of Gases and Vapors on Liquid Surfaces .... 579

29. Form of Gibbs' Equation for Adsorption from a

Gaseous Phase 579

30. The Experiments of Iredale 581

31. The Experiments of Micheli, Oliphant, and Cassel . 583
XII. The Thermal and Mechanical Relations Pertaining to

the Extension of a Surface of Discontinuity 586

32. Need for Unambiguous Specification of the Quanti-

ties Which Are Chosen as Independent Variables. 586

33. Alternative Method of Obtaining the Results in

This Section. Total Surface Energy 588

34. Empirical Relations Connecting a and t. Degree of

Molecular Association in Liquids 592

35. Heat of Adsorption 594

36. Dependence of <r on the "Age" of a Surface 596

XIII. The Influence of Gravity 597

37. The Variation of p, <r, m, m, . . with Depth in a

Liquid. An Apparent Inconsistency in Gibbs'
Argument. The Argument Justified 597

XIV. The StabiUty of Surfaces of Discontinuity 605

38. Conditions for the Stability of a Dynamical

System 605

39. Restricted Character of Such Conditions as Applied

to a Thermodynamical System 607

40. Stability of a Plane Portion of a Dividing Surface

Which Does Not Move 608

41. Three Conclusions Drawn from the Analysis in

Subsection (40) 612

42. Determination of a Condition Which Is Sufficient

though Not Necessary for Stability when the
Dividing Surface Is Not Plane and Is Free to
Move 615

43. Gibbs' General Argument Concerning Stability in

Which the Difficulty Referred to in Subsection
(39) Is Surmounted 617

xxii CONTENTS

44. Illustration of Gibbs' Method by a Special

Problem 619

45. An Approach to this Problem from a Consideration

of the Purely Mechanical Stability of the Surface. 622
XV. The Formation of a Different Phase within a Homo-
geneous Fluid or between Two Homogeneous Fluids . . . 625

46. A Study of the Conditions in a Surface of Dis-

continuity Somewhat Qualifies an Earlier Con-
clusion of Gibbs Concerning the Stable Coexist-
ence of Different Phases 625

47. The Possibility 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 627

48. The Possibility of the Formation of a Homogeneous

Mass between Two Homogeneous Masses 631

XVI. The Formation of New Phases at Lines and Points of

Discontinuity 640

49. The Possible Growth of a Fifth Surface at a Line

of Discontinuity Common to Four Surfaces of
Discontinuity Separating Four Homogeneous
Masses 640

50. The Possible Growth of a New Surface at a Point

of Meeting of a Number of Lines of Discon-
tinuity 642

51. Some General Ideas and Definitions Concerning

the Possibility of a New Homogeneous Mass
Being Formed at a Line of Discontinuity or at a
Point of Concurrence of Such Lines 644

52. The Stability of a New Homogeneous Mass Formed

at a Line of Discontinuity. A Summary of the
Steps in the Argument 648

53. The Details of the Argument Omitted from the

Summary in (52) 650

54. Consideration of the Case When the New Homo-

geneous Mass is Bounded by Spherical Lunes . . . 655

55. The Stability of a New Homogeneous Mass

Formed at the Point of Concurrence of Four

Lines of Discontinuity 657

XVII. Liquid Films 659

56. Some Elementary Properties of Liquid Films.

The Elasticity of a Film 659

57. The Equilibrium of a Film 662

58. Foams. The Draining of a Film. The "Gibbs

Ring" 667

59. The Black Stage of a Soap Film 668

CONTENTS xxiii

XVIII. Surfaces of Solids 670

60. The Surface Energy and Surface Tension of the

Surface of a SoUd 670

61. Contact Angles. The Adhesion of a Liquid to a

Sohd, Heat of Wetting 675

XIX. Discontinuity of Electric Potential at a Surface. Elec-

trocapiUarity 678

62. Volta's Contact Potential between Two Metals

and Its Connection with Thermoelectric and
Photoelectric Phenomena 678

63. Discontinuity of Potential between a Metal and

an Electrolyte 684

64. Gibbs' Comments on Electrode Potentials 687

65. Lippmann's Work on ElectrocapiUarity and Its

Connection with Gibbs' Equation [690] 688

66. The Double-Layer Hypothesis of Helmholtz 691

67. Recent Developments in the Thermodynamical

Treatment of ElectrocapiUarity 692

68. The Reason Why Gibbs' Derivation of His Electro-

capillarity Equation [690] Exhibits It as Equiva-
lent to Lippmann's Equation 697

69. Guggenheim's Electrochemical Potential of an Ion. 698

70. Derivation by Means of the Postulate of "Specific

Adsorption" of Ions of an Equation Combining
Gibbs' Terms for Ions with a Lippmann Term for
Electrons 700

71. Some Brief Remarks on the Fundamental Electri-

cal Equations Used by Stern in His Treatment
of the Distribution of Ions in a Solution Close
to the Cathode of a Capillary Electrometer 704

M. The General Properties of a Perfect Electro-
chemical Apparatus. Electrochemical Thermo-
dynamics (Gibbs I, pp. 338-349; 406-412), H. S.

Harned 709

I. The General Thermodynamics as Explicitly Developed . . 709
II. On the Question of Absorption or Evolution of Heat

During Galvanic Processes 717

III. The Extension of the Theory of Galvanic Cells Not Ex-

plicitly Developed, but Contained ImpUcitly in the
Thermodynamics of Gibbs 720

IV. Developments of Importance to the Theory of the Physi-

cal Chemistry of Solutions since Gibbs 724

Indexes 737

NOTE ON SYMBOLS AND NOMENCLATURE

F. G. DONNAN

In the following Commentary on the thermodynamic writings
of J. Willard Gibbs the symbols used by him for the principal
thermodynamic quantities have been retained in general. Since
the majority of authors have employed symbols which differ con-
siderably from those of Gibbs, and the notation employed varies
in some respects from author to author, a short comparison
Table is given below. There has also existed, and indeed still
exists, a very considerable variation of usage as regards the
names assigned to some of the quantities. It has therefore been
thought desirable to give a correlated list of the principal names
which are, or have been, employed. We shall denote six im-
portant thermodynamic quantities by the numerals 1, 2, 3, 4, 5,
6. The symbols assigned to these six quantities by Gibbs and

TABLE 1
Comparison of Symbols

Author

Thermody

aamic quantities

i
T

T
T
T
T
T
T
T
T
T

(

u
u
u
u

u
u
u

3
V

s
s
s
s

s
s
s

X
U'

W
H

H
H
H

E,
H

Gibbs

-n

H
F
F
A
^
A
F
F
F,
F

Massieu

Helmholtz

Duhem

Planck

-r$

Lewis and Randall

Lorentz

Noyes and Sherrill

Partington

Schottky, Ulich and Wagner

Sackur

Guggenheim

DONNAN

ART. A

various authors are shown in Table 1, whilst the corresponding
names are given in Table 2.

TABLE 2

Names of Quantities

Quantity

2
3

Names employed

Absolute Temperature. Temperature on the Kelvin (ther-
modynamic) scale.

Energy. Total internal Energy.

Entropy.

Total Heat (term used by engineers). Heat Function for
constant pressure (Gibbs).

Heat Function (Partington; Sackur; Milne).

Heat Content (Lewis and Randall; Noyes and Sherrill).

Enthalpy (Kamerlingh Onnes).

Available Energy (Clerk Maxwell).

Free Energy (Helmholtz). Isothermal Potential (Helm-
holtz). Internal Thermodynamic Potential (Duhem).

Free Energy (Planck; Lorentz; Sackur; Partington; Schott-
ky, Ulich and Wagner; most European authors since Helm-
holtz).

Work Content (Noyes and Sherrill). Work Function (Milne).

Helmholtzian Free Energy (Guggenheim).

Thermodynamic Potential at constant Pressure (Duhem).

Free Energy (Lewis and Randall; Noyes and Sherrill; many
authors, American and European, following the lead of the
American School of chemical thermodynamics created by
Noyes and Lewis).

Thermodynamic Potential (Lorentz; Sackur; Partington).

Gibbs' Thermodynamic Potential (Schottky, Ulich and
Wagner) .

Gibbs' Free Energy (Guggenheim).

Notes to Tables

(a) Gibbs, using i^ to denote 5, called —\p the "Force Function for
Constant Temperature."

(b) Massieu called his functions i/' and \p' the "Characteristic Func-
tions" of the system.

(d) As regards nomenclature used at the present time, it is to be
noted that both the quantities 5 and 6 are called Free Energy. This is a
source of confusion to students of thermodynamics. Similar remarks
apply to the use of the symbol F, which may denote either 5 or 6.

SYMBOLS AND NOMENCLATURE

F. Massieu

P. DUHEM

H. VON Helmholtz

M. Planck

A. A. Notes and
M. S. Sherrill

G. N. Lewis and
M. Randall

J. R. Partington

H. A. LORENTZ

O. Sackur

E. A. Milne

W. SCHOTTKY, H. UlICH

and C. Wagner
E. A. Guggenheim

REFERENCES

Comptes rendus de I'acad. des sciences,
Vol. 69, pp. 857; 1057 (1869).

Trait6 61ementaire de Mecanique chimique
fondee sur la Thermodynamique, Vol. 1.
(Paris, 1897.)

Vorlesungen iiber theoretische Physik, Bd.
VI, Theorie der Warme. (Leipzig, 1903.)

Vorlesungen iiber Thermodynamik, Neunte
Auflage. (Berlin and Leipzig, 1930.)

An Advanced Course of Instruction in Chem-
ical Principles. (New York, 1922.)

Thermodynamics and the Free Energy of
Chemical Substances. (New York, 1923.)

Chemical Thermodynamics. (London, 1924.)

Lectures on Theoretical Physics. Vol. II.
(English Translation, London, 1927.)

Lehrbuch der Thermochemie und Thermo-
dynamik. Zweite Auflage von CI. von
Simson. (Berlin, 1928.)

Article F of the present Volume.

Thermodynamik. (Berlin, 1929.)

Modern Thermodynamics by the Methods of
Willard Gibbs. (London, 1933.)

MATHEMATICAL NOTE

JAMES RICE

1. The Method of Variations Used for Determining the Condi-
tions under Which a Function of Several Variables Has a Maximum
or Minimum Value. In the discussion of the conditions for
equihbrium of a system and of the criteria of stabihty of a state
of equihbrium, the following mathematical problem is presented :

To determine the values of the variables Xi, Xi, . . . . Xn for
which a given function of these variables, f(xi, X2, .... Xn) has a
maximum or minimum value, the variables themselves being
subject to a condition such as

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

<t>(qi, qz, qn) = 0,

<f>{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

mq)

:, . . ^.^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

(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

c>4>(q), , e

c)V(g)

.=1 ^Qr "" ■ 2!(^^jfr<bg.dg.
For very small values of 6, this becomes

^r ?s

+ etc. = 0.

yA 5</>(g)

^1 ^^r

^r = 0.

(4)

MATHEMATICAL NOTE

Suppose we multiply (3) by d</)(g)/dgi, (4) by df(q)/dqi and sub-
tract (4) from (3) we obtain

r = 2

dgi dgr ^qi ciqr

^r = 0.

(5)

Now we can certainly choose the n — 1 quantities ^2, ?3, . . • • ^n
in an arbitrary fashion, since on choosing a set we can adjust
the value of ^1 to satisfy (4). It follows that in order to satisfy
(5) for any values of ^2, ^3, . . ■ ■ ^n the following relations must
be true : —

bf(q) /d4>(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

<t>(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

<f>{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

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

a„

ttrn

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

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.

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

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' =

} (12)

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

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

\dtJ

r]v

to
rio

tp
r\Y

tr
Vh

PT
VH

\dtj

—

* 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)

^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.

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]

/ d^e Y _ _ (dp) (dt\ _ /dpV

\dvdr]' \dv/„ \dri/^ xd-q/^

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

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,

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.

WILSON

AET. C

At the critical point Vi = V2, log P = 0. We may take P ^ 1.
Furthermore

(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

bRt/a

b^p/a

1.0

2.0

.296

.0370

2.00

2.11

1.90

.296

.0370

2.00

2.24

1.79

.296

.0368

2.00

2.40

1.68

.295

.0365

2.01

2.60

1.56

.294

.0360

2.02

2.86

1.43

.292

.0351

2.03

3.23

1.29

.290

.0338

2.06

3.79

1.14

.285

.0316

2.10

4.77

.95

.277

.0279

2.17

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 = - (^)^ = R log (v-h)- $' (t), (9)

e = _ ^ + $(^) -t^'{t), (10)

^•^ (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)

7] = R\og(v - h) -{- c log t, (14)

e = - - + d, (15)

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)

^=-- + ^(^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)

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\

\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

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.

{dP/dT)y

2/3

27/32 =

.84

3/4

25/27 =

.93

16/27 =

.59

32/5

5/6

243/250 =

.97

108/125 =

.86

16/3

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

25/32 =

.78

1/2 =

.50

8/5

16/27 =

.59

7/27 =

.26

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

--^ + 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,

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

— 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

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.

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

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 = <t>ii', i")i

where ^{t' , t") is a function of t' and t" alone.
Kelvin defined absolute temperature so that

t' — t"
<l>{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 - «'

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

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

(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

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 <S is one
of the quantities representing the state of the system, it is shown
that all such differential coefficients as

d^S

d'S

dt'

etc..

are zero in the state for which (5r?)« ^ 0.

It is evident that the system cannot be in equilibrium unless
all these quantities have the value 0, for if dS/dt is zero in the
initial state and one of the higher coefficients has a finite value,
dS/dt will have a finite value at a subsequent time. The proof
that they are zero in the state for which (Stj)^ ^ 0 may be stated
in greater detail as follows. If any of the differential coefficients
have finite values, the system must undergo a change, which,
however, may be infinitely slow so long as (677) « ^ 0. But, by an
infinitesimal modification in the circumstances, it will be pos-
sible to produce a state for which (8T])t < 0. Such changes
will then be impossible. That is, an infinitely small modifica-
tion of the circumstances will cause a finite change in the
values of those differential coefficients which previously had
finite values. But this is regarded as impossible. The sys-
tem can therefore continue unchanged in the state for which
(8r))t ^ 0, which must be regarded as a state of equihbrium,
but since there are changes for which (Atj)^ > 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 — <i),
and — </> 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 = <f)(xi, X2,...Xn) be a homogeneous function of xi,
X2,. . .a:„ of the w"* degree;

dy dy dy

Xi — -i- X2— ... + x„ 7- = my. (52)

0X1 00:2 OXn

Now a homogeneous function of the w"" degree is one for which

<j){kxi, kx2, . . . kx„) = k'"<t>{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
€ = <f>(j], V, mi,. . .w„) we have

de dt de de

i = VT-i-v— +mi- — ... +m„ t — >
drj dv dmi dmn

t = r]t — vp -\- mi/xi . . . + mnUn,

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^

dmi • drrin

= 0.

(54)

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),

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]

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.

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<Ba + /3®6 + etc. = /c®,. + X©i + etc., (82) [30]

where <Sa, @6, ©a, ®z, etc., denote the units of mass of the sub-
stances Sa, Sb, Sk, Si, etc., and a, jS, k, X, etc., the numbers of
these units which enter into the relation. Then, as before,
(73) will reduce to

M„26ma + Mb^bMb + etc + Mk'Ednik

+ MiZSmi + etc. ^ 0. (83) [31]

It is evidently possible to give 25Wa, S5m6, ^8mk, ^dnii, etc.,
values proportional to a, 13, —k, —X, etc., and also to the same

this reduces to

MiS5mi + M22dm2 + MsSSjms ^ 0. (78)

The limitation of values of 5m to zero, whenever they refer to parts of
which the component in question is not an actual component, does not
aflfect the range of possible values of SStoi, SSmj and S5wj and may be
disregarded.

100 BUTLER ART. D

values taken negatively; therefore

aMa + ^Mb + etc - KMk — \Mi - etc =0,

or,

aMa + ^Mb + etc = KMk + \Mi -\- etc (84) [33]

The relation between the quantities Ma, Mb, etc., is thus of
the same form as that between the units of the component
substances (82). These relations take a very simple form if we
employ as the unit quantity of each substance, its formula-
weight in grams. Thus if we take as unit quantities of water,
sodium chloride and the hydrate, NaCl-H20 the quantities in
grams represented by the symbols H2O, NaCl and NaCl • H2O,
the relation between these substances is represented qualita-
tively and quantitatively by the equation.

H2O ^- NaCl = NaCl-HaO.

With this choice of units, (84) becomes

■^HjO + -^NaCl = -^NaClHzO-

Therefore the values of mhjOj MNaCi ^^^ A'NaCiH20 for these sub-
stances, in parts of the system of which they are present as
actual components, are related by the equation

MH2O + MNaCl = MNaClHjO'

Similarly, if the substances hydrogen chloride, oxygen, water
and chlorine are components of a system when the unit of quan-
tity of each substance is the quantity (in grams) represented by
its chemical formula (82) becomes

2HC1 + ^02 = H2O -f CI2,

and equation (84) takes the form

2Mhci + hMo, = Mu,o + ^ch-

Thus the values of /i in parts of the system of which these sub-
stances are present as actual components, are related by the

THERMODYNAMIC AL SYSTEM OF GIBBS 101

equation

2mhC1 + 2MO2 — MH20 + MCI2 >

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 <Sa, ®6, ©*, ®z represent one gram of hydrogen chloride,
oxygen, water and chlorine, respectively; and (84) would take
the form

73 Ma + 16 Mb = 18 Mk + 71 Mr,

or,

73 Ha + 16 jUb = 18 Hk + 71 Hi,

where the value of /x for each substance is that in a part of the
system in which it is present as an actual component.

Again, the four substances magnesium chloride, potassium
sulphate, magnesium sulphate, potassium chloride, may be
regarded as components of a solution made by dissolving mag-
nesium chloride and potassium sulphate in water, since the last
two may be formed out of the first two according to the equation

MgCl2 + K2SO4 = MgS04 + 2KC1.

K the units of quantity of the four substances are the quantities
represented by the symbols MgCl2, K2SO4, MgS04 and KCl,
(84) takes the form

-^MgCla + -^K2S04 = -^MgSO* + 2 M^Ch

so that the potentials in the solution are related by the equation

/^MgCh + MK2SO4 — MMgSO* + 2 mkci-

Gibbs shows that if there are r independent relations similar

102 BUTLER

ART. D

to (82) between the components, >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)

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

(I)

(IL)

i
r
1
1

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

= 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,

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" -

-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).

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)

rrix
0

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

■n"

nn"

n'"

mi'"

m<l"

dt,

eo that

dp
dt

7] mi mi — t] mi m^ — t? mi m^
v'" mi' m" - V mi'" ma" - v" mi' m^'"'

(102)

If the system consists of a quantity a of lime and h of carbon
dioxide, together with a quantity (a + h) of calcium carbon-

THERMODYNAMIC AL SYSTEM OF GIBBS 111

ate, we shall have

m/ = mi" = h, and 1712" = W2'" = a,
and (102) reduces to

dp ^ v'" - V - v" ^ _Q_

dt v'" -v' - v" t.Av ^ ^

where Q is the heat absorbed when a quantity a + 6 of calcium
carbonate is dissociated into lime and carbon dioxide at the
same temperature and pressure, and Av is the increase of volume
in the same change. For rj'" — r\ — r\" is the difference be-
tween the entropy of a quantity (a + 6) of calcium carbon-
ate, and that of the quantities a of lime and 6 of carbon di-
oxide. Q = tij]" — v' — v") is thus the heat absorbed in
the dissociation of the calcium carbonate.

When the number of potentials considered in various parts
of the system is n + h, there will be h independent relations
between them, by means of which the variations of h of the
potentials may be eliminated from the equations of the form of
(100) in which they occur. We may thus obtain n + 1 equa-
tions between the n potentials of the independently variable
components of the system as a whole.

IS. Cases in Which the Number of Degrees of Freedom is
Greater Than One* (a) Systems of Two or More Components
in Two Phases. We will consider first the case of two inde-
pendent components in two phases. We shall have two equa-
tions similar to (100), one for each phase:

y' dp = T]' dt -\- mi dni + mz dm,

v"dp = v"dt + mi"dni + m2"dfjL2. (104)

Eliminating d/x2 from these equations, we obtain:

(vW - v'W)dp = Wm^" - v"m2')dt

+ (ini'nh" - mi"rrh')dni, (105)

* Gibbs, I, 99-100.

112 BUTLER ART. D

i.e., the system can undergo two independent variations, in
accordance with (96). Now if the proportions of the two com-
ponents are the same in the two phases, i.e., if

m.\ mi"
nh' nii"

the coefficient of dju in (105) is zero, so that

{v'm" - v"mi')dv = Wni2" - ■n"m')dt, (106)

i.e., the same relation between dp and dt holds, as for a single
component. For example, in the equilibrium between ammo-
nium chloride and its vapor, the latter may contain ammonia
and hydrogen chloride, formed by dissociation. These two
substances may be regarded as the independently variable
components of the system, but if no excess of either of them is
added the ratios of their amounts are the same in both phases.
Then (106) holds, so that the system behaves as if it had a
single component.

When there are n independent components in the two phases,
then in the absence of any restriction on their proportions the
number of degrees of freedom is ^^ = n -f 2 — 2 = n. But
when the quantities of all components are proportional in the
two phases, the equality of the n — 1 ratios of m/, rth', . . . m„'
with the n — 1 ratios of m/', mz', . . . mn" gives n — 1 additional
conditions, so that the number of degrees of freedom is reduced
to one and there is a relation similar to (106) between the
variations of temperature and pressure.

Again, in a system of two components in two phases, at
constant temperature, (105) becomes

dp mi' m^" — mi" W

T~ = ~' T, — -77 r • (107)

dm V m2 — V m^

If the proportions of the two components are the same in the

two phases, the numerator of the fraction on the right is zero, so

that

THERMODYNAMICAL SYSTEM OF GIBBS

113

Thus, at constant temperature, the pressure is in general a
maximum or a minimum when the composition (i.e., the pro-
portions of the two components) of the two phases is identical.
Similarly, it can be shown that, at constant pressure, the tem-
perature of the two coexistent phases is in general a maximum
or a minimum when the composition of the two phases is
identical.

Applying these relations to the equilibrium between a binary
liquid and its vapor, we see that (1) at constant temperature
the vapor pressure is a maximum or a minimum when the
vapor has the same composition as the liquid, and (2) the

% Benzene
Fig. 2

100

temperature at which the two phases are in equilibrium at
constant pressure, i.e., the boiling point, is a maximum or a
minimum when the composition of the two phases is identical.
These rules were arrived at independently by Konowalow.*
As an example of this behavior. Figure 2 shows the boiling
points and compositions of the liquid and vapor phases of
ethyl alcohol and benzene.

Similarly, the temperature at which a binary liquid is in
equilibrium with a solid phase, which may be a solid solution of
the same components (mixed crystals) or a compound of

* Wied. Annalen, 14, 48, (1881).

114

BUTLER

ART. D

invariable composition, is a maximum or a miaimum, for
constant pressure, when the two phases have the same com-
position. Examples of this behavior are shown in Figures

% lodobenzene
Fig. 3

100

V 10 9 Q 7 6 5 4 3 2
Composition of solution ( Hz o/ FeCli)
Fig. 4

3 and 4. Figure 3 shows the compositions of the solutions and
mixed crystals of bromobenzene and iodobenzene which are in
equilibrium with each other (at constant pressure) at different
temperatures. The composition of the two phases is the same

THERMODYNAMIC AL SYSTEM OF GIBBS

115

when the temperature is a minimum. Figure 4 shows the
conditions under which aqueous solutions of ferric chloride are
in equilibrium with various solid compounds of the same
components. A maximum in the temperature-composition
curve occurs when the liquid phase has the same composition
as the solid compound with which it is in equilibrium.*

(6) Systems of Three Components in Three Coexistent Phases.
In this case, we have three equations similar to (100):

v' dp = 7]' dt + mi djxi + tn^' dm + mz dm,
v" dp = If]" dt + mi" dm + m2" dm + mz' dm,
v'"dp = r)"'dt + mi' "dm + m2"'dm + mz"'dm,,

from which, by eliminating dm and dm, we obtain:

(108)

v' mi m^

v" mi" m^"

dp =

v"' mi"' mo'"

7] mi m2
v" mi" m/'
■n'" mi'" m^"

Ml m^ mz
m" irvi' mz"
mi'" W" mz'"

dm-

(109) [132]

When the composition of one of the three phases is such as can
be produced by combining the other two, i.e., if we may take
quantities of the three phases such that

m/ = mi" + mi",

m^ = m" + mi" ,
I

mz = mz + m"';
the last of these determinants is zero, so that when di = 0,

and when dp = 0,

dp
dm

dt_ _
dm

* A more exhaustive discussion of systems of two components in two
phases is given on pages 175-177. Compare also Article H of this vol-
ume.

116 BUTLER ART. D

Since similar equations can be obtained for the other com-
ponents, the pressure will thus in general be a maximum or a
minimum at constant temperature, and the temperature a
maximum or minimum at constant pressure when the foregoing
condition is fulfilled.

For example, the three components water, alcohol, salt may
give rise to a system of the three phases, solid, salt-solution,
vapor. When the composition of the solution is such that it
can be formed by combining quantities of the salt and the
vapor, i.e., when the proportions of alcohol and water in the
vapor are the same as in the solution, the pressure is a maxi-
mum or a minimum at constant temperature. Again, in the
three component system; potassium sulphate, aluminium
sulphate, water; with the three phases, viz., solid potassium
alum, solution, vapor, the vapor pressure is a maximum or a
minimum when the solution can be formed out of the solid salt
and the vapor, i.e., when it contains the two salts in the same
proportions as in the solid phase.

VI. Values of the Potentials in Very Dilute Solutions

16. A Priori Considerations * We may draw some con-
clusions as to the values of the potentials in a homogeneous
mass, when the quantity of one of the components is very
small, from the form of (56). Applying this equation to a
homogeneous mass having two independently variable com-
ponents, we obtain, when t, p and nii are constant

m(^) +n,J^) =0. (110) [210]

When TUi = 0, this equation requires that either

(

P) = 0, (111) [211]

' d\x

(

, . =00. (112) [212]

dm^/ 1, p, m,

*Gibbs, I, 135-138.

THERMODYNAMIC AL SYSTEM OF GIBBS 117

We can distinguish between these possibilities by making use of
a proposition which we shall obtain in a later section,* viz., that
when t, p, and 1712 are constant, ni is an increasing function of
mi. We shall now consider two cases.

(a) Mi Is Capable of Negative AsW ell As Positive Values. Thus
if we regard the hydrate FeCls • 6H2O (*Si) and anhydrous ferric
chloride {S2) as the components of a solution of ferric chloride
and water, the amount of ferric chloride will be negative in
solutions containing a smaller proportion of ferric chloride than
the hydrate itself and positive in solutions containing a greater
proportion. We may add the hydrate Si to solutions for which
the amount of ferric cliloride is either negative or positive. In
both cases ^ti is increased. Therefore ^ui must be a maximum
when the mass consists wholly of Si, i.e., when Wa = 0. There-
fore, if ?ri2 is capable of negative as well as positive values.

(

3, -«.

p, t, mi

when m2 = 0.

(6) rrii Is Capable Only of Positive Values. For example, if
water {Si) and ferric chloride {S2) are regarded as the components
of the solutions, m^ cannot have negative values. The potential
of water {m) must increase when water is added to a ferric
chloride solution, and therefore decrease when ferric chloride is
added to the solution. Thus, in the limiting case when nh =
0, the value of the differential coefficient in (111) cannot be
positive.

Gibbs points out that "if we consider the physical signifi-
cance of this case, viz., that an increase of rrh denotes an
addition to the mass in question of a substance not before con-
tained in it," there does not appear "any reason .... for supposing
that this differential coefficient has generally the value zero." Sup-
pose that we have a mass of water in equilibrium with ice. The
addition of a salt to the water will destroy the possibility of this
equilibrium at the same temperature and pressure and, if the
temperature and pressure are kept constant, the liquid will

See page 167.

118 BUTLER ART. D

dissolve the ice. Similarly the addition of a salt to water
causes a decrease in the pressure of water vapor which is in
equihbrium with the hquid at the same temperature. Both
phenomena show "that m (the potential for water in the liquid
mass) is diminished by the addition of the salt, when the tem-
perature and pressure are maintained constant. Now there
seems to be no a priori reason for supposing that the ratio of
this diminution of the potential for water to the quantity of the
salt which is added vanishes with this quantity. We should
rather expect that, for small quantities of the salt, an effect of
this kind would be proportional to its cause, i.e., that the differ-
ential coefficient in [211] would have a finite negative value for
an infinitesimal value of vi2. That this is the case with respect
to numerous watery solutions of salts is distinctly indicated by
the experiments of Wtillner* on the tension of the vapor yielded
by such solutions, and of Rlidorff f on the temperature at which
ice is formed in them; and unless we have experimental evidence
that cases are numerous in which the contrary is true, it seems
not unreasonable to assume, as a general law, that when nh has
the value zero and is incapable of negative values, the differ-
ential coefficient in [211] will have a finite negative value, and
that equation [212] will therefore hold true." We may observe
that the truth of this law has been confirmed by numerous
more exact experimental investigations.

The change of mi caused by the addition of a small amount
drrh of S2 is evidently inversely proportional to the amount
(mi) of Si, so that we may write, in the limiting case, when
W2 = 0,

P) = - -' (114)

(

where A' is positive and independent of mi.
Then, by (110),

\dm-ijt. p. m,

* Pogg. Ann., 103, 529 (1858); 105, 85 (1858); 110, 564 (1860).
t Pogg. Ann., 114, 63 (1861).

THERMODYNAMIC AL SYSTEM OF GIBBS 119

i.e.,

•>

(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]

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 '

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

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

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)

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).

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)

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- • <l^' ("*)

where

W - v") t

Q =

is the heat absorbed in the melting of unit weight of the solvent
into the solution.!

* The term m^ — • dt, which vanishes when 7112' = 0, is neglected
at

here.

t van't Hoff, Z. physikal. Che?n., 1, 481, (1887).

THERMODYNAMICAL SYSTEM OF GIBBS

127

(c) Lowering of the Vapor Pressure of the Solvent by an
Tnvolatile Solute. Consider a mass of the solution in equilibrium
with the vapor, in which the quantity of the solute is in-
appreciable. At constant temperature we have, for a variation
of the solution,

dp = 0 + ^ • (^/xi' + 7- • dn^', (145)

and for a variation of the vapor

dp = 0 + -jj- ■ dm

(146)

In order to preserve equiHbrium, dm' = dm", so that, sub-
stituting the value of dm given by (146) in (145) and putting

W2' , At

and writing

7" = ^^'

= 71

etc.,

we obtain

At
dp = ~, dp + j^^ dy2 ,

71'

— dp =
Thus, by integration,

71' - 71" M2^^^

^72'.

7i'

Po-P = :;T3^-]^)-72,

(147)

(148)

where po is the value of p when 72' = 0. Since 71" is small in
comparison with 7/, we may write, approximately.

Po - p =

71' M2(^>

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.

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

~ dt

\dtni)

dr,
drrii

131

Substituting the value of tj^i given by (151) and (153) we have

= Ml - XI, (160)

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

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 +

: log pi\

= Mi" +

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.)

ai =

pi/pi"

ai/Ni

at =

cti/Ni

0.000

—

1.000

344.5

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

.000

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.

pi (mm. mercury)

ai/Ni

Cli/N2

92.35

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

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

— 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)

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

= - 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)

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

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]

But according to (44) we have

a — = t a — -\- ^il a — +M2« —

• V V V V

...+Mn^-, (212) [147]

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]

Since

dr]

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]

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]

(^' - 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]

[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

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

^ ^

Phenol %
Fig. 8

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

(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/,

r, /il, . . .Mn-2i lir

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

(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,

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,

> (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),

d?^p

dt;2

dvdmi

dvdm n-i

ART. D

Pn =

drriidv

dm-^

dmidnin-i

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

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, ... <S„,
independently, and without passive resistances, but for which
the phase is completely determined by its temperature and
pressure and its ultimate composition (in respect at least to the
particular substances just mentioned). By the absorption of
the substances Si, ... /S„ independently and without passive
resistances, it is meant that when the absorbing body is in equi-
librium with another containing these substances, it shall be
possible by infinitesimal changes in these bodies to produce the
exchange of all these substances in either direction and inde-
pendently. An exception to the preceding statement may of
course be made for cases in which the result in question is
prevented by the occurrence of some other kinds of change; in
other words, it is assumed that the two bodies can remain in
contact preserving the properties which have been mentioned.

"The term catalysis has been apphed to such action as we are
considering. When a body has the property of reducing
another, without limitation with respect to the proportion of
the two bodies, to a phase of dissipated energy, in regard to a
certain kind of molecular change, it may be called a perfect
catalytic agent with respect to the second body and the kind of
molecular change considered."

OSMOTIC AND MEMBRANE EQUILIBRIA, IN-
CLUDING ELECTROCHEMICAL SYSTEMS

[Gibbs, I, pp. 83-85; 4iS-417]

E. A. GUGGENHEIM

1. Introduction. The power and elegance of the methods of
Willard Gibbs in thermodynamics are nowhere better illustrated
than in their apphcation to membrane equilibria.* Owing to
the form in which he expressed the conditions for chemical
equilibria, the same conditions for the equilibrium between two
phases as regards a given species hold good whether the two

* A list of the most important symbols used, in addition to those used
by Gibbs, is as follows:

E Electromotive force of cell.
F Faraday.

/, Activity coefficient of species St.
/± Mean activity coefficient of electrolyte.

g Osmotic coefficient.
Ni Mol fraction of species St.
P Osmotic pressure.
q+, q- Number of cations and anions per mol of electrolyte.

r Ratio of partial molar volume at infinite dilution of electrolyte to
that of solvent, both at a pressure equal to the mean of those
at either side of membrane.
Vi Partial molar volume of species Si at given temperature, pressure

and composition.
Vi* Partial molar volume of species Si at given temperature, zero

pressure and infinite dilution.
[vi] Partial molar volume of species Si at given temperature, infinite
dilution and at a pressure equal to the mean of those at either
side of the membrane.
Zi Valency, positive or negative, of ionic species Si.
Ki Coefficient of compressibility of species Si at infinite dilution,
[/i,] Potential of ionic species Si.
The suffix 0 always refers to the solvent species, e.g., Vo* is the molar
volume of the pure solvent at zero pressure.

182 GUGGENHEIM art. e

phases be in complete equilibrium or only in partial equilibrium,
that is, in equilibrium as regards this species but not as regards
all the species present.

The general conditions that two phases, denoted respectively
by a single and by a double accent, shall be in complete equilib-
rium are the following. First, in order that the two phases
shall be in thermal equiUbrium the temperatures of the two
phases must be the same, that is,

f ^

t". (1) [19]

Second, in order that the two phases shall be in mechanical or
hydrostatic equilibrium the pressures of the two phases must
be equal, or

P' = P". (2) [20]

Third, in order that the two phases shall be in chemical equi-
librium as regards the various chemical species Si, S2, . . . Sn the
potential of each species must be the same in the two phases, or

Ml = Ml ,

/ n

M2 = M2 ,

/ //

Mn = Mn .

(3) [21]

The essential feature of Gibbs' treatment of equiUbrium is that,
thanks to his invention of the potentials of the chemical species,
the conditions (3) [21] for chemical equilibrium are of a form
analogous to the condition (1) [19] for thermal equilibrium
and to the condition (2) [20] for hydrostatic equiHbrium.

The importance and usefulness of Gibbs' method for the
treatment of membrane equihbria depend on the fact that, pro-
vided two phases are in thermal equilibrium, i.e., (1) [19] is
satisfied, the other equilibrium conditions, namely, (2) [20] for
hydrostatic equilibrium and the several equations of (3) [21]
for chemical equilibrium, are all independent of one another.
In other words, if two phases, denoted respectively by a single
and by a double accent, be separated by a membrane capable

OSMOTIC AND MEMBRANE EQUILIBRIA 183

of supporting an excess of pressure on either side and permeable
to some of the components Sh, Si, . . ., but impermeable to others
Sa, Sb, . . • , the conditions for equihbrium between the two
phases as regards the components Sk, *S., . . •,

w' = w",l

are of exactly the same form as (3) [21].

But the potentials of the components Sa, Sb, . . . , to which
the membrane is impermeable, will in general not be equal,
that is,

Ha 7^ Ma",l

,.'^,."} (5) [77]

Moreover, in general the pressures of the two phases will not be
equal, that is,

p' ^ V"- (6) [77]

The pressure on each phase will be equal and opposite to the
pressure exerted by the phase on the membrane, and so the
resultant force per unit area on the membrane wiU be equal to
the difference between the pressures of the two phases.

2. Proof of General Conditions of Membrane Equilibrium.
The derivation of the general conditions (4) [77] of membrane
equilibrium is given by Gibbs (I, 83). In this proof the
quantities chosen as independent variables are the entropy tj
of each phase, the volume v of each phase, and the quantities
Wi, W2, ... w„ of the various chemical species Si, Sz, ... Sn
in each phase. The corresponding characteristic function is
the energy c. The appropriate form for the general criterion
of the equilibrium is that expressed by [2] (Gibbs, I, 56) .

In accordance with the footnote (Gibbs, I, 90) a somewhat
more familiar derivation of (4) [77] can be obtained by choosing
as independent variable the temperature t instead of the entropy

184

GUGGENHEIM

ART. E

Tj of each phase and by taking for granted the condition for
thermal equiUbrium (1) [19].

With this choice of independent variables the characteristic
function is yp defined by

\p = e — tt].

(7) [87]

Its dependence on the independent variables t, v, mi, nh, . . . m„
is given by

d\p = —r]dt — pdv + fiidmi + tiidm^ . . . + Undrrin. (8) [88]

The condition of membrane equilibrium takes the form

subject to

dyp' + dxp" = 0,

(9.1)

[111]

dt' = dt" = 0,

(9.2)

dv' = dv" = 0,

(9.3)

[73]

drria' = dm" = 0,

drrib = dm" = 0,

(9.4)

[74]

dmh + dmh" = 0,
dmi + dm/' = 0,

(9.5) [75]

Substituting from (8) into (9.1), and using (9.2), (9.3), (9.4) and
(9.5), we obtain

(hh' - Hh")dmH' + (m/ - nHdrrii' + ... - 0. (10)
If mh, mi, ... are independently variable it follows that

lih = fJ'h ,

I //

/it - V-i ,

(11) [77]

OSMOTIC AND MEMBRANE EQUILIBRIA 185

The same form, (11) [77], for the conditions of membrane equi-
Hbrium is thus obtained whether entropy or temperature be
chosen as one of the independent variables. In fact, whatever
choice one makes of independent variables an analogous treat-
ment will lead to the same result, (11) [77].

3. Choice of Independent Comyonents. If the various quantities
mh, rrii, ... are not independently variable but are subject to cer-
tain restrictions expressible in the form of linear relations between
dnih, dnii, . . ., then (10) holds not for any values of dnih, dtUi,
. . . but only for such sets of values of dnih, dnii, ... as conform
with the linear restrictions. Instead of the conditions (11) [77]
one then obtains a smaller number of independent conditions
of the type [78] (Gibbs, I, 83) . The physical meaning of this is
quite simple. The condition for membrane equilibrium is equality
of the potential for those components to which the membrane is
permeable, provided the species chosen as independent compo-
nents include all those which are able to pass the membrane inde-
pendently. An example will make this clear. Suppose the mem-
brane is permeable to methyl alcohol CH4O but not to water H2O.
Then the corresponding condition of membrane equilibrium is

MCH4O = MCH.O- (12)

But from a purely thermodjoiamic standpoint it would be
allowable to choose as independent components methylene
CH2 and water H2O, since these will serve just as well as methyl
alcohol CH4O and water H2O to define the composition of each
phase. With this choice of components both methylene and
water are able to pass through the membrane, not independ-
ently but only in the proportions in which they form methyl
alcohol. Formula (10) in this case is

(mch, - MCH,) dm'cR, + (mhjO - MH20) drn'^^o = 0. (13)
But diucKj and dm^^o ^^^ subject to the restriction

p q '

186 GUGGENHEIM art. e

where p/q is the ratio in which methylene and water combine
to form methyl alcohol. Substituting (14) into C13) we obtain

PMcH, + ^Mh,o = P^CH. + e^ao. (15)

But according to [121] and the definition of the ratio it follows
that (15) is equivalent to

MCH4O = MCH«0, (16)

the same as (12). We see then that the complications discussed
by Gibbs in the paragraph preceding [78] can be avoided if we
always include among the independent components all those
species which can pass through the membrane independently.

4. Choice of Independent Variables. Although the conditions
for any membrane equilibria are completely contained in
Gibbs' formula [77] it is advantageous from a practical point of
view to transform this into a form involving quantities more
directly measurable than the potential n. For this purpose it is
most convenient to choose as independent variables the tem-
perature t, the pressure p and the number mi, nh, . . . nin oi
units of quantity of the various species *Si, S2, ... Sn. The
potentials m, m, ... /in in each phase will then be regarded as
functions of t, p, Wi, nh, • . . Mn.

The manner of dependence of the potentials mij M2, • • • Mn
on the temperature t need concern us very httle as we shall
always deal with systems maintained at a given constant tem-
perature throughout and shall not need to consider tempera-
ture variations. The manner of dependence of the potentials
Mi> ^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

= 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

Nn - ,

(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)

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

p' vh*{1 - hhP' ) + At log N,/

= p"vh*{l - hhP") + At log Nh", (30)

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)

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)

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

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)

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*(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)

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)

Wo 4- 7 , ms

V = moVo -\- 2j ^« ^» = ^0 1'o*. (67)

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)

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].

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)

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 <Si is of the form

[Mi] = Mi*(0 + V^ni - \Kiv) + At log Ni + ZiFV, (75)

where )U»*(0 is for a given solvent a function of the temperature
only, Vi* and Vi*{\ — Kip) are the partial molar volumes of the
ionic species Si at zero pressure and at the pressure p respectively,
Ni is the mol fraction of the species Si, and Zi its valency.
Finally V depends on the "electrical state" of the system, that
is, on the distribution of electric charges at the surface of the
phase, and has the same value for all ionic species. Solutions of
ions behaving in accordance with (75) are called "ideal." In
analogy with ideal solutions of uncharged species it is natural
to define the chemical potential m of the ionic species Si by

/i.- = Mi*(0 + PVi*(l - hiP) + At log Ni, (76)

and to call V the electric potential of the phase.

18. Non-ideal Solutions of Ions. Since all ionic solutions
tend towards ideahty at infinite dilution, it is most convenient to
treat non-ideal solutions by the introduction of activity coeffi-
cients fi just as in the case of non-ideal solutions of uncharged
species. We therefore write formally

[m] = fjii*it) + pvi*(l - ^Kip) + At log Ni

-hAthgfi + ZiFV, (77)

where Mi*(0 is for a given solvent a function of the temperature
only; y,* and Vi*{l — Kip) are the values of the partial molar

202 GUGGENHEIM art. b

volume of the species Si at infinite dilution at the given tem-
perature, and at zero pressure and at the given pressure p respec-
tively; A'",- is the mol fraction of the species Si] Zi its valency; and
fi its activity coefficient which, at given temperature and pres-
sure, tends to unity at infinite dilution. Finally, V has the
same value for all ionic species in the given phase.

Formula (77) will always lead to correct physical results, but
it is partly ambiguous because there is no experimental method
of distinguishing between the last two terms,

At\ogfi + ZiFV. (78)

Thus the activity coefficient of a single ionic species is physically
indeterminate, as in each phase an arbitrary value may be
assigned to V and the value of /» will vary in such a way that
the sum (78) remains invariant. If, however, we consider
combinations of ions with zero net electric charge, the cor-
responding combinations of electrochemical potentials will be
given by

i i i

-\-At^\i\ogNi-^At^\i\ogU (79)

> 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)

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" .^^.

{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

Cu.

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 :

Solution Si containing
Cw++ and large excess
of other ions

Solution ^2 containing
Zn+"'' and large excess
of other ions

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),

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

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,

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 <vL-p + \ V d'p.

J po

Now choose A'p = —Ap, thus restoring the initial pressure.
Then

Ar + A'r < .oAp - lljn + (|)/p - .0) + ...] d'p

Now I 7- ) is negative. Hence
\dp/o

[(A + A')r]^p <0.

4. The Heat Function. We shall now prove that the varia-
tional conditions

(Ax),.p^O (8)

and

(Ae),.„^0 (9)

are equivalent. These criteria are not stated by Gibbs, but
clearly there must be a parallel set of criteria involving the
heat function.

To prove that (8) implies (9) let us suppose there is a
neighbouring state for which

(A€)„„ < 0.

We shall prove that this implies the existence of a neighbouring
state violating (8). Hence if we know that (8) holds, (9) must
also hold.

If this neighboring state is not one of uniform pressure, let
the pressure equalize itself. This can only increase the entropy,
and thus we have a state of the same energy and volume, and
greater entropy. Now remove heat at constant volume until

X, xp, r, ^ND THE CRITERIA OF EQUILIBRIUM 221

the original value of the entropy is restored. The energy can
only decrease in the process. Hence we arrive at a new state of
uniform pressure for which

(Ae),.„ < 0.
Now

and hence in general

Ax = Ae + pAv + vAp + ApAv.

But in our case Av = 0. Hence here

Ax = Ae + vAp.
Consequently

Ax — vAp = Ae < 0.

Now expand or compress adiahatically . For any such process,
the infinitesimal change of energy d'e is given by

d't = —p d'v
and hence for this process

d'x = d'{(: -]r pv) = V d'Pf

whence for the finite change A'

rpo + A'p
A'x = I V d'p.

J PO

Hence

/*P0

Ax + A'x < vAp — j V d'p.

J pa + A'p

ART. F

222 MILNE

Choose the second process such that A'p = — Ap, thus restor-
ing the initial pressure. Then

Ax + A'x < vAp — V d'p

J pa — Ap

But

Hence

< 0.

[(A + A') xl,. V < 0.

This contradicts (8), and so the imposition of (8) must imply
the truth of (9). The proof of the converse may be left to the
reader.

As an example of the application of this criterion we shall
prove that Cp, the specific heat at constant pressure, must be
positive. Divide a homogeneous specimen of the body into two
equal parts, at the same pressure, and take a varied state of
the same total entropy in which one part has been heated at
constant pressure and the other cooled. Then by the properties
of the heat function x already established, we must have, if x
refers to unit mass,

X(77 + Ar?, p) + x{-n - At/, p) > 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

1='-

Hence

Q = 1

1 /dt\

~ Cp

It follows that

> 0.

A similar

argument involving

; the energy

e establishes

that

> 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,

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

^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

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 <i." (This is a
generalisation of the similar result in the particular case ti = t^
estabhshed above.)
We have

^1 — '/'2 = Cl — €2 — (flt/i — ^2^72)

ci - €2 = Ae = AQ + ATF,

where now AQ denotes the heat given up at ^2- Since the
process is reversible and the heat is given up at tz

m — m = AQ/ti.
Hence

AW = A\p -\- (tiTji - tiVi) — t2(vi - V2)
= Alp + r]i{ti — ti).

This result is, of course, somewhat trivial. We may, however,
extend it to include irreversible processes. The following
theorem may be established.

"If the system is brought by any process, reversible or
irreversible, from the state at ti to the state at t^, and not neces-
sarily subject to the condition of only giving up heat when at

230 MILNE ART. F

temperature ^, then the external work performed, A'VF, satisfies
the inequahty

^'W ^AW - {t - h) dv,

the integral being taken along the path in the (rj, t) diagram
actually traversed by the system and AT^'" having the same
meaning as above."

For, along any path whatever, if the differentials which
follow denote positive increments,

d'Q = d'e -\-d'W
and

d'Q denoting the heat given up at t. Hence

d'W ^ - d'€-\-t d'v.

Since d'e and d'r] are the actual increments in the functions e and
7j along the path, we may replace them by de and dt^. Now

Hence

Integrating,

d\J/ = de — tdt] — rjdt,
d'W ^ -# - 7}dL

A'W ^ Arp - vdt

^ A\}/ -\- (tim — ti-qi) — I tdt}
^ AW + t2 (rji - m) - / 'tdr,

on using the result of the first part. Hence
A'W ^AW- \t - k) dr,.

X, ^, i', AND THE CRITERIA OF EQUILIBRIUM 231

Since AQ > 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.

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

^/^\ 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

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-

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.

/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.

f J

/4/n

' ^

/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)

- ^^ + 0.5066 - x^ ^"

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

«r

/^\ 1

I ;<4

< s

'^m 1 1

«:

\ / *

\ /

\. /

>t^ /

^s,„^ /

■ >^'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

01D2O304

—

S1S2S3

—

O102OJ

—

S1S2

L1L2

—

S1S2

L1L2

Types of Univariant Systems; Three Coexisting Phases, and the
Invariant Types from Which They May Be Derived

Derived from

blb203

—

1,2,3

S1S2

—

2,4,5

S1S2

—

3,5

L1L2

—

4,6

5,6

—

L1L2

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<?

,i,^2l-.

30 £0 70 90
COMPOSir/ON

%°

'^C

c '°

^^.—-'^'^

/''

-^ '

^*o

\*^

^*^

-^flO

^^^■v^^^

^^■~~^^

1 1 1

1 ■

K^ 1

t L

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

x" y"

■n'" 1 x'" y'"

ly ^ jy r,jy

T) i. X y

v' 1
v" 1
'"I

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"'

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"'

The equation becomes

dy
dt

AiSiV ~ A 134 17" + Ai2iV"' — A 123 V

A23iV' — Amv" -\- AmV

AmV

HETEROGENEOUS EQUILIBRIUM
It is easy to show that

1 x' y'

269

x" y"

x'" y'"

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

(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

2 2

1
i

\
\ioo

i'
/i

I
j
i
i

\
\

1
1

- 1
1

\
\

*0
30

- ;W

- /' ;'

1 1

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

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"'

mi"

ms"

mz"

jj///

m/"

mz'"

A'V -A"V"

_ A"'V"'

mz"

yiit
ylV

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"')

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

^ 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 :

H" V"
H'" V'"

jjIV ylV

mi
mi

mi
m<i
mi
mi

mz
mz
mz
mz

III

Jjn+l yn+1 fyi^n+l '^^n+l ^g^+l
JJn+2 yn+2 ^,"+2 f}i^^+^ 7^3"+^

mr,
mn

mn
m'7

m
m

n+l

n+2

(A)

m\
m\
mi

m2
m2
m-i

III

mz
mz
mi

Jjn+l ^n+l ^^n+1 ^^n+l

H"+2 ,/j^n+2 rn2"+^ m3"+2

mn
mn
mn

m
mn

n+l

n+2

The equilibrium, P' + P"' + P^^ . . . + P"+i + P^+\
will be stable if this expression is negative, and vice versa.
Also the univariant equilibrium, P" + P'" . . . + P"+i + P"+^
in which P' is the missing phase, is stable when the expression

HETEROGENEOUS EQUILIBRIUM

285

H' V

H" V"
H'" V'"

jlV

Y^y V

mi
mi

?W2

III

W3'
ma

Jjn+l yn+l ^^n+1 ^^"+1 m3"+l
JJn+2 7n+2 ^^71+2 ^^"+2 m3"+2

mn
mn

III

mn
mn

n+1
n+2

///

m2
m2
m2

///

m3'

Jjn+l r/ijn+l m2"+^ m3"+^

JJn+2 ^jn+2 ^^"+2 m3"+2

m„

III

mn
m,

n+l
n+2

(B)

is positive.

The numerators of the two expressions given above are
identical. When a Hnear relation exists between the phases,

pill p

pn+i^ pn+2^ ^YiQ denominators reduce to

mi
mi'

m2
m2

///

mi"+^ m2"+^ m3"+^
mi"+2 m2"+2 m3"+2

and

mi
mi

III

mi
mi

mz
mz

mi"+^ m2"+^ m3"+^
mi"+2 ^2"+^ W3"+2

m^
m.

m„

n+l
n+2

(C)

m,
m„

///

m,,
mn

n+l
n+2

(D)

in which AH denotes the entropy change which takes place
when these n phases, P'", P'^, . . . P"+i, P"+2, react.

286 MOREY

ART. G

It will be observed that these two expressions are identical
except for the first row of the determinants, which in (C) con-
tains the composition terms of phase P' , and in (D) contains
the composition terms of phase P". Hence it is evident that
the numerical values of expressions (A) and (B) will be the
same, i.e., the two curves will be stable in the same direction
from the invariant point, when (C) and (D) have opposite
signs (since (A) and (B) have opposite signs). But (C) and (D)
will have opposite signs only when phases P' and P" lie on
opposite sides of the onefold P'", P'^, . . . P"+\ P"+\ In a
two-component system this onefold is a point; in a three-
component system, a line; in a four-component system, a plane,
etc.

The above may be summarized as follows: When two adjoin-
ing p-t curves (which represent the relation between the
variations in pressure and temperature in two different uni-
variant equilibria between 7i -\- 1 phases in a system of n com-
ponents) coincide, owing to a linear relation being possible
between the compositions of the n phases common to both
equilibria, i.e., to these n phases lying on the onefold n, whose
position is determined by the above Hnear relation, these
equilibria are stable in the same direction from the invariant
point, i.e., their stable portions coincide, when the other two
phases lie on opposite sides of the onefold n. By "the other
two phases" is meant the phases, one in each of the univariant
equilibria, which do not lie on the onefold n. In a two-com-
ponent system, the onefold n is a point; in a three-component
system, a line; in a four-component system, a plane, etc. This
has been proved for the case that a linear relation exists
between the compositions of n of the (n + 2) phases that
coexist at the invariant point. The cases where a linear relation
exists between the composition of (n — 1), {n — 2), ... {n — a),
phases may be regarded as special cases.

3S. Generalizations Concerning p-t Curves. Before illus-
trating the application of the above principles to actual cases,
certain generalizations will be made concerning the p-t curves
from the state of aggregation of the phases. The actual
value of dp/dt for any univariant equilibrium is given by

HETEROGENEOUS EQUILIBRIUM 287

equation (6) [129], which, as will be shown later, is equivalent to

dp _ AH
dt ~ AF '

in which AH is the entropy change, AF the volume change of the
reaction in question. Whenever we have a reaction in which
the vapor does not take part, e.g., the reaction

S' + S'" = S" + L,

the slope of the p-f curve is always very great, because of the
small value of AF. In other words, the p-t curves of all con-
densed systems are almost vertical, and go from the invariant
point to regions of higher pressure, and in almost all cases,
higher temperature. W^ien the reaction is one between solid
phases and vapor, e.g., the dissociation-pressure curve of S',

S' = S" + V,

the 'p-t curve always goes from the invariant point to regions of
lower temperature and pressure; since AF is large (except under
high pressure), the slope of the curve is comparatively small.
In reactions of the type

S' + L = S" + V,

in which both liquid and vapor take part, d-p/dt may be large
or small, positive or negative. We will consider this case in
detail later.

Consider now the application of the above principles to the
determination of the sequence of -p-t curves around an invariant
point.* The method used is based on the fact that the order
of the slopes dp/dt of the various curves at the invariant point
is determined by the masses of the phases which take part in

* The question of the sequence of p-t curves around an invariant
point has been discussed by A. Smits (Proc. Acad. Sci. Amsterdayn,
18, 800-804 (1916)), and by F. A. H. Schreinemakers in the series of
papers beginning with Proc. Acad. Sci. Amsterdam, 18, 116-26 (1916),
and by G. W. Morey and E. D. Williamson, /. Am. Chem. Soc, 40, 59
(1918).

288 MOREY ART. G

the various univariant reactions. This is evident from equation
[129] or from the expanded form of (6) [129] given below. The
method of applying this criterion is by considering what
curves will coincide when we vary the composition of different
phases. If by varying the composition of one phase in a
certain direction n phases get on the onefold (n), then, as proved
above, the p-t curves of the two univariant equilibria formed
by these n phases with each of the other two phases will coin-
cide; these two curves must be adjoining curves, and no other
curves can be between them. By repeating this reasoning,
assuming the composition of the same phase to change in other
directions, or assuming the composition of another phase to
change, the relative positions of the p-t curves, i.e., the order
in which they succeed one another around the invariant point,
can be deduced. The stable and metastable portions can be
distinguished by means of the theorems previously given.

33. Order of the p-t Curves in the Ternary System, H2O-
K2O • SiO^-SiOi. Let us apply the above considerations to the
quintuple points in the ternary system, H2O-K2O • Si02-Si02,
and pay particular attention to the question of the sequence of
the p-t curves around the invariant (quintuple) point. For the
purpose of this discussion, we will combine the above theorems
in regard to the conditions under which p-t curves coincide, and
in regard to the factors which determine whether the curves
coincide stable to stable or stable to metastable, in the following
rule: Whenever in a ternary system three phases lie on a
straight line, the p-t curves of all the ternary univariant
equilibria containing these three phases coincide with each other
and with the p-t curve of the univariant binary equilibrium
between the three phases alone. When the other two phases at
the quintuple point lie on the same side of the line on which
lie the compositions of the three reacting phases, the curves
coincide stable to metastable ; when the other two phases lie on
opposite sides of the line on which lie the compositions of the
three reacting phases, the curves coincide stable to stable.

The compositions of all the phases which are met with in
the ternary system, H2O-K2O • Si02-Si02, are shown in Fig. 6.
In treating this system we will assume that the vapor phase

HETEROGENEOUS EQUILIBRIUM 289

contains H2O only. The presence of K2O in the vapor has been
detected,* but the amount was very small. The boundary
curves show the change in composition of the liquid phase in
the univariant equilibria, S' + S" + L + V, as we pass from
quintuple point to quintuple point, or from quintuple point to
quadruple point, in the limiting binary systems. The com-
position of the liquid phase at each quintuple point is given by
the point of intersection of three boundary curves; these points
are designated by the letters Qi, Q2, etc. The p-t curves
experimentally determined are shown in Fig. 7. Figure 8 is a
diagrammatic representation of the jp-t curves, which is
easier to follow.

At quintuple point Qi we have the three solid phases, K2O • Si02,
K2O • Si02 • ^H20, and K2O -28102, the liquid phase whose com-
position is given by the point Qi, and the vapor phase, whose com-
position is given by the H2O apex of the component triangle (Figs,
5, 6) . Since K2O • SiOa, K2O • SiOa • IH2O and V lie on a straight
line, the curves, K2O • 2Si02 + K2O • SiOs + K2O • Si02 • ^H20 + V
(curve 10a) t and K20-Si02 + KaO-SiOz- h^20 + L + V (curve
106), will coincide; metastably, since the phases, K2O -28102 and
L, lie on the same side of the Hne, V-K20-8i02. These two
curves also coincide with the dissociation-pressure curve of
K2O • 8i02 • IH2O in the binary system, H20-K20-8i02 (curve
10a + 106), Fig. 7, hence their position is as shown.

If the compositions of the phases were such that V, L, and
K2O - 28102 lay on a straight line, the y-t curves of the univariant
equilibrium, KaO-SiOa + KgO- 28102 + L + V (curve 6d), would
coincide, metastably, with the y-t curve, K20-8i02-^H20 +
K2O- 28102 + L + V (curve 6c); if the phases V, L, KgO-SiOg
lay on a straight line, the curve, K2O • 8102 + K2O - 28102 + L
+ V (curve 6d), would coincide, stably, with the curve,
KjO-SiOa + K20-8i02-^H20 + L -f- V (curve 106). Hence
curve, K20-8i02 + K2O- 28102 4- L -|- V {<6d), must lie be-

* Consult the discussion of this point on p. 1210 of the paper: G. W.
Morey and C. N. Fenner, J. Am. Chem. Soc, 39, 1173 (1917).

t The curves are numbered as in Figs. 5, 6, 7, and 8. In Fig. 6 only
curves of the type S' + S" + L + V are shown. In Fig. 8 the p-t curves
of the condensed systems are not numbered; their position is obvious.

290 MOREY ART. G

tween the metastable prolongation of curve, K2O • Si02 • ^H20
+ K2O -28102 + L + V (6c), and the stable portion of KsO-SiOs
+ K20-Si02-|H20 + L + V (10&). The position of the latter
curve being fixed, the position of the curves, K2O-Si02 +
K2O -28102 + L + V (6c/) and K2O • SiOg • IH2O + K20- 28102 +
L + V (6c), must either be as shown at Qi, Fig. 7, or the position
of these curves in regard to the curves, K2O • 28102 + K2O • 8102
+ K20-8102-IH20 + V (10a) and K2O-8IO2 + K2O - 8102 • ^HgO
+ L + V (106), must be reversed. That the latter arrangement
cannot be correct is shown by the fact that If K2O • 8102 - IH2O
- L - K2O • 28IO2 all lay upon a straight line, the curve,
K20-8i02-|H20 + K.O- 28102 + L + V (6c), would comcide
with the curve, K20-8i02 + K2O • 8102 • iH20 + KgO- 28102 + L,
the p-t curve of the condensed system. But such a coincidence is
possible only with the arrangement shown in Fig. 7 ; the reversed
arrangement cannot be the correct one.

In order to show further the relation between the composition
diagram and the p-t diagram, let us consider under what con-
ditions the curves, K2O-SIO2 + K2O - 8102 • JH2O + K2O- 28102
+ L (the p-t curve of the condensed system) and K20-Si02
+ KsO- 28102 + L + V (6d), will coincide stably. For
this coincidence to take place, it is necessary that the
phases, L, K20-8i02 and K20- 28102, lie on a straight line in
the composition diagram (Fig. 6), which Intersects internally
the line, V-K20-8i02-|H20. On reference to Fig. 6 we see
that before the phases can take on the position mentioned above,
it will be necessary for the phases, K20-8i02, K2O • 8102 • IH2O
and K20- 28102, then the phases, V, L, and K20-Si02, to
fall on straight lines. But in the y-t diagram, the first of these
will result in the curves, KsO-SiOg + K2O • 8102 • IH2O +
K2O • 28102 + L and K2O • SiOs + K2O • 8102 • IH2O + K2O • 28102
+ V (10a), approaching each other, coinciding, then again
diverging, having changed places. 8imilarly, as a result of
the second triplet of phases getting on a straight line, the
curves, K20-8i02 + K20- 28102 + L + V {M) and K20-8i02 +
K20-8102-IH20 + L + V (10a) will change places. The
curves, K20-8102 + K20-8102-^H20 + K2O ■ 28102 + Land
K20-Si02 + K2O -28102 + L + V {M), now lie next to each

HETEROGENEOUS EQUILIBRIUM 291

other, their stable portions adjoining, and when the phases, L,
K20-Si02, and K20-2Si02, fall on a straight line these two
curves will coincide in their stable portions.

The quintuple point Qz is exactly similar to Qi, but in-
stead of KzO-SiOa-^HsO, KaO-SiOz, and K2O -28102 we have
K20-Si02-H20, KaO-SiOs-IHaO, and K20-2Si02-H20, respec-
tively. Making these substitutions, the discussion of Qi will

apply to Q3.

Quintuple points Q2 and Q4 also are similar to each other.
Both contain the same three phases, V, K2O -28102 -1120, and
K2O • 28102 • At Q2 we also have the liquid represented by the
point Q2 and the solid phase, K2O • 8i02 • IH2O; at Qi we have the
liquid represented by the point Qi and the solid phase,
K2O • 48102 • H2O. Since in both systems the phases, V,
K2O • 28102 • H2O, and K2O • 28102, He on a straight line, the curve,
V-f K2O • 28102 • H2O + K2O • 28102 + K2O • 8102 • IH2O (7a), which
proceeds from Q2 to lower temperatures and pressures, and the
curve, V + K2O • 28102 • H2O + K20- 28102 + K2O • 48102 • H2O
(76 + 7a), which proceeds from Q4 to lower temperature
and pressure, and the curve, V + L + K2O • 28102 • H2O +
K20- 28102 (76 + 7c), which proceeds from both Q2 and Qa
to higher temperatures and pressures, coincide with each other
and with the dissociation-pressure curve of K2O • 28102 • H2O in
the binary system, HgO-KaO- 28102 (curve 7a + 76 + 7c).
The positions of the other curves that proceed from Qo and Q4
are easily found by the same mode of reasoning as that applied
to the curves at Qi.

The quintuple points Qsa and Qsb* differ from the preceding

* The compound, K2O -48102, was not met with in the original study
of the ternary system, by Morey and Fenner {J. Am. Chem. Soc, 39,
1173 (1917)), but was found later in the study of the anhydrous binary
system by Kracek, Bowen, and Morey (/. Phys. Chem., 33, 1857 (1929)).
The evidence in both studies makes it probable, though not certain,
that K20-4Si02-H20 has a congruent melting point. The relations
around quintuple points Qoa and Qiu are thus in part hypothetical, and
in Fig. 8 the two invariant points are not separated, nor are the two
curves from Qsa and Q&h to the sides of the diagram. The eutectics
containing K2O -48102 in the binary system, K20-8i02, are at 752° and
69 weight per cent silica, and at 764° and 72 per cent silica.

292 MOREY ART. G

ones in that in each the hquicl phase hes inside the triangle formed
by the coexisting soUd phases, and hence they are both eutec-
tics. At Qsa the coexisting phases are V + L + K2O • 4Si02 • H2O
+ K2O • 2Si02 + K2O • 4Si02, and the Uquid hes within the triangle,
K20-4Si02 - K20-4Si02-H20 - K2O -28102; and, similarly, at
Qsb the liquid lies within the triangle, K2O -48102 -1120 —
K2O • 48102 — 8102. 8ince the liquid is symmetrically placed with
regard to the three solid phases, the four univariant equilibria
containing liquid will be stable in the same direction from the
invariant point. 8ince V - K2O • 48102 - H2O - K2O- 48102
is a straight line, the p-t curves of the equilibria, V + L +
K2O- 48102 -HzO + K20- 48102 and V + K2O- 48102 •H2O +
K20- 28102 + K20- 48102, will coincide metastably with each
other, and will coincide with the binary equilibrium, V +
K2O - 48102 - H2O + K20- 48102, the dissociation-pressure curve
of K20- 48102 -HaO. Hence the p-t curve of V + L +
K2O • 48102 - H2O + K2O- 48102, and therefore of all those con-
taining liquid, will go to higher temperatures and pressures. 8ince
only a small change in liquid composition will make K2O - 28102
- L - K2O • 48102 • 520 a straight line, with V and K2O • 48102 on
opposite sides, the curves, V + L + K2O - 28102 + K2O - 48102 • H2O
and L + K2O -28102 + K2O- 48102 + K2O • 48102 • H2O (the
condensed system), will coincide stable to stable, and with a
continuous change in the same direction in the liquid composi-
tion the curves will cross. 8imilar reasoning applied to the
phases, liquid, K20- 48102, and K20- 48102 -1120, shows that the
curve, V + L + K2O -28102 + KaO- 48102 -HaO, must lie
between the curves, L + K20- 28102 + K2O - 48102 - H2O +
K20- 48102 and V + L + K2O - 48102 • H2O + K20- 48102,- and
the latter curve must coincide with the dissociation-pressure
curve of K2O - 48102 - H2O, the equilibrium, V + K2O - 48102 • H2O
H- KsO- 48102.

8imilar reasoning will serve to place the sequence of p-t
curves around the other eutectic, the invariant point Q^h-
The noteworthy feature of the curves proceeding from Qsb is
the rapid rise in pressure in the univariant equilibrium, V + L -F
K20- 48102 •H2O + 8102.

In the preceding discussion it has been shown how the funda-

HETEROGENEOUS EQUILIBRIUM 293

mental thermodynamic equations developed by Gibbs not only
lead to the qualitative generalization known as the Phase Rule,
but also afford a direct and detailed treatment of problems of
heterogeneous equilibrium. Such an analj^tical treatment is
illustrated for systems of two and three components. In
simpler systems it has the advantage of stressing the funda-
mental relationships that determine the course of equilibrium,
in contrast to the graphical method in which these fundamentals
may be overlooked in a geometrical maze. With increasing
number of components the geometrical methods become in-
creasingly involved, and the analytical method outlined above
offers the most hopeful procedure for developing the theory of
phase equilibrium in multi-component systems.

THE GRAPHICAL REPRESENTATION OF EQUI-
LIBRIA IN BINARY SYSTEMS BY MEANS OF
THE ZETA (FREE ENERGY) FUNCTION

[Gibbs, I, pp. 115-129]

F. A. H. SCHREINEMAKERS

I. Introduction

1. In the section entitled Geometrical Illustrations (pp. 115-
129 of the "EquiUbrium of Heterogeneous Substances") Gibbs
indicates how a general geometrical treatment of phase equilib-
ria can be based on the properties of the thermodynamic func-
tions. A full account of this geometrical method and its sub-
sequent developments would require an exposition of the whole
subject of generalised graphical thermodynamics. Since such a
comprehensive treatment is not possible in this Commentary,
it is hoped that the following discussion of certain equilibria in
two-component (binary) systems will serve to illustrate and
explain the important geometrical method initiated by Gibbs,
and introduce the student to the study of graphical thermo-
dynamics based on the properties of the free energy function ^.

II. The r-x Diagram and the f -Curve (Free Energy Curve)

2. Let us represent the composition of a phase containing the
two components W and X thus: x mols X + (1 — x) mols W.
We shall call this quantity, which contains in toto 1 mol, the
unit quantity of the phase. Then m unit quantities of the phase
contain mx mols X and m{l — x) mols W. Now the f-value of a
phase is determined by its temperature t, its pressure p, its
composition x, and its quantity m (units). Unless mentioned
otherwise, however, we shall mean by the f of a phase the ^ of
unit quantity of this phase. The f of w units will then be m^,
provided that the total energy, total entropy and total volume

295

296

SCHREINEMAKERS

ART. H

of the phase are first degree homogeneous functions of the
mass variables. This proviso means that we assume we can
neglect the surface effects which enter into the consideration
of micro-heterogeneous systems. For given t and p, the f of a
phase will depend, therefore, only on its composition. In the
case of a binary system this composition is defined by the
value of X (the composition parameter).

Fig. 1

In Fig. 1, in which WX = 1, the point a represents a phase
containing Wa{= x) mols X and aX(= 1 — a;) mols W. If we
now draw the ordinate aa' equal to the ^ of this phase, we shall
call the point a' the f-point of the phase a. If we give all
compositions, from pure W to pure X, to the phase a, then the
point a runs along the line WX, whilst the point a' traverses a
curve W'a'X', which, at constant t and p is called the f-curve
(free energy curve). Clearly W is the f-point of the pure
substance W and X' the f-point of the pure substance X. It
can be shown that the f-curve touches the lines WW and XX'
at the points W' and X' respectively (for proof see note at the
end of this article).

When all points of WX represent liquids, then W'a'X' is the
f -curve of these hquids, whilst W' and X' are the ^-points of the
pure liquids and a' that of liquid a. When the points of WX
represent vapors (gases), then W'a'X' is the f-curve of these
vapors, whilst W' and X' are the respective f-points of the pure

REPRESENTATION BY ZETA FUNCTION

297

vapors W and X and a' that of the binary vapor a. When
the points of WX represent homogeneous mLxed crystals, then
W'a'X' is the ^-curve of these mixed crystals, whilst W and X'
are the respective ^-points of the pure solid substances W and X
and a' that of the mixed crystalline phase a.

3. We now take two phases A and B with the compositions
Xi mols X + (1 — Xi) mols W, X2 mols X -\- {\ — x^) mols W.
If we bring together mi units of A and nii units of B, and if we
suppose that they continue to exist unchanged beside one
another, then we have a system or phase complex

miA + ?W2-B,

(1)

which may be stable or not. Let its total composition be
represented by x mols X + (1 — x) mols W. Since this system
contains in toto (mi + m2) mols and contains mia;i + 711.2X2 mols
X, we have

X =

mjXi + 702X2
mi + W2

From this follows

7ni{x — x^ = 7n2{x2 — x).

(2)

If we imagine the phases A and B and the system represented
in Fig. 2 by the points a, h and s, then we have Xx = Wa,

298 SCHREINEMAKERS art. h

a-2 = Wb, X = Ws; and x — Xi ^ as, Xo — x = sb. From (2) it
follows that

mi X as = mn X sb. (3)

If we put (compare Fig. 2) sb = ab — as, or as = ah — sb, then

nij mi

as = ; ab, sb = ; ab. (4)

m.i -\- nii ' mi + m2

Thus the position of the point s depends upon the ratio rui'.m^.
When mi = m2, as = sb, so that point s is situated in the middle
of ab; when mi > 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

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

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,

Fig. 11

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)

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,

^ = ^^ - '^^'

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

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

(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

Obs.

Calc.

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

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

M X 10-3

cm.

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)

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

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)

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 —

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,

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

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

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

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

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

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)

/dx\ _ »"(= + «) /p\

(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

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,

(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.

[{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

aieH^'

(43)

whence

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 )

+ 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 '

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)

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 =

+ 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<l' = O.J

At equiUbrium d/x/ = diix' , dii-l = dix2" , whence, if m^'/mi = r'

(
364 KEYES ART. J

and 1712" /mi" = r",

(—, - -^^ dp = (^,- ^)dt +(h- T, ) dp,', (74)
\W2 W2 / \m2 m2 / \r r /

(—> - ^) 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

(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

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

dp . , ^

\p = t-^ v'. (85)

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)

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)]

- 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.

Psat.

Ua + n2

Psat. — P _

Psat.

Us + ^2
W2

(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

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

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.

(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

F = Fi + F2, or

t;/ = viiCi log t + miai log — + rriiHi,

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

= r^-iV

Sri

2J (vi - vx) = 2j «i^i ^°g - = C'-i Zy '' l^s ' ^^^^^ ^^^^1

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

GAS 1

a/x/

GAS Z

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

^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,

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

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

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

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

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,)

(Rty

]

Xip

'§2

(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

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)

1 -

2 Co

1 -B

Ao" r - ^° e ' ' (138)

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.

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(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).

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

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'

en - \ f -
dy

dx"

dy'

6l3 - „ ,

dv dy"
''' - dx' = dx''

622

dv
dy

dy"
dy'

dv
''' ~ dz'

dw dz"
''' = dx'^ dx''

632

dw dz"
dy' dy'

633

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.

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

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

I ^31 + ei3 , 612 — 621 , 631 — en ,

631

—

ei3

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

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") =

(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

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.,

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

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^

06
02

05
04

. (24)

= r^T'^r^.

416

RICE

ART, K

«11

ai2

«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

= 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

■^

: V

At U they are
^^ ~ bx ^'

Fig. 6

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

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

8^^r

in the direction OX. The remaining pair of faces contribute a
resultant force in the direction of OX of amount

dXz

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/— ;

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

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

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)

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.

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

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

<f>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)=

—,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

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

'dYx' 9Fy' dY

(dZx' dZy' dZz' A I . . .

(a'Xx' + ^'Xy' + t'Xz') + av

D£
Ds'

+
+

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

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

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

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 + aB<t>BDSB.

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 <f>{x', y', z') reaches as a limit a value </>ci which
differs finitely from the limit </)c2 which is reached as we ap-
proach C along AB from A. In applying Green's theorem now
we must apply it separately to the two regions and integrate
(d4>/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

Ol2

ai3

^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<iai2 • • • + Azzdas^.

This result depends on the fact that if the constituents of the
determinant \a\, written above, are all altered by infinitesimal
amounts, dan, dan, etc., then the infinitesimal change in the
value of \a\ is equal to the expression on the right-hand side
of the equation just written. Now by equation [355]

de = tdrj + Xx'dan + Xy'da

+ Zz'da.

33,

[400a]

since for the postulated cube ev and riv are identical with «
and t]. Also from [388]

dt = td-q + ridt — pdv — vdp + mdm,

remembering that vv is identical with v.
Equating [400a] and [401] we obtain

■qdt — vdp + mdni = Xx' dan + Xy da^ + . .

+ Zz' dttss + pdv
= {Xx' + An p) dan + {Xy + An p) da^ + . .
+ (Zz- + Azi p) dazz.

[401]

-. [404a]

This is our form of equation [404]. If we then proceed to
equation [405] which holds for a fluid identical in substance
with the solid (so that we are dealing with fusion and solidifica-
tion) we arrive at our form of [406], viz.,

(vf — v) dp — {riF — 7]) dt = (Xx' -{- Anp) dan
+ {Xy' + An p) dan . . . + {Zz' + Azz p) dazz. [406a]
In consequence we find that

dp
dt

[407]

t{vF — v)
Let us recall that p is the fluid pressure on a pair of opposite

STRAINED ELASTIC SOLIDS 479

faces of the solid which is compatible with the given state of
strain an, an, . . . 033. Thus p is a function of an, an, ... 033
and the temperature; dp/dt is therefore the rate of variation of
this pressure with temperature at constant strain, i.e., with the
solid constrained to keep its size and shape (in the state of
strain) unchanged. This is the analogue of the usual equation
for the variation of the melting point with pressure. The
melting point is t at pressure p and strain an, an, . • . ass- At
pressure p -\- dp and the same strain an, a,n, . ■ . clss the melting
point is ^ + dt, the latent heat per unit volume is Q, and so
Q/t(vp — v) is equal to the limit of dp/dt. It is necessary to real-
ize the conditions under which Q is the latent heat of fusion.
From [393] the energy of the solid with the proper pressure p'
on a pair of faces is given by

€ = trj — p'v -{- m'm.

That of the same mass of the fluid in equihbrium with the faces
is given by

Hence

€f = tr\F — p'vf + ni'm.

€f — e = t{r]F — ri) = Q.

As Gibbs points out, if we imagine the cube surrounded entirely
by the fluid so that the conditions are those of the case usually
considered, the quantities e and rj have different values from those
considered above (see equations [396]), and Q is also different
in value.

The more general case considered on page 200 when the fluid is
not identical in substance with the solid can be followed up as
is done by Gibbs, and we arrive at [411] in the form

{

djii (t, p, nir) \

m — v> 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

ai2

«13

^22

«23

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 -

tte

tti

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

tti

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

= Al + Al^^... +A

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

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

«i

0C6

Oi2

CCb

OCi

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 <i){t, E, F, H) to refer to the "true" value of ypv and one
x{t, E, F, H) to refer to the linear function of E, F, H in [444]
which is approximately equal to ypv. These can both be
expanded as series in terms of ri, r2, r^, or rather of ri — ro,
^2 — ro,rz — ro; the discussion centers round the problem of deter-
mining at what power of n — ro, etc., the two series begin to
show a difference. A little thought will show that the series for
X will terminate at fourth order terms. In fact writing for
the moment x for ri — ro, y for ra — ro, z for rs — ro, we see that

X = i + e{{x + roY + (2/ + ro)^ + (z + ro)'}

+ f{{y + roYiz + roY -\- (z-h roYix + ro)^ +

(x + roYiy + roY]
+ h{x + ro) {y + ro) {z + ro).

494 RICE

ART, K

The series therefore involves first powers and squares of x, y, z
and product terms such as xy, xyz, x^y, x^y^. Of course the series
for </) will in general extend beyond such terms and may indeed
be a convergent infinite series. Before proceeding further,
it might be well to point out that Vo is just an ordinary factor
of temperature expansion (linear), resembling in fact the
familiar 1 -\- at oi the text-book of heat. It is necessary to
bear in mind that the state of reference is a state at a given
original temperature. If the solid is warmed (or cooled) to
another temperature without any application of external forces
and creation of stress, straining takes place; for an isotropic
material it is a uniform expansion. This is an excellent illustra-
tion of the necessity of keeping the notions of strain and of
stress clearly separated in the mind. Our instinctive notions
of pulling, pushing, twisting, bending bodies into different shapes
and sizes gives us an unconscious bias towards the idea that
stress must invariably accompany strain and vice-versa, whereas
change of temperature produces strain (change of size at all
events, if not a change of shape which generally accompanies
heating of crystalline material) without stresses being created,
and if we prevent the strain occurring we have to exert external
force on the body with the creation of internal stress, sometimes
of relatively enormous value. (We can all recall the experi-
ment in our lecture course in elementary physics when the
demonstrator fractured the red-hot bar, or the clamps which
held it tightly at its ends, by pouring cold water over it.) If
therefore we alter the temperature of the (isotropic) body and
subject it to external force, the principal elongations with
reference to the unstressed state of reference at this tetnperature
will be Vi/ro, /'2A0, fs/ro] and ipv, regarded as a function of the
temperature and the elongations, can be considered as expanded
by Taylor's theorem in the form of a series in the relatively
small variables (ri/ro) — 1, (r2/ro) — 1, (rz/ro) — 1. This
comes to the same thing as regarding ypv (either its "true"
value (f){t, E, F, H) or its approximative value x{t, E, F, H))
expanded as a series in ri — r^, r^ — ro, rz — ro.

Let <f>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'^<l)/dr'^)o, (d^4)/drdr')o be the common values, assumed accord-
ing to [445] by the various first and second differential coeffi-
cients of (f) with respect to the variables ri, r^, rz. Use a similar
notation for x- Then if we write down

Xo — 4>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 <f) in terms of ri — ro, r2 — ro, n — ro,
we have

* = *, + ( ^

\dr

■) (n - ,-.) + (^) (r. - r.) + ('-*) (ra - r,)

i/o \3r2/o \9'Vo

+ 1 {m (., _ r„)' + (q) in - r„y+ (^) (r. - r.)=
2! \\drVo \drl/o \drl/o

+ 2 (-^) (r2 - ro) (rs - ro) + 2 (^^) (r, - ro) (n - ro)
\dr2dr3/o Xdndri/ q

+ 2 ( — — ) (ri — ro) (r2 — ro) > + higher powers
\dridr2/o J

= <^o + ( — 1 (ri + rg + rs - 3ro)
\ar/o

496

RICE

ART. K

+ 2

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

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

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

5ai2 aai2 aai2 da

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

<j>V' = €v' — trjv — HaTa' — jJ^bTb — etC,

whose differential satisfies the equation

d<f)v' = —rjv'dt + ZiXXx'dan — ^Tadna.
This will lead to the second group of [472].

504 RICE

ABT, K

The function ^v of t, Xx', Xy', . . . Zz', Ta, T^, etc., defined

will give the second set of [473], and a function x^. of t, Xx'>
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..

I.e.,

dXx'

dan ~~

drjv

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) for sufficiently large separation of the
molecules is zero. This makes the value of 0(r), for smaller
values of r, negative, at all events until the critical separation
is reached at which the attractive force is changed into a repul-
sion. There we have the minimum value of 0(r). (Of course,
the numerical value of </>(r) will be greatest at this distance.)
Anj'" further decrease in r will produce an increase in <j){r), which
will very quickly reach zero and even positive values owing to
the enormous resistance to compression exhibited by liquids.

514 RICE ART. L

In terms of 4>{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) is negative for the values of r considered, the con-
tribution of this molecule to the total potential energy is
greater than for a molecule in the interior of the liquid. In
short, if a body of liquid is divided into two portions which are
then separated from one another against mutual attraction we
know that the potential energy of the whole is increased. This
increase is made up of the larger contributions of those mole-
cules which lie near to the two new surfaces created by the
division. This increase can be calculated in terms of 0(r) and
we can thus obtain an expression which represents the "surface
energy," meaning by that the extra energy associated with the
molecules in the surface layer of thickness h over and above
that which would be associated with them if they were all in the
interior of the liquid mass. This is not the place to enter into
the analytical details, but it can be shown that the whole
potential energy of the body of liquid can be written in the form

pV + <tA,

where V and A are the volume and superficial area of the mass ;

* Actually it is the concentration of the vapor or gas phase, rather
than zero.

516 RICE ART. L

p is the expression (2) and c is the expression

Trn^

r^({>(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

Fig. 1

'^^'^

4 ■

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^<l>{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

D
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],

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<t>

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 <r still a function of physical variables as
measured in the homogeneous mass. The writer is not aware
that anyone has attempted to use Thomson's formula (16) or
(18) in numerical calculation. The feature of it just men-
tioned would render it difficult; but if it were possible it would
probably produce some improvement on the results calculated
by the approximate form of Gibbs' relation. To show this
suppose we write x for {—\/Rt) (da/dK); x will be positive
when there is actually a surface excess, i.e., when {dd/dK) is
negative. Equation (18) would then be

P X?' X?

The approximation would be

- = \ -\- X.
P

Clearly, since x is positive, the values of p obtained from the
first of these would be markedly larger than those obtained
from the second if x were not entirely negligible compared to
unity, and it is well-known that even in those experimental
results which show the best accord between observation and
calculation the tendency is for the observed concentration to be
above that calculated by the approximate form of Gibbs'
equation, which the second of the above equations most re-
sembles.

It also merits attention that Thomson's equation can be
readily obtained by the present-day methods of statistical
mechanics in a very direct way. If the reader will look once
more at section IV of this article (Article L) under the heading
"Statistical Considerations" he will observe in equation (10)
how the concentrations in two phases are related in simple cases
to the work required to extract a molecule from one phase and
introduce it into another. Now in the present instance the
solution in bulk may be regarded as the second phase and the

SURFACES OF DISCONTINUITY 549

surface film the first; a is the surface energy per unit area of the
film, meaning by that the energy possessed by the molecules
in unit area of the film in excess of what they would possess
if they were in the body of the fluid. Hence the da- in (18) will
refer to an increase in this, i.e., the work required to extract from
the bulk and bring to the surface a number of molecules given
by N^dp, where N is the number of molecules in unit mass of the
solute; for f is equal to the volume of unit area of the film and
^dp the increase in the mass of the solute in it. Hence, since
R refers to the gas constant for unit mass of the solute,

R^dp = Nk^dp,

and we see that (l/R) {da/dn) is equal to the work required
to bring one molecule from the interior to the surface divided
by k, i.e., to {0(ni) - e{n2)]/k. Thus by (10)

-, = exp

\ Rt dKj '

which is just Thomson's equation. Thus, not only in the form
of the equation but also in the possibility of deducing it in this
way, one might state with some show of reason that it is really
more akin to some recent results obtained by Langmuir and
others than to Gibbs' law.

It should be mentioned as a matter of interest that Warburg
in 1890 made use of an equation, which is virtually Gibbs' ap-
proximate result, in his well-known paper on "Galvanic Polari-
zation" (Ann. d. Physik, 41, 1, (1890)). By means of it he
made some calculations on the forcing of the solute out of the
surface layer in the case of inorganic salts which raise the
surface tension of water and so are desorbed. He used a
thermodynamical argument; in an addendum to the paper he
refers to the earlier proofs of Gibbs and Thomson.

Quite a number of proofs of Gibbs' equation, usually in the
approximate form, have been published from time to time.
(See Swan and Urquhart's paper cited above.) Porter, in the
Trans. Faraday Soc, 11, 51, (1915), has derived an equation for

550 RICE AKT. L

concentrated solutions, viz.,

(1 — acY da

r = — ^ >

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 <x and c.
Langmuir's Adsorption Equation. FrenkeVs Equation

We shall now turn for a moment to one or two empirical
relations between surface tension and bulk concentration in
solutions. For relatively strong solutions of acetic acid Milner
{PhU. Mag., 13, 96 (1907)) found that a formula of the type

0-0 — a = a + jS log c

was satisfied, where ctq is the surface tension of water, cr that of a
solution of concentration c, and a and ^ are constants. Shortly
after, Szyszkowski (Z. physik. Chem., 64, 385, (1908)) sug-
gested a somewhat different form, viz.,

^-1^^ . 5 log (l + -^) ,

where a and 6 are constants. He verified this for solutions of
the shorter-chain normal fatty acids. It was observed that the
constant b had the same value for all the acids, while a was
different for each acid. Its values, however, for two acids dif-
fering by one carbon atom bore a nearly constant ratio, the three
carbon acid having an a 3.4 times larger than the a for the four
carbon acid, and so on. This means that 1 + (c/a) is a larger
quantity for the same concentration the longer the h3^drocarbon
chain in the acid, and so in this homologous series of acids the
diminution of surface tension at a given concentration increases
rapidly in amount as the hydrocarbon chains are lengthened,
which is just an example of a well-known rule due to Traube
that the capillary activity of a member of an homologous series
increases strongly and regularly as we ascend the series. For
by the Gibbs' simple formula

c da
^ ^ ~Rtdc
hao C
lit' c + a

ART. L

552 RICE

Thus to obtain the same surface concentration we require for
each successively higher member of the series a bulk concentra-
tion about one third of that of the previous member, and so the
higher members are more and more "capillary active," to use a
common term which designates the property of causing a lower-
ing of surface tension and being in consequence adsorbed in
excess quantity at the surface. It will be observed that, if c is
large compared to a, Szyszkowski's formula approximates to that
of Milner. A relation has just been found from the former, viz.,

where g is a constant at given temperature and would be in fact
the upper limiting value of r if the law held for extremely high
concentrations. Now this relation is virtually equivalent to an
equation deduced by Langmuir {J. Am. Chem. Soc, 38, 2221,
(1916)) for the adsorption of gases on a solid surface
(plane crystalline). Although not of special interest now, it
may not be amiss to mdicate Langmuir's argument in broad
fashion, inasmuch as Gibbs at a later point in his treatment
deals with the conditions at a surface separating a solid from a
fluid.

Langmuir's special hypothesis is that the molecules of the
gas are "condensed" on the crystalline surface when they strike
it, and do not in fact rebound in an elastic fashion as sometimes
postulated in kinetic theory of gases, except in a minority of
impacts. There is a good deal of evidence that this is actually
the case, and that in general the molecules remain on the
surface for a longer or shorter time depending on the attractive
forces between the solid and the adsorbed layer, and on the tem-
perature. There is therefore a concentration of molecules on
the surface whose amount depends on the average length of time
during which the molecules remain upon it. This state of
affairs obviously resembles what happens when molecules of a
solute pass from the solution into the surface layer and so it
is not surprising that there should be a formal resemblance in
the laws deduced in the two cases. Indeed Langmuir's analysis
could be easily adapted to give a theoretical foundation for

SURFACES OF DISCONTINUITY 553

Szyszkowski's formula in the latter case. A further assumption
is that the adsorbed layer is one molecule thick and that no
further adsorption occurs in a second layer beyond this. This
assumption is also in keeping with what are nowadays believed
to be the conditions at the surface of a solution, a matter to
which we shall devote some attention later on, as it is one on
which Gibbs' equation brings important considerations to bear.
Let a fraction 6 of the surface be covered with adsorbed gas, and
the rate at which molecules evaporate from unit area of the
adsorbed layer be ad, a being a function of the temperature and
depending also on the attractive forces. The rate at which gas
molecules unpinge on unit area of the surface is proportional to
the density of the gas and the average molecular velocity, i.e.,
to p0 (t is the absolute temperature). Since p = pt this rate is
therefore proportional to p/t^. Therefore the rate of condensa-
tion (which by the postulates we take to be comparable with
and proportional to the rate of impact) on unit area of the bare
surface can be written as ^pt~\ where j3 is a constant. We
suppose that no condensation occurs on the top of an adsorbed
layer. (That is the second postulate above and assumes that
the attractive forces of the solid do not extend appreciably
through the first layer, — a reasonable assumption on our present
knowledge.) Thus the rate of condensation on unit area of the
surface of the adsorbed layer will be

^pt-Kl - 9)
since a fraction 1 — 0 is bare. Hence in equilibrium

/3prKl - d) = a9 ,
from which we easily obtain

e =

P + oi^

p + a
where a is a constant depending on attractive forces and tem-

554 RICE

ART. L

perature. If n is the number of molecules actually adsorbed
per unit area at any moment, and Um the maximum number
which could possibly be adsorbed if the unit area were entirely
covered with a monomolecular layer, 6 is n/n,n, and so Langmuir's
result can be written

V
n = Um — I (20)

The result is of considerable theoretical importance in connec-
tion with the so-called "poisoning" of solid catalysts. The
formal similarity of (19) and (20) is obvious, the pressure of the
gas being the analogue of solution concentration in (19). As
stated above, Langmuir's analysis could easily be adapted to
prove (19) and so by the aid of Gibbs' equation to derive
Szyszkowski's relation. Frenkel in the Zeit.f. Physik, 26, 117,
(1924) derives a special functional form for the constant a in
(20). On certain assumptions he shows that the mean length
of time during which a molecule adheres to the surface is equal
to T exp iii/kt), where r is the period of thermal oscillation, at
right angles to the surface, of an adsorbed molecule, u the energy
of desorption, i.e., the energy required to tear an adsorbed
molecule away, and k the gas constant per molecule. Thus the
rate of evaporation from unit area is n/[T exp {u/kt)\ and so the
constant a is equal to r~i exp { — u/kt). Also it can be shown
from the kinetic theory of gases that jS = (2Trmk)~^, where m is
the mass of a molecule. Hence Frenkel's form of Langmuir's
result can be written

n = n„

, (27rm/c)i -I
p + -^ e ^«

For further information on these and similar equations the
reader can consult Chapter V of Rideal's Surface Chemistry
and Chapter VIII of Adam's Physics and Chemistry of Sur-
faces (1930).

SI. Energy of Adsorption
Returning to adsorption at the surfaces of solutions, it has

SURFACES OF DISCONTINUITY 555

already been stated that Thomson's equation has a close kinship
with some equations of Langmuir and others. We can enlarge a
little on this point. The surface film of a liquid is a region
where the potential energy of a molecule of the solute is greater
by a definite amount e than that possessed by the molecule
when in the bulk of the solution. It follows from the funda-
mental statistical law that since r/f is the volume concentration
in the film

~ = c exp

V kt)

€ = - kt log — •

Langmuir has applied this result to Szyszkowski's measure-
ments of the surface tensions of solutions of the fatty acids and
to the adsorptions calculated therefrom. If e„ and €„_i are the
energies of adsorption per molecule for acids with n and n — 1
carbon atoms, respectively, then

E„ — €„_i = — kt

'"K«X-'°^ffL.

assuming that the film thickness f is the same in all cases. In
the case of dilute solutions where c is small compared to a this
result becomes by (19)

€n - €„_i = -^^{log ttn-l " log a„} ,

since g, i.e. hao/Rt, is the same for all the acids. Now, as men-
tioned above, an-i/ctn has an almost constant value about 3.4,
so that log a„_i — log a„ is the same for any pair of successive
acids. Thus the energy of adsorption increases by a constant
amount for each CH2 group added to the hydrocarbon chain of
fatty acids. "This must mean that each CH2 group is situated
in the same relation to the surface as every other such group in
the chain, and this can only be the case if chains lie parallel

556 RICE

ART. L

to the surface. Hence Langmuir concluded that the molecules
lie flat in the surface, in the gaseous adsorbed films."*

Equation (19) is an example of an adsorption law deducible
from statistical considerations. We shall bring these references
to such equations to a conclusion by adapting an argument to
be found in Rideal's Surface Chemistry, p. 71, which leads to
another example of them. Let there be Wi molecules of a solute m
the surface layer of thickness f and area s, and n^ molecules of
solute in a volume V of the solution. If the layer is of the uni-
molecular type, the evidence for which we shall discuss in the
next section, there is a free volume in it of amount sf — n-iV,
where v is the effective volume of one molecule. If we add some
more molecules to the solution there will be a division into two
groups; one whose number is bn^ will be found in the layer, one
whose number is bn^ will be found in the solution. The volume
concentration of the first group will be 5wi/(sf — Uiv), of the
second bUi/V, and these two concentrations will have the ratio
exp(— u/U) where u is the energy of adsorption; i.e.,

sf — iiiv V
X being written for exp{—u/kt). By integration we obtain

log {si; — Hiv) = — —^ + constant

= — \vc + constant ,

where c is the bulk concentration. Hence

s^ — HiV = Ce-^"".
Since Ui is zero when c vanishes, C = s^ and therefore

niv = 5^(1 — 6"^"')

* Quoted from page 128 of Adam's Physics and Chemistry of Sur-
faces. The reader must not interpret "gaseous adsorbed" as meaning
adsorbed from the superincumbent gas. It is a term applied to a special
type of film, of which we shall say more at a later stage.

SURFACES OF DISCONTINUITY 557

rii f

(1 — €-'>''")
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 (<r, c) curve quickly becomes nearly
parallel to the c-axis, and only very doubtful values of r could
be obtained. A rough experimental method gave as the surface
excess 0.4 mgm. per square meter, which Milner regarded as a
"moderately good estimate" for it at the moment of formation
of the bubbles of air which were passed through the oleate solu-
tion; but he was of the opinion that this was "very much less
than the ultimate value of the excess." He concluded that
there was an irreversible process here which actually caused the
solute to come out of solution in the surface in consequence
of excessive adsorption. As we have pointed out above, if
such is the case the theoretical conditions for an application of
Gibbs' equation do not hold under these circumstances.

Actually the first attempts at a quantitative verification of the
equation were made by W. C. M. Lewis at the suggestion of
Donnan (Phil. Mag., 15, 499, (1908) and 17, 466, (1909)). In
one set of experiments an oil-water interface was used and
solutes were chosen so as to be insoluble in the oil phase and
very capillary-active in the aqueous phase. Sodium glyco-
cholate, however, yielded results for the direct measurement of

560

RICE

ART. L

r which were about 80 times as great as those calculated on the
basis of Gibbs' equation. The sodium salt of congo red, methyl
orange and sodium oleate were also tried and exhibited a similar
though less marked discrepancy. Despite the experimental
difficulties of the tests, there was no possibility of ascribing
these results to experimental errors or to the use of the simple
form of the equation. The excessive adsorption was almost
certainly a characteristic of the semi-colloidal solutes employed.
Subsequently Lewis used a solute of much simpler constitution,
and one truly soluble in the aqueous medium, viz. aniline, and
measured the adsorption at a mercury-water interface (Zeit.
physik. Chem., 73, 129, (1910)). The calculated and observed
adsorption values now showed agreement as regards order of
magnitude, both being small multiples of 10"^ grams per sq. cm.
A still more successful test was carried out by Donnan and
J. T. Barker {Proc. Roy. Soc, 85 A, 557, (1911)) who measured
the adsorption of nonylic acid at an air-water surface. The T
was evaluated from the expression { — c/Rt) (dcx/dc) and cal-
culated, first, on the assumption of non-ionization of the acid
and, second, on the assumption of complete ionization. The
table gives the observed and calculated values.

Adsorption of Nontlic Acid at Air-Water Surface

Percentage
Concentration in

r X 10' obs.

r X 10' calc.

Solution

(1)

(2)

0.00243
0.00500
0.00759
0.00806

0.95
1.52
1.09
0.915

0.55
1.14
1.26

0.26
0.57
0.63

Donnan and Barker also measured the adsorption of the
glucoside saponin at an air-water surface; this forms very stable
foams and viscous films at the bounding surface of air bubbles.
There was agreement between the orders of magnitude of r
observed and calculated, but from a substance of this character
little more could be expected, and the results with nonylic acid
are of greater value.

Patrick (Zeit. physik. Chem., 86, 545, (1914)) investigated the

SURFACES OF DISCONTINUITY 561

behavior of mercurous sulphate, saHcylic acid and picric acid
at a mercury-water interface, but the experiments only gave
qualitative results from our point of view, as a quantitative
estimate of F could not be made. Later, Patrick and Bachman
(Journ. Phys. Chem., 30, 134, (1926)) found that the cation
is more readily adsorbed than the anion of a mercurous salt at a
mercury-water interface.

Frumkin in Zeit. physik. Chem., 116, 498, (1925) described a
method for testing the law which differed considerably in the
experimental procedure from those previously used. He worked
with lauric acid, chosen because of its relatively slight solu-
bility in water, and managed to produce a saturated layer of
the acid on the water whose concentration he could measure,
obtaining an adsorption of 5.2 X 10"'^" moles per sq. cm. Using
the ((T, c) curve in the neighborhood of saturation he calculated
r to be 5.7 X lO"'-" moles per sq. cm. He made control experi-
ments to test the accuracy of his measurements and concluded
that the error in the calculated value was not more than 10 per
cent, and that about the same Uncertainty affected the observed
amount. If this is so, Frumkm's measurements constitute one
of the most satisfactory tests yet made.

Reference should also be made to some experiments made by
Bancelin (J. chim. phys., 22, 518, (1925)) on the adsorption
of dyestuffs (at very low bulk concentration) both at liquid-air
and liquid-mercury interfaces. Rather remarkably, Bancelin
obtained fair agreement between calculated and observed
values for these solutes.

Historically, the next important contribution is that of
Schofield (Phil. Mag., 1, 641, (1926)), who observed the adsorp-
tion by mercury of its own ions from solution. However, in this
work we are concerned with somewhat wider issues than those
raised by the Gibbs capillary adsorption equation. Questions
concerning the electric potentials at the surface enter into the
discussion, and we shall postpone deahng with these until we
treat electrocapillarity towards the end of this article.

23. The Experiments of McBain and His Collaborators

The most extensive and exact experimental test of Gibbs'
equation carried out up to the present is that of McBain

562 RICE

ABT. L

and Davies (J. Am. Chem. Soc, 49, 2230, (1927)). Brief
accounts of it will be found in the books by Adam and Rideal.
The substances examined were aqueous solutions of p-toluidine,
of amyl alcohol and of camphor. The method used for deter-
mining r was the bubble method much improved as to accuracy
over previous investigations, an accuracy of a few per cent
being claimed. If this is so, there is no doubt that these experi-
ments have left the whole matter in some doubt. Hitherto,
it had been regarded as very satisfactory that an agreement
in order of magnitude between calculated and observed values
had been reached, in view of the manifest difficulty of the
measurement of the adsorbed amounts. If the claim to high
accuracy made by McBain and his co-workers is justified, this
state of satisfaction is hardly possible any longer. The general
idea of the method is that bubbles of very pure nitrogen satu-
rated with the vapors of the solution are passed up a long
inclined tube of large diameter containing the solution. The
slope of the tube is adjusted so that the time occupied by the
bubbles in passing to the top end of the tube is amply sufficient
to insure that the surface of each bubble has attained the full
adsorption concentration corresponding to the bulk concentra-
tion of the solution, the tube being so large that the adsorption
does not appreciably lower this bulk concentration. At the
top of the inchne the bubbles rise into a vertical tube so narrow
that each bubble fills its diameter. Each bubble in the vertical
tube rapidly overtakes its predecessors and draining is so rapid
that within a few inches there is a continuous column of cylindri-
cal bubbles in contact with one another. At the height at
which draining is found to be sufficiently complete the narrow
tube is curved over and down. The films break in the down-
ward portion of the tube and collapse to a liquid which is caught
and analyzed. For a full account of the very stringent pre-
cautions taken to insure accuracy the literature should be
consulted. It must be admitted that little was left undone in
that direction. Perhaps the only possible source of trouble
has been indicated by Harkins (Colloid Symposium Mono-
graph, 6, 36, (1928)). As bubbles pass along the tube, they
oscillate in shape; this involves an oscillation in the extent of the

SURFACES OF DISCONTINUITY

563

drop surface. Suppose that saturation in adsorption were
attained when the surface is at its maximum value, then when a
subsequent contraction takes place the compression (in two
dimensions) thereby produced might cause some of the adsorbed
material to gather into droplets on the surface, and so more of it
would accumulate on the surface than would correspond to true
adsorption. Be that as it may, the general nature of McBain's
results may be indicated broadly thus :

Firstly, the calculated value of r tends to a maximum as the
bulk concentration increases. Actually this might be antic-
ipated from the equation of Szyszkowski quoted earlier. Thus
according to it

(To — cr

= 6 log

(-3

and

hero
T = -^

Rt c + a

which approaches a limit hao/Rt as c increases.

Secondly, the observed values of T also rise to a maxi-
mum, but during the whole course of events are definitely greater
than r calculated. The table for p-toluidine shows this.

Concentration of Solution

r X 108 obs.

r X lO'calc.

(in grams per liter)

(in grams per sq. cm.)

0.6

2.4

1.5

1.0

6.5

4.7

1.4

10.4

6.6

2.0

12.7

6.8

3.0

13.4

7.1

4.0

13.2

7.3

5.0

13.0

7.5

6.0 (saturated)

The results for camphor show also a discrepancy of about
two to one; while for amyl alcohol the discrepancy is still greater,
amounting to about four or five to one.

At first McBain regarded this discrepancy as due, in part at

564 RICE ART. L

all events, to the approximate character of the expression
( — c/Rt) (da/dc) which was used for T calculated ; but in a later
paper with Wynne-Jones and Pollard (Coll. Symp. Monograph, 6,
57, (1928)) he abandons this explanation, as it was found for p-
toluidine that its partial vapor pressure over an aqueous solution
was directly proportional to the concentration of the solute.
This partial pressure gives a direct measure of the activity of
the dissolved p-toluidine and so there is no difference in value
between c(da/dc) and a{da/da). That being so, McBain
repeated still more decidedly a suggestion which he had already
made tentatively in the first paper, viz., that the situation is
complicated by the existence of surface electrification effects,
and that the omission of any consideration of these vitiates the
theoretical basis of the adsorption equation, as it stands, without
an additional differential term on the right-hand side represent-
ing increase in the energy of this surface electrification when
concentration increases by a differential amount. We cannot
deal with this point now, but will return to it at a later stage
of this commentary. A further point raised by McBain and
Davies (Jioc. cit.) is that in these and similar experiments "seldom
or never have true, two-component systems been actually under
observation, although this is fundamental. Solutions of
electrolytes or substances capable of hydrolysis, such as soap,
cannot be treated as two component systems except in the rare
event that the composition of the adsorbed material is identical
with that of the solute remaining in the solution." The point
of this remark is that we are implicitly using the equation

da = —Tid/jLi — T2dn2

and making Ti zero by adjusting the surface so that we have

da = — r2(i) dn2.

But this is invalid if there are still other components present.
As McBain and Davies say "The component (or components)
actually present, but hitherto ignored, is the gas (or air) in
presence of which the surface tension is measured when bubbles
are produced." If we set Ti for the solvent equal to zero there

SURFACES OF DISCONTINUITY 565

are at least two other components (such as p-tokiidine and
nitrogen); "their adsorption is r2 = —(9o"/ 9^2)^3 and Ts =
— (da/dfxs)^^ each of which is readily measured, although this
has never been done. It is obvious that the two adsorptions

will mutually interfere For example, it has been stated

that the surface tension of mercury is 10 per cent lower in the
presence of one atmosphere of nitrogen than in vacuo; similarly,
nitrogen lowers the surface tension of water by about one per
cent, which would correspond to the adsorption of about 3 per
cent as many molecules of nitrogen as of p-toluidine. However,
such mutual interference cannot explain the high values of the
observed adsorptions."

As it can be stated here that McBain's explanation of the
discrepancies in terms of surface electrification effects has not
been universally accepted, it is clear that the evidence for the
complete quantitative validity of Gibbs' law, as against a rough
qualitative agreement, is far from satisfactory. In reflecting
on the various theoretical steps in the proof one naturally feels
somewhat dubious about the arbitrary placing of the surface
of division in order to get rid of one term in the differential
expression ; in discussing these matters the writer has, for exam-
ple, heard such statements as these: "Nature fixes the surface;
surely we cannot mess it about as we please." There is some-
thing to be said for this instinctive recoil from a procedure
apparently so arbitrary; yet a close investigation leaves us little
hope of evading our difficulties by pressing this instinct into our
service. For instance, let us look at Gibbs' equation [515],
where the strictly placed dividing surface is used, showing us
that

r _ r _ r ^ - y"

■I 2(1) — 1 2 J- 1 / ,/•

7i ~ 7i

In this r2(i) is the surface excess as calculated, while r2 is what
we might call, if we were disposed to press the point we are
presenting, the "true" surface excess, and it would appear that
r2 is greater than r2(i)*provided Fi is positive, which is certainly
in the right direction for an elucidation of the mystery. The
value of 7i", the concentration of the solvent in the vapor phase

566 RICE ART. L

in the gas-liquid experiments, is negligible compared to 7/, so
that r2 exceeds r2(i) by (72' — 72")ri/7i'. Until we know some-
thing about Ti we cannot say whether this is going to improve
matters or not. We shall have occasion in the following
section to return to this point, which we leave for the present.

IX. Gibbs' Equation and the Structure of Adsorbed Films

24- Impermeable or Insoluble Films

On pages 275, 276 Gibbs makes a very brief allusion to
"impermeable films" which may offer an obstacle to the passage
of some of the components from one phase to the other. "Such
may be the case, for example, when a film of oil is spread on a
surface of water, even when the film is too thin to exhibit the
properties of the oil in mass." The latter part of this sentence
is most significant in view of subsequent events. Gibbs con-
tents himself with pointing out that for any component which is
found on both sides of the film, but which cannot pass the film
itself, the potentials on either side cannot be proved to be equal,
and so in the adsorption equation, for example, a single term
such as —Tidni must be replaced by —Tidni — V^dn^, where
Fi and r2 refer to the surface excesses of the same component on
the two sides of the dividing surface and mi and /i2 indicate the
differing potentials in each adjacent phase.

Soon after the existence of "surface tension" became known,
it was discovered that oil films on water reduced this property
very markedly. This is of course quite a different phenomenon
from the lowering by capillary-active soluble substances. It
was Rayleigh who began accurate experimental work on the
thickness of such oil films {Proc. Roy. Soc., 47, 364, (1890)).
Some very important results were discovered by Miss Pockels
who was the first to use the method of "barriers," which by rest-
ing just on the surface of a liquid in a trough and extending
over its whole width could be used to push a surface film in front
of them so that it could be compressed or extended in two
dimensions (Nature, 43, 437, (1891)). •She made the dis-
covery that provided the area of a film formed by a small given
quantity of oil exceeded a certain critical value the surface

SURFACES OF DISCONTINUITY 567

tension did not differ appreciably from that of water, but as
the area was reduced below this value, the surface tension
fell rapidly. Later, Rayleigh (Phil Mag., 48, 321, (1899))
suggested that at this critical area the molecules are just
crowded into a layer one molecule thick; that they are in fact
floating objects which begin to repel one another when coming
into contact in a single layer. This accounts for the first
appearance of a diminution in surface tension at this point; a
barrier moving a small distance in the direction of the pressure
arising from this would gain kinetic energy, presently dissipated
in the general body of the fluid. The corresponding loss of
energy will be found in the fact that the expanding surface
covered by oil will not gain as much surface energy as is lost
at the contracting clean surface, which is merely a statement of
the fact that the oil covered surface has a smaller "surface
tension" than the clean, but does not imply the existence of a
physical tangential pull in the surface. Actually, as Devaux
was the first to point out, some films may acquire the properties
of a two-dimensional solid possessing a tangential rigidity in
the surface which prevents them being blown about into differ-
ing shapes.

£5. The Work of Langmuir and Adam. The Concept of ^'Surface
Pressure." Equations of Condition for Surface Phases

Great improvements in the experimental appliances were
introduced by Langmuir (J. Am. Chem. Soc, 39, 1848, (1917))
so that it became possible to measure these small surface pres-
sures, and his work has been extended with great success by
Adam. In Adam's book, already cited, will be found an
account of his work with references to the numerous papers by
himself and his co-workers. In the most recent form of Adam's
apparatus surface pressures as small as 0.01 dyne per cm. can
be measured. Also a great many tests have been made with
substances which are simpler than oils and whose chemical
constitution is better known. It is possible actually to give the
results in terms of the surface pressure corresponding to the
area of surface covered by a known number of molecules.
Thus, for the normal saturated fatty acids, no trace of surface

568 RICE

ART. L

pressure was discernible until the area per molecule was reduced
to 22 sq. A.* At 20.5 sq. A the pressure was very marked and
increased very rapidly for further decrease. It was a significant
fact that these figures were not altered by using different acids
provided the long-chain molecule contained a sufficient number
of groups. It was this fact which led to the introduction by
Langmuir of his well-known theory that such molecules are
oriented into vertical or nearly vertical positions in the surface,
suggesting that the sectional area of such a molecule is about

20 sq. A. ^ As the volume of a CH2 group is known to be about
29 cubic A, this gives 1.4 A as an approximate measure of the
distance of one carbon atom from the next in the chain, a
measure substantially in agreement with the results obtained
by X-ray analysis. This conception illuminates the whole
subject. At the end of the fatty acid or alcohol molecules
there is the group OH or COOH which is very soluble in water.
This group tends to get into the body of the water, and although
not able to drag the whole of a very long molecule in also,- it
succeeds in "anchoring" the molecule as it were in an almost
upright position. In this oriented state the molecules adhere
laterally, and this adhesion keeps them together as a ''coherent"
film showing no sign of surface pressure as soon as each mole-
cule has about 22 sq. A room for its cross section. Thus there
are "condensed" films close-packed and strongly adhering, and
"liquid-expanded" films in which adhesion and packing are less
marked. In addition Langmuir found that certain films such as
those of the short-chain fatty acids were quite different in
behavior; these appear to lie flat on the surface — the argument
has been given earlier in connection with statistical considera-
tions— and to move about independently, resembling a two-
dimensional gas. Such "gaseous films" appear to exert a
pressure, by reason of a bombardment on the barrier due to
thermal movement, entirely analogous to the three-dimensional
pressure of an ordinary gas. Just as there are no "ideal" gases
so there are no "ideal" gaseous films; nevertheless the laws
which have been discovered to hold between the surface pressure

* 1 A (1 Angstrom unit of length) = 10 ~^ cm.

SURFACES OF DISCONTINUITY 569

of a given amount of gaseous film and the surface area over
which it extends resemble in form the laws for gases, such as
Amagat's and van der Waals'. Actually there appear to be
processes in the surface analogous to fusion and vaporization
and a whole new "two-dimensional" world seems to be open-
ing up.

So far these remarks have been concerned with films of
insoluble or nearly insoluble materials, and have had no direct
connection with adsorption from solutions, but in the paper
already cited Langmuir used Gibbs' equation to indicate that
similar conditions exist in adsorbed films. By using Szysz-
kowski's data on the relation between surface tension and
concentration he calculated from the adsorption equation the
amount adsorbed and thus obtained the area per molecule in
the film for various bulk concentrations of the solutions of the
very short-chain fatty acids, from 3 to 6 carbons in length.
He found that with increasing bulk concentrations this tended
to decrease to a constant value roughly consistent with what
might be regarded as the sectional area of the molecule, thus
suggesting that at the limit of adsorption there exists a close-
packed unimolecular film in the surface. For the most dilute
concentrations the film is, of course, much more sparsely occupied
by the adsorbed solute molecules, and these appear to have the
properties of a gaseous film. This^is easily shown from the
Szyszkowski formula

Langmuir, interpreting o-q — <r as the "surface pressure" (actu-
ally Traube suggested this interpretation for the lowering of
surface tension in these adsorption films long ago), writes it

F = aoh log (l + H

c 1 c^ 1 c^
a 2 a^ 3 o^

= <^o&^7-o7; + o3- etc.

570 RICE ART. L

If c is small compared to a

But

c da aob c

Rt dc Rt c -{- a

or, approximately,

Cob c F
^ ^ Rt a~ Rt

If A is the area occupied by unit mass of the adsorbed solute
then

FA = Rt,

since A = T~^. The analogy between this and the Boyle-
Charles law is obvious, and exactly the same equation can be
obtained by applying to this two dimensional phase the familiar
kinetic argument which derives that law for a three-dimen-
sional gas. These gaseous films, however, deviate in practice
from such an ideal law in the case of larger concentrations for
which the approximations used above are no longer valid. Ac-
tually the deviations resemble the deviations for gases. Thus

Rt " da dF

— = RtV = - 7- = c -7- •
A dc dc

Therefore

Rt^ _ d logF
FA~ d log c

PA _ d log c
Rt ~ dlogF'

From the tables of c and F (or ao — a), the values of FA/Rt
for various concentrations can be calculated and plotted against
F. The curves show a resemblance to the {PV /Rt, P) curves
for gases. (See Rideal's Surface Chemistry, Chapter II,

SURFACES OF DISCONTINUITY 571

page 65.) Indeed an equation analogous to Amagat's has been
shown by Schofield and Rideal (ibid., page 66) to represent with
some exactitude the behavior for all but the most dilute con-
centrations. It is

FiA - B) = xRt ,

where B is the limiting area of the unit mass of molecules when
crowded together in the unimolecular film, and x is a measure of
the lateral molecular cohesion, having a definite value not
greater than unity for each solute, and being smaller the larger
the cohesion. The values of B agree quite well with the values
suggested from other considerations. The equation is well
supported by its application to about a dozen solutes which
include the shorter-chain fatty acids and some alcohols. In so
far as it is valid it leads to an interesting equation as follows.
By the exact Gibbs' equation

dF = — d<x = Vdn .

Therefore

= B + xRt'l^.
t

Integrating,

li =

BF + xRt log F + constant.

If a is the activity of the solute

Rt log <

a = xRt log F + BF + constant

and so

fFB\
a = CF-exp(-).

Rideal states that this equation is in good accord with the

572 RICE

ART. L

precise surface tension measurements of Szyszkowski. A two-
dimensional analogue of van der Waals' equation has also been
suggested, but data do not appear to be available over a wide
enough range of temperature to justify a definite opinion.

26. Unimolecular Layers and the Dividing Surface

This use of Gibbs' equation and the consistency of the
information which it gives concerning the surface structure, is
strong evidence for its validity in the case of substances such
as the shorter-chain fatty acids. Indeed, this conception of the
unimolecular Gibbs layer may throw some light on the dis-
crepancies which have raised doubt concerning its validity.
There does not appear to have been any such idea in Gibbs'
own mind. Possibly he held the view which, with the weight of
Laplace's name behind it, seems to have been prevalent in his
day, viz., that the discontinuous layer, although physically
very thin, is nevertheless many molecules thick and shows a
gradation of properties as it is passed through. Yet if the layer
is really only a molecule or two thick, the placing of the dividing
surface becomes a somewhat perplexing matter. Indeed, the
whole physical theory of placing the "surface of tension" so as
to exclude the Ci 8ci + C2 dc2 terms in the original differential
equation becomes very doubtful. Earlier in this commentary
we have somewhat expanded Gibbs' presentation of this in
order to assist the reader to an understanding of his concise
formulation, and on referring to this again the reader will see
that the basis of it is hardly tenable for a unimolecular layer.
A very significant illustration of the point involved here will be
found in two well-known calculations made by Schofield and
Rideal (Proc. Roy. Soc., 109 A, 57, (1925)) ; they refer to alcohol
and pyridine. The data for the surface tension of mixtures of
water and ethyl alcohol from pure water to pure alcohol were
known from some work of Bircumshaw, and data for the partial
vapor pressure of ethyl alcohol could also be obtained so as to
give the activity and therefore the potential. With the aid of
these the surface excess of alcohol was calculated by the strict
Gibbs' equation for over a dozen mixtures between the extreme
limits. It was found that this excess rose very rapidly until it

SURFACES OF DISCONTINUITY 573

attained a maximum when the mol fraction of the alcohol was
about 0.25, and the value there corresponded to an area of 24
sq. A per alcohol molecule, which indicates a close-packed uni-
molecular layer of these molecules. Thereafter the surface ex-
cess rapidly fell, and when the mol fraction was 0.75 the surface
excess was apparently no greater than it was when the mol
fraction had a value well under 0.1; this value of siu-face excess
was apparently maintained for mixtures still richer in alcohol
right up to alcohol itself. Exactly similar results were obtained
for the surface excess of pyridine at the interface between mer-
cury and mixtures of pyridine and water, care being taken to
neutralize the electric charge which is known to exist normally
at a surface between mercury and water. Now it is highly
improbable that there is really a decrease in the surface excess
with increase in the proportion of alcohol or pyridine, and the
situation shows how troublesome the interpretation of Gibbs'
equation may become in particular cases. We have seen that
it does definitely point to the existence of a unimolecular layer,
and there is also evidence, which we shall touch on later in this
commentary, that at least partial orientation of the molecules
occurs as well (just as in the case of insoluble films). Now it
might happen that with increasing concentration of the alcohol,
the more polar water molecules being replaced by weaker alco-
hol molecules, there would be a decrease in orientation with an
increase in area occupied, caused by each alcohol molecule lying
flatter in the surface. But a more probable explanation has
been given by Rideal and Schofield, viz., that there is formed
below the outer layer of alcohol, a second layer of water. "In
the derivation of Gibbs' equation, the mathematical dividing
membrane X Y was so placed as to make the adsorption of the
water zero — that is, so that the average concentration of
water in volumes above and below XY were exactly equal to
those in the vapor and the liquid at a distance from the sur-
face. If there is a layer of water below the outermost layer
of pure alcohol, this will involve placing the dividing surface,
not below the alcohol molecules, but some distance above the
average level of their lowest points, perhaps more than half-way
up the molecules (owing to the thermal agitation this refers to

574 RICE

ART. L

the average position of the alcohol molecules)."* As the alcohol
in the bulk phase is supposed also to extend up to this dividing
surface with the bulk concentration, /or the purpose of calculating
r for the alcohol, such a gradual creeping outward of the surface
will have the effect of causing only a portion of the outer layer of
alcohol molecules to appear as "adsorbed alcohol." This illus-
trates very forcibly the difficulties that arise when we begin to
"tamper" with the dividing surface for the purpose of getting
rid of a term in the true adsorption equation for a binary mix-
ture, viz., (at constant temperature)

da = —Tidni — T2dn2-

If, however, we keep the dividing surface fixed, say at the depth
of the unimolecular layer, we can use the equation referred to
earlier,

"--('— S).

dfi

(the equation [515] of Gibbs, slightly modified). This in-
cidentally shows us how the right-hand side of the equation
diminishes with increasing alcohol concentration; for with an
accumulation of water molecules in the layer just inside the
fixed dividing surface, Fi is positive and increasing and 72V71'
is also increasing in the bulk phase. This is then a way of
stating the explanation, alternative to that using the moving
surface. It has been suggested by Bradley {Phil. Mag., 7,
142, (1929)) that an additional relation, which with the above
would enable us to determine both Ti and r2 could be obtained
from the alteration in the air-liquid electric potential difference
which is dependent on the electric moments of solvent and solute
molecules in the surface layer; this would of course change with
the change in the composition of the surface. The reader is
referred to this paper for further information.

The difficulty of the situation is clear, and it is possible that
similar considerations may be brought to bear on all the
apparent failures of the Gibbs law. Unfortunately it is not

* Quoted from Adam's book, p. 131.

SURFACES OF DISCONTINUITY 575

easy to see how this can be done in connection with the work of
McBain and his colleagues. There the difficulty is different from
that just dealt with. The experiments on amyl alcohol, for
example, show that the measured amount at maximum adsorp-
tion was so great that if it were packed in a unimolecular layer

the area was only 14 sq. A for each molecule; in the case of
sodium oleate only 1 1 sq. A. It is impossible for these molecules
to be packed so tightly in a layer one molecule thick. It may be
possible, as we have stated earlier, that there may be a uni-
molecular layer with the additional material forced out into
small droplets above it here and there, the unimolecular layer
being the true adsorption agreeing with the adsorption equa-
tion. But clearly these difficulties still await solution. It is
interesting to note that a somewhat similar situation exists in
connection with insoluble oil films. The evidence for uni-
molecular layers is strong, yet there can be no doubt that the
area of an oil film can be reduced until there is no longer room
for all the molecules at their closest possible packing. The
suggestion is that the film gives way under tangential squeezing,
buckles and expels enormous numbers of molecules to form local
ridges, the rest of the film being unchanged. Adam in his book
hazards the opinion that the cases of "polymolecular" films such
as those obtained by Harkins and Morgan {Proc. Nat. Ac. Sci.,
11, 637, (1925)) are really examples of "partially collapsed uni-
molecular films, with the excess material collected into lumps
much thicker than the film itself."

X. Desorption

27. Unimolecular Layers and Negative Adsorption

If a solute raises the surface tension of a solution above that of
the pure solvent, the Gibbs' equation shows that the calculated
value of r2 (Fi being made zero as usual) is negative. This is
interpreted by saying that the surface is poorer in the solute
than the bulk phase or (alternatively) richer in the solvent. In
the nature of things "negative adsorption" cannot reach such
large numerical values as the positive ; obviously it cannot exceed
the bulk concentration of the solute divided by the thickness
of the layer in numerical amount.*

* See Gibbs. I, 274.

576 RICE

ABT. L

The data available are not numerous, and concern aqueous
solutions of familiar inorganic salts such as the chlorides,
nitrates and sulphates of familiar metals. A table of results will
be found on page 74 of Rideal's book; these indicate that for a
given salt the increase in surface tension above that of water
varies in an approximately linear manner with the salt con-
centration. Langmuir has considered these results also from
the point of view of a unimolecular layer. The quantity of
solute which has gone out of the surface film so as to leave it
poorer in the solute than a corresponding volume of solution is
cf per unit area (where c is bulk concentration and f the film
thickness) if a film of pure water one molecule thick exists at the
surface. Hence on this hypothesis cf should equal — V obtained
by the equation

da
da

d{kc)
dc

= — kc ,

since, as we have stated, o-q — o- is approximately equal to —kc,
where A; is a constant. Hence f can be calculated. This should
be the thickness of an adsorbed water layer on the surface. Lang-

muir found f to be from 3.3 to 4.2 A, which is certainly the right
order of magnitude for a water molecule if it is not of an elongated
shape. More recent work by Goard, Harkins and others, using
the accurate form of Gibbs equation, finds varying values for f

which decrease from Langmuir's value between 4 and 5 A at

low bulk concentrations to about 2.5 A at high concentrations.
Adam suggests that this diminution may be due to the increas-
ing tendency of the solute to diffuse into the surface layer as
the bulk concentration increases.

The evidence for the truth of Gibbs' law in connection with
the hypothesis of unimolecular layers is imposing, and one
further remark may be made with reference to the cases of
apparent failure in the attempts to verify it by direct
measurements.

SURFACES OF DISCONTINUITY 577

It has already been stated that recent research shows the
existence in the case of numerous substances of surface films in
different phases, liquid, solid, gaseous, which can occur under
stable conditions at definite temperatures. If this is so the
surface layer may not always be a single phase of the whole
system ; we may have at times to consider it as a system of
phases and treat them thermodynamically just as we treat the
different homogeneous bulk phases. The usual procedure
would show that the potential of each component would have the
same value in each surface phase, still agreeing with the value
of this component in each of the homogeneous bulk phases.
Actually, in the subsection of Gibbs' treatise which deals with
the stability of surfaces of discontinuity, he considers the
possibility of a part of the surface being changed in nature while
the remaining part remains as before, and the entropy, total
masses and volumes of the whole system remain unchanged.
The changed part is to be uniform in nature and still to be in
equilibrium with the adjacent bulk phases. We shall com-
ment on this presently; but in the meantime we may antic-
ipate and mention the conclusion to which Gibbs comes (page
240). If two films of the same components can exist between
the same homogeneous masses, having the same temperature and
potentials as the homogeneous masses have for the components
in those masses, and the same potentials for components only
existing in the surface, then the film which is most stable is the
one with the smaller tension. Consequently in a stable film
consisting of two or more surface phases the surface tension for
each must be the same, for if one phase had a greater surface
tension than the other it would disappear on the slightest dis-
turbance of equilibrium. Suppose, therefore, that with increas-
ing bulk concentration there comes a point when a part of a
hitherto gaseous film begins to condense into a liquid film. This
seems to be a natural way of imagining the creation of a close-
packed unimolecular layer. A small addition of the solute to
the bulk phase would not result in an increase of bulk con-
centration; all the material would go to the surface gradually
increasing the extent of the liquid surface phase which en-
croaches on the gaseous; during this period there would be no

578 RICE ART. L

increase of <j or of the potential of the solute. Supposing now
that <7a is the value of the surface tension and /X2a the value of the
potential of the solute at a concentration below this transforma-
tion point, and ah and )U25 values above it, then {<ja — cb)/
(m26 — /X2a) would be intermediate in value between the surface
concentration of the gaseous jSlm and that of the liquid film.
The essential point is that it would be less than the actual con-
centration in the liquid film. This is a somewhat enlarged
version of an explanation suggested by Rideal in his book, on
pages 51 and 52, to account for the fact that V observed is
nearly always greater than r calculated.

28. The Recent Experiments of McBain and Humphreys on
Slicing Off a Thin Layer at a Surface

Note-. Just as this manuscript is going to press the writer has
read in the Journal of Physical Chemistry, 36, 300 (1932), a
preliminary account by McBain and Humphreys of some fresh
experiments in progress on the determination of the absolute
amount of adsorption at surfaces of solutions, and if subsequent
results follow the indications given by these then it may be
said that very dependable evidence for the truth of Gibbs' law
by du-ect measurement is at last available. The apparatus is
extremely ingenious, and is novel in that for the first time a
static surface is involved and not one which is in motion, as in
the experiments with bubbles; the criticisms levelled against the
latter have been referred to above.

Briefly, the solution is at rest in a shallow trough of silver
surrounded by a saturated atmosphere. The ends of the
trough are paraflSned, so that the solution is made to bulge up
above them without overflowing. A uniform layer 0.05 to 0.1
mm. thick is cut off from a known area of the surface by a small
microtome blade travelling at a speed about 35 feet per second.
This layer is collected in a small silver-lined cylinder, on which
the blade is mounted, and is weighed, its concentration being
then compared with that of the bulk solution by means of a
Lewis interferometer. From the observed difference of con-
centration the adsorption can be calculated. Extraordinary

SURFACES OF DISCONTINUITY

579

precautions appear to have been taken against every conceivable
source of error.

The following results of preliminary trials indicate the very
satisfactory agreement now obtained between T calculated and
observed. It is no longer a matter of agreement of order of
magnitude, or a ratio between 2:1 and 4:1.

Substance

p-Toluidine

Phenol

Caproic Acid

Hydrocinnamic Acid
Hydrocinnamic Acid

Concentra-
tion
(grams per
1000 grams
H2O)

r X 10» obs.

2.00

6.1

1.76

4.6

20.48

4.1

2.59

6.8

3.00

5.1

5.25

6.2

1.5

5.6

4.5

5.4

r X 10* calo.

5.2
4.9
4.8
6.3
6.5
6.3
5.1
7.9

XI. Adsorption of Gases and Vapors on Liquid Surfaces

29. Form of Gihbs' Equation for Adsorption from a Gaseous

Phase

Hitherto we have considered the experimental tests on ad-
sorption from the liquid side of a gas-liquid interface; but we
must make some reference to the work carried out on adsorption
from the other side. In such experiments it is convenient to
replace the potential of the gaseous component by its pressure
in the adsorption equation. The theory of such transformation
of variables is given very completely by Gibbs (I, 264-269).
For our purpose we need only consider the part on pages 267-
268 which leads to equation [581]. We shall suppose that the
single accent phase is the liquid, the double accent the gaseous,
that component 1 is the liquid, ai-i component 2 the gas or
vapor adsorbed. From equation [581] we see that if <r is re-
garded as a function of the temperature and pressure, then

— . >

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"

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 <r is plotted against the
partial pressure of the vapor (in the vapor-air mixture) the result
is a straight line; hence F = kP, where A; is a constant. From a
knowledge oi k, Fg the amount adsorbed when the partial pres-
sure is equal to the maximum vapor pressure at the temperature
of adsorption could be determined. His comment on the re-
sults is as follows: "The fact that a linear relationship holds
between the partial pressure of the vapor and F right up to
the value F„ and also that this quantity changes so rapidly with
increasing temperature, indicate clearly that a limiting condi-
tion, such as would obtain if a closely-packed adsorbed layer
were formed, had not been reached." He also shows from his
calculations that the values for the area occupied by one mole-
cule of benzene is larger than Adam's value for a closely packed
layer of certain benzene derivatives on a water surface. "In
this case, clearly an unsaturated layer is formed." His pro-
portionality factor decreases as the temperature at which the
experiments are performed is raised. This indicates that ad-
sorption is accompanied by an evolution of heat, but we shall
postpone the discussion of this matter until we reach the com-
mentary on the subsection dealing with thermal effects.

584 RICE ART. L

This work was carried out at a water surface. However,
Micheli also experimented with a mercury-air interface, sub-
stantially confirming Iredale's conclusion that at this surface
the amount adsorbed approaches a definite limiting value as the
partial pressure of the hydrocarbon vapor increases, and that
the values of the area per molecule obtained from T, in this
case agreed well with Adam's value for benzene already referred
to and, in the case of pentane, hexane, heptane and octane,
with the values also found by Adam for closely packed mole-
cules of straight-chain alphatic acids or alcohols oriented at a
water-air interface so that the OH or CO OH groups are attached
to the surface. One feature of Iredale's work with benzene,
Micheli did not obtain; this concerned the point of inflexion
obtained by Iredale on his (a, log p) curve for benzene, indicat-
ing a maximum adsorption before saturation was reached, with a
subsequent diminution. Micheli remarks that such an effect
would not be probable, and draws attention to the curves of
Schofield and Rideal concerning the adsorption of ethyl alcohol
from an aqueous solution on which we have already commented
and where a similar apparent maximum exhibits itself. Micheli
states that the observed maxima really indicate that conditions
exist which render the Gibbs equation inapplicable, and is
obviously suggesting that we must look for an explanation of
Iredale's result, if it really exists, along the lines already referred
to in our previous comments on this point.

It is interesting to observe that Gibbs' own footnote on page
235 is concerned with adsorption from a vapor phase. He
quotes some figures of Quincke for the tension of mercury in
contact with air (which he takes to be practically the same as
for contact with its own vapor free of water vapor), and for the
interfacial tension of water and mercury and of water and its
vapor. They are, when expressed in present-day units, 539,
417 and 81 dynes per cm. Assuming that the tension of
mercury in contact with the saturated vapor of water is the sum
of the two latter, which is tantamount to assuming that at
saturation pressure of water vapor the adsorbed film is begin-
ning to have the properties of water condensed in mass, the
reduction in the tension of mercury by adsorption of water

SURFACES OF DISCONTINUITY 585

vapor is 41 dynes per cm. for an increase of water vapor pressure
of 1.75 cms. of mercury. There he leaves the calculation, but
had he known, as we know now from Iredale's and Micheli's
work, that the fall in tension is proportional to the increase of
vapor pressure, he could have finished the calculation for the
amount of the layer adsorbed just at saturation before actual
condensation into a genuine liquid water phase begins. For

da - 41

dp 1.75 X 981 X 13.6
and

dff _ 17 A X 10-« X 41
^ ~ ~ '^ dp~ 1.75 X 981 X 13.6

= 3 X 10~^ (grams per sq. cm.),

since y, the density of water vapor at 20°C, is 17.4 X 10~^ grams
per c.c. This is just the figure for a unimolecular film of water
molecules, but there is no doubt that no such conception was
in his mind. Indeed, the assumption he makes above shows
this. Iredale in one of his papers has some very interesting
remarks to make on the general theory of adsorption and con-
densation, but reference to them will be deferred until we have
commented on the subsection of Gibbs' work which deals with the
formation of a new phase at the interface between two phases,
since Iredale's comments involve the theoretical considerations
in that subsection.

Another very interesting set of experiments were carried out
by Oliphant (Phil. Mag., 6, 422, (1928)). His apparatus was
adapted from one invented for another purpose by Schofield.
He found that an expanding mercury surface selectively adsorbs
carbon dioxide from a mixture of that gas with an excess of
hydrogen or argon, and that at all concentrations above 2 per
cent the carbon dioxide thus selectively adsorbed was constant
at a value about 6.5 X 10^'^ molecules per sq. cm. This very
nearly corresponds to a close-packed unimolecular layer.
Actually, Schofield's method does not involve the yda/dp rule
or the measurement of da/dp. It should be mentioned that

586 RICE ART. L

Bircumshaw (Phil. Mag., 6, 510, (1928)) has found that the
surface tension of mercury in contact with such gases exhibits
some anomahes with lapse of time which have not yet been
explained. Finally, reference should be made here to the ex-
cellent work of H. Cassel and his collaborators on the adsorption
of gases and vapors on mercury and water surfaces (Z. Elektro-
chem. 37, 642 (1931); Z. physik. Chem., Aht. A, 155, 321 (1931);
Trans. Faraday Soc, 28, 177 (1932); Kolloid-Z., 61, 18 (1932)).

XII. The Thermal and Mechanical Relations Pertaining to the
Extension of a Surface of Discontinuity

SS. Need for Unambiguous Specification of the Quantities Which
Are Chosen as Independent Variables

In this subsection Gibbs makes use of the results obtained in
the previous subsection of his work, to which we have already
referred at the beginning of the part of the commentary just
concluded. The results are in equations [578], [580] and [581].
When there is one component in two homogeneous phases and a
surface of discontinuity, o- is a function of t and n (the one
potential involved). The transformation effected at the
bottom of page 265 still leaves it a function of two variables t
and p' — p". If the surface is plane there is only one variable,
t, involved; this is obvious in any case since with only one
component in two phases, say vapor and liquid, p is a function
of tf and of course o- is also.

Equation [580], which refers to two components in two homo-
geneous phases, and equation [581] are framed as if cr were
again a function of two variables, and yet a is originally regarded
as a function of three, viz., t and the potentials ni and m2 of
each component. The reason is clear. Equation [579] shows
that there are really three variables involved, t and the two
pressures; but since the surface is regarded as practically plane,
the difference between the two pressures is ignored. Actually,
since the surface is plane and p' = p", this gives us an equation
between two functions of t, ni, /X2 and thus /Lt2 is a function of t and
Hi and is not an independent variable; so o- is really a function
of t and Ml or t and p. It would be a great assistance to students

SURFACES OF DISCONTINUITY 587

of thermodynamic texts if writers would cultivate the habit of
indicating by bracketed symbols just what quantities are being
considered as the variables upon which the physical properties
being discussed are dependent, — at all events in circumstances
where ambiguity might otherwise easily arise. For example, in
the present instance, a regarded as depending on t, ni, would
be written as cr(f, ni), meaning the function of the variables t, m
which is, for any given values of t and mi, equal to the value of
the surface tension at these values of temperature and the
potential of the first component. On the other hand <t regarded
as depending on t, p would be written as (r(f, p). Of course it
would be implied in such a convention that the functional form
of <T{t, Ml) would not be the same as (r{t, p). Actually, to satisfy
the requirements of a strictly rigorous use of mathematical
symbolism we should write the two functions, which both repre-
sent the same physical quantity, in different ways, say f{t, m)
and g{t, p); but the situation does not really demand such rigor
and there is an advantage in indicating just what physical
quantity is being represented, provided the implication referred
to is kept in mind. Such a symbolism when combined with the
modern partial differential coefficient notation (the use of d
instead of d, not in use when Gibbs wrote his memoir), would
also clearly indicate what quantities are being regarded as con-
stant in any particular differentiation, so that the use of the
subscript after a bracket (the usual method of the thermo-
dynamic texts) would be unnecessary. Thus in equation [593]
{da/dt)p would be da(t, p)/dt and {da/dp)t in [595] would be
d<T{t, p)/dp. In [587] and [592] the differentials are total differ-
ential coefficients. Gibbs makes a special reference to this
point at the top of page 271. With only one component, say
a liquid and its vapor, p is a function of t, and <r can be re-
garded either as a function of p only or as a function of t only
and written accordingly a{p) or ait) as the case may be; so that
in [587] the total differential coefficient symbol would still be
correct and we would write it as

588 RICE

ART. L

In [592] p is a function of t, and so Hs a function of p and we
have

da{t) dt(v)

and the right hand side by a well-known proposition of the
differential calculus is equal to

da(p)

The reader whose acquaintance with mathematical technique
may be limited should not regard these remarks as idle comments
on mere mathematical "niceties." Actually, if the method
suggested were more widely used, and not merely in thermo-
dynamic texts, it would conduce to clarity of exposition and
consequent ease of understanding on the part of the reader.

S3. Alternative Method of Obtaining the Results in This Section.

Total Surface Energy

The methods by which Gibbs arrives at the results of this
section are easy to follow and eminently physical. It may not
be out of place, however, to obtain them by a more analytical
method which will also help to illustrate the remarks just made.

Thus the energy of the whole system consisting of two phases
and surface of discontinuity with n components is a function of
the variables 77, v, s, mi, 1712, . . . nin, since

€ = tri — pv -{- as -\r Mi^i + M2W2 . . . + finmn*
and

de = tdr] — pdv + ads + iJ-idmi + fi^drn^ . . . + fJ-ndnin.

We should write the functional form which represents the energy
in these variables as e(r], v, s, mi, m2, . . . ) but actually, with the
assumption of a practically plane interface, we have an equation

p'{t, Ml, M2, . . .) = p"{t, m, /i2, . . .) .

* See Gibbs, I, 240.

SURFACES OF DISCONTINUITY

589

This gives us one equation between the variables and so we
can reduce them from w + 3 in number to n + 2; the most
convenient set of variables is then r/, v, s, n, Vi, . . . r„_„ where
7-j = mi/nin, n = nii/nin, etc. So we write the function for c as
i{rj. V, s, n, r2, . . . ) and

de{-n, V, s, ri r2, . . . ) = tdr{ — pdv + ads + vi dn + va drj + . . . ,

where vi, v-z, etc. are functions of r], v, s, n, r^, ...
The other three Gibbs functions are then

yP{t, V, S, ri, rg, . . .) = e - tr],
^(t, p, s, n, rz, . . .) = e - tr] + pv,
xiv, P, s, n, rg, ...) = e + pv,
with the differential equations

d\l/{t, V, s, n, r2, . . .) = —vdi — pdv

+ ods ■\- vidn -\- . . . ,
d^{t, p, s, n, ra, . . .) = -vdt -{- vdp

+ <^ds -\- vidri -{■ . . . ,
dxiv, P, s, n, ra, . . .) = tdri + vdp
+ ads -\- vidri -\- . . .

From the second of those we have
9f («, P, s, ri, ra, ...)

= - iC^, P, s, n, ra, ...)

and

d^{t, p, s, n, ra, ...)
ds

By cross differentiation
dr]{t, p, s, ri, ra, .

a{t, p, s, n, ra, ...)*.

dajt, p, s, n, ra, ■ ..)
dt

(22)

Actually <r is only dependent on t, p and n — 2 of the ratios n, ri, .

590 RICE ART. L

This is equation [593] ; the left-hand side is the rate of change of
entropy with increase of surface, while t, p and the composition
of the masses are unchanged (this is the condition stated in
the paragraph preceding [593] in Gibbs), and so is equal to
Q/t. In the right hand side the variables p, n, r^, ... are kept
unchanged in the differentiation; in Gibbs' case no ratios occur
in the variables on which a depends, since he is dealing with two
components only and there would only be one ratio r, and even
this does not appear since we have just stated in the footnote
that in general <r depends on only n — 2 of the n — 1 ratios
ri, r2, ... as well as t and p. Indeed c depends only on n
variables; for we know it can be expressed as a function of
/, m, y.i, ... nn in general, but the assumption of the equality of
pressures in the two phases reduces the number of variables to n.
The addition of Q to cr gives the total energy acquired by the
surface when extended one unit of area if the temperature,
pressure and composition of the phases remain unchanged.
This quantity

<r(f, p, r) - « -^ .

is sometimes called the total surface energy, a being called the
free surface energy. With the exception of a few molten metals,
liquids exhibit a decreasing surface tension with rising tem-
perature, and so as a rule total surface energy is greater than free
surface energy. In many liquids the relation between a and t
is linear, so that the total surface energy does not vary with
temperature. Actually, if the variation is not zero, we can easily
see that the ordinary specific heat of a liquid will vary with the
extent of surface offered by a definite mass of it which will
change with a change of form in the mass. For

^ d^vjt, p, s, r) ^ d dr}{t, p, s, r)
dt ds ds dt

* For brevity let r stand for the series n, u, ... r»_i.

SURFACES OF DISCONTINUITY 591

Now tdrj/dt is the ordinary heat capacity of the mass of the
fluid, and if the left hand side of the equation is not zero, the
specific heat will depend on s.

The fact that extension of the surface of a liquid (all the other
variables remaining constant) involves cooling in most cases (as
is obvious since, in general, heat must be supplied to maintain
the temperature constant) can be seen very easily from mechan-
ical considerations. We can imagine the system of liquid and
vapor to be contained in a flexible but non-expanding enclosure
which will permit a change of extent of surface without altera-
tion in volume, etc. of the two individual phases. In enlarging
the surface some molecules must pass from the interior to the
surface; i.e., must travel through the molecular cushion against
the inward attracting field of force there. This involves an in-
crease of potential energy, and with no supply of energy from
without there must be a diminution of molecular kinetic energy,
which means a fall of temperature.

The equation [593] or the form which we have given it above
can be written in another form involving the total surface
energy. Thus

a(t, p,r) - t = <x{t, p,r) +t —

Also, by the third equation of (22), we see that

dxjt, V, s, r) dyjt, p, s, r)

^s ^=^ ^s + ^(''P'^>'

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)

(^ = ki^-^) (tc-t- d),

where 5 is a small number, approximately 6. The "Eotvos
constant" k, they found to be 2.1. However, later research
has shown that the number k is not a constant for all liquids,
and that the use of this law as a method of measuring degree of
association is not reliable. Other suggestions have been made,
such as one by van der Waals based on thermodynamical
reasoning, viz.,

o-Q

(■ ^ !)■

In this equation n is a constant for all liquids and

o-o =A;(pc^O%

where k is a constant for all liquids and pc and tc are critical
pressure and temperature. Experimental research shows that
this result also is not exact; although n for a number of
common organic liquids does not vary by more than a few per
cent from 1.21. Katayama (Set. Reports Tohoku Imp. Univ.
[1], 4, 373 (1916)) has suggested a modification of Eotvos' law
in the form

•m

^ = \~iir) ^^^ - ^^'

where d is the density of the vapor; and actually an elimination
of tc — t from this and the equation suggested by van der
Waals, taking 7i to be 1.2, gives a relation

a = C{D - dy

594 RICE

ART. L

discovered empirically by McLeod (Trans. Faraday Soc, 19, 38,
(1923)) which holds accurately for a great number of organic
liquids over a wide range of temperature. In it C is a con-
stant, different for each liquid, and the relation is of great value
in comparing densities.

As stated, these relations all have an important bearing on
molecular complexity in liquids, a problem which still awaits
solution. In applying them it is assumed that M changes with
temperature since with increasing temperature polymerized
molecules tend to dissociate into the simple molecules which
exist in vapor, and the assumed truth of the expressions enables
relative values of M to be found at each temperature. Although
too great reliance cannot be placed on the conclusions deduced,
Bennett and Mitchell (Zeit. phijsik. Chem., 84, 475, (1913) and
Bennett {Trans. Chem. Soc, 107, 351, (1915)) have shown that
the ''total molecular surface energy"

(-9

(Mvy

is a better quantity to use for this purpose than the "free
molecular surface energy"

of Eotvos, and that this leads to more consistent conclusions
concerning molecular association.

35. Heat of Adsorption

Returning to Gibbs work, the reader will find on pages 271,
272 a reference to the "amount of heat necessary to keep the
phases from altering while the surface of discontinuity is ex-
tended." If dcr/dt is negative, as appears to be the general
rule, this heat is positive and if not supplied the temperature of
the surface will fall, causing an increase of tension. Actually, if
da/dt were positive, an increase of tension would also occur since
in this case the heat would be negative, so that if transfer of
heat were prevented the surface would warm up. Now this
heat must be carefully distinguished from "heat of adsorption,"

SURFACES OF DISCONTINUITY 595

which is heat required to get rid of adsorbed molecules on the
surface, and bears some resemblance to latent heat. We can
best illustrate its nature by a reference once more to MicheH's
work on adsorption of gases (Phil. Mag. 3, 895 (1927)). As
stated earlier, he showed, if P is the partial pressure of the
vapor, that

r = kP,

and his results also show that for a given vapor over the water
surface the constant k decreases markedly with rise of tem-
perature. Thus for pentane at 25°C., A; is 75 X 10-^; at 35°C.
it is 35.8 X 10-^; for hexane the decrease is from 106 X 10"^ to
55.5 X 10-^ and for heptane from 256 X 10"^ to 115 X 10-^ a
rise of ten degrees roughly halving the value of k in each case.
This means that a rise of temperature causes desorption, the
partial pressure P being kept constant. Thus desorption
requires heat and adsorption is accompanied by an evolution of
heat. We can, of course, use the well-known Clapeyron rela-
tion to obtain this molecular heat of adsorption. Thus from
the equation

d log P„

heats of adsorption can be calculated in the same way as latent
heats are calculated, where P„ is the partial pressure of the
vapor when n mols are adsorbed per unit area and Hn is the heat
of adsorption at constant temperature and pressure at the same
stage of adsorption. If P„, and P„2 are values of P„ at the tem-
peratures ^1 and ti, then as a first approximation

_ R k t2 (log Pm - log Pn2)

h — h

Also, if ki and ki are the values of the constant k for ti and U,
kiPni = k^P n2 , and therefore

_ R h tj (log ^2 - log ki)
ti — ti

596 RICE ART. L

Thus "the heat of adsorption is independent of the particular
value of n, so that each equal increment in the amount adsorbed
is accompanied by the same heat evolution. This is, of course,
only possible when the adsorbed layer is so diffuse that the
amount already adsorbed has no effect on further adsorption."
Micheli calculates the heats of adsorption for pentane, hexane,
and heptane, and finds them to be 13.6, 11.7, 14.6. He notes
that the probable error may amount to 20 per cent and so he
takes the three results to be roughly the same ; at all events they
do not show any sign of increasing with the number of carbon
atoms in the molecule; even an accuracy of only 20 per cent
precludes that possibility. From this he concludes that the
molecules do not lie flat on the surface, for then we should
expect the heat of adsorption to be roughly proportional to the
number of carbon atoms in the molecule. (The reader will
recall a similar line of argument by Langmuir in section VII
of this commentary.) * 'These considerations, then, furnish some
additional support for the conclusion that an end CH3 group
forms the only point of attachment to the water surface."

A good deal of work on heat of adsorption and "heat of
wetting" has been carried out at the interfaces between solids
and gases or vapors, but reference to this will be deferred until
we reach the subsection of Gibbs' treatise which deals with
solid-fluid interfaces.

S6. Dependence of a on the ''Age" of a Surface

With reference to the subject discussed on pp. 272-274 of
Gibbs' work, namely, the effect on the surface tension of creat-
ing a fresh surface, it may not be out of place to mention the
suggestion sometimes made, that because ordinary liquids, even
"pure," are constituted really of different molecules (since they
differ in degree of polymerization or chemical activity) they
should display a surface tension different at a fresh surface from
that which would exist there some time after formation. This
argument is clearly based on the adsorption law and the assump-
tion that there are at least two types of molecules in the liquid,
one of which produces a higher surface tension than the other.
On forming a fresh surface, the composition of the surface layer

SURFACES OF DISCONTINUITY 597

would be identical with that in the interior, but with lapse of time
the molecules affording the greater surface tension would tend
to leave the surface and a fall in surface tension would be
observed. In his book (page 152) Adam criticizes the experi-
mental evidence which has been brought forward to substantiate
the hypothesis on which this argument is built, pointing out
that the purely mechanical effects of the appliances employed
could easily account for the initial elevation of water in a
capillary tube apart from the effect of the postulated increase
of surface tension at the beginning. He states that until
apparatus capable of dealing with liquid surfaces not older than
0.005 sec. has been devised, the question cannot be regarded as
settled. Undoubtedly contamination produces change in
surface tension.

XIII. The Influence of Gravity

S7. The Variation of p, a, m, p.2, • • • with Depth in a Liquid.
An Apparent Inconsistency in Gibbs' Argument. The

Argument Justified

Before proceeding to consider the question of stability of
equilibrium it will be well to dispose of the subject of equilibrium
in itself and to proceed at once to deal with the subject matter
treated on pages 276-287 of Gibbs' treatise which is the natural
continuation of the considerations raised earlier on pages 144-
147. The conditions obtained there still hold, with the addi-
tional important equations [614] and [615]. There is a certain
economy in the notation at the outset of this subsection. In
[599], for instance, djDe^ stands really for the sum of a number
of terms such as

8fDe'y + SfDe"^ + SfDe'"^ + . . .

each referring to one homogeneous mass, while 8j^gzDm^ is a
double sum such as

dfgzi' Dm,' y + bfgzi' Dm-l y ...

+ 8fgz,"Dm/'y + hfgz^'Dm^'y . . .

+ etc.

598 RICE ART. L

Also Sj'Des consists of as many terms as there are surfaces, and
similar remarks apply to Sj'gzDm^. It will be quite sufficient
to limit the system to one with two homogeneous masses and one
surface of discontinuity.

The transformation from the equation [599] to [600] is one
which calls for careful scrutiny on the part of the reader. The
difficulties are hinted at in the beginning of the paragraph
succeeding equation [600], but perhaps the fact that they are
fully met in the transformation may not be so "evident" to
every reader as it was to Gibbs. Take for instance one inclu-
sive term such as Sj'De^ in [599]. (We omit accents and
consider it as referring for the moment to either homogeneous
mass.) We know that

De^ = tDr]^ - pDv + fiiDmi^ + mDmz^

and so Sj'Di^ should apparently be equal to

SftDriy - 8fpDv + SfniDmiy + 8fnJ)miy.

But if we carry the sign of variation, 5, within the sign of
integration, we ought in strict mathematical procedure to write
Sj^tDrjy as J'ditD'qy), bfpDv as J'8(pDv) and so on. Instead
they are written J'tdDn], J'pSDv, etc. Later, near the top of
page 280, XpbDv is transformed back into J'dipDv) —J'SpDv,
and to the unwary this might seem to a veritable "trick"
in order to get the first two terms of equation [611] and
thereafter the equation [612]. The matter seems still more
mystifying when we consider an inclusive term in [599] such as
Sj^gzDmy] for it is written J'SigzDmy) and expanded to
J'gzSDmy + SgSzDmy, and not merely left as equivalent to
the first integral of that sum. However, the solution is not
obscure when pointed out. Looking back to [15] and [497] we
recall that the conditions of equilibrium without gravity are not

8{t7]) - 8{pv) + Simmi) + Sifjuiui) = 0
but

t8r} — p8v + fii8mi + iJizSnii = 0.

SURFACES OF DISCONTINUITY 599

When we take gravity into account p, c, ni, m2 are no longer
constant throughout a given homogeneous mass or on a given
dividing surface; they are now functions of position as well as of
7], V, s, Wi, W2, and vary in value from point to point in one phase.
Considering a given infinitesimal element of volume Dv in the
unvaried state, it will change in size to Dv + ^Dv and will
move to a new adjacent position; the value of p at its new
location will vary not only for the reasons which would cause
variation without influence of gravity and which have been
involved in the earlier treatment of equilibrium, but also because
it has moved to a different position; and the veriest tyro in
hydrostatics knows that if a difference of level is produced pres-
sure will vary. This is where great care is needed; when
p8Dv is written in [600], the 8Dv is multiplied by the pressure
which existed where the element Dv was situated before the varia-
tion was conceived to take place. A similar remark applies to
fynbDmi^, fcdDs, fixiWmi^, etc. Now the term fpWv
should be written

fp'bDv' + fp"bDv", (23)

and considering the first integral in this we can regard it as the
sum of two parts, one in which the varied positions of the Dv'
still remain in the volume occupied by the single accent phase
before variation, and one in which the varied positions of the
Dv' are to be found between the original and varied situations
of the dividing surface. To evaluate the first part we shall for
the moment represent the element Dv' before variation by
Dva and after by Dvh, Dva being situated at x, y, z and Dvb at
X -\- 8x,y -\- 8y,z -{- 8z. The value of the first part is equal to

fpa'iDvb' - DvJ) = fp^Dvi' - fpa'DvJ.

But since the extent of integration is the same for the second of
these as for the first we can write this equal to

fp^Dv,' - fpiIDVi! = /{pa' - Pb')DVi'

= f{v'{^i y, 2) - p'ix -^ 8x,y -h 8y, z + 8z)]Dv'

/I'

' ^ aa; dy ^ dz J

600 RICE ART. L

Thus we have the first part of the integral fp'bDv' in (23).
The second part will be the integral fp'bNDs throughout the
region between the two positions of the dividing surface, for p'
is the pressure which existed where the element bND8 was before
it moved into the region originally occupied by the double
accent phase. Hence in (23) f'p'bDv' is equal to the sum of a
surface integral, and a volume integral, viz.,

In the second integral of (23) we must, in the same way, first
integrate

dv" dp" dp" \ ^ „

throughout the original region occupied by the double accent
phase and then subtract from this the surface integral fp"bNDs.
Thus we find that

fp'bDv' + fp"bDv"
= j{p'- p") bNDs - /(!' bx + I' by + f bz) Dv'

dp" dp" dp" \ ^ „

for which [609] is a condensed form.

With reference to the term fabDsy we see in just the same way
that it is equal to the change produced by the variation in the
integral faDs, minus the value of the integral fbaDs, where
h(T is given by [608]. The term bfaDs consists of two
parts. To see this, imagine normals drawn to the surface s
at points on the boundaries between the various elements Ds.
The normals projecting, as it were, from the boundary of a given
Ds will form a tube which will cut out on the varied position of s
a corresponding element of area whose size isDs[l + (ci + c^) bN]*
All the original elements of s will thus mark out a defined

* See the note on curvature p. 12 of this volume.

SURFACES OF DISCONTINUITY 601

(and much the larger) part of the varied surface. This con-
tributes to the variation of SaDs the amount S^i^i + C2) 8NDs.
But around this large portion of the varied surface there
will be a narrow "rim" bounded by the varied position of
the perimeter and by the line obtained by projecting the perim-
eter of the original surface on the varied position of the
surface. Between an element Dl of the varied perimeter and
the corresponding element of this projection there is a distance
8T, where 8T is the "component of the motion of this element
which lies in the surface and is perpendicular to the perimeter."
Thus this rim can be conceived as consisting of elements of area
dTDl, and we obtain in this way the second contribution to
dJ'aDs, viz., SabTDl; thus we see that

/ cbDs = / o-(ci -f C2)8NDs + / adTDl

f/da- da \ ^

J \oa>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

[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

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

^22

^23 .

. . azn

031

^32

^33 .

. flan

dnl

anz

* 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" - <j')Ds + (m/' - n,')Dm/' + (m." - HK')Dm/' + . . . (25)

This is the expression which occurs just a little below the middle
of page 241, stated for a small portion of the film Ds. If this
is a positive quantity for all changes, infinitesimal or finite, the
system is stable. To discuss instability we must consider two
different cases. The expression (25) may be negative even
when Dwo*', . . . Dmg^', . . . differ by infinitesimal amounts
from Dma^", . . . Dmg^" , . . . and therefore nj, tih, . . . o-' differ
by infinitesimal amounts from Hg", nh", . . . <r". If this be so,

SURFACES OF DISCONTINUITY 611

the system is definitely unstable in the first state. However,
it may be possible that the expression (25) is positive for
infinitesimally small values of a" — a', iij' — nj, nh" — iJ-h, etc.,
but would be negative for finite values of these changes. The
system would satisfy the theoretical conditions of stability
which, as any student of dynamics knows, only compare the
state of a system with other states infinitesimally near it. Yet
the system, as Gibbs points out, would not be stable in the
practical sense; for a disturbance which, while being small,
would be sufficient to carry the system beyond the infinitesi-
mally near states of larger energy would bring it to states of less
energy from which it would not tend to return to the first state.
Perhaps it may not be out of place to remind the reader that the
quantities Dnia^', Drria^", etc., are not variations of mass; they
are the small masses initially and finally present in the small
part of the film. Further, that Drria^" — Drtia^', etc., are not
necessarily small compared to Dma^', Drria^", etc. They are
small compared to the masses in the rest of the film and the
homogeneous masses; that is why we can use them correctly in
the expression (24). But since they are finite changes in
respect to the small portion Ds of the system, they produce
finite changes in the surface tension and the g,h, ... potentials
there, so that we can regard a" — o-', iig" — Hg, nh" — fj-h, etc.,
as finite differences if necessary. This small digression on the
meaning of the D symbol may serve to illuminate the point
about practical instability.

The argument can now be extended to the whole film.
Having effected the change in one small part of the film, we can
carry it out for another small part, changing entropy and masses
there so as to produce the g,h, ... potentials and surface tension,
fjLg", iJLh", . . . cr", which exist in the first small part, and so on.
This is simply the procedure indicated by the integrations on
Gibbs, I, 240. The changed condition in the film is therefore
uniform in nature throughout and is one which could exist in
equilibrium with the homogeneous masses in their practically
unchanged condition. The difference of energy in the whole
system for the two states of the film is

(a" - a')s + w/'(m/' - m/) + mtS"{y^H" - n[) + . . . (26)

612 RICE ART. L

41. Three Conclusions Drawn from the Analysis in Subsection (40)

This disposes of the analytical steps on these pages of Gibbs'
treatise. There are three conclusions based on them. The first
appears at the top of page 240. As presented it is somewhat
elusive, but we can put it as follows. It is possible that the
potentials n/\ iih", . . . which correspond to the masses Wa^",
nih^", . . . mg^", ruh^", . . . may be respectively equal to the
potentials ix/, nh, . . . which correspond to Ma^', nih^', . . .
fn/', nih^', . . . (Of course, the potentials ^a, M6, • • • remain
unchanged in any case.) If this is so, then by (26) {a" — (t')s
must be positive if the single accent state is to be a stable state
of equilibrium; i.e., g" > 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
<t" is different from o-' at all if Ha, Hb, ■ . ■ \i.g\ t^h, ... do not
differ in value from Ha, M6> • • • )"»", 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<r da

—— Aflg + — AHh + . . .
OUg dfXh

1 r av , , av , , av "1

+ 2{^' (^''«)' + w ^^'-'^ • • ■ + %7;^ ^"'^"^ + ■■■■}

-t- higher powers < — TgAfjLg — ThAfXH — . ..

614

Now by [508]

RICE

ART. L

da-

dfXg

da_ _
Olih

Hence, if we neglect the cubes and higher powers, we can write

(AmJ2 + ^-. (A/x,)2 . . . + 2 777- ^^Ji,^^lH + . . . < 0.

dfih^

dfigdnh

Now at the outset of this section of the commentary, on page
606, we dealt with the conditions which render such a quadratic
expression always positive or always negative in value. We see
that in order to comply with the present condition of negativity
a series of determinants beginning with

}

, and so on,

dfihdno duk

will be alternately negative and positive for the values of the
variables Hg, nh, ... which exist in the "single-accent" film,
i.e., ng^\ iJih^' . . . Looking at the question from a purely mathe-
matical point of view, if, in addition to these conditions,

ba da
— , — , . . .

djjLg dnh

were all zero for the same values of Hg, Hh, . . . then a regarded as
a function of Hg, iJ^h, ... would have a maximum value for these
same values of Hg, nh, ... This is the meaning of the cryptic
remark at the end of the paragraph (p. 242). But of course the
"necessary conditions relative to the first differential coefficients"
are not fulfilled; in other words da/dug is not zero for the values
HgS', fXh^', ... oi Hg, fjLh, . . . ; it is equal to — Tg\ and so on. To
be sure, the conditions for the second differential coefficients are
satisfied, but for a reader who is not familiar with the concrete
forms of these conditions, the way in which the conclusion is

SURFACES OF DISCONTINUITY

615

stated in Gibbs' text is somewhat confusing. We have limited
the matter to the second differential coefficients, as that is suffi-
cient to make the meaning of the sentence more apparent to
the reader. (As the order of Hg, ma, m», • • • is immaterial, the
conditions are, that the constituents in the principal diagonal
of the determinant

dfikdug

dfXgdnh dugdm

av av

dfjLhdm

dfjLidug dfiidnh dfifi

and all the minors of the third, fifth, seventh, etc. order, formed
by erasing the necessary number of rows and corresponding
columns, shall be negative, while the minors of the second, fourth,
etc. order formed by similar erasures shall be positive in value.)

4^. Determination of a Condition Which Is Sufficient though Not

Necessary for Stability when the Dividing Surface Is

Not Plane and Is Free to Move

The investigation so far has been limited by the proviso that
the surface is plane and does not move. The removal of this
limitation renders the problem more difficult, although it is
easy to derive a condition which in this case will insure stability,
without actually being necessary for it. Gibbs' treatment of this
occurs at the very end of this subsection, on pages 251, 252, but
it is so relatively simple compared to the other material of the
subsection that the reader may find it helpful to have his
attention directed to it at once. To make the presentation as
direct as possible, consider a system with two homogeneous
masses separated by one surface of discontinuity, the whole
enclosed in a rigid envelop. We can suppose that two fine tubes
inserted through the envelop put each mass in communication

616 RICE

ART. L

with a very large external mass which contains all the compo-
nent substances at suitable temperature and potentials; this is
also enclosed in an external rigid envelop and bounded inter-
nally by the envelop enclosing the system. A movement of
the surface of discontinuity in the system entails in general a
change in the volumes of the homogeneous masses of the system.
This does not involve any change in the potentials of the
various components in them or in the surface (in so far as they
are components in the surface) ; for the amounts of components
withdrawn from or passed into these masses are passed into or
withdrawn from the external mass, and that is so large that the
amounts are relatively too small to affect the potentials in it.
For the two masses we have equations such as these :

Ae' = t At)' - p' Av' + fjiiAmi + . . . ,
Ae" = t At?" - p"Av" + yuAmi" + ...,

and an equation

Ae'" = t At?'" + fnAmi" + ...

for the external mass, since its volume does not change. For
the surface

A^ = t Aijs -\- (tAs + MiAmi-s + . . .

The variations may be finite* since t, ni, ^2, ... remain constant;
p' and p" are not necessarily equal since we are not assuming
the surface to be plane, but since each of them is a definite
function of t, /xi, 1J.2, . . ., each remains constant. Now if

Ae' + Ae" + Ae"' + Ae-^ > 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") = (<r + 8a) (ci + dci + c, + 8c,),

so that the varied state need not he a state of equilibrium as regards
the condition expressed by equation [500].

The energy of the system, depending as it does on the variables
of the system, can, as we have just seen, be expressed as a func-
tion of v'. The energy in the varied state is by Taylor's
theorem

622 RICE ART. L

de , I dH , „

For equilibrium de/dv' must be zero. For stable equilibrium
we must have the additional condition

dH'

The amplification of this condition on page 250 to the form [544]
is easy; in [544] we regard dp' /dv', dp" /dv\ da/dv' as given
by the equations above, and of course ds/dv', d^s/dv'^ can be
calculated from the geometrical form of the system and the
fixed perimeter of the film. Equation [547] is the result for the
special case when one potential only is variable.

45. An Approach to this Problem from a Consideration of the
Purely Mechanical Stability of the Surface

Thus we have learned the general theoretical method of
dealing with stability when sufficient knowledge is available
concerning the functional forms of the various energy functions.
It involves no trouble concerning the mechanical stability of
the surface of discontinuity, which in a manner of speaking
takes care of itself. However, it is interesting to approach
the problem from that angle as well, and this is what Gibbs
does in the pages immediately preceding those on which we
have just commented. Going back we take up this aspect at
the bottom of page 244 where a system just like the one we have
been considering is posited. (We are not assuming a circular
orifice.) Passing by the two short paragraphs at the top of
page 245 (which are unimportant for our present purpose) we
have the relation for equilibrium

p' — p" = o-(ci + C2),

where, as before, p' , p", <r are functions of y' the volume of
one phase. A slight variation of the surface of discontinuity
will cause a change in p' — p", a and Ci + Ci. If there is to
be stability the surface must tend to return to its original

SURFACES OF DISCONTINUITY 623

position and (p' + 8p') — (p" + 8p") must be less than
(o- + 5<r) (ci + 8ci + C2 + 5C2), so that

8{p' - p") < (ci + C2)5<r + a 8(ci + C2).

As every one of the variables can be represented as a function of
v' it follows that, for mechanical stability of the surface,

djci + C2) dp' dp" da

" dv' ^ dv' ~ dv' - ^'' + ^^ 'cb''

Now it can be shown that

where s is the area of the surface, bounded as it is by the edge of
the orifice. (See the note on curvature, p. 10 of this volume.)
Hence it follows that

d^s dp' dp" d(T ds
'^ d7^ ^ di/ ~ ~d7 ~ d^'"dv''

which is just equation [544]. The problem can be completed
as on page 251. Thus we see that the same conclusion is
reached as before when we took no special heed of mechanical
stability and merged that stability, as it were, in the general
method of dealing with stability with reference to the neigh-
boring equilibrium and quasi-equilibrium states. This provides
still further justification for the validity of the general method.
The only point of special importance about the problem on
page 245 concerns the assumed circularity of the orifice. One
then has special values for ds/dv' and d^s/dv'^. These can be
derived from the special geometry of the case as outlined in the
middle of page 245; by the aid of the equations there one can
prove that

and

r — X

ds 2

dv'

irrx^

dv' r

d's

2 dr

2(r - x)

dv" ~

~ r^ dv' '

= — f

TTJ^X^

624 RICE ART. L

and so equation [547] takes on the special form [540] in this
case.

The reader will now find no difficulty in following the matter
on pages 242-244. The special corollary concerning the system
in which "the interior mass and surface of discontinuity are
formed entirely of substances which are components of the
external mass" (of which a drop of water in an indefinitely large
mass of vapor is a good illustration) offers a good example for
applying the sufficient test which is given on page 252, and on
which we have already commented. Thus, the interior volume
being v' and the radius r, let the radius increase to r + 8r. Now it
is a feature of the method, which must not be overlooked, that
As and Av' are not to be taken as SirrSr and iirr^Sr respectively;
that overlooks the higher powers of dr which are vital for the
purpose of the test. Actually, if we merely retain first powers
of 8r, 8s = SttSt, 8v' = ^ivr'^br and 8v" = -^irr'^br', therefore

S(a5s) - S(p5?;) = {<T.87rr - (p' - p")47rr2}5r,
which is zero (as it should be for equilibrium). But

As = 87rr5r + 47r(5r)2,
and

Av' = ^TcrHr + 47rr(5r)2 + y {brY = -Av".

Hence

2(0- As) - 'LivAv)

= 47r«T(5r)2 - 47rr(p' - v") i^rY - J ip' - v") (5r)'

f . 2{8rY\

= ^ivaU8ry - 2{8rY - ^]

= - 47r(T(5r)2

(provided 8r is small compared to r). This is negative for any
sign of 8r. Hence the sufficient test of stabihty is not satisfied.

SURFACES OF DISCONTINUITY 625

Of course this test is not conclusive on the matter; it gives
strong presumptive evidence that the system is not stable, but
as it is not absolutely necessary for stability the matter has
to be cjinched by the necessary test which is actually applied in
the text. This goes beyond the purely mechanical considera-
tions, and uses the fact that p', p" and a do not change if there
is a large enough external mass to draw on to maintain con-
stancy of composition in the phases. Hence if p' — p" = 2cr/r
then p' — p" > 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

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 ^ <tac -\- (^bc, or if ctab < c^c + csc-
Although in the latter case a plane film of C would obviously be
unstable for a reason similar to that just given, a lentiform film
might develop and so a quite definite problem is posited in this
case also. In a paper on emulsification (J. Phys. Chem., 31,
1682, (1927)) Bancroft criticizes the statement that vab cannot
be larger than cac + <^b c, but seems to be under a misapprehen-
sion as to the situation. Gibbs on page 258 does not assert
that as a general rule for three such fluids cab cannot be greater
than (Tac -\- (tbc'i he merely, for the purposes of the problem he is
discussing, rules out of account fluids for which such an in-
equality would be true, presumably (as the writer has pointed
out definitely) on the grounds that the problem does not
exist; it is solved in the very statement of such a condition.
Now if the temperature and potentials have such values that
Pc < VA{t, ij) (and of course < psit, m)), the phase cannot
form under any circumstances ; for if it formed as a plane sheet
between A and B (or as an anticlastic sheet for which Ci + C2

SURFACES OF DISCONTINUITY 633

is zero) p c would have to be equal to Pa or pa, and if in the form
of a lentif orm mass p c would have to be greater than Pa or pa.
Hence A and B in contact would be quite stable as regards the
formation of C in such a range of values of t, ni, IJ.2, . . . If we
now consider the range of values of these quantities for which
Pc(t, m) ^ PA{t, ijl), we have to deal with the two cases
which arise; (1) when ct^bC^ m) = (^Acit, m) + <rBc{t, m);
(2) when CAsit, m) < <^Ac{t, fx) + (Tscit, /x).

(1) If pc(t, m) = PA{t, m) there would just be equilibrium with
a plane sheet of C between A and B, since the surface tensions
between A and C, and B and C would just balance the surface
tension between A and B in the portion where A and B meet.
On the other hand if we varied t, ni, 1x2, ■ ■ • to values t', ^i/,
H2, ... such that pc(^', mO > ??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 <tac and cbc since
(Tab = (Tac + (Jbc* Hence the edge tends to move outward,
i.e., the mass C tends to increase and in so doing draws on the
masses A and B for material, and so alters the phases in such a
way as to bring them to such values that the equality of p c to
Pa will be restored. We see that in this case there is a tendency
for the mass C to form between A and B.

(2) If (Tab < (J AC + (Tbc the argument of the previous para-
graph breaks down. Clearly, no plane sheet of C can form
between A and B when pc = Pa, the force equivalent to ctab
being too small to pull it out, as it were, against the force equiv-

* As is well-known, this is a convenient way of dealing with the fact
that if an outward displacement of the edge were made there would be a
diminution of free surface energy.

634 RICE ART. L

alent to oac + (Tbc If, however, the temperature and poten-
tials are such that pc > 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
<tac and ffBc being less than their numerical sum could pos-
sibly be equal and opposite to the force equivalent to ctab-
However, the argument on pages 259, 260 of the original shows
that the existence of such a lentiform mass would yield a
system of greater energy than the one from which it starts.
Hence in general there would be no tendency to form it. The
mathematical steps of the argument will offer no trouble pro-
vided the reader notes one or two points. Let us designate by X'
the center of the surface EH'F, and by X" that of the surface
EH"F. The cosine of the angles between EI and the tangent
to EH'F at E is (r' — x')/r'. The area of the spherical cap,
represented by EH'F in Gibbs' Figure 10 and denoted by Sac, is
known to be 2x(l - cos e')r"^, where d' is the angle EX'H';
so that, since cos 6' = (r' — x')/r', the area is 2Trr'x'. The
volume of the spherical sector standing on Sac with its centre
at X' is ^Sac-t' = ^irr'^x'. The volume of the cone standing
on the base Sab (i.e., the circle with EF as diameter) is
f Sab-X'I = ^rR^ir' — x'). Hence the volume of the spheri-
cal segment between Sab and Sac, being equal to the difference
of the sector and cone, is as given in [566].

So far we have maintained the condition pA{t, n) = psit, n).
If, however, this condition be abandoned, and if the functions
are such that in general pA{t, fj.) > 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 <tab = o-ac + (Tbc, a thin film of C would just be
in equilibrium between the surfaces of A and B, which would
have a curvature given by Ci + C2 = (Pa — Pb)I(Tab provided
that

, . (TBcit, fl) PA(t, m) + (TAcit, (J.) PB{t, H)

Vc^t, n) = — , [571]

as proved on page 262. If pc(t, n) were less than this critical
value the film would not form. If the values of i, /xi, m2, • • •

SURFACES OF DISCONTINUITY

635

were in the range for which pc{t, m) is greater than the right-
hand side of the above equation, the film would form, tending to
get into the lens shape at first and then, as its growth drew on
the adjacent masses A and B for material and modified the
potentials so as to restore the condition given by equation [571],
would spread out in the film again. If, as in (2) of previous
paragraphs, the phases are in such a condition of temperature
and potentials that cab < oac + obc, we can show that a mass

Fig. 5

of C will not tend to form on the surface between A and B,
curved as before to the radius given above, even when p c(^ m)
is greater than the critical value on the right hand of [571].
This requires a repetition of the proof on pages 260, 261 with the
surface DEIFG in Figure 10 of Gibbs regarded as curved and
not plane ; the adaptation of it to this wider geometrical con-
dition is given on page 263 (see Fig. 5). The area represented
by EH'F is Sac, by EH"F Sbc, by EIF Sab. Va is the volume
represented by EIFH', Vb by EIFH", and Vc is the sum of these.

636 RICE

ART, L

The geometry of the figure is not so simple now, and we cannot
make a direct calculation of W as on page 261. The device
which Gibbs uses is stated with such conciseness in the sentences
toward the bottom of page 263 that the implications involved
in them had better be more fully expounded. If the state
indicated did form in a natural way, it would happen in some
such fashion as this. Beginning at an initial stage of tem-
perature and potentials to, mio, M20, . . . for which

. (TBcito, Mo) PA(to, /Xo) + (TAcito, JUq) Vsito, Mo)

Pc{to, Mo) = — ^ >

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

<TidTi -\- (X28T2 + asdTz + aidTi = 0

for all possible displacements 00'. That is condition [615].
Since the components of the displacements are actually parallel
to the lines OA, OB, OC, OD it appears that this is just the same
as the well-known "virtual work" condition for the equilibrium
of four coplanar forces which could be conceived to exist in the
plane of the paper, with magnitudes ai, 0%, az, a and with direc-
tions along the four hues.* Or for that matter we could
consider the system of conceptual forces "swung round"
through a right angle so that their directions would be at right
angles to the four surfaces as Gibbs conceives them to be drawn;
such a change in orientation would not affect their equilibrium,
if it existed before the change. Gibbs' Figure 13 is the usual
polygon -of -forces diagram drawn on this principle. Now sup-
pose that two masses of the liquids A and C were brought into
contact with one another and were found to have a surface
tension larger than that represented by the length of ay in
Figure 13; the condition represented in Figure 11 would be per-
fectly stable, since free energy does not tend to increase. If,
however, this tension were less than that represented by 0:7,
the condition would be practically unstable; but to come to a
definite conclusion in that case one would have to go more
fully into changes in the several components and potentials in
the four homogeneous masses occasioned by the development
of the surface represented by O'O". Smiilar considerations in
relation to the diagonal /3§ would govern the possible growth
of a surface between the masses B and D.

* The reader must guard against the inference that the surface
tensions are really tangential forces in the surfaces. We have already
referred on p. 510 of this article to the convenience, but the physical
unreality, of this conception.

642 RICE

ART, L

50. The Possible Growth of a New Surface at a Point of Meeting
of a Number of Lines of Discontinuity

We might have a system in which there is more than one Hne
of discontinuity, these hnes meeting at a point. The latter
half of page 289 has a very concise statement about the stability
of such a system as regards the development of fresh surfaces
at the point. Any reader who is not trained in solid geometry
or lacks the power to visualize diagrams in space may require
some assistance here. Let us begin with the simplest case of
four different fluid masses. In this case there will be six
surfaces of discontinuity, and four lines of discontinuity. The
easiest way to realize this is to drive three nails into a drawing
board, calling them X, Y, Z. Attach three threads to them
which can be drawn tight and knotted at a point 0 above the
board. A fourth thread, tied to the other three at 0, is stretched
tight and tied to another nail U, in a support above 0. One
can then see that we can have one mass of fluid in the pyramid
OYZU, one in OZXU, one in OX YU and one in OXYZ. Let
us call these masses A, B, C, D, respectively. The surface
between B and C is OXU; between C and A, OYU; between A
and B, OZU; between A and D,OYZ; between B and D, OZX;
between C and D, OXY. There are four lines of discontinuity
OX, OY, OZ, OU. Since the surfaces OXY, OXZ, OXU
meeting in the line OX are in equilibrium, three forces having
magnitudes proportional to (tcd, ctbd and (Tbc, and directions
normal to these surfaces, are in equilibrium, and can be repre-
sented by the sides of a triangle whose corners we shall name
/3, 7, 5, the side yb representing aco, 5/3 representing aoB, &y
representing gbc In the same way, if the surfaces OYX,
OYU, OYZ meeting in OF are in equilibrium, three forces
odc, (tca, (Tad normal to these surfaces can be represented by the
sides of a triangle 8ya, where a is a fourth point not in the plane
of I3y8. The figure a^y8 is a tetrahedron, and it will now be
easy for the reader to see that the equilibrium of the other two
triads of ^rfaces, viz., OZX, OZY, OZU and OUX, OUY,OUZ
is related in a similar way to the triangles fiba and a^y. In
short, the tetrahedron a^yb is a geometrical representation of

SURFACES OF DISCONTINUITY 643

the whole state of equilibrium if it exists. The six edges of the
tetrahedron are perpendicular to the corresponding surfaces
and represent by their lengths the six surface tensions. The
four sides of the tetrahedron, viz., the triangles ^y8, ya8, a^b,
a/37 are perpendicular to OX, OY, OZ, OU, respectively, and if
the tetrahedron ajSyd were drawn with the point 0 inside it,
the four points a, /?, 7, 8 would be respectively situated in the
masses A, B,C, D. It is hoped that in this way the reader may
grasp the meaning of the earlier sentences of the paragraph,
where the "closed solid figure" is the tetrahedron in our illus-
tration for four masses. (There is a small misprint in the
second sentence of the paragraph. Beginning after the second
comma of the sentence it should read "the edges to the sur-
faces of discontinuity, and the sides to the lines in which
these surfaces meet." Notice that "edge" refers to a line of the
representative tetrahedron, and "side" to a triangular face of
this tetrahedron; "line" and "surface" are retained for the
physical lines and surfaces of discontinuity in the system.)
After this is grasped, consider a greater number of masses whose
dividing surfaces intersect in pairs in lines all of which meet in
one point 0. Any group of four masses which have six dividing
surfaces between them, say, A, B, C, D can be represented as
above by a tetrahedron a^y8. Suppose there is another mass
A ', which has three dividing surfaces with the masses B, C, D,
but has no dividing surface with A, having only the point 0
in common with A. The condition for equilibrium of these
surfaces is bound up with a tetrahedron q:'/375 where a' is on the
opposite side of Py8 to a. All the edges of this double tetra-
hedron will have the right directions and lengths to corre-
spond to the surfaces and their tensions. If now a new sur-
face were to develop at 0 between A and A' and to be in
equilibrium, the normal to this new surface would be parallel to
aa' and the tension of the surface A A' would be represented by
aa', so that for stability with respect to such a formation the
tension of the surface between two masses of A and A' would
have to be greater than that represented by aa'.

644

RICE

ART. L

51 . Some General Ideas and Definitions Concerning the Possibility
of a NeiD Homogeneous Mass Being Formed at a Line
of Discontinuity or at a Point of Concur-
rence of Such Lines

Of course bulk phases might develop at a line of discontinuity
or at a point where such lines meet. Gibbs considers the first of
these possibilities in the subsection beginning on page 289, the
second in the subsection beginning on page 297. The argument
in each case runs on very much the same lines as in the treat-
ment in pages 258-264 of the possible formation of a new phase
between two phases, although it might not appear so on first

Fig. 6

reading. We shall recast the argument in pages 289-297 so as
to bring out this feature.

First of all there must be certain relations between the surface
tensions in order that the problem may not be trivial. In the
first instance aBcit, n), crcAit, m), o-^b(^, M),must satisfy conditions
of equilibrium, which necessitates any one of them being less
than the sum of the other two. Now we assume that we know
of a phase D and that we know for it the functional forms of
(TAoit, fi), (TbdH, m), (Tcoit, m) as well as Voit, m)- The values of
(Tbc, <rcA, (Tab determine the angles at which the surfaces B-C,
C-A, A-B meet where no phase D exists. If the phase D is
formed and is in equilibrium, (Tad, obd, (Tab will have to satisfy

SURFACES OF DISCONTINUITY 645

certain conditions; so also will (Tbd,(^cd, (Tbc, and aco, ctad, oca.
For instance, if oab > 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 <tbc, <tcd, (^db', its angles must be the
supplements of the angles between the tangents or normals at a.
(The sides of this triangle are not parallel to the normals to the
surfaces at a unless da is a straight line.) Similarly we can
construct triangles 7Q! 5" and a^8"'. There are various ways in
which the lines a8", ab'", etc. can fall. If the lines da, dh, dc
are straight and abc a curvilinear triangle convex inwards,
they fall as in Gibbs' Figure 16; if convex outwards they fall as in
Figure 8 of this text. Another case is shown later in Figure 9.
Only in special cases when the angles of the triangle abc are

648 RICE ART. L

together equal to two right angles (not necessarily confined to
rectilinear triangles) can the situation for equilibrium be repre-
sented as in Gibbs' Figure 15. The case represented by Figure 16
is said by Gibbs to be one in which the tensions of the new sur-
faces "are too small to be represented by the distances of an in-
ternal point from the vertices of the triangle representing the
tensions of the original surfaces," as is the case in Figure 15.
The cases represented in Figures 8 and 9 of this text are said to
be of the type in which the tensions of the new surfaces are
too large to be represented as in Gibbs' Figure 15.

52. The Stability of a New Homogeneous Mass Formed at a Line

of Discontinuity. A Summary of the Steps

in the Argument

Having laid down these general ideas and definitions Gibbs
proceeds to the argument concerning the stability of a mass
formed in this way. It is long and detailed, covering more
than four pages, and it may be well for the reader first
to glance through a summary of the steps, with certain details
left out which can be filled in later. (In following such details
at first, one is apt to lose the thread of the argument.)

The first step is on page 292 and concerns equilibrium, stable
or not. It is shown that if Ws and Wv are the two quantities
defined in [626] and [627] then if the system is in equilibrium

Ws = 2Wy.

(Notice that a similar type of numerical relation holds for cog-
nate quantities in cases of equilibrium treated previously. See
equations [563], [564], [569] of pages 260, 261.) It is also shown
that for equilibrium the quantity Ws — Wv must be at a maxi-
mum or minimum value as compared with any configuration
(equilibrium or not) of the surfaces adjacent to the equilibrium
configuration, i.e., so long as tensions and pressures are main-
tained unchanged at the values corresponding to the tempera-
ture and potentials throughout the system.

In the second step it is shown that, since for stable equilibrium
Ws — Wv must be at a minimum value as compared with
adjacent configurations, there is instability if Wv is a positive

SURFACES OF DISCONTINUITY 649

quantity (and therefore also Ws, since Ws = 2Wv). If Wv is
negative the system is stable. (One can hardly say that this is
"shown." It can be inferred from the proposition that Wv°^Ws^,
proved on page 293, but the inference is not an obvious one; and
on the face of it there appears to be a puzzling contradiction
between this proposition and [633]. The contradiction, of
course, is only apparent; but the reader is asked to defer these
difficult points until later and to proceed along the general
line of argument.)

The third step shows how these ideas are to be applied to any
given set of circumstances. If the pressures and tensions are
known, the figure ahcd can be constructed for the appropriate
configuration of equilibrium, if it exists. For since the
relative magnitudes of the tensions determine all the angles
round the points a, h, c, d we can find the angles of each
of the curvilinear triangles hcd, cad, abd, abc. Also since
Pd — Pa = ^ffAo/rAD, . . ■, Pb — Pc = 2cTBc/rBc, . . . , we can calcu-
late the six radii of the curvilinear sides. The angles and radii
are sufficient data to construct the various triangles, if they are
consistent with the possibility of a construction ; if they are not,
of course no such equilibrium configuration exists, and the
problem of stability does not arise. If the construction is
possible it shows us that the relative magnitudes of the quan-
tities Vd, Va, Vb, Vc (which are the areas of the curvilinear
triangles abc, hcd, cad, abd) i.e., the volumes of the mass D per
unit length normal to the plane of the paper, and the parts
into which it is divided by the surfaces B-C, C-A, A-B) can
be determined. These can therefore be taken as known in
terms of the tensions and pressures. An inspection is now
made of the quantity

VaVa + VbPb + VcPc
Vd

If the pressure po is greater than this it is obvious that Wv as
defined in [627] is positive, and from the second step the equilib-
rium of the mass D is unstable so that a disturbance producing
a small increase in it would result in a tendency for it to increase
still further. If it so happened that this volume Vd were small

650 RICE ART. L

enough it would mean that the equiUbrium of the Hne of dis-
continuity at d, without any formation of the phase D, would be
at least practically unstable ; for if a small filament of the phase
D should be formed a little greater than Vd in size per unit length
the formation of more of the phase would tend to occur.
On the other hand, if po happened to be less than the expression
written above, Wv would be negative, and the equilibrium of
this filament of the phase D would be stable; any small dis-
turbance increasing it would not tend to cause further growth
but the filament would tend to return to its equilibrium size.
Were Vd small enough this would be tantamount to saying that
the equilibrium of the original Une of discontinuity was stable.
On pages 294-296 Gibbs goes into more detail concerning this
for each of the three special cases where the tensions can be
represented as in his Figure 15, or are too small to be so repre-
sented, or are too large.

53. The Details of the Argument Omitted from the Summary

in {52)

Let us now return and fill in the omitted details. We know
from earlier parts of Gibbs' treatise that when the values of tem-
perature and potentials remain constant, so that all the tensions
and pressures are determined, the equilibrium of any configura-
tion is determined by the test that for any deformation of the
configuration to an adjacent configuration, equilibrium or not,
the variation

S(t5s - 2p5v = 0,

and if the equilibrium is stable the variation

2o-As - SpAv > 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{t,iJ^) to be changed to slightly
different functions of t, m, H2, . . ., say (rBc'{t,n), . . . aAD'(t,fx),
while the pressures still retained the same functional forms as
before. This would involve a slightly different configuration,
causing a change in the areas to Sbc + dsBc, ■ • • Sad + dsAo,
and in the volumes to Va + dvA, ... if equilibrium is to be
preserved. For this configuration we should have

. Ws = (Tad'(Sad + dSAo) + . . . —(TBciSBC + dsac) — . . .

W/ = Pd(vd + dvo) — Pa(va + dvA) — . . .,

652 RICE ART. L

so that

(W/ - Ws) - {Wy' - Wy) = {cad' - <rAD)SAD + CAD'dSAD + • . .

— ((Tflc — (TbcjSbc — Cbc CISbc — • • •

— Pd dvo -{■ PAdvA-{- ...
or, at the Hmit,

dWs — dWy = (Tad dSAD + Sad d(TAD + . • .

— (Tbc dSBc — Sbc d(TBc — • • •

- Pd dvo + PAdvA + ...

But since [629] is true for any small change in the configuration
it is true for the change indicated by dsac etc., so that

(TAD'dSAD I • • • (TBcdSBC • • •

- Pd dvo + PAdvA -\- . . . =0,
and from this it follows that

d{Ws — Wy) = SADdcAD + . . . — SBcd(TBC — . . .

which is equation [630]. Now this change in Ws — Wy
accompanies small changes in the functional forms which express
aBc, etc. in terms of t, jjli, H2, • • • but not in the forms for Pa,
etc. Suppose these changes to be of such a nature that the
tensions all diminish in the same ratio, the pressures of course
not altering. Since

Pd -

-Pa =

(Tad
Tad

- Pc

(Tbc
Tbc

etc..

and

Pd -

(Tad

Tad'

- Pc

(Tbc
Tbc'

etc..

it appears

that

TAi

0 • '"XD =

(Tad

• (Tad,

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

((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

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 <xad, etc., as does Ws, but a different Sad] in fact

Sad Sbc

- - - - etc.

Sad Sbc

On the other hand

Wv'

Va' Vb' Vc' Vd'

Wv ~

Va Vb Vc Vd

Hence on account of the similarity of the figures

Wv' Ws"

Wv W,

(As mentioned earlier there is no contradiction here with [633]
since Ws 9^ 2Wv'-) Expressed in another way

Wv + AWv _ (Ws + AWsy

Wv ~ Ws""

AWv AWs

= 2 4-

Wv Ws ^

Since Ws = 2Wv, it follows that

/AWsV
\Ws)

/AWsV

Neglecting quantities of the second order b{Ws — Wv) is zero,
as it should be for equilibrium ; but if we retain higher quantities,
A{Ws - TFy) < 0 if T^y is positive, and A{Ws - Wv) > 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

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]

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

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

= exp

i-t)

11
or

kt(\og n — log n') = e<t>.

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.

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 = <l>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 — - )

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 <tos due to ordinary molecular causes, where
(To is the surface tension with the surface uncharged and s the
area of surface, and in addition an amount of electrical poten-
tial energy ^QV where Q is the charge and V the potential of the
conductor. (Note that there is no hypothesis of a potential
discontinuity of amount V at the surface and a double layer of
charge.) Were the form of the conductor to change so as to
increase the surface by an amount 5s and heat to be supplied
reversibly so as to maintain the temperature constant, the
increase in the energy due to molecular causes is crods, but since
the same charge Q is on the surface its surface density will be
reduced and there will be a fall in electric potential energy, for
further separation of similarly charged particles always involves
decrease of potential energy. Hence the actual increase in
surface energy at constant temperature is less than aoSs which
means that the surface tension of the charged surface is less
than ffQ. The total surface energy is e,(s, V, t), a, function of
area, potential and temperature, and o-(s, V, t), the surface
tension, is defined in the usual way as dcg/ds. A change to a
new state of equilibrium with the variables at the values
s + 8s,V -\- 8V,t produces a change in the total energy given by

9e„ dea ^,

But this must be

equal to

ads + V8Q,

i.e.,

dQ dQ

a8s + V^8s + V-8V,

where Q(s, V, t) is the electric charge on the surface. Hence

des , ^dQ

Ts=^^^'^'

dV dV

690 RICE ART. L

By cross-differentiation

dV ds ^ dVds dVds

dsdV

Therefore

aa _ dQ
dV ~ ~ ds

This is the result which Lippmann appHed to the cathode
mercury surface of his electrometer. In the usual form of the
experiment a steady current is established in a potentiometer
wire, the positive end of which is attached to the large mercury
surface of the electrometer; a wire from the mercury in the
capillary tube goes to the travelling contact maker on the
potentiometer. As the contact slides away from the positive
end towards the negative, so that the potential V of the mercury
cathode above the electrolyte is lowered, it is observed that a at
first increases and then, passing a maximum, decreases until a
state of affairs is reached at which the polarization of the
cathode is unable to prevent a flow of current under the external
E.M.F. and equilibrium ceases to be possible. If E represents
this applied E.M.F., i.e., the P.D. between the positive end of
the potentiometer wire and the contact in any state of equilib-
rium, then V = Vq — E, where Va is the excess of potential of
the mercury above that of the electrolyte in the "natural state"
(i.e., when the applied E.M.F. is zero) ; and if E„i is the value of
this apphed E.M.F. in the state of maximum surface tension,
then Vm = Vo — Em, where 7™ is the P.D. between mercury and
electrolyte in this state. Since initially da/dE is positive,
da/dV is negative and so dQ/ds is positive. Now dQ/ds
measures the increase of charge required for unit increase in the
area of surface, the potential being kept constant; in other
words the charge per unit area; it is also in general a function of
s, V, t, just as Q is, and we write it q{s, V, t). Thus initially

SURFACES OF DISCONTINUITY 691

there is a positive charge on the mercury cathode surface. At
the point of maximum a, where da/dV vanishes, q is zero, and
on further increase in E, q becomes negative. If we write
Lippmann's result in the form

one sees that it is equivalent to Gibbs' equation [690], although
at the first glance it would seem as if there were a difference of
sign between the two results; for V" — V is the applied electro-
motive force and so [690] becomes

da^ ly,

dE ~ Oa

Since Tif/aa is the excess ionic charge at the surface, a contra-
diction apparently arises. This disappears, however, on a
little thought; one must bear in mind that Gibbs considered the
transport of electricity in terms of ions, e.g., hydrogen ions;
these only travel from one discontinuous layer to the other;
Fa' represents the excess of (hydrogen) ions in the layer of the
electrolyte adjacent to the mercury represented by the singly
accented symbols, i.e., the cathode. Thus, as Gibbs points out,
there will be a defect of hydrogen ions in this layer in the natural
state, since by his equation Tj is negative if da' /BE is positive.
This involves a negative charge in this layer which is the
counterpart of the positive charge on the mercury surface;
for of course the region of discontinuity is uncharged as a whole.

66. The Double-Layer Hypothesis of Helmholtz

It was in fact this phenomenon of the double layer of charge
which Helmholtz emphasized. Holding as he did decided views
in favor of Volta's hypothesis of contact potentials, he pointed
out that a discontinuity of potential could only exist between
metal and electrolyte for the same reason as between two metals
in contact, viz., by a condenser-like action arising from equal
and opposite charges segregated in adjacent layers of the two

692 RICE ART. L

materials.* Assuming that the distribution is actually on
the surfaces in analogy to the distribution in an ideal plane
condenser, it appears that Q = csV, where c is a constant, viz.,
the capacity of the double-layer condenser per unit area. Hence

and

Thus

ais, F, t) = (Tr, - IcV^
or

a{s,E,t) = am- hc{V, - E)\

This leads to two results: (1) that the graph of a and E should
be a parabola; (2) that Em = Fq. The first conclusion is
certainly only true in a very limited number of cases, while
the second, although it has served for some time as the basis for
a method of determining absolute electrode potentials, is unques-
tionably not exact. It was Helmholtz who suggested the
method in question. It consisted in measuring the E.M.F. of
a cell with one electrode of mercury and the other of the metal
whose P.D. against a given salt was required; the desired P.D.
was then calculated on the assumption that the potential at the
mercury electrode was that given by the value of Em, obtained
as above. Shortly after, he suggested the use of the dropping
electrode, a method based on a similar physical picture of the
phenomenon.

67. Recent Developments in the Thermodynamical Treatment of

Electrocapillarity

Since those days the developments of the theory have been
along two main lines. We can do little more than make verv

♦ Monatsber. Akad. Wiss., Berlin, 945 (1881). Cf. A. Konig, Ann.
Phys. u. Chem., 16, 31 (1882). See also Planck, Ann. Phys., 44, 385 (1891).

SURFACES OF DISCONTINUITY 693

brief reference to them in our limited space; so we shall have to
be content with giving a few of the most important references
and then conclude with some remarks, which, it is hoped,
will enable the reader to study these papers more critically
than he otherwise might do.

One line of advance has carried forward a formal development
of Gibbs' thermodynamic treatment of the phenomena at
charged interfaces. Consult for example:

Gouy: Ann. phijs., 7, 129 (1917).

Frumkin: Z. physik. Chem., 103, 55 (1923).

Frumkin: Z. Physik, 35, 792 (1926).

Frumkin: Ergeb. der exakt. Naturwiss., 7, 235 (1928).

O. K. Rice: /. Phys. Chem., 30, 1348 (1926).

Butler: Proc. Roy. Soc, A, 112, 129 (1926); 113, 594(1927).

A good summary of this work will be found in Chapter VII
of Newman's book and in an article contributed by Frumkin
to the Colloid Symposium Annual, Vol. VII, pp. 89-104.

On the other hand the unsatisfactory nature of the conclu-
sions deduced from Helmholtz's condenser-layer theory of the
distribution of the charge, and his lack of suggestions as to the
manner in which they would be kept apart, has given rise to
theories, based on statistical treatment, of "diffuse layers" of
double charge. The interested reader can consult the following
papers.

Goiiy: Ann. chim. phys., 29, 145 (1903); 8, 291 (1906); 9, 75 (1906).

Gouy: Ann. phys., 6, 5 (1916); 7, 129 (1917).

Chapman: Phil. Mag., 26, 475 (1913).

Herzfeld: Phijsik. Z., 21, 28, 61 (1920).

Frumkin: Phil. Mag., 40, 363 (1920).

Stern: Z. Elektrochem., 30, 508 (1924).

O. K. Rice: /. Phys. Chem., 30, 1501 (1926).

This development of theory has been occasioned by the
deviation of the ascertained facts from the simple conclusions
derived from the combination of Helmholtz's ideas with Lipp-
mann's result. We can only mention here one or two of the
most important of these deviations. (In the experimental work
the solution in contact with the mercury electrodes is generally
saturated with an appropriate mercury salt to ensure that the

ujILIIRARY

694

RICE

ART. L

anode surface is unpolarizable. Thus a potassium chloride
solution is saturated with mercurous chloride; a sulphate with
mercurous sulphate, and so on.) The simple parabolic graph
for 0- and E is very far from being the rule. Thus while curve I
(Fig. 13) shows that an iV/20 solution of KCl nearly fits a para-
bola, a similar solution of KI (Curve II) is too steep in its
ascending portion; its maximum is lower than that for KCl and
corresponds to a larger value of E; beyond the maximum it
gradually approaches and merges into the KCl graph. Accord-
ing to the simple Helmholtz view, the mercury in its natural
state ought to be higher in potential than the KCl solution by

an amount represented by OP, about 0.6 volt; but higher than
the KI solution by OQ, about 0.8 volt. Now if this were so
we would expect to find that a cell containing these two solu-
tions with a mercury electrode in each would give a P.D. of
0.2 volt; but it is known that the P.D. is much smaller than
this. If then we assume that because the curve is "normal"
for KCl there really is a P.D. of 0.6 volt between mercury and
KCl in the natural state, we must admit from the evidence of
the cell just mentioned that the mercury must also in the
natural state be above the KI solution by practically the same
amount. Hence, at the maximum state for the latter solution
(represented by Q), when according to Lippmann's result the

SURFACES OF DISCONTINUITY 695

mercury surface has no charge and according to Gibbs (even
apart from the fact that the charges in mercury and electrolyte
must compensate one another) the electrolyte layer has no
excess or defect of ions, it follows that the solution should be
higher than the mercury by about 0.2 volt. Of course we can-
not be sure that our assumption for the KCl is correct, which
only makes matters still more ambiguous; for it is clear that
the situation renders doubtful the whole basis of the various
methods hitherto employed to measure an absolute elec-
trode P.D.

An explanation for this behavior has been offered on thermo-
dynamic grounds as follows. We have seen that in the natural
state positive ions (cations), such as hydrogen ions, will be in
defect in the electrolyte layer of the discontinuous region, while
negative ions (anions) would preponderate. It is assumed
therefore that in this state there are present anions which are
capillary-active, in the sense defined in the earlier part of this
article; i.e., they tend to lower the surface tension and are
"specifically adsorbed" at the surface of the solution on that
account. Now, in so far as this has any meaning, it apparently
assumes that the negative charge of these adsorbed anions will
be to some extent neutralized by the positive charge on the
cations in the electrolyte layer. The corresponding positive
charge on the mercury will exert the usual depressing effect on
the surface tension represented in the Lippmann equation by
— q 8V. But in addition to this, these anions will exert a still
further depressing effect represented by an additional term of
the Gibbs type — TSix (not an equivalent term). Whether this
"combination" effect can be derived from a really theoretical
treatment we shall consider presently. Of course it is part of
the assumed state of affairs that the cations are capillary-
inactive and are therefore not "specifically" adsorbed, their
presence in the layer is determined by the external electrical
influence. Without the specific adsorption of the anions it is
assumed that we would have the "normal" parabolic curve;
with the adsorption we have an additional depression and the
curve begins lower down than the normal. As the E.M.F. rises
the electric field drives the active anions out of, and brings

696 RICE

ART. L

inactive cations into, the electrolyte layer, so that the depression
of 0- below the "normally" depressed value due to the charge
grows less ; the actual curve gets nearer to the ideal. Even when
the exact neutralization of charge is just attained there are still
some anions ui the layer, balanced as regards charge by cations
and, with no charge on the mercury surface, still exerting some
depressing effect. At the maximum, the specifically adsorbed
anions have nearly disappeared from the layer, so that there are
practically only inactive cations with a corresponding negative
charge on the mercury, producing a normal depressing effect
on the surface tension with a very small specific anionic effect;
presently all the anions will have left the layer of solution and
thereafter the effect is normal; the curve merges with the ideal
curve. It would appear that at the maximum the surface of the
mercury should not be uncharged but should have a small
negative charge and the electrolyte should be a little above the
mercury in potential.

Certain solutions exhibit an opposite effect, producing a
curve practically normal to begin with, but falling below the
ideal as E increases. This could obviously be accounted for by
a hypothesis of active cations with inactive anions. Also
there are solutions for which the curve rises like the normal
curve, then falls under it and later on merges into the ideal
curve once more on its descending branch.

Certain deductions from this view have been verified. Thus,
for a solution involving only inactive ions, the P.D. between
it and mercury in the natural state should equal the value of
Em', hence if a cell were constructed with mercury electrodes in
two such solutions, its E.M.F. should be the difference of the
observed values of each Em. This has been found to be so.
Also, if we were to make a cell with mercury electrodes and two
solutions each of which involves active anions, we should find
that its E.M.F. is equal to the difference between the values
of E for the same surface tension provided this value of the
tension falls on the normal parts of the graphs in their final
descending portions; for at such a stage the specific effect of
adsorbed ions has disappeared and only the "purely electric"
effect is remaining. This has also been found to be true.

SURFACES OF DISCONTINUITY 697

This hypothesis, as we have pointed out previously, involves a
combination of a Gibbs term and a Lippmann term in the
expression for da, which are not equivalent to one another, but
complementary. Thermodynamical deductions of this equa-
tion will be found in the references mentioned above. The
most complete theoretical treatment is given in Butler's papers
in the first list of these references. In the writer's opinion it
suffers somewhat by an unnecessary complication, the intro-
duction of a second "surface tension," denoted by 7 in the paper.
The writer will give a statement of the theory without introduc-
ing this additional conception, at the same time making a critical
reference to one feature of such proofs.

68. The Reason Why Gibbs' Derivation of His Electrocapillary

Equation [690] Exhibits It as Equivalent to

Lippmann's Equation

In the first place it may be well to point out once more
just exactly how Gibbs' deduction of [690] comes to be equiva-
lent to Lippmann's result, and not complementary to it like the
"Gibbs terms" in more recent formulae for da. It simply arises
from the fact that in Lippmann's proof "electricity" is a
"component" of the mercury whose "chemical potential"
corresponds to V, the electric potential. We can actually make
the proof of Lippmann's result correspond in every mathemati-
cal detail to the manner in which Gibbs derives his adsorption
equation. Calling e* the energy of the mercury surface we
write

S(S = fSjjs 4- 0-55 -{- V8Q

as the condition of equilibrium of this surface, V corresponding
to n^ and Q to m^, the potential and quantity of the component,
"electric charge." By the usual reasoning based on the fact
that an increase of s requires, for equilibrium conditions at the
same t, a, V, proportional increases in e^, s, and Q we see that

e^ = tri^ -\- as -{- VQ.
Hence

des = t dT]S + T)S dt -\- a ds + s da -{- VdQ + Q dV.

698 RICE ART. L

Therefore

ri^dt + sda -Y QdV = Q
or

da = — r]s dt — q dV,

i.e., at constant temperature,

da-

i7 = - ^'

Gibbs' own proof just carries through the same steps for the
"surface" of the solution, the component being the hydrogen ion
whose quantity in the electrolyte is supposed just to neutralize
the charge on the mercury (the apparent difficulty about the
sign has already been explained) and the chemical potential of
the ion is supposed to alter by the amount /3 6F where 5F is the
alteration of the electric potential of the solution and /3 the
reciprocal of the electrochemical equivalent a. Let us turn our
attention for a moment to this latter assumption.

69. Ouggenheim's Electrochemical Potential of an Ion

If one conceives an ion to be transferred from one solution to
another (in both of which it is an actual component) across the
interface, we can easily prove in the same manner as that in
which Gibbs derives his equations [687], [688], that

V + om' = V" + afx",

where the electrochemical equivalent a is a positive quantity
for cations and negative for anions. We can write this in the
form

where /3 is the reciprocal of a, the "chemo-electrical" equivalent
as we might call it. Actually it is the quantity n + fiV which
is the physically important and significant "intensity factor" in
the expression for the energy transferred from one phase to the

SURFACES OF DISCONTINUITY 699

other by the passage of the mass 8m of an ion, viz. (^t + ^V)8m.
It appears that in the transfer the division of the energy into
two parts nSm and /375w(or V8e) is of no practical importance.
The writer need not discuss the point fully here, since the reader
can find in a paper by E. A. Guggenheim (Jour, of Phys. Chem.,
33, 842, (1929)) some very interesting remarks on it which
will repay careful consideration. We shall consequently
replace the expression /x + fiV by M, referring to it, as is
suggested by Guggenheim in his paper, by the name "electro-
chemical potential" of the ionic component. This quantity M
has equal values in equilibrium conditions for an ion on opposite
sides of an interface if the ion is an actual component of both
phases. It has of course the same value in the region of dis-
continuity if it is an actual component of this region. If it
exists at the interface and in one bulk phase only, the electro-
chemical potential has the same value in each, a value which
cannot be greater than the value in the bulk phase in which it
is only a possible component.

Now it appears on reading parts of the literature that some
authors take it for granted that if the electric potential of a
phase is altered by the amount 57, then the quantity ^ + /3F or
M must alter by the amount /3 8V. This is no doubt based on
an implicit assumption that /x does not change in the meanwhile;
but this view seems to the writer to be too narrow and based on
the artificial splitting of the real chemical potential of the ion,
its electrochemical potential as we call it, into a "purely
chemical" and a "purely electric" part, which can vary
independently of one another. Even on this physically non-
significant analysis, one cannot guarantee that a change in V
will not alter the concentration of the ion and therefore the n
of the ion. The truth is, that the only chemical potential of an
ion of which we have any direct cognizance is the quantity we
have denoted by M, and we actually would have preferred still
to use the symbol m for it, but for the possibility of confusion
with the terminology of other writers. In fact, the electric po-
tentials of the phases are now to be reckoned among the thermo-
dynamic variables of the system, and the electrochemical
potentials of the ions (although to be quite exact the term

700 RICE ART. L

chemical potentials should still be used) are dependent on the
values of these as well as on the other variables. If a change
takes place in the electric potentials and a new state of equilib-
rium results, the M quantities change so as to preserve the same
equalities and inequalities as before. The real physical signifi-
cance of the equivalence of M and ju + j87 can be expressed by-
saying that, if all the electric potentials of the various phases of
a system in equilibrium are increased by the same amounts,
then the system still remains in equilibrium, no transference of
ions (or other components) takes place, and the electrochemical
potentials are all effectively unchanged and at their original
values. But if the changes of electric potential in the various
phases are not equal, no general statement about the changes
in the various phases can be made beyond the one concerned
with the preservation of equalities, etc., in the case of a varied
state of equilibrium.

70. Derivation by Means of the Postulate of ''Specific Adsorption"

of Ions of an Equation Combining Oibbs* Terms for

Ions with a Lippmann Term for Electrons

In accordance with this we write the elementary change of
energy in a homogeneous mass in the form

5e' = tdr]' - p8v' + Mi8mi + M^ bm^ + . . . ,
and in a surface in the form

5es = tbr)S J^ o-^g 4- M^bmi^ + M2 bm^^ + . . .

As before, we prove that

da = - rjsdt - Ti dMi - T2 dMi - ...

Since in general each homogeneous mass is uncharged as a whole,
and also each surface of discontinuity, it is clear that

TiiSi + r2/32 + TsPs + . . . =0,

for this expression is the whole charge per unit area on the ions
in the region of discontinuity. Hence

da = — -qadt — Vx dni — Fa d^ — . . .,

SURFACES OF DISCONTINUITY 701

which is, of course, the expression Gibbs uses just before [689].
But in reahty we can show just as readily that

d(T = - T]sdt - Vi dNi - Vi dNi - . . . ,

where A^i, A^2, etc. , are any quantities differing from Mi, M^, etc. ,
by amounts proportional to the various chemo-electrical equiva-
lents. This is in fact one aspect of the statement made above
concerning a system which has the electric potentials of all
its phases raised or lowered by the same amount.

So far we have considered solutions. If a metal, such as
mercury, is one of the phases, then we regard it as a phase with
two components, electrons and mercury ions Now the present
theory of metals considers the electric charge of a piece of metal
to be measured by the excess of the electrons in it above the
positive metallic ions, or the deficiency under; and quite simply
the "chemical potential" of the electron is just fi^V, where /S^ is
the chemo-electrical equivalent of an electron, viz., the
negative quotient of the electron-charge number by the mass
of the electron, i.e., —1.77 X 10*. In consequence, if the elec-
trical potential of mercury changes by 5F the chemical potential
of an electron changes by /S^SF or BM^ = ^JV. The region of
discontinuity between the mercury and the electrolyte is now
treated in the usual way. We must, of course, define the
position of the ideal dividing surface in order to give a definite
meaning to surface excess of any component. Various defining
conditions have been employed by different authors. For our
purpose the one used by Butler seems to be the simplest; this
places the surface so that the excess of mercury ions on the
mercury side of the surface is zero, i.e., so that the excess or
deficiency of electrons in the mercury measures the electric
charge on it ; in other words, if T^ is the excess of electrons per
unit area, q, the electric charge per unit surface, is equal to
fiiTf* There may of course be an excess concentration of
mercury ions on the electrolyte side of the surface, measured, as
usual, by the amount of these ions in excess of the amount that
would be in the electrolyte if the concentration of them were the

Note that /3e is an essentially negative number.

702 RICE ART. L

same right up to the surface as throughout the solution. We
will denote this excess per unit area by Ti. There will of course
be other ions present; positive ions such as those of other metals
and of hydrogen; negative ions such as sulphion, chlorion, etc.
The total charge on all these ions, positive and negative, must
be equal and opposite to the charge on the mercury side of the
surface, so that if there is a deficiency of electrons in the mercury
the negative ions must preponderate in the solution part
of the discontinuous region; i.e.,

Te/?, + TiiSi + r2i82 + . . . = 0,

where 2, .... refer to ions other than the mercury ions. We
now have the equation

d<T = - rjadt - Te dM^ - Ti dMi - Tg dMi - . . • ,

or, at constant temperature,

da = - ^,V, dV - Ti dMi - T2 dMi - ...

= - qdV - Ti dMi - Ta dikfg - • • .

This formula exhibits the Lippmann term —q dV(q is the charge
per unit area on the mercury) and Gibbs terms in addition for
the various ions present in excess or deficiency on the solution
side of the dividing surface. These are the specifically adsorbed
ions, cations or anions, whose influence causes the deviations
from the simple normal state of affairs covered by the Lippmann
term alone. Thus the simple criterion that at the condition
for maximum a the charge should be zero is not necessarily
true, since for that condition it is the expression

dMi dMi

which is zero. If we assume that 8M1, 8M2, etc., are all altered
by ^idV, jSa^F, etc., respectively, we would, on account of the
fact that Sr/3 = 0, obtain the result that da is always zero,
which is absurd. Or we might assume that some of the MrS
alter by ^rSV (say the Mi for the mercury ion because it is a

SURFACES OF DISCONTINUITY 703

component of the mercury, while assuming that the Mr's of the
other ions do not alter). This would require that when da is
zero q + ^iTi should be zero, and would imply that in this
condition the charge on the mercury is just balanced by the
electric charge on the excess mercury ions in the solution part
of the region of discontinuity, and that there are no anions in
this part, or if there are, their charge is neutralized by other
cations situated there also. The truth is, however, that such
assumptions are not a necessary feature of the analysis. In
simple electrostatic theory, a change of electric potential
involves a difference of "charge" on the surface of a metal.
We make the hypothesis that this is occasioned by excess or
defect of electrons. Such electrons are a component of the
mercury alone. The mercury ions may travel in or out of the
solution across the interface. Other ions do not leave the
solution. The change in the concentration of the mercury ions
in the solution occasions changes of concentrations in the other
ions in the solution, but it does not necessarily follow that these
changes produce a change in each Mr which is exactly
equal to /3r5F. Indeed, electrocapillary curves constitute the
experimental evidence which should enable us to trace the actual
changes in the MrS, had we sufficient knowledge of the distri-
bution of the various ions in the solution layer adjacent to
the electrode. It may seem peculiar that changes in the very
small region adjacent to the capillary cathode of the electro-
meter should be responsible for changes in the Mr throughout
the whole solution, for of course the M of any ion in the solution
must equal its M in the surface layer; but we must not overlook
the fact that the electrometer is only part of a complete circuit
containing a voltaic cell, and we must not forget the existence
of the large mercury anode. It is assumed that it is not
polarized, i.e., that its surface has on it the normal positive
charge; it is not electrically changed. Now this might be quite
consistent with a different distribution of ions in the layer of
solution adjacent to it; fewer cations and fewer anions in this
layer could still provide just the correct negative charge in this
layer to balance the unchanged positive charge on the mercury-
anode. On changing the external E.M.F. by 8E(=—8V)

704 RICE

ART. L

there is a flow of current for a moment. Electrons go round the
external part of the circuit towards the cathode to remove some
of the deficiency there; some mercury ions leave the layer ad-
jacent to the anode; some anions enter this layer and, together
with some of the anions already present there, are discharged
and supply electrons to the mercury anode to maintain the elec-
tron flow in the main circuit; for we have supposed that there
might be fewer anions as well as fewer cations in this layer and
yet the electrical conditions remain unchanged. Thus there
would be relatively quite considerable exchanges of ions between
this larger layer and the solution, which would occasion differ-
ences of concentration and electrochemical potentials in the
main body of the solution. This main body would, of course,
still be uncharged as a whole, but this again is quite consistent
with the existence of fewer cations and fewer anions in it. It
is not contended that the physical processes are just those pic-
tured, but the theory must somehow or other justify some
changes in the electrochemical potentials of the ions in the main
body of the solution if we assume changes in those in the layer
of electrolyte adjacent to the cathode, as we clearly do when
we assert the validity of an expression such as

da = -qdV - i:V dM.

One can hardly see how there are to be such changes in the Mr
of the ions in the solution if the concentrations are to remain
unchanged; and we have just seen that certain changes in con-
centrations are quite consistent with unchanged purely electric
conditions of the solution as a whole and of the anode surface.

71 . Some Brief Remarks on the Fundamental Electrical Equations

Used by Stern in His Treatment of the Distribution

of Ions in a Solution Close to the Cathode

of a Capillary Electrometer

It is clear that the electrocapillary curves are insufficient in
themselves to unravel the complexities of the situation, without
some theory of the distribution of the ions in the layer of solution
adjacent to the cathode. This question is dealt with in the
second list of references given above. The most exhaustive

SURFACES OF DISCONTINUITY 705

treatment will be found in the paper by Otto Stern. In the
space available the writer can only hope to try to throw some
light for the beginner on the fundamental equations used. Re-
garding the surface of the mercury as the origin from which
the distances z of parallel planes in the solution are measured,
we represent the electric potential at a plane distant z from the
cathode surface by \p(z), or briefly xp* The quantity \p changes
continuously, from the value xpo at the cathode, to zero well out
in the solution, i.e., practically at s = oo. If we denote the
concentration of a positive ion at z by Ci(z), and of a negative
by C2(z), then the concentrations in the solution are Ci(oo)
and Ci(x,). These are equal if we adopt as a simple view
that there are only two kinds of equi-valent ions, so that we write

Cl(oo) = C2(00) = C.

Statistical theory then shows that

C.(.)=Cexp[-^^}
Ciiz) = C exp + -^ J'

where F is the numerical value of the charge on a gram-equiva-
lent of ions, and R is the universal gas constant, t being the
absolute temperature. Hence the electric charge density p at
the position z in the solution is given by

p{z) = F[Ci{z) - C,{z)\

r r F^p{z)i vF^p{z)y

= FC

In addition to this there is a well-known theorem of Poisson
connecting the potential of a distribution of electric charge with
the density of this charge. It is

aV av 9V 47r ^ . ^

* It has been referred to as V hitherto in conformity with Gibbs'
notation. The alteration is made to conform to Stern's symbol.

706

RICE

ART. L

where ^{x, y, z) is the potential at the point x, y, z, and D is
the dielectric constant of the medium. In the present instance,
since \l/ depends only on z, this simplifies to

dV 47r , ^

This result is introduced into the previous one and in this way
solutions for 4^(z) in terms of z can be found. For details the
reader is referred to the literature.

One or two results, however, can be indicated in a general

0 L

Fig. 14

way by means of graphs. Thus suppose we have a graph of
ypiz) before us (Fig. 14), then wherever p{z) is positive, d}\{//dz'^
is negative, i.e., dip/dz is decreasing with increasing z, or the
slope of the graph is decreasing. (This means, decreasing in the
algebraic sense; so that if the slope is negative as in the region
OL in the figure, the numerical value of the slope is increasing;
in the region LM, the slope is increasing algebraically although
in the first portion of it the numerical value of the slope is
decreasing.) Thus in the figure p is positive in the region OL,
negative in the region LM and positive once more in the region

SURFACES OF DISCONTINUITY 707

MN, fading off to zero. P and Q are points of inflection in the
curve where the sign of (P4'/dz^ changes, that quantity being
zero at each of them, so that p is zero at the planes L and M.
Also, since

4:Trp{z) d^\p(z)

D dz"

it follows that

Jz = z, \dz /,

\dz

\dz Jz = z,

where Ei is the intensity of electric force at the plane L, OL being
equal to Zi, and OM to 22. (It is well known that the electric
intensity is measured by the gradient of the potential, and has
the direction in which the potential is decreasing. We are

assuming the graph to start from zero slope.) Now / pdz

is the charge per unit area between the planes 2 = 0 and 2 = 2i.
Hence this charge is DEi/iir. The charge between L and M
per unit area is negative and is equal to

pdz,

which works out as D{Ei + E^l^ic numerically, where Ei is the
numerical value of the intensity of force at the plane M (directed
towards the plane at 0.) Finally the charge beyond the plane
M is positive and numerically equal to DE^/^tt. The theory
attributes the positive charge DEi/4:Tr to the mercury surface.
To do so we imagine that OL is very small and that the graph
turns down very suddenly and steeply at first, so that this
portion of the graph is really in the mercury. The changes in
the solution may be more gradual. The graph we have drawn
would suit a picture in which there is a layer of negative ions in
the region LM and a layer of (fewer) positive ions beyond it;

708

RICE

ART. L

this is a picture employed by some writers. The original
Helmholtz idea would be pictured by a graph such as that in
Fig. 15, curved extremely near the beginning and end of the
graph, and a straight steep portion between, sharp bends being
the rule at both ends. In the straight portion dif/dz does not
change, so that d'^^p/dz'^ is zero there and there is no charge; the
positive and negative charges are concentrated in extremely thin
layers resembling a condenser distribution. The previous
graph gives a picture of a practically plane distribution for the
positive charge on the mercury surface and a "diffuse layer" of

Y (Jy^

M z
Fig. 16

charge in the solution, such as Goiiy first suggested. Sugges-
tions have also been made that there may be a diffuse layer
in the mercury also.

One last picture (Fig. 16) will show that we might conceive q
not to be zero, and yet there might exist no difference of poten-
tial between mercury and solution, as the graph has risen to
the same level as at the beginning.

THE GENERAL PROPERTIES OF A PERFECT
ELECTROCHEMICAL APPARATUS. ELEC-
TROCHEMICAL THERMODYNAMICS

[Gibbs, I, pp. S88-S49; 406-^12]

H. S. HARNED

Introduction

The importance of the contribution of Gibbs to the thermo-
dynamics of galvanic cells resides in the exactness, completeness,
and simpHcity of his method of treatment. In less than three
printed pages, he has set down the complete thermodynamic
theory, and has pointed out the fundamental relations between
the electromotive force and those basic thermodynamic func-
tions which have proved to be of such immense value to subse-
quent physico-chemical investigations.

In the following discussion, the thermodynamics of galvanic
cells, so far as explicitly treated by Gibbs, will be developed,
both by the use of the general functions and by the method of
a reversible cycle. Secondly, the arguments of Gibbs regarding
the heat suppUed to or withdrawn from galvanic cells during
their charge or discharge at constant temperature will be pre-
sented. In a third section, further ramifications of the theory
of this subject not explicitly stated, but contained implicitly in
Gibbs' general thermodynamics, will be discussed. Finally,
the role of Gibbs' fundamental contributions in the subsequent
development of the theory of solutions will be briefly outlined.

I. The General Thermodynamics as Explicitly Developed

Certain combinations of two or more pairs of electrical con-
ducting surfaces in electrical contact constitute a galvanic cell.
Not all such cells, however, may be subjected to numerical

709

710 HARMED

ART. M

treatment by the methods of thermodynamics, but only those
cells which fulfil the following conditions:

(1) No changes must take place without the passage of the

current.

(2) Every change which takes place during the passage of

the current may be reversed by reversing the direction
of the current.

These conditions define the "perfect electrochemical appa-
ratus," or the reversible galvanic cell.

The first condition excludes cells containing metal to fiquid
surfaces which react chemically, such as Volta's in which
alternate copper and zinc plates were separated by a fibrous
material moistened with sulphuric acid. The second condition
makes possible the measurement of the reversible electrical work
of the cell, and, concomitantly, the change in thermodynamic
potential, f, or the change of work content, i/', which accom-
panies the physical or chemical changes occurring in the cell.

Since this second condition is necessary for every direct
measurement of changes in f or \p, its more careful considera-
tion, particularly in reference to cell measurement, will help to
clarify further discussion. A reversible process is one in which
every successive state is a state of equilibrium. The maximum
or reversible work is that obtainable from this ideal reversible
process. Thus, the evaporation of a liquid against an external
pressure just equal to its vapor pressure is a reversible process,
and the work done by the vapor is the reversible work.

Let us consider a cell which has proved of considerable im-
portance in recent physical chemistry, and which has the char-
acteristics necessary for the present discussion, namely,

Pt 1 H2 (1 atm.) 1 HCl (m) | AgCl 1 Ag.

This consists of a hydrogen electrode, at one atmosphere pres-
sure, in contact with a hydrochloric acid solution at a concentra-
tion m, which is also in contact with a silver-silver chloride
electrode. All these substances will remain unchanged after
the solution has become saturated with the slightly soluble
silver chloride. If we connect the terminals, this cell will dis-
charge, positive current will flow from the hydrogen electrode

ELECTROCHEMICAL THERMODYNAMICS 711

to the silver-silver chloride electrode within the cell, and the
chemical changes corresponding to the passage of one faraday
of electricity F, when summed up will correspond to the
reaction

iH2 (1 atm.) + AgCl -> 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)

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)

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 — '

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,

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

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

log

y'mi
y"m2

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)

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.

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