Ebook Thermodynamics and chemistry Part 1

THERMODYNAMICS
AND
CHEMISTRY
SECOND EDITION

HOWARD DEVOE


Thermodynamics
and Chemistry
Second Edition
Version 7a, December 2015

Howard DeVoe
Associate Professor of Chemistry Emeritus
University of Maryland, College Park, Maryland


The first edition of this book was previously published by Pearson Education, Inc. It was
copyright ©2001 by Prentice-Hall, Inc.
The second edition, version 7a is copyright ©2015 by Howard DeVoe.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs
License, whose full text is at
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S HORT C ONTENTS

Biographical Sketches

15

Preface to the Second Edition

16

From the Preface to the First Edition

17

1 Introduction

19

2 Systems and Their Properties

27

3 The First Law

56

4 The Second Law

101

5 Thermodynamic Potentials

134

6 The Third Law and Cryogenics

149

7 Pure Substances in Single Phases

163

8 Phase Transitions and Equilibria of Pure Substances

192

9 Mixtures

222

10 Electrolyte Solutions

285

11 Reactions and Other Chemical Processes

302

12 Equilibrium Conditions in Multicomponent Systems

366

13 The Phase Rule and Phase Diagrams

418

14 Galvanic Cells

449

Appendix A Definitions of the SI Base Units

470

4


5

S HORT C ONTENTS

Appendix B Physical Constants

471

Appendix C Symbols for Physical Quantities

472

Appendix D Miscellaneous Abbreviations and Symbols

476

Appendix E Calculus Review

479

Appendix F Mathematical Properties of State Functions

481

Appendix G Forces, Energy, and Work

486

Appendix H Standard Molar Thermodynamic Properties

504

Appendix I

507

Answers to Selected Problems

Bibliography

511

Index

520

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


C ONTENTS

Biographical Sketches

15

Preface to the Second Edition

16

From the Preface to the First Edition

17

1 Introduction

1.1 Units . . . . . . . . . . . . . . . . . . .
1.1.1 Amount of substance and amount
1.2 Quantity Calculus . . . . . . . . . . . .
1.3 Dimensional Analysis . . . . . . . . . .
Problem . . . . . . . . . . . . . . . . . . . . .

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19
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26

2 Systems and Their Properties
2.1 The System, Surroundings, and Boundary . . . .
2.1.1 Extensive and intensive properties . . . .
2.2 Phases and Physical States of Matter . . . . . . .
2.2.1 Physical states of matter . . . . . . . . .
2.2.2 Phase coexistence and phase transitions .
2.2.3 Fluids . . . . . . . . . . . . . . . . . . .
2.2.4 The equation of state of a fluid . . . . . .
2.2.5 Virial equations of state for pure gases . .
2.2.6 Solids . . . . . . . . . . . . . . . . . . .
2.3 Some Basic Properties and Their Measurement .
2.3.1 Mass . . . . . . . . . . . . . . . . . . .
2.3.2 Volume . . . . . . . . . . . . . . . . . .
2.3.3 Density . . . . . . . . . . . . . . . . . .
2.3.4 Pressure . . . . . . . . . . . . . . . . . .
2.3.5 Temperature . . . . . . . . . . . . . . .
2.4 The State of the System . . . . . . . . . . . . .

2.4.1 State functions and independent variables
2.4.2 An example: state functions of a mixture
2.4.3 More about independent variables . . . .

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27
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6

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7

C ONTENTS

2.4.4 Equilibrium states . . . . .
2.4.5 Steady states . . . . . . . .
2.5 Processes and Paths . . . . . . . . .
2.6 The Energy of the System . . . . .
2.6.1 Energy and reference frames
2.6.2 Internal energy . . . . . . .
Problems . . . . . . . . . . . . . . . . .

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48
50
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52

53
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55

3 The First Law
3.1 Heat, Work, and the First Law . . . . . . . . . . . . . .
3.1.1 The concept of thermodynamic work . . . . . .
3.1.2 Work coefficients and work coordinates . . . . .
3.1.3 Heat and work as path functions . . . . . . . . .
3.1.4 Heat and heating . . . . . . . . . . . . . . . . .
3.1.5 Heat capacity . . . . . . . . . . . . . . . . . . .
3.1.6 Thermal energy . . . . . . . . . . . . . . . . . .
3.2 Spontaneous, Reversible, and Irreversible Processes . . .
3.2.1 Reversible processes . . . . . . . . . . . . . . .
3.2.2 Irreversible processes . . . . . . . . . . . . . . .
3.2.3 Purely mechanical processes . . . . . . . . . . .
3.3 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Heating and cooling . . . . . . . . . . . . . . .
3.3.2 Spontaneous phase transitions . . . . . . . . . .
3.4 Deformation Work . . . . . . . . . . . . . . . . . . . .
3.4.1 Gas in a cylinder-and-piston device . . . . . . .
3.4.2 Expansion work of a gas . . . . . . . . . . . . .
3.4.3 Expansion work of an isotropic phase . . . . . .
3.4.4 Generalities . . . . . . . . . . . . . . . . . . . .
3.5 Applications of Expansion Work . . . . . . . . . . . . .
3.5.1 The internal energy of an ideal gas . . . . . . . .
3.5.2 Reversible isothermal expansion of an ideal gas .
3.5.3 Reversible adiabatic expansion of an ideal gas . .
3.5.4 Indicator diagrams . . . . . . . . . . . . . . . .
3.5.5 Spontaneous adiabatic expansion or compression

3.5.6 Free expansion of a gas into a vacuum . . . . . .
3.6 Work in a Gravitational Field . . . . . . . . . . . . . . .
3.7 Shaft Work . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Stirring work . . . . . . . . . . . . . . . . . . .
3.7.2 The Joule paddle wheel . . . . . . . . . . . . . .
3.8 Electrical Work . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Electrical work in a circuit . . . . . . . . . . . .
3.8.2 Electrical heating . . . . . . . . . . . . . . . . .
3.8.3 Electrical work with a galvanic cell . . . . . . .
3.9 Irreversible Work and Internal Friction . . . . . . . . . .
3.10 Reversible and Irreversible Processes: Generalities . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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56
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Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


8

C ONTENTS

4 The Second Law
4.1 Types of Processes . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Statements of the Second Law . . . . . . . . . . . . . . . . . . . . .
4.3 Concepts Developed with Carnot Engines . . . . . . . . . . . . . . .
4.3.1 Carnot engines and Carnot cycles . . . . . . . . . . . . . . .
4.3.2 The equivalence of the Clausius and Kelvin–Planck statements
4.3.3 The efficiency of a Carnot engine . . . . . . . . . . . . . . .
4.3.4 Thermodynamic temperature . . . . . . . . . . . . . . . . . .
4.4 Derivation of the Mathematical Statement of the Second Law . . . .
4.4.1 The existence of the entropy function . . . . . . . . . . . . .
4.4.2 Using reversible processes to define the entropy . . . . . . . .

4.4.3 Some properties of the entropy . . . . . . . . . . . . . . . . .
4.5 Irreversible Processes . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Irreversible adiabatic processes . . . . . . . . . . . . . . . .
4.5.2 Irreversible processes in general . . . . . . . . . . . . . . . .
4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Reversible heating . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Reversible expansion of an ideal gas . . . . . . . . . . . . . .
4.6.3 Spontaneous changes in an isolated system . . . . . . . . . .
4.6.4 Internal heat flow in an isolated system . . . . . . . . . . . .
4.6.5 Free expansion of a gas . . . . . . . . . . . . . . . . . . . . .
4.6.6 Adiabatic process with work . . . . . . . . . . . . . . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 The Statistical Interpretation of Entropy . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Thermodynamic Potentials
5.1 Total Differential of a Dependent Variable . . .
5.2 Total Differential of the Internal Energy . . . .
5.3 Enthalpy, Helmholtz Energy, and Gibbs Energy

5.4 Closed Systems . . . . . . . . . . . . . . . . .
5.5 Open Systems . . . . . . . . . . . . . . . . .
5.6 Expressions for Heat Capacity . . . . . . . . .
5.7 Surface Work . . . . . . . . . . . . . . . . . .
5.8 Criteria for Spontaneity . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .

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6 The Third Law and Cryogenics
6.1 The Zero of Entropy . . . . . . . . . . . . . . . . . . . .
6.2 Molar Entropies . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Third-law molar entropies . . . . . . . . . . . . .
6.2.2 Molar entropies from spectroscopic measurements
6.2.3 Residual entropy . . . . . . . . . . . . . . . . . .
6.3 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Joule–Thomson expansion . . . . . . . . . . . . .
6.3.2 Magnetization . . . . . . . . . . . . . . . . . . .

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Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook

9

C ONTENTS

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Pure Substances in Single Phases
7.1 Volume Properties . . . . . . . . . . . . . . .
7.2 Internal Pressure . . . . . . . . . . . . . . . .
7.3 Thermal Properties . . . . . . . . . . . . . . .
7.3.1 The relation between CV;m and Cp;m . .
7.3.2 The measurement of heat capacities . .
7.3.3 Typical values . . . . . . . . . . . . .
7.4 Heating at Constant Volume or Pressure . . . .
7.5 Partial Derivatives with Respect to T, p, and V
7.5.1 Tables of partial derivatives . . . . . .
7.5.2 The Joule–Thomson coefficient . . . .
7.6 Isothermal Pressure Changes . . . . . . . . . .
7.6.1 Ideal gases . . . . . . . . . . . . . . .
7.6.2 Condensed phases . . . . . . . . . . .
7.7 Standard States of Pure Substances . . . . . .
7.8 Chemical Potential and Fugacity . . . . . . . .
7.8.1 Gases . . . . . . . . . . . . . . . . . .
7.8.2 Liquids and solids . . . . . . . . . . .
7.9 Standard Molar Quantities of a Gas . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .

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180
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181
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185
188

8 Phase Transitions and Equilibria of Pure Substances
8.1 Phase Equilibria . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Equilibrium conditions . . . . . . . . . . . . . . .
8.1.2 Equilibrium in a multiphase system . . . . . . . .
8.1.3 Simple derivation of equilibrium conditions . . . .
8.1.4 Tall column of gas in a gravitational field . . . . .
8.1.5 The pressure in a liquid droplet . . . . . . . . . .
8.1.6 The number of independent variables . . . . . . .
8.1.7 The Gibbs phase rule for a pure substance . . . . .
8.2 Phase Diagrams of Pure Substances . . . . . . . . . . . .
8.2.1 Features of phase diagrams . . . . . . . . . . . . .
8.2.2 Two-phase equilibrium . . . . . . . . . . . . . . .
8.2.3 The critical point . . . . . . . . . . . . . . . . . .
8.2.4 The lever rule . . . . . . . . . . . . . . . . . . . .
8.2.5 Volume properties . . . . . . . . . . . . . . . . .
8.3 Phase Transitions . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Molar transition quantities . . . . . . . . . . . . .
8.3.2 Calorimetric measurement of transition enthalpies
8.3.3 Standard molar transition quantities . . . . . . . .
8.4 Coexistence Curves . . . . . . . . . . . . . . . . . . . . .
8.4.1 Chemical potential surfaces . . . . . . . . . . . .

8.4.2 The Clapeyron equation . . . . . . . . . . . . . .
8.4.3 The Clausius–Clapeyron equation . . . . . . . . .

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192
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192
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209
211
211
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213
213
214
215
218

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Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


10

C ONTENTS

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9 Mixtures
9.1 Composition Variables . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Species and substances . . . . . . . . . . . . . . . . . . . . . .
9.1.2 Mixtures in general . . . . . . . . . . . . . . . . . . . . . . . .
9.1.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.4 Binary solutions . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.5 The composition of a mixture . . . . . . . . . . . . . . . . . .
9.2 Partial Molar Quantities . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Partial molar volume . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 The total differential of the volume in an open system . . . . . .
9.2.3 Evaluation of partial molar volumes in binary mixtures . . . . .
9.2.4 General relations . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.5 Partial specific quantities . . . . . . . . . . . . . . . . . . . . .
9.2.6 The chemical potential of a species in a mixture . . . . . . . . .
9.2.7 Equilibrium conditions in a multiphase, multicomponent system
9.2.8 Relations involving partial molar quantities . . . . . . . . . . .
9.3 Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Partial pressure . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.2 The ideal gas mixture . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 Partial molar quantities in an ideal gas mixture . . . . . . . . .
9.3.4 Real gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Liquid and Solid Mixtures of Nonelectrolytes . . . . . . . . . . . . . .
9.4.1 Raoult’s law . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.3 Partial molar quantities in ideal mixtures . . . . . . . . . . . .
9.4.4 Henry’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.5 The ideal-dilute solution . . . . . . . . . . . . . . . . . . . . .
9.4.6 Solvent behavior in the ideal-dilute solution . . . . . . . . . . .
9.4.7 Partial molar quantities in an ideal-dilute solution . . . . . . . .
9.5 Activity Coefficients in Mixtures of Nonelectrolytes . . . . . . . . . .
9.5.1 Reference states and standard states . . . . . . . . . . . . . . .
9.5.2 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.3 Real mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.4 Nonideal dilute solutions . . . . . . . . . . . . . . . . . . . . .
9.6 Evaluation of Activity Coefficients . . . . . . . . . . . . . . . . . . . .
9.6.1 Activity coefficients from gas fugacities . . . . . . . . . . . . .
9.6.2 Activity coefficients from the Gibbs–Duhem equation . . . . .
9.6.3 Activity coefficients from osmotic coefficients . . . . . . . . .
9.6.4 Fugacity measurements . . . . . . . . . . . . . . . . . . . . . .
9.7 Activity of an Uncharged Species . . . . . . . . . . . . . . . . . . . .
9.7.1 Standard states . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.2 Activities and composition . . . . . . . . . . . . . . . . . . . .
9.7.3 Pressure factors and pressure . . . . . . . . . . . . . . . . . . .
9.8 Mixtures in Gravitational and Centrifugal Fields . . . . . . . . . . . .

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222
222
222
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223

224
225
225
226
228
230
232
234
235
235
237
238
239
239
239
242
245
245
247
248
249
252
254
255
257
257
258
258
260
261

261
264
265
267
269
269
271
272
274

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


11

C ONTENTS

9.8.1 Gas mixture in a gravitational field . . . . . . . . . . . . . . . . . . 274
9.8.2 Liquid solution in a centrifuge cell . . . . . . . . . . . . . . . . . . 276
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10 Electrolyte Solutions
10.1 Single-ion Quantities . . . . . . . . . . . . . . . . . . . . .
10.2 Solution of a Symmetrical Electrolyte . . . . . . . . . . . .
10.3 Electrolytes in General . . . . . . . . . . . . . . . . . . . .
10.3.1 Solution of a single electrolyte . . . . . . . . . . . .
10.3.2 Multisolute solution . . . . . . . . . . . . . . . . .
10.3.3 Incomplete dissociation . . . . . . . . . . . . . . .
10.4 The Debye–H¨uckel Theory . . . . . . . . . . . . . . . . . .
10.5 Derivation of the Debye–H¨uckel Equation . . . . . . . . . .
10.6 Mean Ionic Activity Coefficients from Osmotic Coefficients

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Reactions and Other Chemical Processes
11.1 Mixing Processes . . . . . . . . . . . . . . . . . . . . . . .
11.1.1 Mixtures in general . . . . . . . . . . . . . . . . . .
11.1.2 Ideal mixtures . . . . . . . . . . . . . . . . . . . . .
11.1.3 Excess quantities . . . . . . . . . . . . . . . . . . .
11.1.4 The entropy change to form an ideal gas mixture . .
11.1.5 Molecular model of a liquid mixture . . . . . . . . .
11.1.6 Phase separation of a liquid mixture . . . . . . . . .
11.2 The Advancement and Molar Reaction Quantities . . . . . .
11.2.1 An example: ammonia synthesis . . . . . . . . . . .
11.2.2 Molar reaction quantities in general . . . . . . . . .
11.2.3 Standard molar reaction quantities . . . . . . . . . .
11.3 Molar Reaction Enthalpy . . . . . . . . . . . . . . . . . . .
11.3.1 Molar reaction enthalpy and heat . . . . . . . . . . .
11.3.2 Standard molar enthalpies of reaction and formation
11.3.3 Molar reaction heat capacity . . . . . . . . . . . . .
11.3.4 Effect of temperature on reaction enthalpy . . . . . .
11.4 Enthalpies of Solution and Dilution . . . . . . . . . . . . .
11.4.1 Molar enthalpy of solution . . . . . . . . . . . . . .
11.4.2 Enthalpy of dilution . . . . . . . . . . . . . . . . .
11.4.3 Molar enthalpies of solute formation . . . . . . . . .
11.4.4 Evaluation of relative partial molar enthalpies . . . .
11.5 Reaction Calorimetry . . . . . . . . . . . . . . . . . . . . .
11.5.1 The constant-pressure reaction calorimeter . . . . .
11.5.2 The bomb calorimeter . . . . . . . . . . . . . . . .
11.5.3 Other calorimeters . . . . . . . . . . . . . . . . . .
11.6 Adiabatic Flame Temperature . . . . . . . . . . . . . . . .
11.7 Gibbs Energy and Reaction Equilibrium . . . . . . . . . . .
11.7.1 The molar reaction Gibbs energy . . . . . . . . . . .

11.7.2 Spontaneity and reaction equilibrium . . . . . . . .

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285
286
288
291
291
292
293
294
297
299
301

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312
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315
318
318
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323
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326
327
328
333
333
335
340
341
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342

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


12

C ONTENTS

11.7.3 General derivation . . . . . . . . . . . . . . . . . .
11.7.4 Pure phases . . . . . . . . . . . . . . . . . . . . . .
11.7.5 Reactions involving mixtures . . . . . . . . . . . . .
11.7.6 Reaction in an ideal gas mixture . . . . . . . . . . .
11.8 The Thermodynamic Equilibrium Constant . . . . . . . . .
11.8.1 Activities and the definition of K . . . . . . . . . .
11.8.2 Reaction in a gas phase . . . . . . . . . . . . . . . .
11.8.3 Reaction in solution . . . . . . . . . . . . . . . . .
11.8.4 Evaluation of K . . . . . . . . . . . . . . . . . . .
11.9 Effects of Temperature and Pressure on Equilibrium Position

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Equilibrium Conditions in Multicomponent Systems
12.1 Effects of Temperature . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Variation of i =T with temperature . . . . . . . . . . . . .
12.1.2 Variation of ıi =T with temperature . . . . . . . . . . . . .
12.1.3 Variation of ln K with temperature . . . . . . . . . . . . . .
12.2 Solvent Chemical Potentials from Phase Equilibria . . . . . . . . .
12.2.1 Freezing-point measurements . . . . . . . . . . . . . . . .
12.2.2 Osmotic-pressure measurements . . . . . . . . . . . . . . .
12.3 Binary Mixture in Equilibrium with a Pure Phase . . . . . . . . . .
12.4 Colligative Properties of a Dilute Solution . . . . . . . . . . . . . .

12.4.1 Freezing-point depression . . . . . . . . . . . . . . . . . .
12.4.2 Boiling-point elevation . . . . . . . . . . . . . . . . . . . .
12.4.3 Vapor-pressure lowering . . . . . . . . . . . . . . . . . . .
12.4.4 Osmotic pressure . . . . . . . . . . . . . . . . . . . . . . .
12.5 Solid–Liquid Equilibria . . . . . . . . . . . . . . . . . . . . . . .
12.5.1 Freezing points of ideal binary liquid mixtures . . . . . . .
12.5.2 Solubility of a solid nonelectrolyte . . . . . . . . . . . . . .
12.5.3 Ideal solubility of a solid . . . . . . . . . . . . . . . . . . .
12.5.4 Solid compound of mixture components . . . . . . . . . . .
12.5.5 Solubility of a solid electrolyte . . . . . . . . . . . . . . . .
12.6 Liquid–Liquid Equilibria . . . . . . . . . . . . . . . . . . . . . . .
12.6.1 Miscibility in binary liquid systems . . . . . . . . . . . . .
12.6.2 Solubility of one liquid in another . . . . . . . . . . . . . .
12.6.3 Solute distribution between two partially-miscible solvents .
12.7 Membrane Equilibria . . . . . . . . . . . . . . . . . . . . . . . . .
12.7.1 Osmotic membrane equilibrium . . . . . . . . . . . . . . .
12.7.2 Equilibrium dialysis . . . . . . . . . . . . . . . . . . . . .
12.7.3 Donnan membrane equilibrium . . . . . . . . . . . . . . .
12.8 Liquid–Gas Equilibria . . . . . . . . . . . . . . . . . . . . . . . .
12.8.1 Effect of liquid pressure on gas fugacity . . . . . . . . . . .
12.8.2 Effect of liquid composition on gas fugacities . . . . . . . .
12.8.3 The Duhem–Margules equation . . . . . . . . . . . . . . .
12.8.4 Gas solubility . . . . . . . . . . . . . . . . . . . . . . . . .
12.8.5 Effect of temperature and pressure on Henry’s law constants

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343
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349

349
352
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356
359

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385
386
386

389
391
391
391
394
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395
395
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399
399
400
404
405
407

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


13

C ONTENTS

12.9 Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
12.10 Evaluation of Standard Molar Quantities . . . . . . . . . . . . . . . . . . 410
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
13 The Phase Rule and Phase Diagrams
13.1 The Gibbs Phase Rule for Multicomponent Systems . . .
13.1.1 Degrees of freedom . . . . . . . . . . . . . . . . .
13.1.2 Species approach to the phase rule . . . . . . . . .

13.1.3 Components approach to the phase rule . . . . . .
13.1.4 Examples . . . . . . . . . . . . . . . . . . . . . .
13.2 Phase Diagrams: Binary Systems . . . . . . . . . . . . .
13.2.1 Generalities . . . . . . . . . . . . . . . . . . . . .
13.2.2 Solid–liquid systems . . . . . . . . . . . . . . . .
13.2.3 Partially-miscible liquids . . . . . . . . . . . . . .
13.2.4 Liquid–gas systems with ideal liquid mixtures . . .
13.2.5 Liquid–gas systems with nonideal liquid mixtures .
13.2.6 Solid–gas systems . . . . . . . . . . . . . . . . .
13.2.7 Systems at high pressure . . . . . . . . . . . . . .
13.3 Phase Diagrams: Ternary Systems . . . . . . . . . . . . .
13.3.1 Three liquids . . . . . . . . . . . . . . . . . . . .
13.3.2 Two solids and a solvent . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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418
418
419

419
421
422
425
425
426
430
431
433
436
439
441
442
443
445

14 Galvanic Cells
14.1 Cell Diagrams and Cell Reactions . . . . . . .
14.1.1 Elements of a galvanic cell . . . . . . .
14.1.2 Cell diagrams . . . . . . . . . . . . . .
14.1.3 Electrode reactions and the cell reaction
14.1.4 Advancement and charge . . . . . . . .
14.2 Electric Potentials in the Cell . . . . . . . . . .
14.2.1 Cell potential . . . . . . . . . . . . . .
14.2.2 Measuring the equilibrium cell potential
14.2.3 Interfacial potential differences . . . .
14.3 Molar Reaction Quantities of the Cell Reaction
14.3.1 Relation between r Gcell and Ecell, eq .
14.3.2 Relation between r Gcell and r G . .
14.3.3 Standard molar reaction quantities . . .

14.4 The Nernst Equation . . . . . . . . . . . . . .
14.5 Evaluation of the Standard Cell Potential . . .
14.6 Standard Electrode Potentials . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .

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449
449
449
450
451
451
452
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454
455
457
458
459

461
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464
464
467

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Appendix A Definitions of the SI Base Units

470

Appendix B Physical Constants

471

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


14

C ONTENTS

Appendix C Symbols for Physical Quantities

472

Appendix D Miscellaneous Abbreviations and Symbols
476
D.1 Physical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
D.2 Subscripts for Chemical Processes . . . . . . . . . . . . . . . . . . . . . . 477
D.3 Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
Appendix E Calculus Review
E.1 Derivatives . . . . . .
E.2 Partial Derivatives . .
E.3 Integrals . . . . . . . .
E.4 Line Integrals . . . . .

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479
479
479
480
480

Appendix F Mathematical Properties of State Functions
F.1 Differentials . . . . . . . . . . . . . . . . . . . .
F.2 Total Differential . . . . . . . . . . . . . . . . . .
F.3 Integration of a Total Differential . . . . . . . . .
F.4 Legendre Transforms . . . . . . . . . . . . . . . .

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481
481
481
483
484

Appendix G Forces, Energy, and Work
G.1 Forces between Particles . . . . . . . . . . . . .
G.2 The System and Surroundings . . . . . . . . . .
G.3 System Energy Change . . . . . . . . . . . . . .
G.4 Macroscopic Work . . . . . . . . . . . . . . . .
G.5 The Work Done on the System and Surroundings
G.6 The Local Frame and Internal Energy . . . . . .
G.7 Nonrotating Local Frame . . . . . . . . . . . . .
G.8 Center-of-mass Local Frame . . . . . . . . . . .
G.9 Rotating Local Frame . . . . . . . . . . . . . .
G.10 Earth-Fixed Reference Frame . . . . . . . . . .

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486
487
490
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502
503

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Appendix H Standard Molar Thermodynamic Properties

504

Appendix I

507

Answers to Selected Problems

Bibliography

511

Index

520

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


B IOGRAPHICAL S KETCHES

Benjamin Thompson, Count of Rumford
James Prescott Joule . . . . . . . . . .

Sadi Carnot . . . . . . . . . . . . . . .
Rudolf Julius Emmanuel Clausius . . .
William Thomson, Lord Kelvin . . . . .
Max Karl Ernst Ludwig Planck . . . . .
Josiah Willard Gibbs . . . . . . . . . .
Walther Hermann Nernst . . . . . . . .
William Francis Giauque . . . . . . . .
´
Benoit Paul Emile
Clapeyron . . . . . .
William Henry . . . . . . . . . . . . . .
Gilbert Newton Lewis . . . . . . . . . .
Peter Josephus Wilhelmus Debye . . . .
Germain Henri Hess . . . . . . . . . . .
Franc¸ois-Marie Raoult . . . . . . . . .
Jacobus Henricus van’t Hoff . . . . . .

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63
85
106
109
114
116
138
150
159
217
250
270
295
321
379
382


P REFACE TO THE S ECOND E DITION

This second edition of Thermodynamics and Chemistry is a revised and enlarged version
of the first edition published by Prentice Hall in 2001. The book is designed primarily as a
textbook for a one-semester course for graduate or undergraduate students who have already

been introduced to thermodynamics in a physical chemistry course.
The PDF file of this book contains hyperlinks to pages, sections, equations, tables,
figures, bibliography items, and problems. If you are viewing the PDF on a screen, the links
are present, although they are not marked in any special way. If you click on a reference to
a page number, equation number, and so on, it will take you to that location.
Scattered through the text are sixteen one-page biographical sketches of some of the
historical giants of thermodynamics. A list is given on the preceding page. The sketches
are not intended to be comprehensive biographies, but rather to illustrate the human side of
thermodynamics—the struggles and controversies by which the concepts and experimental
methodology of the subject were developed.
The epigraphs on page 18 are intended to suggest the nature and importance of classical thermodynamics. You may wonder about the conversation between Alice and Humpty
Dumpty. Its point, particularly important in the study of thermodynamics, is the need to pay
attention to definitions—the intended meanings of words.
I welcome comments and suggestions for improving this book. My e-mail address appears below.

Howard DeVoe


16


F ROM THE P REFACE TO THE F IRST
E DITION

Classical thermodynamics, the subject of this book, is concerned with macroscopic aspects of the
interaction of matter with energy in its various forms. This book is designed as a text for a onesemester course for senior undergraduate or graduate students who have already been introduced to
thermodynamics in an undergraduate physical chemistry course.
Anyone who studies and uses thermodynamics knows that a deep understanding of this subject
does not come easily. There are subtleties and interconnections that are difficult to grasp at first. The
more times one goes through a thermodynamics course (as a student or a teacher), the more insight

one gains. Thus, this text will reinforce and extend the knowledge gained from an earlier exposure
to thermodynamics. To this end, there is fairly intense discussion of some basic topics, such as the
nature of spontaneous and reversible processes, and inclusion of a number of advanced topics, such
as the reduction of bomb calorimetry measurements to standard-state conditions.
This book makes no claim to be an exhaustive treatment of thermodynamics. It concentrates
on derivations of fundamental relations starting with the thermodynamic laws and on applications
of these relations in various areas of interest to chemists. Although classical thermodynamics treats
matter from a purely macroscopic viewpoint, the book discusses connections with molecular properties when appropriate.
In deriving equations, I have strived for rigor, clarity, and a minimum of mathematical complexity. I have attempted to clearly state the conditions under which each theoretical relation is valid
because only by understanding the assumptions and limitations of a derivation can one know when
to use the relation and how to adapt it for special purposes. I have taken care to be consistent in the
use of symbols for physical properties. The choice of symbols follows the current recommendations
of the International Union of Pure and Applied Chemistry (IUPAC) with a few exceptions made to
avoid ambiguity.
I owe much to J. Arthur Campbell, Luke E. Steiner, and William Moffitt, gifted teachers who
introduced me to the elegant logic and practical utility of thermodynamics. I am immensely grateful
to my wife Stephanie for her continued encouragement and patience during the period this book
went from concept to reality.
I would also like to acknowledge the help of the following reviewers: James L. Copeland,
Kansas State University; Lee Hansen, Brigham Young University; Reed Howald, Montana State
University–Bozeman; David W. Larsen, University of Missouri–St. Louis; Mark Ondrias, University
of New Mexico; Philip H. Rieger, Brown University; Leslie Schwartz, St. John Fisher College; Allan
L. Smith, Drexel University; and Paul E. Smith, Kansas State University.

17


A theory is the more impressive the greater the simplicity of its
premises is, the more different kinds of things it relates, and the more
extended is its area of applicability. Therefore the deep impression

which classical thermodynamics made upon me. It is the only physical
theory of universal content concerning which I am convinced that,
within the framework of the applicability of its basic concepts, it will
never be overthrown.
Albert Einstein
Thermodynamics is a discipline that involves a formalization of a large
number of intuitive concepts derived from common experience.
J. G. Kirkwood and I. Oppenheim, Chemical Thermodynamics, 1961
The first law of thermodynamics is nothing more than the principle of
the conservation of energy applied to phenomena involving the
production or absorption of heat.
Max Planck, Treatise on Thermodynamics, 1922
The law that entropy always increases—the second law of
thermodynamics—holds, I think, the supreme position among the laws
of Nature. If someone points out to you that your pet theory of the
universe is in disagreement with Maxwell’s equations—then so much
the worse for Maxwell’s equations. If it is found to be contradicted by
observation—well, these experimentalists do bungle things sometimes.
But if your theory is found to be against the second law of
thermodynamics I can give you no hope; there is nothing for it but to
collapse in deepest humiliation.
Sir Arthur Eddington, The Nature of the Physical World, 1928
Thermodynamics is a collection of useful relations between quantities,
every one of which is independently measurable. What do such
relations “tell one” about one’s system, or in other words what do we
learn from thermodynamics about the microscopic explanations of
macroscopic changes? Nothing whatever. What then is the use of
thermodynamics? Thermodynamics is useful precisely because some
quantities are easier to measure than others, and that is all.
M. L. McGlashan, J. Chem. Educ., 43, 226–232 (1966)

“When I use a word,” Humpty Dumpty said, in rather a scornful tone,
“it means just what I choose it to mean—neither more nor less.”
“The question is,” said Alice,“whether you can make words mean
so many different things.”
“The question is,” said Humpty Dumpty, “which is to be master—
that’s all.”
Lewis Carroll, Through the Looking-Glass


C HAPTER 1

I NTRODUCTION

Thermodynamics is a quantitative subject. It allows us to derive relations between the
values of numerous physical quantities. Some physical quantities, such as a mole fraction,
are dimensionless; the value of one of these quantities is a pure number. Most quantities,
however, are not dimensionless and their values must include one or more units. This
chapter reviews the SI system of units, which are the preferred units in science applications.
The chapter then discusses some useful mathematical manipulations of physical quantities
using quantity calculus, and certain general aspects of dimensional analysis.

1.1 UNITS
There is international agreement that the units used for physical quantities in science and
technology should be those of the International System of Units, or SI (standing for the
French Syst`eme International d’Unit´es). The Physical Chemistry Division of the International Union of Pure and Applied Chemistry, or IUPAC, produces a manual of recommended symbols and terminology for physical quantities and units based on the SI. The
manual has become known as the Green Book (from the color of its cover) and is referred
to here as the IUPAC Green Book. This book will, with a few exceptions, use symbols recommended in the third edition (2007) of the IUPAC Green Book;1 these symbols are listed
for convenient reference in Appendices C and D.
The SI is built on the seven base units listed in Table 1.1 on the next page. These base
units are independent physical quantities that are sufficient to describe all other physical

quantities. One of the seven quantities, luminous intensity, is not used in this book and is
usually not needed in thermodynamics. The official definitions of the base units are given
in Appendix A.
Table 1.2 lists derived units for some additional physical quantities used in thermodynamics. The derived units have exact definitions in terms of SI base units, as given in the
last column of the table.
The units listed in Table 1.3 are sometimes used in thermodynamics but are not part
of the SI. They do, however, have exact definitions in terms of SI units and so offer no
problems of numerical conversion to or from SI units.
1 Ref.

[36]. The references are listed in the Bibliography at the back of the book.

19


CHAPTER 1 INTRODUCTION
1.1 U NITS

20
Table 1.1 SI base units

Physical quantity

SI unit

Symbol

length
mass
time

thermodynamic temperature
amount of substance
electric current
luminous intensity

meter a
kilogram
second
kelvin
mole
ampere
candela

m
kg
s
K
mol
A
cd

a or

metre

Table 1.2 SI derived units
Physical quantity

Unit

Symbol

Definition of unit

force
pressure
Celsius temperature
energy
power
frequency
electric charge
electric potential
electric resistance

newton
pascal
degree Celsius
joule
watt
hertz
coulomb
volt
ohm

N
Pa
ı
C
J
W

Hz
C
V

1 N D 1 m kg s 2
1 Pa D 1 N m 2 D 1 kg m 1 s 2
t =ı C D T =K 273:15
1 J D 1 N m D 1 m2 kg s 2
1 W D 1 J s 1 D 1 m2 kg s 3
1 Hz D 1 s 1
1C D 1As
1 V D 1 J C 1 D 1 m2 kg s 3 A 1
1 D 1 V A 1 D 1 m2 kg s 3 A 2

Table 1.3 Non-SI derived units
Physical quantity

a

volume
pressure
pressure
pressure
energy
a or

litre

Unit
liter

bar
atmosphere
torr
calorie c

b or

l

c or

Symbol
b

L
bar
atm
Torr
cal d

thermochemical calorie

Definition of unit
1 L D 1 dm3 D 10 3 m3
1 bar D 105 Pa
1 atm D 101,325 Pa D 1:01325 bar
1 Torr D .1=760/ atm D .101,325/760/ Pa
1 cal D 4:184 J
d or

calth

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


CHAPTER 1 INTRODUCTION
1.1 U NITS

21
Table 1.4 SI prefixes

Fraction

Prefix

Symbol

Multiple

Prefix

Symbol

1

deci
centi
milli
micro
nano

pico
femto
atto
zepto
yocto

d
c
m

n
p
f
a
z
y

10
102
103
106
109
1012
1015
1018
1021
1024

deka
hecto

kilo
mega
giga
tera
peta
exa
zetta
yotta

da
h
k
M
G
T
P
E
Z
Y

10
10
10
10
10
10
10
10
10
10

2
3
6
9
12
15
18
21
24

Any of the symbols for units listed in Tables 1.1–1.3, except kg and ı C, may be preceded
by one of the prefix symbols of Table 1.4 to construct a decimal fraction or multiple of the
unit. (The symbol g may be preceded by a prefix symbol to construct a fraction or multiple
of the gram.) The combination of prefix symbol and unit symbol is taken as a new symbol
that can be raised to a power without using parentheses, as in the following examples:
1 mg D 1
1 cm D 1

1 cm3 D .1

10

3g

10

2m

10

2 m/3

D1

10

6 m3

1.1.1 Amount of substance and amount
The physical quantity formally called amount of substance is a counting quantity for particles, such as atoms or molecules, or for other chemical entities. The counting unit is
invariably the mole, defined as the amount of substance containing as many particles as the
number of atoms in exactly 12 grams of pure carbon-12 nuclide, 12 C. See Appendix A for
the wording of the official IUPAC definition. This definition is such that one mole of H2 O
molecules, for example, has a mass of 18:0153 grams (where 18:0153 is the relative molecular mass of H2 O) and contains 6:02214 1023 molecules (where 6:02214 1023 mol 1 is
the Avogadro constant to six significant digits). The same statement can be made for any
other substance if 18:0153 is replaced by the appropriate atomic mass or molecular mass
value.
The symbol for amount of substance is n. It is admittedly awkward to refer to n(H2 O)
as “the amount of substance of water.” This book simply shortens “amount of substance” to
amount, a common usage that is condoned by the IUPAC.2 Thus, “the amount of water in
the system” refers not to the mass or volume of water, but to the number of H2 O molecules
in the system expressed in a counting unit such as the mole.

2 Ref.

[118]. An alternative name suggested for n is “chemical amount.”

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook

CHAPTER 1 INTRODUCTION
1.2 Q UANTITY C ALCULUS

22

1.2 QUANTITY CALCULUS
This section gives examples of how we may manipulate physical quantities by the rules of
algebra. The method is called quantity calculus, although a better term might be “quantity
algebra.”
Quantity calculus is based on the concept that a physical quantity, unless it is dimensionless, has a value equal to the product of a numerical value (a pure number) and one or
more units:
physical quantity = numerical value units
(1.2.1)
(If the quantity is dimensionless, it is equal to a pure number without units.) The physical
property may be denoted by a symbol, but the symbol does not imply a particular choice of
units. For instance, this book uses the symbol for density, but can be expressed in any
units having the dimensions of mass divided by volume.
A simple example illustrates the use of quantity calculus. We may express the density
of water at 25 ı C to four significant digits in SI base units by the equation
102 kg m

D 9:970

3

(1.2.2)

and in different density units by the equation
D 0:9970 g cm

3

We may divide both sides of the last equation by 1 g cm
=g cm

3

D 0:9970

(1.2.3)
3

to obtain a new equation
(1.2.4)

Now the pure number 0:9970 appearing in this equation is the number of grams in one
cubic centimeter of water, so we may call the ratio =g cm 3 “the number of grams per
cubic centimeter.” By the same reasoning, =kg m 3 is the number of kilograms per cubic
meter. In general, a physical quantity divided by particular units for the physical quantity is
a pure number representing the number of those units.
Just as it would be incorrect to call “the number of grams per cubic centimeter,”
because that would refer to a particular choice of units for , the common practice of
calling n “the number of moles” is also strictly speaking not correct. It is actually the
ratio n=mol that is the number of moles.

In a table, the ratio =g cm 3 makes a convenient heading for a column of density
values because the column can then show pure numbers. Likewise, it is convenient to use
=g cm 3 as the label of a graph axis and to show pure numbers at the grid marks of the
axis. You will see many examples of this usage in the tables and figures of this book.

A major advantage of using SI base units and SI derived units is that they are coherent.
That is, values of a physical quantity expressed in different combinations of these units have
the same numerical value.

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


CHAPTER 1 INTRODUCTION
1.2 Q UANTITY C ALCULUS

23

For example, suppose we wish to evaluate the pressure of a gas according to the ideal
gas equation3
pD

nRT
V

(1.2.5)
(ideal gas)

In this equation, p, n, T , and V are the symbols for the physical quantities pressure, amount
(amount of substance), thermodynamic temperature, and volume, respectively, and R is the
gas constant.
The calculation of p for 5:000 moles of an ideal gas at a temperature of 298:15 kelvins,
in a volume of 4:000 cubic meters, is
pD

.5:000 mol/.8:3145 J K 1 mol

4:000 m3

1 /.298:15 K/

D 3:099

103 J m

3

(1.2.6)

The mole and kelvin units cancel, and we are left with units of J m 3 , a combination of
an SI derived unit (the joule) and an SI base unit (the meter). The units J m 3 must have
dimensions of pressure, but are not commonly used to express pressure.
To convert J m 3 to the SI derived unit of pressure, the pascal (Pa), we can use the
following relations from Table 1.2:
1J D 1Nm

1 Pa D 1 N m

2

(1.2.7)

When we divide both sides of the first relation by 1 J and divide both sides of the second
relation by 1 N m 2 , we obtain the two new relations
1 D .1 N m=J/

.1 Pa=N m

2

/D1

(1.2.8)

The ratios in parentheses are conversion factors. When a physical quantity is multiplied
by a conversion factor that, like these, is equal to the pure number 1, the physical quantity
changes its units but not its value. When we multiply Eq. 1.2.6 by both of these conversion
factors, all units cancel except Pa:
p D .3:099
D 3:099

103 J m

3/

.1 N m=J/

.1 Pa=N m

103 Pa

2

/
(1.2.9)

This example illustrates the fact that to calculate a physical quantity, we can simply

enter into a calculator numerical values expressed in SI units, and the result is the numerical
value of the calculated quantity expressed in SI units. In other words, as long as we use
only SI base units and SI derived units (without prefixes), all conversion factors are unity.
Of course we do not have to limit the calculation to SI units. Suppose we wish to
express the calculated pressure in torrs, a non-SI unit. In this case, using a conversion factor
obtained from the definition of the torr in Table 1.3, the calculation becomes
p D .3:099

103 Pa/

D 23:24 Torr

.760 Torr=101; 325 Pa/
(1.2.10)

3 This

is the first equation in this book that, like many others to follow, shows conditions of validity in parentheses immediately below the equation number at the right. Thus, Eq. 1.2.5 is valid for an ideal gas.
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


CHAPTER 1 INTRODUCTION
1.3 D IMENSIONAL A NALYSIS

24

1.3 DIMENSIONAL ANALYSIS
Sometimes you can catch an error in the form of an equation or expression, or in the dimensions of a quantity used for a calculation, by checking for dimensional consistency. Here
are some rules that must be satisfied:
both sides of an equation have the same dimensions

all terms of a sum or difference have the same dimensions
logarithms and exponentials, and arguments of logarithms and exponentials, are dimensionless
a quantity used as a power is dimensionless
In this book the differential of a function, such as df , refers to an infinitesimal quantity.
If one side of an equation is an infinitesimal quantity, the other side must also be. Thus,
the equation df D a dx C b dy (where ax and by have the same dimensions as f ) makes
mathematical sense, but df D ax C b dy does not.
Derivatives, partial derivatives, and integrals have dimensions that we must take into
account when determining the overall dimensions of an expression that includes them. For
instance:
the derivative dp= dT and the partial derivative .@p=@T /V have the same dimensions
as p=T
the partial second derivative .@2 p=@T 2 /V has the same dimensions as p=T 2
R
the integral T dT has the same dimensions as T 2
Some examples of applying these principles are given here using symbols described in
Sec. 1.2.
Example 1. Since the gas constant R may be expressed in units of J K 1 mol 1 , it has
dimensions of energy divided by thermodynamic temperature and amount. Thus, RT has
dimensions of energy divided by amount, and nRT has dimensions of energy. The products
RT and nRT appear frequently in thermodynamic expressions.
Example 2. What are the dimensions of the quantity nRT ln.p=p ı / and of p ı in
this expression? The quantity has the same dimensions as nRT (or energy) because the
logarithm is dimensionless. Furthermore, p ı in this expression has dimensions of pressure
in order to make the argument of the logarithm, p=p ı , dimensionless.
Example 3. Find the dimensions of the constants a and b in the van der Waals equation
pD

nRT
V nb

n2 a
V2

Dimensional analysis tells us that, because nb is subtracted from V , nb has dimensions
of volume and therefore b has dimensions of volume/amount. Furthermore, since the right
side of the equation is a difference of two terms, these terms have the same dimensions
as the left side, which is pressure. Therefore, the second term n2 a=V 2 has dimensions of
pressure, and a has dimensions of pressure volume2 amount 2 .
Example 4. Consider an equation of the form
Â
Ã
@ ln x
y
D
@T p
R
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook


CHAPTER 1 INTRODUCTION
1.3 D IMENSIONAL A NALYSIS

25

What are the SI units of y? ln x is dimensionless, so the left side of the equation has the
dimensions of 1=T , and its SI units are K 1 . The SI units of the right side are therefore
also K 1 . Since R has the units J K 1 mol 1 , the SI units of y are J K 2 mol 1 .

Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version: www.chem.umd.edu/thermobook