Main Thermal Physics: Thermodynamics and Statistical Mechanics for Scientists and Engineers

# Thermal Physics: Thermodynamics and Statistical Mechanics for Scientists and Engineers

In Thermal Physics: Thermodynamics and Statistical Mechanics for Scientists and Engineers, the fundamental laws of thermodynamics are stated precisely as postulates and subsequently connected to historical context and developed mathematically. These laws are applied systematically to topics such as phase equilibria, chemical reactions, external forces, fluid-fluid surfaces and interfaces, and anisotropic crystal-fluid interfaces.

Statistical mechanics is presented in the context of information theory to quantify entropy, followed by development of the most important ensembles: microcanonical, canonical, and grand canonical. A unified treatment of ideal classical, Fermi, and Bose gases is presented, including Bose condensation, degenerate Fermi gases, and classical gases with internal structure. Additional topics include paramagnetism, adsorption on dilute sites, point defects in crystals, thermal aspects of intrinsic and extrinsic semiconductors, density matrix formalism, the Ising model, and an introduction to Monte Carlo simulation.

Throughout the book, problems are posed and solved to illustrate specific results and problem-solving techniques.

• Includes applications of interest to physicists, physical chemists, and materials scientists, as well as materials, chemical, and mechanical engineers
• Suitable as a textbook for advanced undergraduates, graduate students, and practicing researchers
• Develops content systematically with increasing order of complexity
• Self-contained, including nine appendices to handle necessary background and technical details
Year: 2015
Edition: 1
Language: english
Pages: 610
ISBN 10: 0128033045
ISBN 13: 9780128033043
File: PDF, 7.57 MB

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```Thermal Physics

Thermal Physics
Thermodynamics and Statistical Mechanics
for Scientists and Engineers
Robert F. Sekerka
Carnegie Mellon University
Pittsburgh, PA 15213, USA

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier
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ISBN: 978-0-12-803304-3

Dedication

To Care . . . .
who cared about every word
and helped me write what I meant to say
rather than what I had written

v

•••
•••
•••
Tableof Contents
About the Cover
Preface

PART I
1

2

3

xv

XVII

Thermodynamic s

1

Intr oduct ion

3

1.1

Temper ature

3

1.2

Thermodynamics Versus Statistical Mechanics

5

1.3

Classification of State Variabl es

6

1.4

Energy in Mechanics

8

1.5

Elementary Kinetic Theory

12

First Law of Thermodynamics

15

2.1

Statement of the First Law

15

2.2

Quasistatic Work

17

2.3

Heat Capacities

19

2.4

Work Due to Expansion of an Ideal Gas

24

2.5

Enthalpy

28

Second Law of Thermod ynamics

31

3.1

Statement of the Second Law

32

3.2

Carnot Cycle and Engines

35

3.3

Calculation of th e Entropy Chang e

39

3.4

Combined First and Second Laws

41

3.5

Statistical Interpretation of Entrop y

47

vii

viii Table of Contents

4

5

6

7

8

Third Law of Thermodynamics

49

4.1

Statement of the Third Law

49

4.2

Implications of the Third Law

50

Open Systems

53

5.1

Single Component Open System

53

5.2

Mu lticomponent Open Systems

55

5.3

Euler Theorem of Homogeneous Functions

59

5.4

Chemical Potential of Real Gases, Fugacity

64

5.5

Legen dre Transformations

67

5.6

Partial Molar Quantities

71

5.7

Entropy of Chemical Reaction

75

Equilibrium and Thermodynamic Potentia ls

79

6.1

Entropy Criterion

79

6.2

Energy Criterion

84

6.3

Other Equilibrium Criteria

88

6.4

Summary of Criteria

92

Requirements for Stabi lity

95

7.1

Stability Requirements for Entrop y

95

7.2

Stability Requirements for Internal Energy

100

7.3

Stability Requirements for Other Potentials

102

7.4

Consequences of Stability Requiremen ts

105

7.5

Extension to Many Variables

106

7.6

Principles of Le Chatlier and Le Chatlier-Braun

107

Monocomponent

Phase Equi librium

8.1

Clausius-Clapeyron

8.2

Sketches of the Thermodynamic

8.3

Phas e Diagram in th e v, p Plane

109

110

Equation
Function s

115
118

Table of Contents

9

10

11

12

13

ix

Two -Phase Equilibrium for a van der Waa ls Fluid

121

9.1

van der Waals Equation of State

121

9.2

Thermodynamic Functions

124

9.3

Phase Equilibrium and Miscibility Gap

127

9.4

Gibbs Free Energy

131

Binary Solutions

137

10.1 Thermodynamics of Binary Solutions

137

10.2 Ideal Soluti ons

142

10.3 Phase Diagram for an Ideal Solid and an Ideal Liquid

145

10.4 Regular Solution

148

10.5 General Binary Solutions

153

Externa l Forces and Rotating Coordinate Systems

155

11.1 Conditions for Equilibrium

155

11.2 Uniform Gravitational Field

157

11.3 Non-Uniform Gravitational Field

164

11.4 Rotating Systems

164

11.5 Electric Fields

166

Chemica l Reactions

167

12.1 Reactions at Constant Volume or Pressure

168

12.2 Standard States

171

12.3 Equilibr ium and Affinity

173

12.4 Explicit Equilibrium Conditions

175

12.5 Simul taneous Reactions

182

Thermodynam ics of Fluid-Fluid Interf aces

185

13.1 Planar Interfaces in Fluids

186

13.2 Curved Interfaces in Flu ids

197

x Table of Contents

13.3 Interface Junctions and Contact Angles

202

13.4 Liquid Surface Shape in Gravity

205

14 Thermodynamics of Solid-Flu i d Interfaces

215

14.1 Planar Solid-Fluid Interfaces

216

14.2 Anisotropy of y

22 1

14.3 Curved Solid-Fluid Interfaces

227

14.4 Faceting of a Large Planar Face

233

14.5 Equilibrium Shape from the~ -Vector

236

14.6 Herring Formula

240

14.7 Legendre Transform of the Equilibrium Shape

241

14.8 Remarks About Solid-Solid Interfaces

242

PART II

Statistical Mechanics

15 Entropy and Information Theory
15.1 Entropy as a Measure of Disorder
15.2

Boltzmann Eta Theorem

16 Microcanonical Ensemble

245
247
247
251

257

16.1 Fundamenta l Hypothesis of Statistical Mechanics

258

16.2 Two-Stat e Subsystems

261

16.3 Harmonic Oscillators

265

16.4 Ideal Gas

267

16.5 Multicomponent

Ideal Gas

17 Classical Microcanonical Ensemble

273

277

17.1 Liouville's Theorem

278

17.2 Classical Microcanonical Ensemble

280

Tab le of Contents

18

Distinguishable Part icles with Negligible
Interaction Energies
18.1

Derivation of the Boltzmann Distribution

18.2 Two-State Subsystems
18.3

Harmonic Oscillators

18.4 Rigid Linear Rotator

19 Canonical Ensemble
19. 1 Three Derivations
19.2

21

285
285
289
293
303

305
305
312

19.3 Classica l Idea l Gas

313

19.4 Maxwell-Boltzmann Distribution

317

19.5 Energy Dispersion

320

19.6 Paramagnetism

321

19.7

20

Factorizat ion Theorem

xi

Partition Function and Densit y of States

330

Classical Canonical Ensemble

337

20.1

Classical Ideal Gas

338

20.2

Law of Dulong and Petit

342

20.3

Averaging Theorem and Equipartition

343

20.4

Virial Theorem

346

20.5

Virial Coefficients

348

20.6

Use of Canonical Transforma tions

354

20.7

Rotating Rigid Polya tomi c Molecu les

356

Grand Canonical Ensemble

359

21.1

Derivation from Microcanonical Ensemble

360

21.2

Ideal Systems: Orbitals and Factorization

368

xii Table of Contents

22

21.3 Classical Ideal Gas with Internal Structure

380

21.4 Multicomponent

388

21.5 Pressure Ensemb le

389

Entropy for Any Ensemb le

397

22.1 General Ensemble

397

22.2

23

405
406

23.2 The Functions hv(A, a)

408

23.3 Virial Expansio ns for Ideal Fermi and Bose Gases

410

23.4 Heat Capacity

412

Bose Condensat ion

413

Bosons at Low Temperatures

413

24.2 Thermodynamic Functions

416

Condensate Region

421

24.3

Degenerate Fermi Gas

425

25.1

Ideal Fermi Gas at Low Temperatures

425

25.2

Free Electron Model of a Metal

428

25.3 Thermal Activation of Electrons

429

25.4 Pauli Paramagnetism

433

25.5

Landau Diamagnetism

436

25.6 Thermionic Emission

439

Semiconductors

442

25.7

26

402

Integr al Formulae

24.l

25

Summation over Energy Levels

Un ified Treatment of Idea l Fermi, Bose, and Classical Gases
23.1

24

Systems

Quantum Statistics

451

26.1 Pure States

451

26.2

Statistical States

453

Table of Contents

26.3 Random Phases and External Influ ence

454

26.4 Time Evoluti on

455

26.5 Densit y Operators for Specific Ensembles

456

26.6 Examp les of th e Density Matrix

459

Indi stinguishable Particles

465

26.7

27

Ising Model

469

27.1

470

Ising Mod el, Mean Fie ld Treatment

27.2 Pair Sta tistics

477

27.3

479

Soluti on in One Dimension for Zero Field

27.4 Transfer Matrix

480

27.5 Oth er Methods of Soluti on

483

27.6 Monte Carlo Simulation

484

PARTIll
A

B

C

D

xiii

Appendices

495

Stirl ing 's Approximation

497

A.l

Elem entary Mot ivation ofEq. (A.l )

498

A.2

Asymptotic Series

499

Use of Jacobians to Convert Partial Derivatives

503

B.l

Properties of Jacobians

503

B.2

Connect ion to Thermody namic s

504

Differential Geometry of Surfaces

509

C.l

Alterna tive Formulae for ~ Vector

509

C.2

Sur face Differe nti al Geome try

511

C.3

~

516

C.4

Herring Form ula

Vector for Genera l Surfaces

Equi librium of Two-State Systems

518

523

xiv Table of Contents

E

F

G

H

Aspects of Canonical Transformations

529

E.1

Necessary and Sufficient Conditions

530

E.2

Restricted Canonical Transformations

534

Rotation of Rigid Bodies

537

El

Moment oflnertia

537

F.2

Angular Momentum

539

F.3

Kinetic Energy

540

F.4

Time Derivatives

540

F.5

Rotating Coordinate System

541

F.6

Matrix Formulation

544

F.7

Canonical Variables

546

F.8

Quantum Energy Levels for Diatomic Molecule

547

Thermodynamic Perturbation Theory

549

G.1

Classical Case

549

G.2

Quantum Case

550

Selected Mathematical Relations

553

H.1

Bernoulli Numbers and Polynomials

553

H.2

Euler-Maclaur in Sum Formula

554

Creation and Annihilation

559

1.1

Harmonic Oscillator

559

1.2

Boson Operators

560

1.3

Fermion Operators

562

1.4

Boson and Fermion Number Operators

563

References
Index

Operators

565
569

About the Cover
To represent the many scientists who have made major contributions to the foundations of
thermodynamics and statistical mechanics, the cover of this book depicts four significant
scientists along with some equations and graphs associated with each of them.
• James Clerk Maxwell (1831-1879) for his work on thermodynamics and especially the
kinetic theory of gases, including the Maxwell relations derived from perfect differentials and the Maxwell-Boltzmann Gaussian distribution of gas velocities, a precursor of
ensemble theory (see Sections 5.2, 19.4, and 20.1).
• Ludwig Boltzmann (1844-1906) for his statistical approach to mechanics of many
particle systems, including his Eta function that describes the decay to equilibrium
and his formula showing that the entropy of thermodynamics is proportional to the
logarithm of the number of microscopic realizations of a macrosystem (see Chapters
15–17).
• J. Willard Gibbs (1839-1903) for his systematic theoretical development of the thermodynamics of heterogeneous systems and their interfaces, including the definition
of chemical potentials and free energy that revolutionized physical chemistry, as well
as his development of the ensemble theory of statistical mechanics, including the
canonical and grand canonical ensembles. The contributions of Gibbs are ubiquitous
in this book, but see especially Chapters 5–8, 12–14, 17, 20, and 21.
• Max Planck (1858-1947, Nobel Prize 1918) for his quantum hypothesis of the energy of
cavity radiation (hohlraum blackbody radiation) that connected statistical mechanics
to what later became quantum mechanics (see Section 18.3.2); the Planck distribution
of radiation flux versus frequency for a temperature 2.725 K describes the cosmic
microwave background, first discovered in 1964 as a remnant of the Big Bang and later
measured by the COBE satellite launched by NASA in 1989.
The following is a partial list of many others who have also made major contributions
to the field, all deceased. Recipients of a Nobel Prize (first awarded in 1901) are denoted
by the letter “N” followed by the award year. For brief historical introductions to thermodynamic and statistical mechanics, see Cropper [11, pp. 41-136] and Pathria and Beale [9,
pp. xxi-xxvi], respectively. The scientists are listed in the order of their year of birth:
Sadi Carnot (1796-1832); Julius von Mayer (1814-1878); James Joule (1818-1889);
Hermann von Helmholtz (1821-1894); Rudolf Clausius (1822-1888); William Thomson,
Lord Kelvin (1824-1907); Johannes van der Waals (1837-1923, N1910); Jacobus van’t
Hoff (1852-1911, N1901); Wilhelm Wien (1864-1928, N1911); Walther Nernst (18641941, N1920); Arnold Sommerfeld (1868-1951); Théophile de Donder (1872-1957); Albert
xv

xvi

About the Cover

Einstein (1879-1955, N1921); Irving Langmuir (1881-1957, N1932); Erwin Schrödinger
(1887-1961, N1933); Satyendra Bose (1894-1974); Pyotr Kapitsa (1894-1984, N1978);
William Giauque (1895-1982, N1949); John van Vleck (1899-1980, N1977); Wolfgang Pauli
(1900-1958, N1945); Enrico Fermi (1901-1954, N1938); Paul Dirac (1902-1984, N1933);
Lars Onsager (1903-1976, N1968); John von Neumann (1903-1957); Lev Landau (19081968, N1962); Claude Shannon (1916-2001); Ilya Prigogine (1917-2003, N1977); Kenneth
Wilson (1936-2013, N1982).

Preface
This book is based on lectures in courses that I taught from 2000 to 2011 in the Department
of Physics at Carnegie Mellon University to undergraduates (mostly juniors and seniors)
and graduate students (mostly first and second year). Portions are also based on a
course that I taught to undergraduate engineers (mostly juniors) in the Department of
Metallurgical Engineering and Materials Science in the early 1970s. It began as class notes
but started to be organized as a book in 2004. As a work in progress, I made it available
on my website as a pdf, password protected for use by my students and a few interested
colleagues.
It is my version of what I learned from my own research and self-study of numerous
books and papers in preparation for my lectures. Prominent among these sources were
the books by Fermi , Callen , Gibbs [3, 4], Lupis , Kittel and Kroemer , Landau
and Lifshitz , and Pathria [8, 9], which are listed in the bibliography. Explicit references
to these and other sources are made throughout, but the source of much information is
beyond my memory.
Initially it was my intent to give an integrated mixture of thermodynamics and statistical mechanics, but it soon became clear that most students had only a cursory understanding of thermodynamics, having encountered only a brief exposure in introductory
physics and chemistry courses. Moreover, I believe that thermodynamics can stand on
its own as a discipline based on only a few postulates, or so-called laws, that have stood
the test of time experimentally. Although statistical concepts can be used to motivate
thermodynamics, it still takes a bold leap to appreciate that thermodynamics is valid,
within its intended scope, independent of any statistical mechanical model. As stated by
Albert Einstein in Autobiographical Notes (1946) :
“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 on 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.”
Of course thermodynamics only allows one to relate various measurable quantities to
one another and must appeal to experimental data to get actual values. In that respect,
models based on statistical mechanics can greatly enhance thermodynamics by providing
values that are independent of experimental measurements. But in the last analysis, any
model must be compatible with the laws of thermodynamics in the appropriate limit of
xvii

xviii

Preface

sufficiently large systems. Statistical mechanics, however, has the potential to treat smaller
systems for which thermodynamics is not applicable.
Consequently, I finally decided to present thermodynamics first, with only a few
connections to statistical concepts, and then present statistical mechanics in that context.
That allowed me to better treat reversible and irreversible processes as well as to give a
thermodynamic treatment of such subjects as phase diagrams, chemical reactions, and
anisotropic surfaces and interfaces that are especially valuable to materials scientists and
engineers.
The treatment of statistical mechanics begins with a mathematical measure of disorder,
quantified by Shannon [48, 49] in the context of information theory. This measure is
put forward as a candidate for the entropy, which is formally developed in the context
of the microcanonical, canonical, and grand canonical ensembles. Ensembles are first
treated from the viewpoint of quantum mechanics, which allows for explicit counting of
states. Subsequently, classical versions of the microcanonical and canonical ensembles
are presented in which integration over phase space replaces counting of states. Thus,
information is lost unless one establishes the number of states to be associated with a
phase space volume by requiring agreement with quantum treatments in the limit of high
temperatures. This is counter to the historical development of the subject, which was
in the context of classical mechanics. Later in the book I discuss the foundation of the
quantum mechanical treatment by means of the density operator to represent pure and
statistical (mixed) quantum states.
Throughout the book, a number of example problems are presented, immediately
followed by their solutions. This serves to clarify and reinforce the presentation but also
allows students to develop problem-solving techniques. For several reasons I did not
provide lists of problems for students to solve. Many such problems can be found in
textbooks now in print, and most of their solutions are on the internet. I leave it to teachers
to assign modifications of some of those problems or, even better, to devise new problems
whose solutions cannot yet be found on the internet.
The book also contains a number of appendices, mostly to make it self-contained but
also to cover technical items whose treatment in the chapters would tend to interrupt the
flow of the presentation.
I view this book as an intermediate contribution to the vast subjects of thermodynamics and statistical mechanics. Its level of presentation is intentionally more rigorous
and demanding than in introductory books. Its coverage of statistical mechanics is much
less extensive than in books that specialize in statistical mechanics, such as the recent
third edition of Pathria’s book, now authored by Pathria and Beale , that contains
several new and advanced topics. I suspect the present book will be useful for scientists,
particularly physicists and chemists, as well as engineers, particularly materials, chemical,
and mechanical engineers. If used as a textbook, many advanced topics can be omitted
to suit a one- or two-semester undergraduate course. If used as a graduate text, it could
easily provide for a one- or two-semester course. The level of mathematics needed in most
parts of the book is advanced calculus, particularly a strong grasp of functions of several

Preface xix

variables, partial derivatives, and infinite series as well as an elementary knowledge of
differential equations and their solutions. For the treatment of anisotropic surfaces and
interfaces, necessary relations of differential geometry are presented in an appendix. For
the statistical mechanics part, an appreciation of stationary quantum states, including
degenerate states, is essential, but the calculation of such states is not needed. In a few
places, I use the notation of the Dirac vector space, bras and kets, to represent quantum
states, but always with reference to other representations; the only exceptions are Chapter
26, Quantum Statistics, where the Dirac notation is used to treat the density operator, and
Appendix I, where creation and annihilation operators are treated.
I had originally considered additional information for this book, including more of my
own research on the thermodynamics of inhomogeneously stressed crystals and a few
more chapters on the statistical mechanical aspects of phase transformations. Treatment
of the liquid state, foams, and very small systems were other possibilities. I do not address
many-body theory, which I leave to other works. There is an introduction to Monte Carlo
simulation at the end of Chapter 27, which treats the Ising model. The renormalization
group approach is described briefly but not covered in detail. Perhaps I will address some
of these topics in later writings, but for now I choose not to add to the already considerable
bulk of this work.
Over the years that I shared versions of this book with students, I received some
valuable feedback that stimulated revision or augmentation of topics. I thank all those
students. A few faculty at other universities used versions for self-study in connection with
courses they taught, and also gave me some valuable feedback. I thank these colleagues
as well. I am also grateful to my research friends and co-workers at NIST, where I have
been a consultant for nearly 45 years, whose questions and comments stimulated a lot
of critical thinking; the same applies to many stimulating discussions with my colleagues
at Carnegie-Mellon and throughout the world. Singular among those was my friend and
fellow CMU faculty member Prof. William W. Mullins who taught me by example the love,
joy and methodologies of science. There are other people I could thank individually for
contributing in some way to the content of this book but I will not attempt to present
such a list. Nevertheless, I alone am responsible for any misconceptions or outright errors
that remain in this book and would be grateful to anyone who would bring them to my
attention.
In bringing this book to fruition, I would especially like to thank my wife Carolyn for
her patience and encouragement and her meticulous proofreading. She is an attorney,
not a scientist, but the logic and intellect she brought to the task resulted in my rewriting
a number of obtuse sentences and even correcting a number of embarrassing typos and
inconsistent notation in the equations. I would also like to thank my friends Susan and
John of Cosgrove Communications for their guidance with respect to several aesthetic
aspects of this book. Thanks are also due to the folks at my publisher Elsevier: Acquisitions Editor Dr. Anita Koch, who believed in the product and shepherded it through
technical review, marketing and finance committees to obtain publication approval;
Editorial Project Manager Amy Clark, who guided me though cover and format design as

xx

Preface

well as the creation of marketing material; and Production Project Manager Paul Prasad
Chandramohan, who patiently managed to respond positively to my requests for changes
in style and figure placements, as well as my last-minute corrections. Finally, I thank
Carnegie Mellon University for providing me with an intellectual home and the freedom
to undertake this work.
Robert F. Sekerka
Pittsburgh, PA

1
Introduction
Thermal physics deals with the quantitative physical analysis of macroscopic systems.
Such systems consist of a very large number, N , of atoms, typically N ∼ 1023 . According
to classical mechanics, a detailed knowledge of the microscopic state of motion (say,
position ri and velocity vi ) of each atom, i = 1, 2, . . . , N , at some time t, even if attainable,
would constitute an overwhelmingly huge database that would be practically useless.
More useful quantities would be averages, such as the average kinetic energy of an atom
in the system, which would be independent of time if the system were in equilibrium.
We might also be interested in knowing such things as the volume V of the system or
the pressure p that it exerts on the walls of a containing vessel. In other words, a useful
description of a macroscopic system is necessarily statistical and consists of knowledge of
a few macroscopic variables that describe the system to our satisfaction.
We shall be concerned primarily with macroscopic systems in a state of equilibrium.
An equilibrium state is one whose macroscopic parameters, which we shall call state variables, do not change with time. We accept the proposition, in accord with our experience,
that any macroscopic system subject to suitable constraints, such as confinement to a
volume and isolation from external forces or sources of matter and energy, will eventually
come to a state of equilibrium. Our concept, or model, of the system will dictate the
number of state variables that constitute a complete description—a complete set of state
variables—of that system. For example, a gas consisting of a single atomic species might be
described by three state variables, its energy U, its volume V , and its number of atoms N .
Instead of its number of atoms, we usually avoid large numbers and specify its number
of moles, N := N /N A where NA = 6.02×1023 molecules/mol is Avogadro’s number.1
The state of a gas consisting of two atomic species, denoted by subscripts 1 and 2, would
require four variables, U, V , N1 , and N2 . A simple model of a crystalline solid consisting of
one atomic species would require eight variables; these could be taken to be U, V , N , and
five more variables needed to describe its state of shear strain.2

1.1 Temperature
A price we pay to describe a macroscopic system is the introduction of a state variable,
known as the temperature, that is related to statistical concepts and has no counterpart
in simple mechanical systems. For the moment, we shall regard the temperature to be an
notation A := B means A is defined to be equal to B, and can be written alternatively as B =: A.
is true if the total number of unit cells of the crystal is able to adjust freely, for instance by means of
vacancy diffusion; otherwise, a total of nine variables is required because one must add the volume per unit cell to
the list of variables. More complex macroscopic systems require more state variables for a complete description,
but usually the necessary number of state variables is small.
1 The

2 This

Thermal Physics. http://dx.doi.org/10.1016/B978-0-12-803304-3.00001-6
Copyright © 2015 Elsevier Inc. All rights reserved.

3

4 THERMAL PHYSICS

empirical quantity, measured by a thermometer, such that temperature is proportional to
the expansion that occurs whenever energy is added to matter by means of heat transfer.
Examples of thermometers include thermal expansion of mercury in a long glass tube,
bending of a bimetallic strip, or expansion of a gas under the constraint of constant pressure. Various thermometers can result in different scales of temperature corresponding to
the same physical states, but they can be calibrated to produce a correspondence. If two
systems are able to freely exchange energy with one another such that their temperatures
are equal and their other macroscopic state variables do not change with time, they are
said to be in equilibrium.
From a theoretical point of view, the most important of these empirical temperatures is
the temperature θ measured by a gas thermometer consisting of a fixed number of moles
N of a dilute gas at volume V and low pressure p. This temperature θ is defined to be
proportional to the volume at fixed p and N by the equation
θ :=

p
V,
RN

(1.1)

where R is a constant. For variable p, Eq. (1.1) also embodies the laws of Boyle, Charles,
and Gay-Lussac. Provided that the gas is sufficiently dilute (small enough N /V ), experiment shows that θ is independent of the particular gas that is used. A gas under such
conditions is known as an ideal gas. The temperature θ is called an absolute temperature
because it is proportional to V , not just linear in V . If the constant R = 8.314 J/(mol K),
then θ is measured in degrees Kelvin, for which one uses the symbol K. On this scale,
the freezing point of water at one standard atmosphere of pressure is 273.15 K. Later,
in connection with the second law of thermodynamics, we will introduce a unique
thermodynamic definition of a temperature, T, that is independent of any particular
thermometer. Fermi [1, p. 42] uses a Carnot cycle that is based on an ideal gas as a working
substance to show that T = θ, so henceforth we shall use the symbol T for the absolute
temperature.3

Example Problem 1.1. The Fahrenheit scale ◦ F, which is commonly used in the United States,
the United Kingdom, and some other related countries, is based on a smaller temperature
interval. At one standard atmosphere of pressure, the freezing point of water is 32 ◦ F and the
boiling point of water is 212 ◦ F. How large is the Fahrenheit degree compared to the Celsius
degree?
The Rankine scale R is an absolute temperature scale but based on the Fahrenheit degree. At
one standard atmosphere of pressure, what are the freezing and boiling points of water on the
Rankine scale? What is the value of the triple point of water on the Rankine scale, the Fahrenheit
scale and the Celsius scale? What is the value of absolute zero in ◦ F?

3 The Kelvin scale is defined such that the triple point of water (solid-liquid-vapor equilibrium) is exactly
273.16 K. The Celsius scale, for which the unit is denoted ◦ C, is defined by T(◦ C) = T(K) − 273.15.

Chapter 1 • Introduction

5

Solution 1.1. The temperature interval between the boiling and freezing points of water at

one standard atmosphere is 100 ◦ C or 212 − 32 = 180 ◦ F. Therefore, 1 ◦ F = 100/180 = 5/9 ◦ C =
(5/9) K. The freezing and boiling points of water are 273.15 × (9/5) = 491.67 R and 373.15 ×
(9/5) = 671.67 R. The triple point of water is 273.16 × (9/5) = 491.688 R = 32.018 ◦ F = 0.01 ◦ C.
The value of absolute zero in ◦ F is −(491.67 − 32) = −459.67 ◦ F.

In the process of introducing temperature, we alluded to the intuitive concept of
heat transfer. At this stage, it suffices to say that if two bodies at different temperatures
are brought into “thermal contact,” a process known as heat conduction can occur that
enables energy to be transferred between the bodies even though the bodies exchange
no matter and do no mechanical work on one another. This process results in a new
equilibrium state and a new common temperature for the combined body. It is common
to say that this process involves a “transfer of heat” from the hotter body (higher initial
temperature) to the colder body (lower initial temperature). This terminology, however,
can be misleading because a conserved quantity known as “heat” does not exist.4 We
should really replace the term “transfer of heat” by the longer phrase “transfer of energy
by means of a process known as heat transfer that does not involve mechanical work” but
we use the shorter phrase for simplicity, in agreement with common usage. The first law
of thermodynamics will be used to quantify the amount of energy that can be transferred
between bodies without doing mechanical work. The second law of thermodynamics will
then be introduced to quantify the maximum amount of energy due to heat transfer
(loosely, “heat”) that can be transformed into mechanical work by some process. This
second law will involve a new state variable, the entropy S, which like the temperature
is entirely statistical in nature and has no mechanical counterpart.

1.2 Thermodynamics Versus Statistical Mechanics
Thermodynamics is the branch of thermal physics that deals with the interrelationship of
macroscopic state variables. It is traditionally based on three so-called laws (or a number
of postulates that lead to the same results, see Callen [2, chapter 1]). Based on these
laws, thermodynamics is independent of detailed models involving atoms and molecules.
It results in criteria involving state variables that must be true of systems that are in
equilibrium with one another. It allows us to develop relationships among measurable
quantities (e.g., thermal expansion, heat capacity, compressibility) that can be represented
by state variables and their derivatives. It also results in inequalities that must be obeyed by
any naturally occurring process. It does not, however, provide values of the quantities with
which it deals, only their interrelationship. Values must be provided by experiments or by
models based on statistical mechanics. For an historical introduction to thermodynamics,
see Cropper [11, p. 41].
4 Such

a quantity was once thought to exist and was called caloric.

6 THERMAL PHYSICS

Statistical mechanics is based on the application of statistics to large numbers of atoms
(or particles) that obey the laws of mechanics, strictly speaking quantum mechanics, but
in limiting cases, classical mechanics. It is based on postulates that relate certain types of
averages, known as ensemble averages, to measurable quantities and to thermodynamic
state variables, such as entropy mentioned above. Statistical mechanics can be used to
rationalize the laws of thermodynamics, although it is based on its own postulates which
were motivated by thermodynamics. By using statistical mechanics, specific models can
be analyzed to provide values of the quantities employed by thermodynamics and measured by experiments. In this sense, statistical mechanics appears to be more complete;
however, it must be borne in mind that the validity of its results depends on the validity
of the models. Statistical mechanics can, however, be used to describe systems that are
too small for thermodynamics to be applicable. For an excellent historical introduction to
statistical mechanics, see Pathria and Beale [9, pp. xxi-xxvi].
A crude analogy with aspects of mathematics may be helpful here: thermodynamics is
to statistical mechanics as Euclidean geometry is to analytic geometry and trigonometry.
Given the few postulates of Euclidean geometry, which allow things such as lengths
and angles to be compared but never measured, one can prove very useful and general
theorems involving the interrelationships of geometric forms, for example, congruence,
similarity, bisections, conditions for lines to be parallel or perpendicular, and conditions
for common tangency. But one cannot assign numbers to these geometrical quantities.
Analytic geometry and trigonometry provide quantitative measures of the ingredients of
Euclidean geometry. These measures must be compatible with Euclidean geometry but
they also supply precise information about such things as the length of a line or the size
of an angle. Moreover, trigonometric identities can be quite complicated and transcend
simple geometrical construction.

1.3 Classification of State Variables
Much of our treatment will be concerned with homogeneous bulk systems in a state of
equilibrium. By bulk systems, we refer to large systems for which surfaces, either external
or internal, make negligible contributions. As a simple example, consider a sample in the
shape of a sphere of radius R and having volume V = (4/3)πR3 and surface area A = 4πR2 .
If each atom in the sample occupies a volume a3 , then for a  R, the ratio of the number
of surface atoms to the number of bulk atoms is approximately
r=

4π(R/a)2
∼ 3(a/R)  1.
(4/3)π(R/a)3 − 4π(R/a)2

(1.2)

For a sufficiently large sphere, the number of surface atoms is completely negligible
compared to the number of bulk atoms, and so presumably is their energy and other
properties. More generally, for a bulk sample having N atoms, roughly N 2/3 are near the
surface, so the ratio of surface to bulk atoms is roughly r ∼ N −1/3 . For a mole of atoms,
we have N ∼ 6 × 1023 and r ∼ 10−8 . In defining bulk samples, we must be careful to

Chapter 1 • Introduction

7

exclude samples such as thin films or thin rods for which one or more dimension is small
compared to others. Thus, a thin film of area L2 and thickness H  L contains roughly
N ∼ L2 H/a3 atoms, but about 2L2 /a2 of these are on its surfaces. Thus, the ratio of surface
to bulk atoms is r ∼ a/H which will not be negligible for a sufficiently thin film. We must
also exclude samples that are finely subdivided, such as those containing many internal
cavities.
From the considerations of the preceding paragraph, atoms of bulk samples can be
regarded as being equivalent to one another, independent of location. It follows that
certain state variables needed to describe such systems are proportional to the number
of atoms. For example, for a homogeneous sample, total energy U ∝ N and total
volume V ∝ N , provided we agree to exclude from consideration small values of N that
would violate the idealization of a bulk sample.5 State variables of a homogeneous bulk
thermodynamic system that are proportional to its number of atoms are called extensive
variables. They are proportional to the “extent” or “size” of the sample. For a homogeneous
gas consisting of three atomic species, a complete set of extensive state variables could
be taken to be U, V , N1 , N2 , and N3 , where the Ni are the number of moles of atomic
species i.
There is a second kind of state variable that is independent of the “extent” of the sample. Such a variable is known as an intensive variable. An example of such a variable would
be a ratio of extensive variables, say U/V , because both numerator and denominator are
proportional to N . Another example of an intensive variable would be a derivative of some
extensive variable with respect to some other extensive variable. This follows because a
derivative is defined to be a limit of a ratio, for example,
dU
U(V + V ) − U(V )
= lim
.
V →0
dV
V

(1.3)

If other quantities are held constant during this differentiation, the result is a partial
derivative ∂U/∂V , which is also an intensive variable, but its value will depend on which
other variables are held constant. It will turn out that the pressure p, which is an intensive
state variable, can be expressed as
p=−

∂U
∂V

(1.4)

provided that certain other variables are held constant; these variables are the entropy
S, an extensive variable alluded to previously, as well as all other extensive variables of a
remaining complete set. Another important intensive variable is the absolute temperature
T, which we shall see can also be expressed as a partial derivative of U with respect to the
entropy S while holding constant all other extensive variables of a remaining complete set.
Since the intensive variables are ratios or derivatives involving extensive variables, we
will not be surprised to learn that the total number of independent intensive variables is
one less than the total number of independent extensive variables. The total number of
5 The

symbol ∝ means “proportional to.”

8 THERMAL PHYSICS

independent intensive variables of a thermodynamic system is known as its number of
degrees of freedom, usually a small number which should not be confused with the huge
number of microscopic degrees of freedom 6N for N particles that one would treat by
classical statistical mechanics.
In Chapter 5, we shall return to a systematic treatment of extensive and intensive
variables and their treatment via Euler’s theorem of homogeneous functions.

1.4 Energy in Mechanics
The concept of energy is usually introduced in the context of classical mechanics. We
review such considerations briefly in order to shed light on some aspects of energy that
will be important in thermodynamics.

1.4.1 Single Particle in One Dimension
A single particle of mass m moving in one dimension, x, obeys Newton’s law
m

d2 x
= F,
dt 2

(1.5)

where t is the time and F(x) is the force acting on the particle when it is at position x. We
introduce the potential energy function


V (x) = −

x

F(u) du,

(1.6)

x0

which is the negative of the work done by the force on the particle when the particle
moves from some position x0 to position x. Then the force F = −dV /dx can be written
in terms of the derivative of this potential function. We multiply Eq. (1.5) by dx/dt
to obtain
m

dx d2 x
dV dx
= 0,
+
dt dt 2
dx dt

which can be rewritten as
d
dt




1
mv 2 + V = 0,
2

(1.7)

(1.8)

where the velocity v := dx/dt. Equation (1.8) can then be integrated to obtain
1
mv 2 + V = E,
2

(1.9)

where E is independent of time and known as the total energy. The first term in Eq. (1.9)
is known as the kinetic energy and the equation states that the sum of the kinetic and
potential energy is some constant, independent of time. It is important to note, however,
that the value of E is undetermined up to an additive constant. This arises as follows: If
some constant V0 is added to the potential energy V (x) to form a new potential Ṽ := V +V0 ,
the same force results because

Chapter 1 • Introduction

−

d
dṼ
dV
= − (V + V0 ) = −
= F.
dx
dx
dx

9

(1.10)

Thus, Eq. (1.9) could equally well be written
1
mv 2 + Ṽ = Ẽ,
2

(1.11)

where Ẽ is a new constant. Comparison of Eq. (1.11) with Eq. (1.9) shows that Ẽ = E + V0 ,
so the total energy shifts by the constant amount V0 . Therefore, only differences in energy
have physical meaning; to obtain a numerical value of the energy, one must always
measure energy relative to some well-defined state of the particle or, what amounts to
the same thing, adopt the convention that the energy in some well-defined state is equal
to zero. In view of Eq. (1.6), the potential energy V (x) will be zero when x = x0 , but the
choice of x0 is arbitrary.
In classical mechanics, it is possible to consider more general force laws such as F(x, t)
in which case the force at point x depends explicitly on the time that the particle is at
point x. In that case, we can obtain (d/dt)(1/2)mv 2 = F v where F v is the power supplied
by the force. Similar considerations apply for forces of the form F(x, v , t) that can depend
explicitly on velocity as well as time. In such cases, one must solve the problem explicitly
for the functions x(t) and v (t) before the power can be evaluated. In these cases, the total
energy of the system changes with time and it is not possible to obtain an energy integral
as given by Eq. (1.9).

1.4.2 Single Particle in Three Dimensions
The preceding one-dimensional treatment can be generalized to three dimensions with a
few modifications. In three dimensions, where we represent the position of a particle by
the vector r with Cartesian coordinates x, y, and z, Eq. (1.5) takes the form
m

d2 r
= F,
dt 2

(1.12)

where F(r) is now a vector force at the point r. The mechanical work done by the force on
the particle along a specified path leading from rA to rB is now given by


Wr A to r B =

path

F · dr.

(1.13)

According to the theorem of Stokes, one has




(∇ × F) · dA =

F · dr,

closed loop,

(1.14)

where the integral on the right is a line integral around a closed loop and the integral on
the left is over an area that subtends that loop. For a force such that ∇ × F = 0, we see
that the line integral around any closed loop is equal to zero. Thus, if we integrate from A
to B along path 1 and from B back to A along some other path 2 we get zero. But the latter
integral is just the negative of the integral from A to B along path 2, so the integral from A
to B is the same along path 1 as along path 2. For such a force, it follows that the work

10

THERMAL PHYSICS

WAB =

rB
rA

F · dr,

any path,

(1.15)

is independent of path and depends only on the end points. Such a force is called a
conservative force and may be represented as the gradient of a potential


V (r) = −

r
r0

F(r ) · dr

(1.16)

such that F = −∇V . In this case, it follows that the work


WAB = −

rB
rA



∇V · dr = −

rB
rA

dV = V (rA ) − V (rB ).

(1.17)

For such a conservative force, we can dot the vector v := dr/dt into Eq. (1.12) to obtain
m

dr d2 r dr
·
· ∇V = 0.
+
dt dt 2
dt

(1.18)

Then by noting that
dr d2 r
d
(1/2)m v · v = m
·
dt
dt dt 2

and

dr
dV
=
· ∇V ,
dt
dt

(1.19)

we are led immediately to Eq. (1.8) and its energy integral Eq. (1.9) just as in one
dimension, except now v 2 = v · v in the kinetic energy.

1.4.3 System of Particles
We next consider a system of particles, k = 1, 2, . . . , N , having masses mk , positions rk , and
velocities vk = drk /dt. Each particle is assumed to be subjected to a conservative force
Fk = −∇k V (r1 , r2 , . . . , rN ),

(1.20)

where ∇k is a gradient operator that acts only on rk . Then by writing Newton’s equations
in the form of Eq. (1.12) for each value of k, summing over k and proceeding as above, we
obtain
d
[T + V ] = 0,
dt

(1.21)

where the total kinetic energy
T :=

N

1
mk vk · vk
2

(1.22)

k=1

and
N

 drk
dV
=
· ∇k V .
dt
dt

(1.23)

k=1

Furthermore, we can suppose that the forces on each particle can be decomposed into
internal forces Fi due to the other particles in the system and to external forces Fe , that is,

Chapter 1 • Introduction

11

F = Fi + Fe . Since these forces are additive, we also have a decomposition of the potential,
V = V i + V e , into internal and external parts. The integral of Eq. (1.21) can therefore be
written in the form
T + V i + V e = E,

(1.24)

where E is the total energy constant. This suggests a related decomposition of T which we
proceed to explore.
We introduce the position vector of the center of mass of the system of particles,
defined by
R :=

N
1 
mk rk ,
M

(1.25)

k=1

where M :=

N

k=1 mk

is the total mass of the system. The velocity of the center of mass is
V :=

N
1 
dR
=
mk vk .
dt
M

(1.26)

k=1

The kinetic energy relative to the center of mass, namely T i , can be written
N

T i :=

1
1
mk (vk − V) · (vk − V) = T − MV 2 .
2
2

(1.27)

k=1

Eq. (1.27) may be verified readily by expanding the left-hand side to obtain four terms
and then using Eq. (1.26). The term (1/2)MV 2 is recognized as the kinetic energy associated with motion of the center of mass of the system. Equation (1.24) can therefore
be written
T i + Vi +

1
MV 2 + V e = E.
2

(1.28)

The portion of this energy exclusive of the kinetic energy of the center of mass and the
external forces, namely U = T i + V i , is an internal energy of the system of particles and
is the energy usually dealt with in thermodynamics. Thus, when energies of a thermodynamic system are compared, they are compared under the assumption that the state of
overall motion of the system, and hence its overall motional kinetic energy, (1/2)MV 2 ,
is unchanged. This is equivalent to supposing that the system is originally at rest and
remains at rest. Moreover, it is usually assumed that there are no external forces so the
interaction energy V e is just a constant. Thus, the energy integral is usually viewed in the
form
U =: T i + V i = E −

1
MV 2 − V e =: U0 ,
2

(1.29)

where U0 is a new constant. If such a system does interact with its environment, U is
no longer a constant. Indeed, if the system does work or if there is heat transfer from its
environment, U will change according to the first law of thermodynamics, which is taken
up in Chapter 2.

12

THERMAL PHYSICS

Sometimes one chooses to include conservative external forces in the energy used in
thermodynamics. Such treatments require the use of a generalized energy that includes
potential energy due to conservative external forces, such as those associated with gravity
or an external electric field. In that case, one deals with the quantity
Ũ =: T i + V i + V e = E −

1
MV 2 .
2

(1.30)

In terms of chemical potentials, which we shall discuss in Chapter 12, such external
forces give rise to gravitational chemical potentials and electrochemical potentials that
play the role [6, p. 122] of intrinsic chemical potentials when external fields are present.
It is also possible to treat uniformly rotating coordinate systems by including in the
thermodynamic energy the effective potential associated with fictitious centrifugal forces
[7, p. 72].

1.5 Elementary Kinetic Theory
More insight into the state variables temperature T and pressure p can be gained by
considering the elementary kinetic theory of gases. We consider a monatomic ideal gas
having particles of mass m that do not interact and whose center of mass remains at rest.
Its kinetic energy is
T =

N

N

k=1

k=1

1  drk drk
1
m
·
=
mv2k .
2
dt
dt
2

(1.31)

If the gas is in equilibrium, the time average T of this kinetic energy is a constant. This
kinetic energy represents the vigor of motion of the atoms, so it is natural to suppose that
it increases with temperature because temperature can be increased by adding energy due
to heat transfer. A simple and fruitful assumption is to assume that T is proportional to the
temperature. In particular, we postulate that the time average kinetic energy per atom is
related to the temperature by6
N

1
1 
3
T =
mv2k = kB T ,
N
2N
2

(1.32)

k=1

where kB is a constant known as Boltzmann’s constant. In fact, kB = R/NA where R is
the gas constant introduced in Eq. (1.1) and NA is Avogadro’s number. We shall see that
Eq. (1.32) makes sense by considering the pressure of an ideal gas.
The pressure p of an ideal gas is the force per unit area exerted on the walls of a
containing box. For simplicity, we treat a monatomic gas and assume for now that each
atom of the gas has the same speed v , although we know that there is really a distribution
of speeds given by the Maxwell distribution, to be discussed in Chapter 19. We consider
6 If
i

the center of mass of the gas were not at rest, Eq. (1.27) would apply and T would have to be replaced by

T . In other words, the kinetic energy (1/2)MV 2 of the center of mass makes no contribution to the temperature.

Chapter 1 • Introduction

13

an infinitesimal area dA of a wall perpendicular to the x direction and gas atoms with
velocities that make an angle of θ with respect to the positive x direction. In a time dt,
all atoms in a volume v dt dA cos θ will strike the wall at dA, provided that 0 < θ < π/2.
Each atom will collide with the wall with momentum m v cos θ and be reflected with the
same momentum,7 so each collision will contribute a force (1/dt)2m v cos θ, which is the
time rate of change of momentum. The total pressure (force per unit area) is therefore
p=


1 n(v dt dA cos θ)(2m v cos θ)
= nm v 2 cos2 θ = nm vx2 ,
2
dA dt

(1.33)

where n is the number of atoms per unit volume and the angular brackets denote an
average over time and all θ. The factor of 1/2 arises because of the restriction 0 < θ < π/2.
Since the gas is isotropic, vx2 = vy2 = vz2 = (1/3) v 2 . Therefore,8
p=

1
2 T
NR
nm v 2 = n
T,
= nkB T =
3
3 N
V

(1.34)

where Eq. (1.32) has been used. Equation (1.34) is the well-known ideal gas law, in
agreement with Eq. (1.1) if the absolute temperature is denoted by T. In the case of an ideal
gas, all of the internal energy is kinetic, so the total internal energy is U = T . Eq. (1.34)
therefore leads to p = (2/3)(U/V ), which is also true for an ideal monatomic gas.
These simple relations from elementary kinetic theory are often used in thermodynamic examples and are borne out by statistical mechanics.

7 Reflection with the same momentum would require specular reflection from perfectly reflecting walls, but
irrespective of the nature of actual walls, one must have reflection with the same momentum on average to avoid
a net exchange of energy.
8 If we had accounted for a Maxwell distribution of speeds, this result would still hold provided that we
interpret v 2 to be an average of the square of the velocity with respect to that distribution. See Eqs. (20.28-20.30)
for details.

2
First Law of Thermodynamics
The first law of thermodynamics extends the concept of energy from mechanical systems
to thermodynamic systems, specifically recognizing that a process known as heat transfer
can result in a transfer of energy to the system in addition to energy transferred by
mechanical work. We first state the law and then discuss the terminology used to express
it. As stated below, the law applies to a chemically closed system, by which we mean that
the system can exchange energy with its environment by means of heat transfer and work
but cannot exchange mass of any chemical species with its environment. This definition is
used by most chemists; many physicists and engineers use it as well but it is not universal.
Some authors, such as Callen  and Chandler , regard a closed system as one that
can exchange nothing with its environment. In this book, we refer to a system that can
exchange nothing with its environment as an isolated system.

2.1 Statement of the First Law
For a thermodynamic system, there exists an extensive function of state, U, called the
internal energy. Every equilibrium state of a system can be described by a complete
set of (macroscopic) state variables. The number of such state variables depends on the
complexity of the system and is usually small. For now we can suppose that U depends
on the temperature T and additional extensive state variables needed to form a complete
set.1 Alternatively, any equilibrium state can be described by a complete set of extensive
state variables that includes U. For a chemically closed system, the change U from an
initial to a final state is equal to the heat, Q, added to the system minus the work, W , done
by the system, resulting in2
U = Q − W .

(2.1)

Q and W are not functions of state because they depend on the path taken during the
process that brings about the change, not on just the initial and final states. Eq. (2.1)
1 There are other possible choices of a complete set of state variables. For example, a homogeneous isotropic
fluid composed a single chemical component can be described by three extensive variables, the internal energy
U, the volume V , and the number of moles N. One could also choose state variables T, V , and N and express U
as a function of them, and hence a function of state. Alternatively, U could be expressed as a function of T, the
pressure p, and N. In Chapter 3, we introduce an extensive state variable S, the entropy, in which case U can be
expressed as a function of a complete set of extensive variables including S, known as a fundamental equation.
2 In agreement with common usage, we use the terminology “heat transferred to the system” or “heat added
to the system” in place of the longer phrase “energy transferred to the system by means of a process known as
heat transfer that does not involve mechanical work.”

Thermal Physics. http://dx.doi.org/10.1016/B978-0-12-803304-3.00002-8
Copyright © 2015 Elsevier Inc. All rights reserved.

15

16

THERMAL PHYSICS

actually defines Q, since U and W can be measured independently, as will be discussed
in detail in Section 2.1.1.
If there is an infinitesimal amount of heat δQ transferred to the system and the system
does an infinitesimal amount of work δ W , the change in the internal energy is
dU = δQ − δ W ,

infinitesimal change.

(2.2)

For an isolated system, U = 0, and for such a system, the internal energy is a
constant.

2.1.1 Discussion of the First Law
As explained in Chapter 1, the term internal energy usually excludes kinetic energy of
motion of the center of mass of the entire macroscopic system, as well as energy associated
with overall rotation (total angular momentum). The internal energy also usually excludes
the energy due to the presence of external fields, although it is sometimes redefined to
include conservative potentials. We will only treat thermodynamic systems that are at rest
with respect to the observer (zero kinetic energy due to motion of the center of mass or
total angular momentum). For further discussion of this point, see Landau and Lifshitz
[7, p. 34].
We emphasize that W is positive if work is done by the system on its environment.
Many authors, however, state the first law in terms of the work W = −W done by the
environment on the system by some external agent. In this case, the first law would read
U = Q + W . This is especially common3 in Europe  and Russia .
The symbol  applied to any state function means the value of that function in the final
state (after some process) minus the value of that function in the initial state. Specifically,
U := U(final state) − U(initial state). As mentioned above, Q and W are not state
functions, although their difference is a state function. As will be illustrated below, Q and
W depend on the details of the process used to change the state function U. In other words,
Q and W depend on the path followed during a process. Therefore, it makes no sense to
apply the  symbol or the differential symbol d to Q or W . We use δQ and δ W to denote
infinitesimal transfers of energy to remind ourselves that Q and W are not state functions.
Some authors [6, 12] use a d with a superimposed strikethrough (d) instead of δ.
The first law of thermodynamics is a theoretical generalization based on many experiments. Particularly noteworthy are the experiments of Joule who found that for two
states of a closed thermodynamic system, say A and B, it is always possible to cause a
transition that connects A to B by a process in which the system is thermally insulated, so
δQ = 0 at every stage of the process. This also means that Q = 0 for the whole process.
3 Fermi  uses the symbol L for the work done by the system; note that the Italian word for work is ‘lavoro’
(cognate labor). The introductory physics textbook by Young and Freedman  also states the first law of
thermodynamics in terms of the work done by the system. Landau and Lifshitz  use the symbol R ≡ −W
(‘rabota’) to denote the work done on the system. Chandler  and Kittel and Kroemer  use W ≡ −W to
denote the work done on the system. This matter of notation and conventions can cause confusion, but we have
to live with it.

Chapter 2 • First Law of Thermodynamics 17

Thus by work alone, either the transformation A → B or the transformation B → A is
possible. Since the energy change due to work alone is well defined in terms of mechanical
concepts, it is possible to establish either the energy difference UA − UB or its negative
UB − UA . The fact that one of these transformations might be impossible is related to
concepts of irreversibility, which we will discuss later in the context of the second law of
thermodynamics.
According to the first law, as recognized by Rudolf Clausius in 1850, heat transfer
accounts for energy received by the system in forms other than work. Since U can be
measured and W can be determined for any mechanical process, Q is actually defined by
Eq. (2.1). It is common to measure the amount of energy due to heat transfer in units of
calories. One calorie is the amount of heat necessary to raise the temperature of one gram
(10−3 kg) of water from 14 ◦ C to 15 ◦ C at standard atmospheric pressure. The mechanical
equivalent of this heat is 1 calorie = 4.184 J = 4.184 × 107 erg. The amount of heat required
to raise the temperature by T of an arbitrary amount of water is proportional to its mass.
It was once believed that heat was a conserved quantity called caloric, and hence the
unit calorie, but no such conserved quantity exists. This discovery is usually attributed
to Count Rumford who noticed that water used to cool a cannon during boring would
be brought to a boil more easily when the boring tool became dull, resulting in even
less removal of metal. Thus, “heat” appears to be able to be produced in virtually
unlimited amounts by doing mechanical work, and thus cannot be a conserved quantity.
Therefore, we must bear in mind that heat transfer refers to a process for energy transfer
and that there is actually no identifiable quantity, “heat,” that is transported. From an
atomistic point of view, we can think of conducted heat as energy transferred by means
of microscopic atomic or molecular collisions in processes that occur without the transfer
of matter and without changing the macroscopic physical boundaries of the system under
consideration. Heat can also be transferred by radiation that is emitted or absorbed by a
system.
We can enclose a system of interest and a heat source of known heat capacity (see
Section 2.3) by insulation to form a calorimeter, assumed to be an isolated system, and
allow the combined system to come to equilibrium. The temperature change of the heat
source will allow determination of the amount of energy transferred from it (or to it) by
means of heat transfer and this will equal the increase (or decrease) in energy of the system
of interest.4

2.2 Quasistatic Work
If a thermodynamic system changes its volume V by an amount dV and does work against
an external pressure pext , it does an infinitesimal amount of work
δ W = pext dV .

(2.3)

4 If the heat source changes volume, it could exchange work with its environment and this would have to be
taken into account.

18

THERMAL PHYSICS

This external pressure can be established by purely mechanical means. For example, an
external force F ext acting on a piston of area A would give rise to an external pressure pext =
F ext /A. Note that Eq. (2.3) is valid for a fluid system even if the process being considered
is so rapid and violent that an internal pressure of the system cannot be defined during
the process. This equation can also be generalized for a more complex system as long as
one uses actual mechanical external forces and the distances through which they displace
portions of the system, for example, pushing on part of the system by a rod or pulling on
part of a system by a rope.
If an isotropic system (same in all directions, as would be true for a fluid, a liquid or a
gas) expands or contracts sufficiently slowly (hence the term “quasistatic”) that the system
is practically in equilibrium at each instant of time, it will have a well-defined internal
pressure p. Under such conditions, p ≈ pext and the system will do an infinitesimal
amount of work
δ W = p dV ,

quasistatic work.

(2.4)

Note that δ W and dV are positive if work is done by the system and both are negative if
work is done on the system by an external agent.
Eq. (2.4) applies only to an idealized process. For an actual change to take place, we
need p to be at least slightly different from pext to provide a net force in the proper
direction. This requires (p − pext ) dV > 0. Thus pext dV < p dV which, in view of Eq. (2.3),
may be written
δ W < p dV ,

actual process.

(2.5)

For the case of quasistatic work, it will be necessary for p to be slightly greater than pext
for the system to expand (dV > 0); conversely, it will be necessary for p to be slightly less
than pext for the system to contract. These small differences are assumed to be second
order and are ignored in writing Eq. (2.4). Consistent with this idealization, a process of
quasistatic expansion can be reversed to a process of quasistatic contraction by making
an infinitesimal change in p. Therefore, quasistatic work is also called reversible work.5
We can combine Eq. (2.4) with Eq. (2.5) to obtain
δ W ≤ p dV

(2.6)

with the understanding that the inequality applies to all actual processes (which are irreversible) and the equality applies to the idealized process of reversible quasistatic work.
For a finite change of V , the quasistatic work can be computed by integration:


W=

p dV ,

quasistatic work.

(2.7)

path

To evaluate this integral, we must specify the path that connects the initial and final states
of the system. It makes no sense to write this expression with lower and upper limits of
5 A process involving quasistatic work will be reversible only if all other processes that go on in the system are
reversible. For example, an irreversible chemical reaction would be forbidden.

Chapter 2 • First Law of Thermodynamics 19

II

V2 , p2

p
I

V1 , p1
V1

V

V2

FIGURE 2–1 Illustration of quasistatic work for a system whose states can be represented by points in the V , p
plane. The system makes a quasistatic transition from a state at V1 , p1 to a state V2 , p2 by two different paths, I and
II. According to Eq. (2.7), the quasistatic work is the area under each curve and is obviously greater for path II. The
difference in work is the area between the paths. Since U for the two paths is the same, the difference in the heat
Q for the two paths is also equal to the area between the paths.

integration unless the path is clearly specified. For a system whose equilibrium states
can be represented by points in the V , p plane, the quasistatic work is represented by the
area under the curve that represents the path that connects the initial and final states,
as illustrated in Figure 2–1. Since the areas under two curves that connect the same two
end points can be different, the quasistatic work W clearly depends on the path. Since
Q = U + W and U is independent of path, Q also depends on path.
If work and heat are exchanged with a system, it is important to recognize that the
internal energy of the system will not be partitioned in any way that allows part of it to be
associated with heat and part with work. That is because work and heat refer to processes
for changing the energy of a system and lose their identity once equilibrium is attained
and the energy of the system is established. On the other hand, other state variables of the
system can differ depending on the relative amounts of heat and work that bring about
the same change of internal energy. For example, consider two alternative processes in
which the internal energy of an ideal gas is increased by exactly the same amount, the
first by means of only work done by a constant external pressure pext and the second by
means of only heat transfer. In the case of only work, the volume of the gas will necessarily
be decreased but in the case of only heat transfer, the volume of the system will not
be changed. Therefore, the two processes result in different thermodynamic states, even
though both result in the same internal energy.

2.3 Heat Capacities
We can define heat capacities for changes in which the work done by the system is the
quasistatic work given by Eq. (2.4). In that case, the first law takes the form
dU = δQ − p dV .

(2.8)

20

THERMAL PHYSICS

The heat capacity at constant volume, CV , is defined to be the ratio of the infinitesimal
amount of heat δQ needed to raise the temperature by an infinitesimal amount dT while
holding the volume constant, namely


CV :=

δQ
dT





=
V

∂U
∂T



,

(2.9)

V

where the last expression, a partial derivative at constant volume, follows from Eq. (2.8).
The heat capacity is an extensive quantity and should not be confused with the specific
heat, which is the heat capacity per unit mass, which is intensive.6
The heat capacity at constant pressure, Cp , is defined to be the ratio of the infinitesimal amount of heat δQ needed to raise the temperature by an infinitesimal amount dT
while holding the pressure constant, namely


Cp :=

δQ
dT





=
p

∂U
∂T





+p
p

∂V
∂T



,

(2.10)

p

where the last expression again follows from Eq. (2.8). Note that the partial derivatives of
U in Eqs. (2.9) and (2.10) are not the same because different variables are held constant.
Thus


∂U
∂T





=
p

∂U
∂T





+
V

∂U
∂V

 
T

∂V
∂T



.

(2.11)

p

Example Problem 2.1. The specific heat of silver at 20 ◦ C is 0.0558 cal g−1 K−1 . Here we
ignore the small difference between constant volume and constant pressure for this condensed
phase. What is the heat capacity of 3 kg of silver? How many Joules of energy are needed to raise
the temperature of 3 kg of silver from 15 ◦ C to 25 ◦ C?

Solution 2.1. The heat capacity of 3 kg of silver is 3000 × 0.0558 = 167 cal K−1 . The temperature interval is 10 K so the energy required is 1670 cal × 4.184 J/cal = 6990 J. We only keep three
significant figures because the specific heat was only given to three figures.

2.3.1 Heat Capacity of an Ideal Gas
One mole of an ideal gas obeys the equation of state
pV = RT ,

one mole of ideal gas,

(2.12)

where T is the absolute temperature and R is the gas constant. Equation (2.12) is essentially a definition of an ideal gas, based on experiments for real dilute gases that obey
Eq. (1.1) that was used to define the empirical temperature θ. For such a real dilute gas,
Joule conducted experiments in which the gas was confined originally to a subvolume V1
of an insulated rigid container having overall volume V2 . The remainder of the volume,
V2 − V1 was initially evacuated (see Figure 3–2). In these experiments, Q = 0 because the
6 Analogous intensive quantities such as the heat capacity per atom, per molecule, or per mole are often used.

Chapter 2 • First Law of Thermodynamics 21

container is insulated and W = 0 because the overall container having volume V2 is rigid.
Therefore, by the first law, the internal energy U remains constant. In the experiments, the
gas was allowed to expand internally from V1 to V2 . Joule observed that the temperature T
of the gas remained practically unchanged during the process. More accurate experiments
were performed later by Thomson (Lord Kelvin) and Joule by causing the gas to expand
through a porous plug until a steady state is reached and measuring the temperature of the
exiting gas directly. For hydrogen, there was hardly appreciable change in temperature; see
the treatise by Planck [15, p. 50] for details. Therefore, the internal energy of such a dilute
gas is practically independent of its volume. For an ideal gas, we shall assume that U is
strictly independent of its volume, V , and therefore only a function of T. We shall see later
that this conclusion can be derived by applying the second law of thermodynamics for a
gas that obeys Eq. (2.12).7
Therefore, for an ideal gas, the second term on the right-hand side of Eq. (2.11) is
zero. The second term on the right of Eq. (2.10) can be evaluated by means of Eq. (2.12),
resulting in8
Cp = CV + R,

one mole of ideal gas.

(2.13)

We observe that Cp is larger than CV by an amount needed to supply the work p dV
done by the gas as it expands at constant pressure. The value of CV depends on the type
of gas under consideration and can be derived by means of statistical mechanics. For
a mole of gas, we shall see that each translational or rotational degree of freedom of a
gas molecule, made up of atoms that are considered to be point particles, contributes an
amount R/2 to CV . For a monatomic gas, each atom has three translational degrees of
freedom, translation along x, y, and z, so CV = 3R/2. A diatomic gas molecule would have
six total degrees of freedom (three translational degrees for each atom) but the distance of
separation of the two atoms remains practically constant due to strong chemical bonds.
The atoms of a diatomic gas can execute vibrations along the line joining them, but these
vibrations are hardly excited except at very high temperatures.9 Thus only five degrees of
freedom are usually active (three translational and two rotational) and CV = 5R/2 for a
diatomic gas. Similarly, if we neglect vibrational degrees of freedom for polyatomic gases,
six degrees of freedom are usually active (three translational and three rotational) and
CV = 3R. This leads to the values listed in Table 2–1.
7 By calculating derivatives of the entropy, it can be shown that dU = C dT + (Tα/κ − p)dV , where α is the
V
T
isobaric compressibility and κ is the isothermal compressibility. From the ideal gas law, α = 1/T and β = 1/p, so
the coefficient of dV vanishes and U depends only on T.
8 As defined by Eqs. (2.9) and (2.10), the heat capacities C and C are extensive. Thus they depend not only on
V
p
the substance under consideration but also on the amount of that substance. One can obtain intensive quantities
by dividing by the number of moles or the mass. These intensive quantities depend only on the substance under
consideration. Here we deal with one mole, which is equivalent to dividing the extensive heat capacities by the
number of moles being considered.
9 If partially excited, the contribution of a vibrational degree of freedom would depend on temperature. If fully
excited, a vibrational degree of freedom would contribute R/2 for kinetic energy and R/2 for potential energy, for a
total of R. Polyatomic gases with linear molecules behave somewhat like diatomic molecules insofar as rotational
degrees of freedom are concerned. See Section 21.3 for a detailed discussion of ideal gases with internal structure.

22

THERMAL PHYSICS

Table 2–1 Heat
of Ideal Gases

Capacities

per

Mole

Molecule

CV

Cp

γ = Cp /CV

monatomic
diatomic
polyatomic

3R/2
5R/2
3R

5R/2
7R/2
4R

5/3 ≈ 1.67
7/5 = 1.40
4/3 ≈ 1.33

It is assumed that the atoms are point particles, translational and
rotational degrees of freedom are totally excited, and vibrational
degrees of freedom of diatomic and polyatomic gases are not
excited.

2.3.2 General Relationship of Cp to CV
By means of a general result of thermodynamics, it will turn out that Cp is always larger
than CV . For the moment, we state this result without proof but will derive it after we
cover the second and third laws of thermodynamics. First, we need to define two other
measurable quantities:
isobaric coefficient of thermal expansion:
α :=

1
V



isothermal compressibility:
1
κT := −
V

∂V
∂T





∂V
∂p

,

(2.14)

p


.

(2.15)

T

The signs in Eqs. (2.14) and (2.15) have been chosen so that κT is positive and α is usually
positive.10 The general result (see Eq. (5.32)) is
Cp = CV +

TV α 2
.
κT

(2.16)

From the form of Eq. (2.16), we observe that Cp ≥ CV for any substance, which is not
obvious from their definitions. From stability considerations, it will be shown in Section
7.4 that Cp ≥ CV ≥ 0. For an ideal gas, we readily calculate from Eq. (2.12) that α = 1/T
and κT = 1/p, in which case Eq. (2.16) becomes Eq. (2.13) for N = 1 mole of gas. For
condensed phases, |α|  1/T and κT  1/p, but the second term in Eq. (2.16) is quadratic
in α so the difference between Cp and CV is very small. Thus, the difference between
Cp and CV is very important for gases but small and often negligible for liquids and
solids.
10 This agrees with our intuition and with experiment. It can be proven from general thermodynamic stability
considerations (see Chapter 7) that κT is positive. α is usually positive but negative values of α are possible, for
example, for water below about 4 ◦ C.

Chapter 2 • First Law of Thermodynamics 23

Example Problem 2.2. The equation of state for one mole of a van der Waals fluid is
(p + a/v 2 )(v − b) = RT ,
where p is the pressure, v is the volume per mole, T is the temperature, and a and b are
constants. Calculate the following quantities and show that they agree with the results for an
ideal gas in the limit a = b = 0:

(a)
(b)
(c)
(d)

The isothermal compressibility, κT = −(1/v )(∂v /∂p)T
The isobaric coefficient of thermal expansion, α = (1/v )(∂v /∂T )p
The molar heat capacity difference, (Cp − CV )/N
Show directly that (∂p/∂T )v = α/κT . Why is this true?

Solution 2.2. We first take the differential of the given equation to obtain




a
2ab
dv = R dT .
dp v − b + p − 2 + 3
v
v

(a)
κT = −

1
v







∂v
1
2 a b −1
1
a
=
→
v −b p− 2 + 3
∂p T
v
p
v
v

for an ideal gas,

(b)
1
α=
v






∂v
a
R
2 a b −1
R
1
p− 2 + 3
=
→
=
∂T p
v
pv
T
v
v

for an ideal gas,

(c)
(Cp − CV )/N =

(d)



a
T v α2
R2 T
2 a b −1
R2 T
 p−
= 
+
→
=R
κT
pv
v2
v3
v −b


for an ideal gas,



  
R
∂v
α
∂p
∂v
=
/
=
.
=−
∂T v
v −b
∂T p
∂p T
κT

This relation is generally true, not just true for the van der Waals fluid, as can be seen from the
differential

dv =


 
∂v
∂v
dT +
dp.
∂T p
∂p T

24

THERMAL PHYSICS

2.4 Work Due to Expansion of an Ideal Gas
We calculate the work due to expansion of one mole of an ideal gas that obeys the equation
of state Eq. (2.12). For simplicity, we will further assume that the gas has a constant heat
capacity CV at constant volume. According to Eq. (2.9), this results in
dU = CV dT ;

U = CV T + constant.

(2.17)

2.4.1 Reversible Isothermal Process
For a reversible isothermal process, the path in the V , p plane is an equilateral hyperbola,
pV = constant, where the value of the constant depends on T. We assume that this path
joins two states that satisfy p1 V1 = p2 V2 , so the quasistatic work is


W=



T = constant

p dV = RT

V2
V1

dV
= RT ln(V2 /V1 ),
V

one mole.

(2.18)

For V2 > V1 the gas expands and does positive work, as shown in Figure 2–2. For the reverse
transformation from V2 to V1 , the gas contracts and does negative work; in this case, the
environment of the gas does positive work on the gas. Since U depends only on T, we have
U1 = U2 so U = 0. Therefore, by the first law, Q = W for this process.

2.4.2 Reversible Isobaric Expansion Followed by Isochoric Transformation
We assume the path to be a reversible expansion from V1 to V2 at constant pressure
p1 (isobaric expansion) followed by lowering the pressure to p2 at constant volume
V2 (isochoric transformation). This is illustrated by the dashed line in Figure 2–3. The
quasistatic work is

W = p1

V2
V1


dV +

V2
V2

p dV = p1 (V2 − V1 )

(2.19)

because the second integral is zero. The temperature will change throughout this process.
In general, the end points will have different temperatures, T1 = p1 V1 /R and T2 = p2 V2 /R,
and the change in internal energy will be U = CV (T2 − T1 ). If the end points happen
to satisfy p1 V1 = p2 V2 , then T1 = T2 , but during the process T will not be constant. In
general, Q = U + W , but if T1 = T2 , then U = 0 and Q = W . Then the work given by
Eq. (2.19) can also be written as RT1 (V2 − V1 )/V1 = RT2 (V2 − V1 )/V1 and is greater than
that given by Eq. (2.18) with T = T1 = T2 . The reader is invited to prove this statement
mathematically.

2.4.3 Isochoric Transformation Followed by Reversible Isobaric
Expansion
We assume the path to consist of lowering the pressure to p2 at constant volume V1
followed by reversible expansion from V1 to V2 at constant pressure p2 . This is illustrated
by the dot-dashed line in Figure 2–3. The quasistatic work is

Chapter 2 • First Law of Thermodynamics 25

V1 , p1

V1 , p1

p

p

V2 , p2
V2 , p2
V1

V

V2

V1

FIGURE 2–2 Illustration of quasistatic work for isothermal expansion of one mole of an ideal gas. The
system makes a quasistatic transition from a state
at V1 , p1 to a state V2 , p2 such that pV = RT . The
work done by the gas is equal to the area under
the curve.


W=

V1
V1

V2

FIGURE 2–3 Illustration of quasistatic work for one
mole of an ideal gas. The dashed line represents
an isobaric expansion at pressure p1 followed by an
isochoric transformation at V2 . The dot-dashed line
represents an isochoric transformation at V1 followed
by an isobaric transformation at pressure p2 . The
full line represents an isothermal transformation from
V1 , p1 to V2 , p2 , which is only possible if V1 p1 = V2 p2 .


p dV + p2

V

V2

V1

dV = p2 (V2 − V1 ),

(2.20)

which is clearly smaller than that given by Eq. (2.19). If the end points happen to be at
the same temperature, the work given by Eq. (2.20) can be written RT1 (V2 − V1 )/V2 =
RT2 (V2 − V1 )/V2 and is less than that given by Eq. (2.18) with T = T1 = T2 .

2.4.4 Reversible Adiabatic Expansion
We assume that the gas is perfectly insulated from its surroundings so that δQ = 0 at
each stage of the process. Such processes are called adiabatic processes.11 We allow the
gas to expand quasistatically, and therefore reversibly, from a state V2 , p2 to a state V3 , p3 .
Applying the first law to each stage of this process gives
CV dT = −p dV ,

(2.21)

which by Eq. (2.12) may be rewritten in the form
CV

dT
dV
+R
= 0,
T
V

one mole of ideal gas.

(2.22)

authors use the word adiabatic to mean that δQ = 0 and that the process is reversible, but we use
adiabatic to mean only δQ = 0. An irreversible adiabatic process is illustrated in Section 2.4.5. See Eq. (3.13) for
the entropy change of an adiabatic process.
11 Some

26

THERMAL PHYSICS

Taking the logarithmic derivative of Eq. (2.12) gives dT/T = dp/p + dV /V which allows
Eq. (2.22) to be recast in the form
CV

dp
dV
+ (CV + R)
= 0.
p
V

(2.23)

Eq. (2.23) is a differential equation for the path in the V , p plane. It can be integrated to
give
ln p + γ ln V = constant,

(2.24)

where γ := (CV + R)/CV = Cp /CV > 1. Exponentiating Eq. (2.24) gives a more usual form
γ

γ

pV γ = p2 V2 = p3 V3 = K = constant.

(2.25)

The path of the system is represented by the solid line in Figure 2–4. The quasistatic work
is therefore

W =K

V3
V2

V −γ dV = K



V
p3 V3 − p2 V2
V 1−γ  3
= CV (T2 − T3 ).
=
1 − γ V2
1−γ

(2.26)

We have labored to produce this result which, however, could have been derived simply
by applying the first law with Q = 0 to give W = −U = −CV (T3 − T2 ). Nevertheless, we
see clearly how the quasistatic work integral depends on path.
Just as Eq. (2.23) is a differential equation for the path in the V , p plane, Eq. (2.22)
is the differential equation of the path in the T, V plane. It could be integrated directly, but the same result can be obtained by substitution of Eq. (2.12) into Eq. (2.25)
to obtain
TV γ −1 = constant.

(2.27)

V2 , p2

p

V 3 , p3
V3

V

V3∗ , p3
V3∗

FIGURE 2–4 The solid line represents a reversible adiabatic expansion for one mole of an ideal gas for γ = 5/3.
The dotted line represents, for the sake of comparison, an isothermal expansion from the same initial state. The
point at V3∗ , p3 is the final state for an irreversible adiabatic expansion at constant external pressure p3 . In this
irreversible case, the system starts in the state V2 , p2 , “leaves the page” as it progresses through non-equilibrium
states, and “reenters the page,” ultimately coming to equilibrium at the state V3∗ , p3 , which is represented by a
circled point.

Chapter 2 • First Law of Thermodynamics 27

Note that γ − 1 = R/CV , consistent with Eq. (2.22). Similarly,
T
= constant.
p(γ −1)/γ

(2.28)

2.4.5 Irreversible Adiabatic Expansion
Here again we assume that the gas is perfectly insulated from its surroundings so that
δQ = 0 at each stage of the process. We start out at the same state V2 , p2 as for the
reversible adiabatic process treated above, but we allow the gas to expand suddenly
against a constant reduced external pressure p3 that is chosen to have the same value as
p3 for the final state of the reversible adiabatic expansion considered above. During this
expansion, the pressure of the gas is not well-defined, so we cannot represent this process
by a path in Figure 2–4. Because this process is irreversible, it will come to equilibrium
in a state having temperature T3∗ and volume V3∗ different from those for the reversible
case. The work done will be W = p3 (V3∗ − V2 ) and the change in internal energy will be
U = CV (T3∗ − T2 ). Since Q = 0 we will have W = −U, which becomes
p3 (V3∗ − V2 ) = CV (T2 − T3∗ ).

(2.29)

By using Eq. (2.12), we can write p3 V3∗ = RT3∗ and p3 V2 = RT2 p3 /p2 , in which case Eq. (2.29)
can be written in the form
T3∗
CV + R p3 /p2
= 1 − q + qr,
=
T2
CV + R

(2.30)

where r := p3 /p2 and q := (γ − 1)/γ . In this same notation, Eq. (2.28) for the reversible
adiabatic expansion leads to
T3
= rq .
T2

(2.31)

We shall see that T3∗ > T3 . This is illustrated in Figure 2–5.
1.4
T3∗ /T2

1.2

T3 /T2

0.5

1 r

1.5

2

0.8
0.6
FIGURE 2–5 Graphs of T3 /T2 for a reversible adiabatic process, Eq. (2.30), and T3∗ /T2 for an irreversible adiabatic
process, Eq. (2.31), versus r = p3 /p2 for q = 2/5, which corresponds to γ = 5/3. The straight line corresponds to
T3∗ /T2 and shows that T3∗ > T3 for r = 1.

28

THERMAL PHYSICS

We first note for r = 1 that T3∗ = T3 = T2 as expected. Then we take derivatives with
respect to r to obtain
d
dr



T3∗
T2


= q;

d
dr



T3
T2


= qr q−1 .

(2.32)

These derivatives are also equal for r = 1, so the curve represented by Eq. (2.31) is tangent
to the line represented by Eq. (2.30) at r = 1. Since q − 1 = −1/γ is negative, we see that
the slope of a graph of T3∗ versus r is less than that of T3 versus r for any r < 1. Moreover,
the slope of a graph of T3∗ versus r is greater than that of T3 versus r for any r > 1. Hence,
T3∗ > T3 for any r = 1, which means that the irreversible adiabatic expansion results in
a final state with greater temperature than the reversible adiabatic expansion. The same
would be true for contraction, in which case V3 < V2 and r > 1. For the end points of the
two processes, Eq. (2.12) can be written p3 V3 = RT3 and p3 V3∗ = RT3∗ . Taking the ratio of
these equations gives
V3∗
T∗
= 3.
V3
T3

(2.33)

From this result, we see that V3∗ > V3 . In summary, irreversible adiabatic expansion or
contraction against a constant external pressure p3 results in a different final state (larger
temperature and volume) than a reversible adiabatic expansion to a final state12 having
pressure p3 .

2.5 Enthalpy
The enthalpy (sometimes called the heat function) is defined by
H := U + pV .

(2.34)

Since U, p, and V are all functions of state, H is also a function of state. In general,
dH = dU + p dV + V dp.

(2.35)

For quasistatic work such that Eq. (2.8) holds, Eq. (2.35) becomes
dH = δQ + V dp.

(2.36)

By combining Eqs. (2.10) and (2.36), we obtain

Cp :=

δQ
dT




=
p

∂H
∂T


.

(2.37)

p

12 The final state depends on the details of the irreversible process. Here we have considered only a specific
case and demonstrated that the final state is different from that for a reversible adiabatic process. Later we
shall introduce a new state variable S, the entropy, in which case it can be shown that the entropy change for
a reversible adiabatic process is zero but that for an irreversible adiabatic process is positive.

Chapter 2 • First Law of Thermodynamics 29

Comparison of Eq. (2.37) with Eq. (2.9) shows that H plays the same role at constant p as U
does at constant V . We will see that this role is very general after developing the second law
and studying Legendre transformations. In essence, the dependence of U on V is replaced
by the dependence of H on p. Thus, if Q = 0 and W = 0, we have U = 0, so energy
is conserved and U is a constant. From Eq. (2.36) we see that for δQ = 0 and constant p
we have dH = 0, so H is a constant. Actually, a less restrictive condition than constant
p suffices for finite changes. If Q = 0 and the only work done by the system is against a
pressure reservoir with constant pressure pr , the first law gives U = −pr V which can
be written in the form
(U + pr V ) = 0.

(2.38)

Then if p = pr in the initial and final states of the system, Eq. (2.38) becomes
H = 0;

Q = 0 and p = pr in initial and final states.

(2.39)

Example Problem 2.3. We saw above that the internal energy, U, of an ideal gas was independent of volume, V , and therefore only a function of temperature, T . Use this information


together with the definition of H to show that ∂H/∂p T = 0, which means that the enthalpy of
an ideal gas is a function of only the temperature, T .
Solution 2.3. We take the partial derivative of Eq. (2.34) while holding T constant to obtain


∂H
∂p




=
T

∂U
∂V

 
T

∂V
∂p




+V +p
T

∂V
∂p


.

(2.40)

T

For an ideal gas, (∂U/∂V )T = 0 so the first term on the right vanishes. From the ideal gas law
V = NRT /p we obtain



∂V
∂p


=−
T

NRT
V
=− .
p2
p

(2.41)

Hence the last two terms on the right of Eq. (2.40) cancel, and we are left with



∂H
∂p


= 0,

ideal gas.

(2.42)

T

Actually, Eq. (2.42) follows from more elementary considerations. Substitution of the ideal gas
law into Eq. (2.34) for one mole gives H = U(T ) + RT , so we see immediately that H depends
only on T . Differentiation with respect to T gives our former result Cp = CV + R.

Example Problem 2.4. As heat is supplied to ice at temperature 0 ◦ C and atmospheric

pressure, the ice melts to become water, still at its melting point 0 ◦ C, until all of the ice has
melted. The heat needed to melt the ice is 80 cal/g. How much does the enthalpy change if one

30

THERMAL PHYSICS

mole of ice is melted? Show that this is equivalent to an effective heat capacity that is a Dirac
delta function at the melting point.
What would you have to know to calculate the corresponding change of the internal energy
U and what would that change be?

Solution 2.4. Integration of Eq. (2.36) at constant p gives an enthalpy change H = Q.
One mole of ice has a mass of 18 g, so H = 18 g/mol × 80 cal/g = 1440 cal/mol. Since
the temperature does not change during melting, an effective heat capacity can be defined
formally by

Cpeff = H δ(T − TM ),

(2.43)

where TM is the melting point and δ(T − TM ) is the Dirac delta function. Equation (2.43) can be
justified by integration from TM − to TM + . From a different perspective, a graph of H versus
T has a discontinuous step at TM whose formal derivative is a delta function. See Section 3.4.1
for a more thorough discussion.
From Eq. (2.15) at constant p we obtain U = H − pV so we would have to know
V to evaluate U. We can estimate V as follows: The volume of ice shrinks about 9% on
melting and its density is about 1 g/cm3 . So for one mole, V ≈ −0.09 × 1 cm3 /g × 18 g/mol =
−1.6 cm3 /mol = −1.6 × 10−6 m3 /mol. One standard atmosphere is p = 1.01 × 105 N/m2 . Thus
pV = −0.16 J/mol = −0.04 cal/mol, which is a negligible correction. So U ≈ H for melting
of ice. This is typical for melting of condensed phases. On the other hand, for the water-steam
transition, V ≈ 2.24 × 10−2 m3 /mol, roughly 1000 times larger in magnitude than for melting.
So for evaporation, pV ≈ 6 cal/mol. But for the evaporation transition, H = 9720 cal/mol so
the difference between U and H is larger but still practically negligible.

3
Second Law of Thermodynamics
Even though the first law of thermodynamics is obeyed, there are additional limitations
on processes that can occur naturally. The second law of thermodynamics deals quantitatively with these limitations and is expressed in terms of an inequality that is obeyed
by changes of a new state function, the entropy S, which is postulated to exist. These
limitations are due to the fact that all natural processes in thermodynamic systems are
irreversible. The boundary between natural processes and processes that are forbidden by
thermodynamics can be characterized in terms of idealized processes that are reversible.
For an idealized reversible process, which is hypothetical, the entropy change obeys an
equality and this allows the entropy change to be calculated. If a system, by virtue of
suitable constraints, is such that all natural processes are forbidden by the second law,
it is in a state of thermodynamic equilibrium. This leads to a criterion for thermodynamic
equilibrium in terms of the entropy.
Historically, the entropy function was discovered by studying limitations that occur
during the process of transformation of heat into work, even though energy is conserved. Theoretically, these processes were imagined to be accomplished by engines
that exchange heat with external heat sources, do mechanical work, and return to their
original thermodynamic state after each cycle. These processes were assumed to obey the
following postulates [1, p. 30]:
Postulate of Kelvin: “A transformation whose only final result is to transfer into work
heat extracted from a source which is at the same temperature throughout is impossible.”

Postulate of Clausius: “A transformation whose only final result is to transfer heat from
a body at a given temperature to a body at a higher temperature is impossible.”
These historical postulates forbid the existence of a process in which a virtually infinite
amount of work can be obtained by extracting with 100% efficiency heat from a huge
thermal source (e.g., the ocean). An engine that would accomplish such a process is
sometimes called a perpetual motion machine of the second kind.1 In fact, many people
have come up with clever ideas and claims of such perpetual motion machines and have
attempted to patent them, but careful analysis has always shown that some irreversible
1A

perpetual motion machine of the first kind is one that would violate the conservation of energy itself,
which is already ruled out by the first law of thermodynamics.

Thermal Physics. http://dx.doi.org/10.1016/B978-0-12-803304-3.00003-X
Copyright © 2015 Elsevier Inc. All rights reserved.

31

32

THERMAL PHYSICS

process occurs such that their efficiency cannot exceed the theoretical efficiency (see
Eq. (3.27)) allowed by the second law.
Fermi [1, pp. 31-34] has shown that the postulates of Kelvin and Clausius are equivalent.
The key phrase in each of them is “only final result.” One can certainly transfer heat from a
refrigerator to a room at higher temperature, but other things must change in the process,
for example, work must be expended by a motor. Based on these postulates, a Carnot
engine, which is a hypothetical reversible engine, and other imagined irreversible engines
can be used [1, chapter IV] to develop a logical process that leads to a classical formula for
the entropy (see Eq. (3.33)).
Rather than dwell on this historical justification of the second law, we shall state it as a
postulate in very general terms and then relate it to its historical roots.

3.1 Statement of the Second Law
For a thermodynamic system, there exists a function of state, S, called the entropy. S is a
function of a complete set of extensive state variables that includes the internal energy,
U. For all other extensive variables held fixed, S is a monotonically increasing function of
the internal energy U. For a homogeneous system, S is an extensive function and its slope
∂S/∂U = 1/T, where the positive quantity T is the absolute temperature.2 If the system is
a composite system, S is the sum of the entropies of its constituent subsystems.
An isolated system is a chemically closed system for which δQ = 0 and δ W = 0, so
dU = 0 and U is a constant. Therefore also Q = 0, W = 0, and U = 0. For an isolated
system, changes of S obey the inequality
S ≥ 0,

isolated system, allowed changes,

(3.1)

where the inequality corresponds to a natural irreversible process and the equality corresponds to a hypothetical idealized reversible process.
If the entropy of an isolated system is a maximum subject to its internal and external
constraints, all natural irreversible processes are forbidden by Eq. (3.1) so the system is in
a state of equilibrium. This leads to the following equilibrium criterion:
Entropy criterion for equilibrium: The criterion for an isolated thermodynamic system to
be in internal equilibrium is that its total entropy be a maximum with respect to variation
of its internal extensive parameters, subject to external constraints and any remaining
internal constraints. Isolation constitutes the external constraints of chemical closure,
perfect thermal insulation and zero external work, which require the internal energy to
be constant.
For example, consider an isolated composite system consisting of two subsystems
having different temperatures and separated by an insulating wall (internal constraint). If
2 For a homogeneous system, the absolute thermodynamic temperature is defined by a partial derivative
1/T : = ∂S/∂U or alternatively by T = ∂U/∂S, where all other members of the complete set of extensive variables
are held constant. Thus T exists independent of any particular measuring device (thermometer). See Fermi
[1, p. 45] for a related discussion in terms of the Carnot cycle.

Chapter 3 • Second Law of Thermodynamics 33

the wall is then allowed to conduct heat (removal of an internal constraint), the energies of
the two systems will change until the temperatures are equalized and a new equilibrium,
corresponding to a state of higher entropy, is established.
In Chapter 6 we will discuss the application of this entropy criterion for equilibrium
and deduce from it several alternative and useful criteria for equilibrium.

3.1.1 Discussion of the Second Law
The second law of thermodynamics is a postulate. The fact that it is believed to be true is
based on extensive experimental testing. It can be rationalized on the basis of statistical
mechanics, which of course is based on its own postulates. It can also be derived, as is
done in classical thermodynamics for chemically closed systems, from other postulates
of Kelvin or Clausius, as stated above. In order to make contact with the historical
development of the second law and to derive equations that allow calculation of the
entropy, we first digress to apply Eq. (3.1) to a composite system consisting of sources of
heat and work.
We consider an isolated composite system having total entropy Stot and apply Eq. (3.1)
in the form
Stot ≥ 0,

isolated system, allowed changes.

(3.2)

We assume that our composite system consists of a chemically closed system of interest
having entropy S, a heat source having entropy Ss , and a purely mechanical system capable
only of exchanging work. By definition, there is no entropy associated with this purely
mechanical system, so the total entropy of our composite system is
Stot = S + Ss .

(3.3)

The heat source is assumed to be a homogeneous thermodynamic system whose only
function is to exchange heat; it does no work, has a fixed number of moles of each
chemical component, a temperature Ts and an internal energy Us . Thus dSs = (1/Ts )dUs by
definition of the absolute temperature of the heat source. We denote by δQ a small amount
of heat extracted from the source.3 From the first law we have −δQ = dUs , so dSs = −δQ/Ts .
Thus dStot = dS − δQ/Ts and for infinitesimal changes, Eq. (3.2) becomes
dS ≥

δQ
,
Ts

chemically closed system, allowed changes.

(3.4)

In Eq. (3.4), the term chemically closed system pertains to the system of interest, having
entropy S. The inequality pertains to a natural irreversible process and the equality
pertains to an idealized reversible process. Thus
dS >

δQ
,
Ts

chemically closed system, natural irreversible changes.

(3.5)

3 δQ is assumed to be so small and the heat source has, by definition, a sufficiently large heat capacity that it
remains practically unchanged during this process.

34

THERMAL PHYSICS

For reversible heat flow, which is an idealization that separates irreversible heat flow from
forbidden heat flow, Ts can differ only infinitesimally from T, the temperature of the
system, so we have
dS =

δQ
,
T

chemically closed system, idealized reversible changes.

(3.6)

Equations (3.5) and (3.6) are sometimes offered as a statement of the second law, although
the distinction between Ts and T is not always made.4
If our system of interest were simply another heat source capable of no other change,
we would have dS = dU/T by definition of its absolute temperature. Then δ W = 0 so
dU = δQ from the first law and we would have dS = δQ/T. For spontaneous heat conduction, a natural irreversible process, we would need


dStot

1
1
−
= dS + dSs = δQ
T
Ts



> 0,

(3.7)

which results in δQ(Ts − T) > 0. This means that spontaneous heat conduction, with no
other change, occurs only from a higher temperature to a lower temperature, in agreement
with our intuition and the postulate of Clausius stated above.
For finite changes, we can integrate Eq. (3.4) to obtain


S ≥

δQ
,
Ts

chemically closed system, allowed changes,

(3.8)

where the equality sign is for a reversible process and requires Ts = T. Our system of
interest can do work (on the mechanical subsystem) of amount


W = −U +

δQ,

(3.9)

provided that Eq. (3.8) is satisfied. We emphasize that our system of interest is not
isolated, so its entropy can be made to decrease by extracting heat reversibly. Therefore,
if a chemically closed system is not isolated, its entropy can increase or decrease, and the
process that brings about this
 change can be either reversible or irreversible, depending
on the relationship of S to δQ/Ts for that process.
In classical thermodynamics, one often speaks of heat reservoirs. A heat reservoir is a
heat source with such a large heat capacity that its temperature remains constant.5 If the
heat source in Eq. (3.8) is replaced by a heat reservoir of temperature Tr from which an
amount of heat Qr is extracted, we obtain
S ≥

Qr
,
Tr

chemically closed system, allowed changes.

(3.10)

the footnote on page 48 of Fermi  for further discussion of Ts . Some books [5, 16] write dS > δQ/T
which is more restrictive than Eq. (3.5); such an equation applies to a process in which the heat conduction
between the heat source and the system of interest is reversible but other processes that take place within the
system of interest are irreversible.
5 For example, if a heat source has a constant heat capacity C and an amount of heat Q is extracted from it,
r
r
its temperature would change by Tr = − Qr /Cr . For a reservoir, Cr is assumed to be so large that Tr can be
made arbitrarily small, and therefore zero for all practical purposes.
4 See

Chapter 3 • Second Law of Thermodynamics 35

If the heat source consists of a number of such reservoirs, Eq. (3.8) becomes
S ≥

 Qr
Tr

r

,

chemically closed system, allowed changes

and Eq. (3.9) is replaced by
W = −U +



Qr .

(3.11)

(3.12)

r

If the amounts of heat Qr in Eqs. (3.11) and (3.12) are very small, the sums can be replaced
by integrals, and the result is essentially the same as Eqs. (3.8) and (3.9).
A system surrounded by perfectly insulating walls requires δQ = 0 and is said to be
adiabatic. For an adiabatic system, Eq. (3.8) becomes
S ≥ 0,

chemically closed adiabatic system, allowed changes.

(3.13)

But Eq. (3.9) yields W = −U, so such a system is not isolated and can still do work. Chandler [12, p. 8] s```
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