Introductory Statistical Thermodynamics

378 Pages·2011·2.728 MB·English

by  Nils Dalarsson, Mariana Dalarsson, Leonardo Golubovic

Introductory Statistical
Thermodynamics
by
Nils Dalarsson and Mariana Dalarsson
Royal Institute of Technology, Stockholm, Sweden
Leonardo Golubovic´
West Virginia University, Morgantown, USA
AMSTERDAM•BOSTON•HEIDELBERG•LONDON
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Introductorystatisticalthermodynamics/editedbyN.Dalarsson,M.Dalarsson[and]L.Golubovic´.
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Includesbibliographicalreferencesandindex.
ISBN978-0-12-384956-4(hardback)
1.Statisticalthermodynamics.I.Dalarsson,M.(Mirjana)II.Dalarsson,N.(Nils)
III.Golubovic´,L.(Leonardo)
QC311.5.E442011
536’.7–dc22
2010042085
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vii

Contents

Preface xi

1 Introduction 1

Part One Quantum Description of Systems 3

2 Introduction and Basic Concepts 5
2.1 Systems of Identical Particles 5
2.2 Quantum Description of Particles *) 6
2.3 Problems with Solutions *) 9

3 Kinetic energy of Translational Motion 11
3.1 Hamiltonian of Translational Motion 11
3.2 Schrödinger Equation for Translational Motion *) 12
3.3 Solution of the Schrödinger Equation *) 12
3.4 Normalization of the Wave function *) 15
3.5 Quantized Energy of Translational Motion 16
3.6 Problems with Solutions *) 17

4 Energy of Vibrations 19
4.1 Hamiltonian of Vibrations 19
4.2 Solution of the Schrödinger equation 20
4.3 Quantized Energy of Vibrations 21
4.4 Hermite Polynomials *) 22
4.5 Normalization of the Wave Function *) 23
4.6 Problems with Solutions *) 25

5 Kinetic Energy of Rotations 29
5.1 Hamiltonian of Rotations 29
5.1.1 Kinetic Energy and Hamiltonian Operator 29
5.1.2 Angular Momentum Operator *) 30
5.2 Solution of the Schrödinger equation *) 32
5.3 Quantized Energy of Rotations 37
5.4 Legendre Polynomials *) 38
5.5 Normalization of the Wave function *) 42
5.6 Spin Angular Momentum *) 46
5.7 Problems with Solutions *) 47
viii

Part Two Thermodynamics of Systems 49

6 Number of accessible states and Entropy 51
6.1 Introduction and Definitions 51
6.2 Calculation of the Number of accessible States 52
6.2.1 Classical Number of Accessible States 54
6.2.2 Number of Accessible States for Bosons 55
6.2.3 Number of Accessible States for Fermions 56
6.3 Problems with Solutions 57

7 Equilibrium States of Systems 59
7.1 Equilibrium Conditions 59
7.2 Occupation Numbers of Energy Levels 61
7.3 Concept of Temperature 63
7.4 Problems with Solutions 65

8 Thermodynamic Variables 71
8.1 Free Energy and the Partition Function 71
8.2 Internal Energy: Caloric State Equation 74
8.3 Pressure: Thermal State Equation 76
8.4 Classification of Thermodynamic Variables 79
8.5 Problems with Solutions 80

9 Macroscopic Thermodynamics 87
9.1 Changes of States. Heat and Work 87
9.2 Laws of Thermodynamics 89
9.2.1 Zeroth Law of Thermodynamics 89
9.2.2 First Law of Thermodynamics 89
9.2.3 Second Law of Thermodynamics 90
9.2.4 Third Law of Thermodynamics 90
9.3 Open Systems 91
9.4 Thermal Properties of Systems 93
9.4.1 Isobaric Expansion 93
9.4.2 Isochoric Expansion 94
9.4.3 Isothermal Expansion 94
9.4.4 Relation between Thermal Coefficients 96
9.5 Caloric Properties of Systems 96
9.5.1 Specific Heat at Constant Volume c 97
V
9.5.2 Specific Heat at Constant Pressure c 98
p
9.5.3 Relation between Specific Heats 98
9.6 Relations between Thermodynamic Coefficients 101
9.7 Problems with Solutions 104

10 Variable Number of Particles 119
10.1 Chemical Potential 119
10.2 Thermodynamic Potential 121
10.3 Phases and Phase Equilibrium 124
10.3.1 Latent Heat 124
10.3.2 Clausius-Clapeyron Formula 125
10.4 Problems with Solutions 127
ix

Part Three Ideal and non-ideal gases 137

11 Ideal Monoatomic Gases 139
11.1 Continuous Energy Spectrum 139
11.2 Continuous Partition Function 141
11.3 Partition Function of Ideal Monoatomic Gases 143
11.4 Kinetic Theory of Ideal Monoatomic Gases 144
11.4.1 Maxwell-Boltzmann Speed Distribution 144
11.4.2 Most probable Speed of Gas Particles 145
11.4.3 Average Speed of Gas Particles 145
11.4.4 Root-Mean-Square Speed of Gas Particles 146
11.4.5 Average Kinetic Energy and Internal Energy 148
11.4.6 Equipartition Theorem 148
11.5 Thermodynamics of Ideal Monoatomic Gases 150
11.5.1 Caloric-State Equation 150
11.5.2 Thermal-State Equation 151
11.5.3 Universal and Particular Gas Constants 152
11.5.4 Caloric and Thermal Coefficients 154
11.6 Ideal Gases in External Potentials 155
11.6.1 General Maxwell-Boltzmann distribution 155
11.6.2 Harmonic and Anharmonic Oscillators 165
11.6.3 Classical limit of Quantum Partition Function 169
11.7 Problems with Solutions 174

12 Ideal Diatomic Gases 187
12.1 Rotations of Gas Particles 190
12.2 Vibrations of Gas Particles 192
12.3 Problems with Solutions 197

13 Non-ideal Gases 203
13.1 Partition Function for Nonideal Gases 203
13.2 Free Energy of Nonideal Gases 204
13.3 Free Energy of Particle Interactions 206
13.4 van der Waals Equation 209
13.5 Caloric-State Equation for Non-ideal Gases 210
13.6 Specific Heats for Non-ideal Gases 211
13.7 Problems with Solutions 213

14 Quasi-static Thermodynamic Processes 229
14.1 Isobaric Process 229
14.2 Isochoric Process 231
14.3 Isothermal Process 232
14.4 Adiabatic Process 234
14.5 Polytropic Process 236
14.6 Cyclic Processes: Carnot Cycle 238
14.7 Problems with Solutions 242
x

Part Four Quantum Statistical Physics 257

15 Quantum Distribution Functions 259
15.1 Entropy Maximization in Quantum Statistics 259
15.1.1 The Case of Bosons 259
15.1.2 The Case of Fermions 260
15.2 Quantum Equilibrium Distribution 261
15.3 Helmholtz Thermodynamic Potential 265
15.4 Thermodynamics of Quantum Systems 266
15.5 Evaluation of Integrals *) 269
15.6 Problems with Solutions 272

16 Electron Gases in Metals 287
16.1 Ground State of Electron Gases in Metals 288
16.2 Electron Gases in Metals at Finite Temperatures 289
16.3 Chemical Potential at Finite Temperatures 294
16.4 Thermodynamics of Electron Gases 295
16.5 Problems with Solutions 297

17 Photon Gas in Equilibrium 301
17.1 Planck Distribution 301
17.2 Thermodynamics of Photon Gas in Equilibrium 304
17.3 Problems with Solutions 308

18 Other examples of Boson Systems 313
18.1 Lattice Vibrations and Phonons 313
18.1.1 Vibration Modes 314
18.1.2 Internal Energy of Lattice Vibrations 321
18.2 Bose-Einstein Condensation 332
18.3 Problems with Solutions 336

19 Special Topics 349
19.1 Ultrarelativistic Fermion Gas 349
19.1.1 Ultrarelativistic Fermion Gas T/T <<1 353
F
19.1.2 Ultrarelativistic Fermion Gas T/T >>1 356
F
19.2 Thermodynamics of the Expanding Universe 357
19.2.1 Internal Energy of Elementary-Particle Species 358
19.2.2 Entropy per Volume Element 359
19.3 Problems with Solutions 361

A Physical constants 365

Bibliography 367
Index 369
Preface
The aim of this book is to discuss basic theoretical results of the statistical physics
and thermodynamics, which cover the fundamental physical concepts used for the
macroscopic description of systems with very large numbers of constituent parti-
cles. The macroscopic concepts used in classical thermodynamics are derived from
the microscopic theories of constituent particles, i.e., quantum mechanics and statis-
tical mechanics. However, in literature, the subject of classical thermodynamics is
frequentlyintroducedfromapurelymacroscopicpointofview.Suchanapproachis
conceptuallymoredifficult,anditsreasoningisoftenhardtofollowbymanyphysics
andengineeringstudents.Inparticular,thesignificanceandtherealphysicalmeaning
of the fundamental concept of entropy are difficult, if not impossible, to explain on
purelymacroscopicgrounds.
Therefore, we chose to approach the subject from the microscopic point of view.
Itisbasedonthefactthatallmacroscopicsystemsconsistofmicroscopicconstituent
particles (molecules, atoms, or elementary particles) obeying the laws of quantum
mechanics.Usingbasicpostulatesofstatisticalmechanicsandelementaryresultsfrom
quantum mechanics, we derive the general thermodynamic description of physical
systemsatmacroscopiclevel.Thisdescriptionislargelyindependentonthedetailsof
themicroscopicmodelsdescribingtheinteractionsoftheparticlesinvariousphysical
systems.
Despitetheinitialmicroscopicapproachtothesubject,thecentralpartofthebook
consistsofphysicalconsiderationsonapurelymacroscopiclevel.Byadoptingsome
simplequantum-mechanicalmodelsoftheconstituentparticlesofasystem,thebook
shows how one can calculate macroscopic thermodynamic quantities on the basis of
therelevantmicroscopicresults.Thebookisfocusedonthestudyofsystemsinther-
modynamicequilibrium.However,thestatisticalapproachtothediscussionofvarious
equilibriumsituationsprovidesthestudentswiththeneededpreparationfortheexten-
siontothediscussionofnonequilibriumsystems.
The basic plan of the book is the following: The first part of the book covers
themicroscopicmodelsofconstituentparticles.Itdiscussesthreebasicmodelsfrom
quantum mechanics. Some sections of the first part of the book are of a purely
quantum-mechanicalandmathematicalsignificance.Althoughnotdirectlyrelatedto
thetraditionalsubjectsofthermodynamics,theyareconsideredinsomedetailinorder
to establish a complete understanding of the quantum description of macroscopic
systemsofparticles.
The second part of the book is devoted to the detailed derivation of the basic
notions of classical statistical mechanics and the general laws of macroscopic ther-
modynamicsthatrelatevariousthermodynamicquantities.Thethirdpartofthebook
xii Preface
coverstheapplicationofthegeneralmacroscopiclaws,derivedinthesecondpart,to
thephysicallyinterestingcasesoftheidealandnonidealgases.Finally,inthefourth
part of the book, a discussion of quantum statistical mechanics and some relativistic
phenomenaispresented.Thequantumrelativisticthermodynamicsincludesastudyof
somemacroscopicphenomenaintheexpandingearlyuniverse.
Somesectionsofthebookcoverpurelymathematicaland/orquantum-mechanical
concepts that are not directly related to what is traditionally the scope of courses of
Statistical Physics and Thermodynamics. These sections are marked by an asterisk
and can be omitted by the instructors adopting a less theoretical and mathematical
approach to the subject. These sections can also be omitted by readers who have
already taken a Quantum Mechanics course prior to taking the course in Statisti-
calPhysicsandThermodynamics,andarefamiliarwiththequantum-mechanicaland
mathematicalconceptscoveredinthesesections.
Thebookisintendedasatextforanintroductoryone-semestercourseinthermo-
dynamics for undergraduate students of physical sciences or engineering. However,
it can also be used as material for an introductory graduate course in thermodynam-
ics. The book has evolved from a set of lecture notes originally prepared by one of
theauthors,N.Dalarsson,buthasbeensubsequentlyexpandedwithanumberofnew
topicsoverthelastfifteenyears.
Itistheintentionofthebooktoprovidethereaderswithahighlevelofdetailin
derivations of all equations and results. All algebraic manipulations are outlined in
greatdetail,suchthattheycanbefollowedbyaninterestedcollegestudentwithvery
littleornoriskofevergettinglost.Itisourexperiencethatacommonshowstopper
forayoungcollegestudent,tryingtomasterasubject,arethephrasesintheliterature
claiming that something can be derived from something else by some “straightfor-
wardalthoughsomewhattediousalgebra.”Ifastudentcannotreadilyreproducethat
“straightforward” algebra, which often is the case, the usual reaction under the time
pressureofthestudiesistoaccepttheclaimasafact.Andfromthatpointon,through-
outtherestofthecourse,thedeeperunderstandingofthesubjectislost.
Thebookcontainsanumberofsolvedproblems.Theyhavebeenselectedtoillus-
tratethetheoreticalconceptsdiscussedthroughoutthebook.Thesolutionsaregiven
in detail to help master the theoretical concepts from the book. It is common in the
literature to include the problems at the end of each chapter and to outline the solu-
tionsorjustprovideanswersattheendofthebook.Inthepresentbook,wechoosea
differentapproachandaddanumberofproblemswithdetailedsolutionsattheendof
eachchapter.Furthermore,anumberofdetailedtheoreticalderivationsthroughoutthe
bookcanalsobeusedashomeworkorexamproblems.
Thereisanumberofadvancedbooksonstatisticalthermodynamics,andwehave
benefitedfromsomeofthose.Thesesourcesarelistedinthebibliographyattheend
of the book, as well as a few other books recommended as suitable further reading.
Regretfully,inanintroductorybooksuchasthisone,itwasneitherpossibletoinclude
anextensivelistofalloriginalreferencesandmajortextbooksandmonographsonthe
subjectnortomentionallthepeoplethathavecontributedtoourunderstandingofthe
subject.
Preface xiii
Wehopethatourreaderswillfindthatwehavefulfilledtheobjectiveofproviding
aself-containedand self-explanatorybook,whichprovidesasmooth introductionto
thisimportantsubjectandthattheywillenjoyreadingthisbookasmuchasweenjoyed
writingit.
N.Dalarsson,M.Dalarsson,andL.Golubovic´
Stockholm,Sweden,andMorgantown,WV,USA,June2010
1
Introduction
Thesubjectofthisbookisadiscussionofsystemsconsistingofmanyparticles.Good
examples for this are atomic and molecular gases; however, the theoretical concepts
we will develop can also be applied to the studies of liquids, solids, electromagnetic
radiation(photons),crystallatticevibrations(phonons),etc.Essentially,allphysical,
engineering,chemical,orbiologicalsystemsofpracticalinterestconsistofmanycon-
stituentparticles.Thus,theconceptsdevelopedinthisbookareusefulforthedescrip-
tionofalargenumberofnaturalphenomena.
The first systematic approach to the investigation of macroscopic systems from a
macroscopic empirical point of view began in the nineteenth century. The subject of
empirical thermodynamics was developed before the discovery of the atomic nature
ofmatter.TheproposalthatheatisaformofenergywasfirstputforwardbyRumford
(1798) and Davy (1799). It was generally accepted after the experimental work of
Joule(1843–1849).Thefirsttheoreticalexplanationoftheprinciplesofheatengines
wasgivenbytheFrenchengineerCarnot(1824).Aconsistentformofthetheoretical
thermodynamics was formulated by Clausius and Kelvin (1850), and it was greatly
developedandgeneralizedbytheAmericanphysicistGibbs(1876–1878).
The microscopic approach to thermodynamics started with the work of Clausius,
Maxwell, and Boltzmann. Maxwell formulated the distribution law of molecular
velocities (1859), while Ludwig Boltzmann derived his fundamental transport equa-
tion (1872). Thereafter, Chapman and Enskog (1916–1917) developed systematic
methods for solving the Boltzmann equation. The modern equilibrium statistical
mechanics was also initiated in the works of Boltzmann (1872). It was then signifi-
cantlygeneralizedbytheworkofGibbs(1902).Thediscoveryofquantummechanics
introducedsomechanges;however,thebasicframeworkoftheGibbstheorysurvived
theadventofquantummechanics.
Inanattempttodiscusscommonsystemsinvolvingmanyparticles,suchasgases,
one realizes that the laws of quantum mechanics are needed to adequately describe
themotionsoftheindividualconstituentparticlesofthesesystems.Ontheotherhand,
theatomicnuclei,oftenbeingthepartoftheseconstituentparticles(suchasatomsor
molecules), are normally not affected by their motion. So, the nuclear forces play a
secondaryrole.Likewise,thegravitationalforcesbetweentheconstituentparticlesare
alsogenerallynegligible.Thus,themajorforcesaffectingtheconstituentparticlesin
the common systems such as gases, liquids, and solids are only the well-understood
electromagneticinteractions,suchasthefamiliarelectrostaticforces.
IntroductoryStatisticalThermodynamics.DOI:10.1016/B978-0-12-384956-4.00001-X
Copyright(cid:13)c 2011Elsevier,Inc.Allrightsreserved.