Statistical Thermodynamics for Pure and Applied Sciences: Statistical Thermodynamics

691 Pages·2021·8.877 MB·English

by  Frederick Richard Wayne McCourt

Frederick Richard Wayne McCourt
Statistical
Thermodynamics
for Pure and
Applied Sciences
Statistical Thermodynamics
Statistical Thermodynamics for Pure and Applied
Sciences
Frederick Richard Wayne McCourt
Statistical Thermodynamics
for Pure and Applied
Sciences
Statistical Thermodynamics
FrederickRichardWayneMcCourt
UniversityofWaterloo
Waterloo,ON,Canada
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Preface
Theconceptsofthermodynamicsandstatisticalmechanicsarefundamentalphysical
conceptsthatunderliemuchofchemicalengineering,chemistry,materialsscience,
and physics. Thermodynamics treats matter as a continuum and is therefore not
concerned with the atomic or molecular nature of its constituents, while statistical
mechanics is concerned with the effective ways to deal with very large numbers
ofconstituents(atomsormolecules)usingstatisticalconcepts,suchasprobability,
averages,andfluctuations,coupledwiththefundamentallawsofphysicsthatgovern
individual atoms and molecules and their interactions at the microscopic level.
Statistical thermodynamics combines these two subdisciplines of science in order
to be able to understand and predict macroscopic properties of specific material
systemsintermsoftheattributesoftheirmicroscopicconstituents.
While the present book does focus mainly upon the employment of statistical
mechanical concepts and methodology to evaluate and interpret thermodynamic
properties in terms of atomic and/or molecular attributes of the bulk matter
constituents,italsoexaminesanumberofothertopics,suchassymmetryeffectsin
molecularspectra(inChap.6),anintroductiontotransportphenomena(inChap.7),
paramagnetism, magnetic susceptibilities of lanthanides, and an introduction to
ferromagnetism (in Chap. 8), and electrons in metals and semiconductors, and the
Bose–Einsteincondensation(inChap.10).
Thestructureofthisbookisasfollows:
Basic background material, including equations of state for classical and quan-
tumidealgasesandconceptsderivedfrommathematicalstatistics,suchasensem-
bles, fluctuations, statistical average, variance, and standard deviation, is reviewed
inChap.1.
Chapter2providesashorttreatmentofmacroscopicthermodynamics.Forthose
whohavetakenapreviouscourseonthermodynamics,itmayserveasareminderof
thoseaspectsofthermodynamics(includingadiscussionofthermodynamicengine
cyclesthatmaybemorerelevantforengineers)thatwillbeneededinlaterchapters.
Forthosewhohavenothadpreviousexposuretothermodynamics,itwillserveto
introducetherelevantthermodynamicconcepts.
v
vi Preface
Chapter 3 introduces and illustrates the main statistical ensembles that are
employed in treating thermodynamic systems, while Chap. 4 makes the relevant
general connections between statistical values and a number of thermodynamic
functionsandconcepts.
Chapter 5 focuses upon atomic systems and examines the contributions to
thermodynamic functions and properties associated with the various degrees of
freedom possessed by individual atoms. This chapter also considers an ensemble
ofsimpleharmonicoscillatorsanditsapplicationtoobtainexpressionsfortheheat
capacitiesofcrystallinemonatomicsolids.
The focus of Chap. 6 is upon molecular systems, and it examines the con-
tributions to thermodynamic functions and properties associated with the various
degreesoffreedompossessedbyindividualmolecules.Thischapteralsoincludesa
discussionofintensityalternationsinmolecularspectrathatarisebecauseofnuclear
interchangesymmetryinhighlysymmetricmolecules.
Chapter 7 covers the employment of classical statistical mechanics to extend
thethermodynamicsofgaseoussystemsbeyondtheidealgaslimitusingthevirial
equationofstateandincludesdiscussionoftheLiouvilleandBoltzmannequations
thatprovidealead-intononequilibriumphenomena.
Chapter 8 extends thermodynamics to include electric and magnetic effects in
polarizable media. Thorough treatments are given of the thermodynamics of spin
systems and magnetic susceptibilities of paramagnetic salts. An introduction to
ferromagnetismandtheIsingmodelisalsoprovided.
Chapter 9 briefly discusses chemical equilibrium and the activated complex
modelforchemicalkinetics.
Chapter 10 deals with the quantum statistics associated with the behaviours of
systemsoffermions(withhalf-odd-integerspins)andbosons(withzeroandinteger
spins). Specifically, quantum statistics is applied to fermion systems, such as the
3Heisotopeandelectronsinmetalsorsemiconductors,andtobosonsystems,such
asthe4Heisotopeandphotons.Thechapterconcludeswithabriefintroductionto
thedensitymatrix.
Seven appendices covering the aspects of combinatorial analysis, multivariate
calculusandinfiniteseries,theStirlingapproximation,atomicandmolecularterm
symbols,sphericalandsymmetrictopmolecules,andreviewsofrelevantsolidstate
andHamiltonianmechanicsthatmayproveusefultoreadershavebeendeveloped
forcompleteness.
Although this book deals only with a small fraction of equilibrium statistical
mechanics, its aim has been to provide a preparation that will suffice for further
adventures into this exciting realm of science. The material covered here is
fairly traditional, and bits of it can be found in many texts written around the
subject.Twoaspectsofthesubjectmattercoveredinthisintroductiontostatistical
thermodynamics are (1) to provide ‘derivations’ of the laws of thermodynamics
and(2)toprovideanintroductiontotheimportantareasofFermi–DiracandBose–
Einsteinstatisticsthatnecessarilyplayrolesatthelevelofnanoscalesystems,where
quantummechanicsreigns.
Preface vii
IamgratefultoDr.KevinBishopforcarryingoutatmyrequestthecalculations
for and preparations of Figs. 1.4, 1.5, and 7.2 for Lennard–Jones argon and to
Dr. Lee Huntington for preparing Figs. 10.2 and 10.11. I am also grateful to my
colleague,ProfessorPierre-NicholasRoy,forhishelpandencouragement,aswell
as his testing of previous versions of this book in his statistical thermodynamics
coursefornanotechnologyengineeringstudents.Finally,Iammostgratefultomy
wife,JanetL.McCourt,forhersteadfastencouragementandsupportthroughoutthe
yearsduringwhichthisbookwasdeveloped.
Waterloo,ON,Canada FrederickR.W.McCourt
Contents
1 BasicBackgroundMaterial ............................................... 1
1.1 Introduction........................................................... 1
1.2 TheIdealGas......................................................... 3
1.2.1 ClassicalIdealGasEquationofState:Microscopic
Derivation................................................... 4
1.2.2 QuantumIdealGasEquationofState ..................... 9
1.3 Fluctuations........................................................... 11
1.4 StatisticalEnsembles................................................. 15
1.4.1 EnsembleAverages......................................... 18
1.4.2 VarianceandStandardDeviation .......................... 23
1.5 ProblemsforThisChapter........................................... 25
References.................................................................... 26
2 MacroscopicThermodynamics ........................................... 27
2.1 BasicThermodynamicDefinitions .................................. 28
2.2 AnIntroductiontoThermodynamics................................ 30
2.2.1 InternalEnergy,Entropy:TheFirstTwoLaws............ 33
2.3 NewThermodynamicStateFunctions .............................. 40
2.4 ExpressionforHeatCapacityDifference ........................... 48
2.5 ThermodynamicEngines ............................................ 52
2.5.1 TheCarnotEngineandEngineCycle ..................... 54
2.5.2 ReverseCarnotEngine(RefrigerationCycle)............. 64
2.5.3 Curzon–AhlbornEndoreversibleEngineCycle........... 66
2.5.4 TheOttoandDieselEngineCycles........................ 70
2.5.5 CounterclockwiseOttoCycles............................. 78
2.6 TheSecondLawandStability....................................... 82
2.7 ExtensiontoMulticomponentSystems ............................. 88
2.8 ThermodynamicsofRealGases..................................... 96
2.8.1 TheVanderWaalsModel.................................. 98
2.8.2 TheVirialEquationofState ............................... 101
2.8.3 Fugacity..................................................... 103
ix
x Contents
2.8.4 TheLawofCorrespondingStates ......................... 107
2.8.5 Joule–ThomsonInversion.................................. 114
2.9 ProblemsforThisChapter........................................... 121
References.................................................................... 128
3 Ensembles:SystemsofParticles .......................................... 129
3.1 MicroscopicConfigurations ......................................... 129
3.1.1 IllustrationoftheRoleoftheBasicPostulate............. 131
3.2 TheCanonicalEnsemble:ClosedSystems ......................... 139
3.2.1 TranslationalStates:ContinuumApproximation ......... 146
3.2.2 SummaryofFormsfortheCanonicalPartition
Function..................................................... 151
3.2.3 ExtensiontoN-ParticleSystems........................... 152
3.3 TheGrandEnsemble:OpenSystems ............................... 155
3.4 TheIsothermal-IsobaricEnsemble.................................. 160
3.5 ProblemsforThisChapter........................................... 163
References.................................................................... 167
4 MeanValuesandThermodynamics...................................... 169
4.1 CanonicalEnsemble:ClosedSystems .............................. 169
4.1.1 ThermodynamicsfromtheCanonicalEnsemble.......... 173
4.1.2 CanonicalPartitionFunctionforaMixture ............... 180
4.1.3 MicrocanonicalEnsemble:IsolatedSystem............... 182
4.2 GrandEnsemble:OpenSystems .................................... 184
4.2.1 ThermodynamicsfromtheGrandEnsemble.............. 186
4.2.2 FluctuationsintheNumberofParticles................... 194
4.3 TheIsothermal–IsobaricEnsemble.................................. 199
4.4 InterconnectionsBetweenEnsembles............................... 203
4.5 ProblemsforthisChapter............................................ 204
References.................................................................... 211
5 AtomicSystems............................................................. 213
5.1 GroundElectronicTermAtoms ..................................... 213
5.1.1 TheCanonicalPartitionFunction.......................... 214
5.1.2 TheIsothermal–IsobaricPartitionFunction............... 218
5.1.3 TranslationalVersusInternalStateContributions......... 219
5.2 WhyIstheChemicalPotentialNegative?........................... 221
5.3 ElectronicandNuclearSpinStates.................................. 224
5.3.1 SettingtheStage:ExcitedElectronicStates............... 224
5.3.2 ExcitedElectronicStateContributions.................... 228
5.3.3 Nuclear Spin Contributions to the Canonical
PartitionFunction........................................... 230
5.3.4 TheFullCanonicalAtomicPartitionFunction............ 231
5.4 AtomicGasThermodynamicFunctions ............................ 231
5.4.1 AtomsHavingNoThermallyAccessibleExcited
ElectronicStates............................................ 231
5.4.2 InfluenceofExcitedElectronicStates..................... 233
Contents xi
5.5 SimpleHarmonicOscillatorEnsembles ............................ 235
5.5.1 TheSimpleHarmonicOscillator .......................... 235
5.5.2 Interlude:DegreesofFreedom............................. 240
5.6 MonatomicSolids.................................................... 242
5.7 ProblemsforThisChapter........................................... 251
References.................................................................... 256
6 MolecularSystems.......................................................... 257
6.1 Introduction........................................................... 257
6.2 DiatomicMolecules.................................................. 260
6.2.1 SettingtheStage............................................ 260
6.2.2 TheDiatomicVibrationalPartitionFunction ............. 264
6.2.3 TheDiatomicRotationalPartitionFunction .............. 271
6.2.4 ElectronicDegreeofFreedom............................. 303
6.2.5 OverallSummaryforDiatomicMolecules................ 306
6.2.6 DirectEvaluationofInternalStateContributions......... 307
6.3 ExtensiontotheIdealPolyatomicGas.............................. 310
6.3.1 The Rigid-Rotor Simple Harmonic Oscillator
Approximation.............................................. 311
6.3.2 Summary of the RR-SHO Approximation for
PolyatomicMolecules...................................... 312
6.3.3 BeyondtheRR-SHOApproximation...................... 315
6.4 MolecularSpectra:NuclearSpinEffects............................ 321
6.4.1 DiatomicRotationalSpectra............................... 321
6.4.2 PolyatomicMolecularSpectra............................. 330
6.5 Third-LawEntropyandResidualEntropy.......................... 336
6.6 EffectofHinderedRotationalMotions ............................. 341
6.6.1 SettingtheStage............................................ 342
6.6.2 InternalRotationinEthane................................. 343
6.7 ProblemsforThisChapter........................................... 347
References.................................................................... 355
7 ClassicalStatisticalMechanics............................................ 357
7.1 Introduction........................................................... 357
7.1.1 EnergyEquipartitioninClassicalMechanics............. 360
7.1.2 DealingwithIntermolecularInteractions.................. 367
7.2 TheVirialEquationofState......................................... 368
7.2.1 SecondVirialCoefficientforMonatomicGases.......... 372
7.3 BeyondtheIdealGas ................................................ 376
7.4 AnApproximateDescriptionforDenseFluids..................... 379
7.5 TheLiouvilleandBoltzmannEquations............................ 382
7.5.1 TheLiouvilleEquation..................................... 382
7.5.2 InterludeonBinaryCollisionKinematics................. 390
7.5.3 ScatteringCrossSectionConcept.......................... 398
7.5.4 TheBinaryCollisionTermandtheBoltzmann
Equation..................................................... 400