The Physical Chemist’s Toolbox

1112 Pages·2023·50.507 MB·English

by  Robert M. Metzger

The Physical Chemist’sToolbox
The Physical Chemist’s Toolbox
Second Edition
Robert M. Metzger
Department of Chemistry and Biochemistry
The University of Alabama
Tuscaloosa and Birmingham, Alabama, USA
Thissecondeditionfirstpublished2023
©2023JohnWiley&Sons,Inc.
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v
Contents
Foreword xvii
PrefaceandPhilosophy xix
AbouttheCompanionWebsite xxiii
1 FundamentalParticles,FundamentalForces,andMathematicalTools 1
1.0 Introduction 1
1.1 FundamentalForces,ElementaryParticles,Nuclei,andAtoms 2
1.2 ForceOne:Gravitation 16
1.3 ForceThree:WeakForce 17
1.4 ForceFour:StrongForce 17
1.5 ReviewofMathematicalConcepts 24
1.5.1 MathematicsandStatistics 24
1.5.2 Functions,Equations,andFunctionals 24
1.5.3 Quadratic,Cubic,andQuarticEquations 24
1.5.4 PartialDerivatives 27
1.5.5 Jacobians 30
1.5.6 Lineintegral 30
1.5.7 TheGauss–Green–StokesTheorem 30
1.5.8 Techniquesofintegration 31
1.5.9 DifferentialForms 33
1.5.10 InfiniteSeries 33
1.5.11 SumandIntegralKillers:TheKroneckerDeltaandtheDiracDeltaFunction 34
1.5.12 OrdinaryDifferentialEquations(ODEs) 35
1.5.13 TheLagrangeMethodofUndeterminedMultipliers 37
1.5.14 PartialDifferentialEquations(PDEs) 38
1.6 Mechanics,Vectors,Tensors,andDeterminants 39
1.6.1 ForcesandLinearMomentum 39
1.6.2 DimensionalAnalysis 39
1.6.3 UnitSystems 39
1.6.4 Vectors 41
1.6.5 Determinants 42
1.6.6 Matrices 44
1.6.7 Tensors 45
1.6.8 SimilarityTransformations 45
1.6.9 Eigenvalue-EigenvectorProblem 46
1.6.10 PlanarRotations 47
1.6.11 EulerianRotations 48
1.6.12 CovariantandContravariantVectorsandTensors 49
1.6.13 Covariant 49
1.6.14 MomentofInertiaandAngularMomentum 50
1.6.15 DerivativeOperators:“SkiSlopes,”“Hernias,”and“Curls” 51
1.6.16 Gradient 52
vi Contents
1.6.17 Divergence 52
1.6.18 Curl 52
1.6.19 Gauss–Green–StokesTheorem 53
1.6.20 Combinatorics 54
1.7 Hooke’sLaw,One-dimensionalChain,DispersionRelations,andStress–StrainTensors 55
1.7.1 LongitudinalMechanicalWavesSubjectedtoHooke’sLaw 57
1.7.2 LongitudinalElasticWavesina1DArrayofEqualMassesandSprings 57
1.7.3 One-dimensionalChainwithTwoKindsofAtoms:ABandGapAppears 59
1.7.4 StressandStrainTensors 61
1.8 Lagrange’sFunctionandHamilton’sFunction 62
1.9 ForceTwo:Electromagnetism 63
1.9.1 Coulomb’sLawandAmpère’sLaw 63
1.9.2 MaterialMediaandTheirReactiontoExternalFields 65
1.9.3 Maxwell’sEquations 66
1.9.4 ScalarandVectorPotentials 70
1.9.5 PolarizationofElectromagneticWaves 71
1.9.6 Multipoles 72
1.9.7 ElectricDipoleMoments 73
1.9.8 ElectricMultipoleMoments,PolarizabilitiesandHyperpolarizabilities 73
1.10 TheSizeofFundamentalParticlesandthePhysicalMeaningofQuantumNumbers 74
1.11 SpecialRelativity 75
1.12 FeynmanDiagrams 79
1.13 GeneralRelativity,GravityasaCurvatureofSpace-TimeandtheSizeoftheUniverse 80
1.14 ElementsofOptics 83
1.14.1 JonesVector 88
1.14.2 AnisotropicIndicesofRefraction 88
1.14.3 Mirrors 89
1.14.4 PrismsandGratings 92
1.15 Transforms 94
1.15.1 ConvolutionTheorem 97
1.15.2 LaplaceTransforms 98
1.15.3 WaveletTransform 100
1.16 ContourIntegrationandKramers–KronigRelations 100
1.17 ErrorAnalysis 102
1.17.1 Errors 102
1.17.2 PropagationofErrors 102
1.17.3 GaussianNormalErrorProbabilityFunction 103
1.17.4 BinomialDistribution 104
1.17.5 PoissonDistribution 104
1.17.6 LeastSquaresorLinearRegressionAnalysis 104
1.18 Statistics 105
1.19 GeneralReferences 107
References 137
End-of-ChapterProblems 138
2 QuantumMechanics 147
2.0 Introduction 147
2.1 QuantumPostulates 147
2.1.1 Bras,Kets,andHermitianOperators 151
2.2 QuantumMechanicsoftheFreeElectron 154
2.3 TheParticleinaBox 155
2.3.1 TunnelingorBarrierPenetrationorScatteringinOneDimension 157
2.3.2 RadioactiveDecay 160
Contents vii
2.3.3 GamowCalculation 161
2.4 TheHarmonicOscillator 162
2.5 TheHamiltonianfortheOne-electronAtominaCentralField 165
2.6 TheRigidRotor 173
2.7 TheHamiltonianandEigenfunctionsfortheN-electronAtomorMolecule 174
2.8 SpaceQuantization:theAllowedOrientationsofOrbitalandSpinAngularMomenta 175
2.9 AufbauforAtoms 176
2.10 LewisOctets 177
2.11 PromotionandHybridizationofAtomicOrbitals 177
2.12 BondinginHydrogenMolecule:ValenceBondvs.MolecularOrbitalandVariationalApproaches 178
2.13 AufbauforMolecules:σ andπBonding 179
2.14 PerturbationTheory 182
2.15 TheHartree–FockMethod 184
2.16 TheRoothaan–HallMatrixFormulationoftheHartree–FockProblem 185
2.17 ImplementationoftheHartree–FockMethod 189
2.18 ConfigurationInteraction 190
2.19 DensityFunctionalTheory(DFT) 191
2.20 MolecularMechanics 194
2.21 TheHückelProblem,or“Simple”HückelMolecularOrbitalTheory(SHMO) 194
2.22 ExtendedHückelTheory 199
2.23 Pariser–Parr–PopleTheory 199
2.24 NeglectofDifferentialOverlap(NDO)Methods 200
2.25 MagneticMoments,Landég-factor,LarmorPrecession,Spin–OrbitCoupling,andThomasPrecession 204
2.25.1 LarmorPrecession 206
2.25.2 Theg=2Puzzle,Stated 208
2.25.3 Spin–OrbitInteraction 208
2.25.4 Theg=2PuzzleSolved:ThomasPrecession 209
2.26 MoreTermsoftheHamiltonianOperatorforaMany-ElectronAtomorMolecule 211
2.27 “VanderWaals”InteractionsinMolecules 214
2.28 AtomicStructure:LS(Russell–Saunders)vs.jjCoupling 216
2.29 MolecularSpectroscopy 219
2.30 EinsteinAandBCoefficients 223
2.31 AbsorptionofLight:Beer–Bouguer–LambertLaw,orBeer’sLaw 225
2.32 Time-DependentPerturbationTheory:TheRabiFormula 227
2.33 Fermi’sGoldenRule 229
2.34 Photon–MoleculeInteraction–TheHamiltonian 232
2.35 TransitionMomentsandEinsteinCoefficients 235
2.35.1 OscillatorStrength 236
2.36 QuantumElectrodynamics 236
2.37 GeneralRadiativeTransitions 237
2.38 Staticvs.ResonantDetection 239
2.39 StaticElectric–DipoleSelectionRulesfortheOne-ElectronAtom 239
2.40 StaticElectric-DipoleSelectionRulesfortheHarmonicOscillator 240
2.41 LifetimesfromResonanceLineshapes 241
2.42 LightScattering 241
References 271
End-of-ChapterProblems 273
3 Thermodynamics 277
3.0 ReviewofThermodynamics 277
3.1 TheThree(PlusOne)LawsofThermodynamics 277
3.1.1 ZerothLawofThermodynamics(Transitivity) 277
3.1.2 FirstLawofThermodynamics(ConservationofEnergy–“YouCan’tWin”) 277
viii Contents
3.1.3 SecondLawofThermodynamics(“YouCannotEvenBreakEven”) 278
3.1.4 ThirdLawofThermodynamics 278
3.2 UsefulAuxiliaryFunctions:Enthalpy,HelmholtzFreeEnergy,andGibbsFreeEnergy 279
3.3 PerfectDifferentials(Two-Forms) 279
3.4 UsefulMeasurables:ThermalExpansivity,HeatCapacity,Joule–ThomsonandIsothermalThompson
Coefficients,andtheChemicalPotential 281
3.5 GibbsPhaseRule 283
3.6 CrystallineandAmorphousSolids 284
3.7 Liquids 284
3.8 PerfectGasLaw,thePVTSurface,thevanderWaalsEquationandVirialEquations 285
3.9 ArrheniusAssumption 290
3.10 MoreAboutGases:Maxwell–BoltzmannDistribution,CollisionFrequency,MeanFreePath,andGaseous
Effusion 291
3.11 MoreAboutLiquids 292
3.11.1 Osmosis 293
3.11.2 Superfluid 294
3.12 MoreAboutSolids 294
3.12.1 MagneticSolids 294
3.12.2 Electrets 295
3.13 LiquidCrystals 295
3.14 Two-componentLiquid–VaporPhaseDiagrams 295
3.15 Two-componentSolid–LiquidPhaseDiagramsforSolid–LiquidEquilibria 298
3.16 TheChemicalPotential,IdealSolutions,andColligativeProperties 298
3.16.1 Freezing-TemperatureDepression 299
3.17 Two-DimensionalVersionofthePerfectGasLaw 300
3.17.1 MicellesandLiposomes 304
3.18 ContactAngleandSurfaceTensionMeasurements 306
3.19 AdiabaticandDiathermalWallsandFixed-TemperatureBaths 308
3.20 ThermodynamicEfficiency:TheCarnot,Otto,Diesel,andRankineCycles 308
3.21 InternationalStandardsforTime,Mass,Length,Temperature,andBrightness 310
3.22 StandardStatesandEnthalpiesandGibbsFreeEnergiesofFormation 311
3.23 BondEnthalpies 313
3.24 Electronegativity 318
3.25 ReachingforHighandLowTemperatures 320
3.26 AttainmentofHighandLowPressures 321
References 321
End-of-ChapterProblems 322
4 StatisticalMechanics 327
4.0 Introduction 327
4.1 ReplicasandEnsembles,Fermions,Bosons,andBoltzons 328
4.1.1 FermionPostulate 329
4.1.2 BosonPostulate 329
4.1.3 BoltzonPostulate 329
4.2 CB,FD,andBEDistributions,andtheMicrocanonicalEnsemble 330
4.2.1 MaximumProbability 331
4.3 Canonical,GrandCanonical,Isothermal-Isobaric,andGeneralizedEnsembles 333
4.4 LinksBetweenthePartitionFunctionsandSomeThermodynamicFunctions 338
4.5 HeatCapacities 338
4.5.1 Translation 339
4.5.2 Rotation 339
4.5.3 NuclearSpinEffectsonRotation 340
4.5.4 Vibration 341
Contents ix
4.5.5 ElectronicExcitation 341
4.5.6 EinsteinandDebyeTheoriesoftheLow-temperatureHeatCapacityofSolids[4.2] 342
4.5.7 DebyeTheoryoftheHeatCapacityofSolids 342
4.6 Black-BoxRadiation,andtheBirthofQuantumMechanics 343
4.7 ElectronicHeatCapacity:Drudevs.Fermi–Dirac 346
4.8 MagneticSusceptibilities 349
4.8.1 GeneralPhenomenology 349
4.8.2 DiluteEnsembleofParamagneticIons 350
4.8.3 Diamagnetism 352
4.8.4 Ferromagnetism 355
4.9 ElectricSusceptibilities 356
4.10 UniversalTheoryofCriticalPhenomena 360
References 360
End-of-ChapterProblems 361
5 Kinetics,Equilibria,andElectrochemistry 363
5.0 Introduction 363
5.1 Energetics,ReactionCoordinate,TransitionStates,Intermediates,andCatalysis 363
5.2 ClassificationofReactionTypes 367
5.3 First-OrderandUnimolecularReactions 368
5.3.1 Carbon-14Dating 368
5.4 Second-Order(Unmixed)andUnmixedBimolecularReactions 369
5.5 Second-Order(Mixed)andMixedBimolecularReactions 369
5.6 Third-Order(Unmixed)andUnmixedTermolecularReactions 370
5.7 ReversibleReactions 370
5.8 ConsecutiveReactions 371
5.9 TheSteady-StateApproximationandtheRate-DeterminingStep 372
5.10 ApproximationMethods:theMichaelis–MentenEquation 372
5.11 ChainReactions.TheReactionofHydrogenandBromineatHighTemperature 375
5.12 UsingLaplaceTransformstoSolveKineticsEquations 377
5.13 ReactionRateTheoriesandEnergySurfaces 378
5.14 MarcusTheoryofElectronTransfer 380
5.15 EquilibriainAqueousSolution:pH 383
5.16 EquilibriainNonaqueousSolvents 391
5.17 LewisAcidsandLewisBases 392
5.18 Electrochemistry:ElectrodePotentialsandtheNernstEquation 393
5.19 Gouy–ChapmanDouble-LayerTheory 405
5.20 Nernst–PlanckandCottrellEquations 406
FurtherReading 409
References 409
End-of-ChapterProblems 410
6 Symmetry 411
6.0 Symmetry 411
6.1 SymmetryinCrystals 411
6.2 SymmetryOperationsandPointGroups 412
6.3 GroupTheoryandCharacterTables 418
6.4 BravaisLattices 427
6.5 The32CrystallographicPointGroups 432
6.6 The17PlaneGroups 433
6.7 The230CrystallographicSpaceGroups 434
6.8 ListingofElements,SimpleCompounds,andTheirCrystalStructures 443
6.9 TheWigner–SeitzCell 455
x Contents
6.10 ReciprocalLattice 456
6.11 Symmetryof2DSurfaces 458
6.12 DescentofSymmetry 459
6.13 CovariantandContravariantTransformations 459
6.13.1 Covariant 459
6.13.2 Contravariant 460
6.13.3 FourbyFour 461
6.13.4 Generationof230SpaceGroupsUsing4by4Matrices 462
6.14 ExampleofDescentofSymmetry:VO 462
2
References 466
7 SolidStatePhysics 469
7.0 Introduction 469
7.1 ElectricalResistance,HallEffect,DrudeModel,Tunneling,andtheLandauerFormula 469
7.2 Fermi–DiracStatisticsforElectronGas:SommerfeldModel 479
7.3 X-rayDiffraction 484
7.4 QuantumNumbersinaMacroscopicSolid:BlochWaves 487
7.5 BlochWavesinOneDimensionandDispersionRelations 488
7.6 BandStructures 490
7.7 TheoreticalMethodsforComputingWavefunctionsinSolids 495
7.7.1 TheTight-BindingMethod 495
7.7.2 CellularMethod 497
7.7.3 BandStructurefortheMuffin-tinPotential 498
7.7.4 AugmentedPlaneWaves(APW) 498
7.7.5 OrthogonalizedPlaneWaves(OPW) 498
7.7.6 HubbardHamiltonian 498
7.8 MixedValenceandOne-DimensionalInstabilities 498
7.9 DefectsandMobileExcitationsinSolidsandMolecules 500
7.10 Superconductivity 501
7.11 LatticeEnergies:Madelung,Repulsion,Dispersion,Dipole–dipole,andOthers 505
References 515
8 ElectricalCircuits,Amplifiers,andComputers 519
8.0 Introduction 519
8.1 ElectricalComponents 519
8.2 SimpleCircuitswithNoRectificationorAmplification 521
8.2.1 Resistors 522
8.2.2 Capacitance 522
8.2.3 Capacitors 522
8.2.4 Inductance 523
8.2.5 Inductors 523
8.2.6 Kirchhoff’sRulesforCircuits 523
8.2.7 SeriesRLCCircuit 524
8.3 VacuumTubeDiode 528
8.4 VacuumTubeTriode 530
8.5 ConductioninPureandDopedSiandGe 533
8.6 RectificationinpnJunctionDiodesorRectifiers 536
8.7 pnpandnpnTransistors 540
8.8 Small-SignalTheoryforTransistors 546
8.9 Large-SignalBehaviorofJunctionTransistors 549
8.10 UnipolarorField-EffectTransistors(FET) 550
8.11 JFETs 554
8.12 OperationalAmplifiers 554