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Thermodynamics and Statistical Mechanics

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Thermodynamics and Statistical Mechanics Thermodynamics and Statistical Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 7 1.1 IntendedAudience.................................. 7 1.2 MajorSources.................................... 7 1.3 Why Study Thermodynamics? ........................... 7 1.4 AtomicTheoryofMatter.............................. 7 1.5 What is Thermodynamics? . ........................... 8 1.6 NeedforStatisticalApproach............................ 8 1.7 MicroscopicandMacroscopicSystems....................... 10 1.8 Classical and Statistical Thermodynamics . .................... 10 1.9 ClassicalandQuantumApproaches......................... 11 2 Probability Theory 13 2.1 Introduction . .................................. 13 2.2 What is Probability? . ........................... 13 2.3 Combining Probabilities . ........................... 13 2.4 Two-StateSystem.................................. 15 2.5 CombinatorialAnalysis............................... 16 2.6 Binomial Probability Distribution . .................... 17 2.7 Mean,Variance,andStandardDeviation...................... 18 2.8 Application to Binomial Probability Distribution . ................ 19 2.9 Gaussian Probability Distribution . .................... 22 2.10 CentralLimitTheorem............................... 25 Exercises ....................................... 26 3 Statistical Mechanics 29 3.1 Introduction . .................................. 29 3.2 SpecificationofStateofMany-ParticleSystem................... 29 2 CONTENTS 3.3 Principle of Equal A Priori Probabilities . .................... 31 3.4 H-Theorem..................................... 32 3.5 RelaxationTime................................... 36 3.6 Reversibility and Irreversibility ........................... 36 3.7 Probability Calculations . ........................... 37 3.8 BehaviorofDensityofStates............................ 38 Exercises ....................................... 39 4 Heat and Work 43 4.1 BriefHistoryofHeatandWork........................... 43 4.2 MacrostatesandMicrostates............................ 45 4.3 MicroscopicInterpretationofHeatandWork.................... 45 4.4 Quasi-StaticProcesses................................ 46 4.5 Exact and Inexact Differentials........................... 48 Exercises ....................................... 51 5 Statistical Thermodynamics 53 5.1 Introduction . .................................. 53 5.2 ThermalInteractionBetweenMacrosystems.................... 53 5.3 Temperature..................................... 56 5.4 MechanicalInteractionBetweenMacrosystems.................. 59 5.5 GeneralInteractionBetweenMacrosystems.................... 60 5.6 Entropy....................................... 62 5.7 PropertiesofEntropy................................ 67 5.8 UsesofEntropy................................... 68 5.9 EntropyandQuantumMechanics.......................... 70 5.10 Laws of Thermodynamics . ........................... 73 Exercises ....................................... 73 6 Classical Thermodynamics 77 6.1 Introduction . .................................. 77 6.2 IdealGasEquationofState............................. 78 6.3 SpecificHeat..................................... 81 6.4 CalculationofSpecificHeats............................ 83 6.5 IsothermalandAdiabaticExpansion........................ 84 6.6 Hydrostatic Equilibrium of Atmosphere . .................... 85 6.7 IsothermalAtmosphere............................... 86 6.8 AdiabaticAtmosphere................................ 88 6.9 InternalEnergy................................... 91 6.10 Enthalpy....................................... 92 6.11 HelmholtzFreeEnergy............................... 93 6.12 GibbsFreeEnergy.................................. 94 6.13 GeneralRelationBetweenSpecificHeats...................... 95 CONTENTS 3 6.14 FreeExpansionofGas................................ 97 6.15 VanderWaalsGas.................................. 99 6.16 Joule-Thompson Throttling . ...........................101 6.17 HeatEngines.....................................104 6.18 Refrigerators.....................................110 Exercises .......................................111 7 Applications of Statistical Thermodynamics 117 7.1 Introduction . ..................................117 7.2 Canonical Probability Distribution . ....................117 7.3 Spin-1/2Paramagnetism...............................119 7.4 SystemwithSpecifiedMeanEnergy........................122 7.5 CalculationofMeanValues.............................123 7.6 Partition Function ..................................125 7.7 IdealMonatomicGas................................126 7.8 Gibb’sParadox....................................129 7.9 GeneralParamagnetism...............................131 7.10 Equipartition Theorem . ...........................136 7.11 Harmonic Oscillators . ...........................138 7.12 SpecificHeats....................................140 7.13 SpecificHeatsofGases...............................142 7.14 SpecificHeatsofSolids...............................146 7.15 MaxwellVelocityDistribution...........................153 7.16 Effusion.......................................159 7.17 Ferromagnetism...................................162 Exercises .......................................167 8 Quantum Statistics 177 8.1 Introduction . ..................................177 8.2 SymmetryRequirementsinQuantumMechanics..................177 8.3 IllustrativeExample.................................179 8.4 FormulationofStatisticalProblem.........................181 8.5 Fermi-DiracStatistics................................182 8.6 PhotonStatistics...................................183 8.7 Bose-EinsteinStatistics...............................184 8.8 Maxwell-BoltzmannStatistics............................185 8.9 QuantumStatisticsinClassicalLimit........................186 8.10 Quantum-MechanicalTreatmentofIdealGas....................189 8.11 DerivationofvanderWaalsEquationofState...................191 8.12 PlanckRadiationLaw................................193 8.13 Black-BodyRadiation................................196 8.14 Stefan-BoltzmannLaw...............................198 8.15 Conduction Electrons in Metal ...........................199 4 CONTENTS 8.16 SommerfeldExpansion...............................203 8.17 White-DwarfStars..................................208 8.18 ChandrasekharLimit................................210 8.19 NeutronStars....................................211 8.20 Bose-Einstein Condensation . ...........................212 Exercises .......................................218 9 Multi-Phase Systems 227 9.1 Introduction . ..................................227 9.2 Equilibrium of Isolated System ...........................227 9.3 Equilibrium of Constant-Temperature System . ................228 9.4 Equilibrium of Constant-Temperature Constant-Pressure System .........230 9.5 Stability of Single-Phase Substance . ....................232 9.6 Equilibrium Between Phases . ...........................236 9.7 Clausius-ClapeyronEquation............................237 9.8 PhaseDiagrams...................................238 9.9 VaporPressure....................................239 9.10 PhaseTransformationsinVanderWaalsFluid...................240 Exercises .......................................244 A Physical Constants 251 B Classical Mechanics 253 B.1 GeneralizedCoordinates...............................253 B.2 GeneralizedForces.................................253 B.3 Lagrange’sEquation.................................254 B.4 GeneralizedMomenta................................256 B.5 CalculusofVariations................................257 B.6 ConditionalVariation................................259 B.7 Multi-Function Variation . ...........................261 B.8 Hamilton’sPrinciple.................................262 B.9 Hamilton’sEquations................................262 C Wave Mechanics 265 C.1 Introduction . ..................................265 C.2 Photoelectric Effect.................................266 C.3 Electron Diffraction.................................267 C.4 RepresentationofWavesviaComplexNumbers..................267 C.5 Schr¨odinger’sEquation...............................269 C.6 Probability Interpretation of Wavefunction . ....................270 C.7 WavePackets....................................272 C.8 Heisenberg’sUncertaintyPrinciple.........................275 C.9 StationaryStates...................................276 CONTENTS 5 C.10Three-DimensionalWaveMechanics........................278 C.11 Simple Harmonic Oscillator . ...........................281 C.12 Angular Momentum . ...........................283 6 CONTENTS Introduction 7 1 Introduction 1.1 Intended Audience These lecture notes outline a single-semester course intended for upper-division physics majors. 1.2 Major Sources The textbooks that I have consulted most frequently while developing course material are: Fundamentals of Statistical and Thermal Physics: F. Reif (McGraw-Hill, New York NY, 1965). Introduction to Quantum Theory: D. Park, 3rd Edition (McGraw-Hill, New York NY, 1992). Classical and Statistical Thermodynamics: A.S. Carter (Prentice-Hall, Upper Saddle River NJ, 2001). 1.3 Why Study Thermodynamics? In a nutshell, thermodynamics is the study of the internal motions of many-body systems. Virtu- ally all physical entities that we encounter in everyday life are many-body systems of some type or other (e.g., solids, liquids, gases, and even electromagnetic radiation). Not surprisingly, therefore, thermodynamics
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