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Contents I Preliminaries 1 1 Introduction 2 1.1 What is a mole? 3 1.2 The thermodynamic limit 4 1.3 The ideal gas 6 1.4 Combinatorial problems 7 1.5 Plan of the book 9 Exercises 12 2 Heat 13 2.1 A definition of heat 13 2.2 Heat capacity 14 Exercises 17 3 Probability 18 3.1 Discrete probability distributions 19 3.2 Continuous probability distributions 20 3.3 Linear transformation 21 3.4 Variance 22 3.5 Linear transformation and the variance 23 3.6 Independent variables 24 Further reading 27 Exercises 27 4 Temperature and the Boltzmann factor 30 4.1 Thermal equilibrium 30 4.2 Thermometers 31 4.3 The microstates and macrostates 33 4.4 A statistical definition of temperature 34 4.5 Ensembles 36 4.6 Canonical ensemble 36 4.7 Applications of the Boltzmann distribution 40 Further reading 44 Exercises 44 II Kinetic theory of gases 45 5 The Maxwell–Boltzmann distribution 46
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Contents
I Preliminaries 1
1 Introduction 21.1 What is a mole? 31.2 The thermodynamic limit 41.3 The ideal gas 61.4 Combinatorial problems 71.5 Plan of the book 9Exercises 12
2 Heat 132.1 A definition of heat 132.2 Heat capacity 14Exercises 17
3 Probability 183.1 Discrete probability distributions 193.2 Continuous probability distributions 203.3 Linear transformation 213.4 Variance 223.5 Linear transformation and the variance 233.6 Independent variables 24Further reading 27Exercises 27
4 Temperature and the Boltzmann factor 304.1 Thermal equilibrium 304.2 Thermometers 314.3 The microstates and macrostates 334.4 A statistical definition of temperature 344.5 Ensembles 364.6 Canonical ensemble 364.7 Applications of the Boltzmann distribution 40Further reading 44Exercises 44
II Kinetic theory of gases 45
5 The Maxwell–Boltzmann distribution 46
xii Contents
5.1 The velocity distribution 465.2 The speed distribution 47
5.2.1 〈v〉 and 〈v2〉 485.2.2 The mean kinetic energy of a gas molecule 485.2.3 The maximum of f(v) 49
5.3 Experimental justification 49Exercises 52
6 Pressure 546.1 Molecular distributions 55
6.1.1 Solid angles 556.1.2 The number of molecules travelling in a certain
direction at a certain speed 556.1.3 The number of molecules hitting a wall 56
6.2 The ideal gas law 566.3 Dalton’s law 58Exercises 59
7 Molecular effusion 627.1 Flux 627.2 Effusion 64Exercises 67
8 The mean free path and collisions 688.1 The mean collision time 688.2 The collision cross-section 698.3 The mean free path 71Exercises 72
III Transport and thermal diffusion 73
9 Transport properties in gases 749.1 Viscosity 749.2 Thermal conductivity 799.3 Diffusion 819.4 More-detailed theory 84Further reading 86Exercises 87
10 The thermal diffusion equation 8810.1 Derivation of the thermal diffusion equation 8810.2 The one-dimensional thermal diffusion equation 8910.3 The steady state 9210.4 The thermal diffusion equation for a sphere 9210.5 Newton’s law of cooling 9510.6 The Prandtl number 9710.7 Sources of heat 98Exercises 99
Contents xiii
IV The first law 103
11 Energy 10411.1 Some definitions 104
11.1.1 A system in thermal equilibrium 10411.1.2 Functions of state 104
11.2 The first law of thermodynamics 10611.3 Heat capacity 108Exercises 111
12 Isothermal and adiabatic processes 11412.1 Reversibility 11412.2 Isothermal expansion of an ideal gas 11612.3 Adiabatic expansion of an ideal gas 11712.4 Adiabatic atmosphere 117Exercises 119
V The second law 121
13 Heat engines and the second law 12213.1 The second law of thermodynamics 12213.2 The Carnot engine 12313.3 Carnot’s theorem 12613.4 Equivalence of Clausius and Kelvin statements 12713.5 Examples of heat engines 12713.6 Heat engines running backwards 12913.7 Clausius’ theorem 130Further reading 133Exercises 133
14 Entropy 13614.1 Definition of entropy 13614.2 Irreversible change 13614.3 The first law revisited 13814.4 The Joule expansion 14014.5 The statistical basis for entropy 14214.6 The entropy of mixing 14314.7 Maxwell’s demon 14514.8 Entropy and probability 146Exercises 149
15 Information theory 15315.1 Information and Shannon entropy 15315.2 Information and thermodynamics 15515.3 Data compression 15615.4 Quantum information 158Further reading 161Exercises 161
xiv Contents
VI Thermodynamics in action 163
16 Thermodynamic potentials 16416.1 Internal energy, U 16416.2 Enthalpy, H 16516.3 Helmholtz function, F 16616.4 Gibbs function, G. 16716.5 Availability 16816.6 Maxwell’s relations 170Exercises 178
17 Rods, bubbles and magnets 18217.1 Elastic rod 18217.2 Surface tension 18517.3 Paramagnetism 186Exercises 192
18 The third law 19318.1 Different statements of the third law 19318.2 Consequences of the third law 195Exercises 198
VII Statistical mechanics 199
19 Equipartition of energy 20019.1 Equipartition theorem 20019.2 Applications 203
19.2.1 Translational motion in a monatomic gas 20319.2.2 Rotational motion in a diatomic gas 20319.2.3 Vibrational motion in a diatomic gas 20419.2.4 The heat capacity of a solid 205
19.3 Assumptions made 20519.4 Brownian motion 207Exercises 208
20 The partition function 20920.1 Writing down the partition function 21020.2 Obtaining the functions of state 21120.3 The big idea 21820.4 Combining partition functions 218Exercises 219
21 Statistical mechanics of an ideal gas 22121.1 Density of states 22121.2 Quantum concentration 22321.3 Distinguishability 22421.4 Functions of state of the ideal gas 22521.5 Gibbs paradox 228
