Thermodynamics and Statistical Mechanics An Integrated Approach

Thermodynamics and Statistical Mechanics

An Integrated Approach

This textbook brings together the fundamentals of the macroscopic and microscopic aspects of thermal physics by presenting thermodynamics and statistical mechanics as complementary theories based on small numbers of postulates. The book is designed to give the instructor flexibility in structuring courses for advanced undergraduates and/or beginning graduate students and is written on the principle that a good text should also be a good reference.

The presentation of thermodynamics follows the logic of Clausius and Kelvin while relating the concepts involved to familiar phenomena and the modern student?s knowledge of the atomic nature of matter. Another unique aspect of the book is the treatment of the mathematics involved. The essential mathematical concepts are briefly reviewed before using them, and the similarity of the mathematics to that employed in other fields of physics is emphasized.

The text gives in-depth treatments of low-density gases, harmonic solids, magnetic and dielectric materials, phase transitions, and the concept of entropy. The microcanonical, canonical, and grand canonical ensembles of statistical mechanics are derived and used as the starting point for the analysis of fluctuations, blackbody radiation, the Maxwell distribution, Fermi-Dirac statistics, Bose-Einstein condensation, and the statistical basis of computer simulations.

Preface xiii

Part I Elements of Thermal Physics 1

1. Fundamentals 3

1.1 PVT Systems 3

1.2 Equilibrium States 6

1.3 Processes and Heat 10

1.4 Temperature 12

1.5 Size Dependence 13

1.6 Heat Capacity and Specific Heat 14

Problems 17

2. First Law of Thermodynamics 19

2.1 Work 19

2.2 Heat 21

2.3 The First Law 21

2.4 Applications 22

Problems 26

3. Properties and Partial Derivatives 27

3.1 Conventions 27

3.2 Equilibrium Properties 28

3.3 Relationships between Properties 34

3.4 Series Expansions 40

3.5 Summary 41

Problems 42

4. Processes in Gases 45

4.1 Ideal Gases 45

4.2 Temperature Change with Elevation 48

4.3 Cyclic Processes 50

4.4 Heat Engines 52

Problems 58

5. Phase Transitions 61

5.1 Solids, Liquids, and Gases 61

5.2 Latent Heats 65

5.3 Van der Waals Model 67

5.4 Classification of Phase Transitions 70

Problems 72

6. Reversible and Irreversible Processes 75

6.1 Idealization and Reversibility 75

6.2 Nonequilibrium Processes and Irreversibility 76

6.3 Electrical Systems 79

6.4 Heat Conduction 82

Problems 86

Part II Foundations of Thermodynamics 89

7. Second Law of Thermodynamics 91

7.1 Energy, Heat, and Reversibility 91

7.2 Cyclic Processes 93

7.3 Second Law of Thermodynamics 95

7.4 Carnot Cycles 98

7.5 Absolute Temperature 100

7.6 Applications 103

Problems 107

8. Temperature Scales and Absolute Zero 109

8.1 Temperature Scales 109

8.2 Uniform Scales and Absolute Zero 111

8.3 Other Temperature Scales 114

Problems 115

9. State Space and Differentials 117

9.1 Spaces 117

9.2 Differentials 121

9.3 Exact Versus Inexact Differentials 123

9.4 Integrating Differentials 127

9.5 Differentials in Thermodynamics 129

9.6 Discussion and Summary 134

Problems 136

10. Entropy 139

10.1 Definition of Entropy 139

10.2 Clausius’ Theorem 142

10.3 Entropy Principle 145

10.4 Entropy and Irreversibility 148

10.5 Useful Energy 151

10.6 The Third Law 155

10.7 Unattainability of Absolute Zero 156

Problems 158

Appendix 10.A. Entropy Statement of the Second Law 158

11. Consequences of Existence of Entropy 165

11.1 Differentials of Entropy and Energy 165

11.2 Ideal Gases 167

11.3 Relationships Between CV, CP, BT , BS, and αV 170

11.4 Clapeyron’s Equation 172

11.5 Maximum Entropy, Equilibrium, and Stability 174

11.6 Mixing 178

Problems 184

12. Thermodynamic Potentials 185

12.1 Internal Energy 185

12.2 Free Energies 186

12.3 Properties From Potentials 188

12.4 Systems in Contact with a Heat Reservoir 193

12.5 Minimum Free Energy 194

Problems 197

Appendix 12.A. Derivatives of Potentials 197

13. Phase Transitions and Open Systems 201

13.1 Two-Phase Equilibrium 201

13.2 Chemical Potential 206

13.3 Multi-Component Systems 211

13.4 Gibbs Phase Rule 214

13.5 Chemical Reactions 215

Problems 217

14. Dielectric and Magnetic Systems 219

14.1 Dielectrics 219

14.2 Magnetic Materials 224

14.3 Critical Phenomena 229

Problems 233

Part III Statistical Thermodynamics 235

15. Molecular Models 237

15.1 Microscopic Descriptions 237

15.2 Gas Pressure 238

15.3 Equipartition of Energy 243

15.4 Internal Energy of Solids 246

15.5 Inactive Degrees of Freedom 247

15.6 Microscopic Significance of Heat 248

Problems 253

16. Kinetic Theory of Gases 255

16.1 Velocity Distribution 255

16.2 Combinatorics 256

16.3 Method of Undetermined Multipliers 258

16.4 Maxwell Distribution 260

16.5 Mean-Free-Path 265

Problems 267

Appendix 16.A. Quantum Distributions 267

17. Microscopic Significance of Entropy 273

17.1 Boltzmann Entropy 273

17.2 Ideal Gas 274

17.3 Statistical Interpretation 278

17.4 Thermodynamic Properties 279

17.5 Boltzmann Factors 284

Problems 286

Appendix 17.A. Evaluation of I3N 286

Part IV Statistical Mechanics I 289

18. Ensembles 291

18.1 Probabilities and Averages 291

18.2 Two-Level Systems 293

18.3 Information Theory 295

18.4 Equilibrium Ensembles 298

18.5 Canonical Thermodynamics 302

18.6 Composite Systems 305

Problems 308

Appendix 18.A. Uniqueness Theorem 308

19. Partition Function 311

19.1 Hamiltonians and Phase Space 311

19.2 Model Hamiltonians 312

19.3 Classical Canonical Ensemble 316

19.4 Thermodynamic Properties and Averages 318

19.5 Ideal Gases 322

19.6 Harmonic Solids 326

Problems 328

20. Quantum Systems 331

20.1 Energy Eigenstates 331

20.2 Quantum Canonical Ensemble 333

20.3 Ideal Gases 334

20.4 Einstein Model 337

20.5 Classical Approximation 341

Problems 344

Appendix 20.A. Ideal Gas Eigenstates 344

21. Independent Particles and Paramagnetism 349

21.1 Averages 349

21.2 Statistical Independence 351

21.3 Classical Systems 353

21.4 Paramagnetism 357

21.5 Spin Systems 360

21.6 Classical Dipoles 365

Problems 367

Appendix 21.A. Negative Temperature 367

22. Fluctuations and Energy Distributions 371

22.1 Standard Deviation 371

22.2 Energy Fluctuations 375

22.3 Gibbs Paradox 376

22.4 Microcanonical Ensemble 380

22.5 Comparison of Ensembles 386

Problems 391

23. Generalizations and Diatomic Gases 393

23.1 Generalized Coordinates 393

23.2 Diatomic Gases 397

23.3 Quantum Effects 402

23.4 Density Matrices 405

23.5 Canonical Ensemble 408

Problems 410

Appendix 23.A. Classical Approximation 410

Part V Statistical Mechanics II 415

24. Photons and Phonons 417

24.1 Plane Wave Eigenstates 417

24.2 Photons 421

24.3 Harmonic Approximation 425

24.4 Phonons 429

Problems 434

25. Grand Canonical Ensemble 435

25.1 Thermodynamics of Open Systems 435

25.2 Grand Canonical Ensemble 437

25.3 Properties and Fluctuations 438

25.4 Ideal Gases 441

Problems 443

26. Fermions and Bosons 445

26.1 Identical Particles 445

26.2 Exchange Symmetry 447

26.3 Fermi–Dirac and Bose–Einstein Statistics 452

Problems 456

Appendix 26.A. Fermions in the Canonical Ensemble 457

27. Fermi and Bose Gases 461

27.1 Ideal Gases 461

27.2 Fermi Gases 465

27.3 Low Temperature Heat Capacity 466

27.4 Bose Gases 469

Problems 472

28. Interacting Systems 475

28.1 Ising Model 475

28.2 Nonideal Gases 481

Problems 487

29. Computer Simulations 489

29.1 Averages 489

29.2 Virial Formula for Pressure 490

29.3 Simulation Algorithms 496

A. Mathematical Relations, Constants, and Properties 501

A.1 Partial Derivatives 501

A.2 Integrals and Series 501

A.3 Taylor Series 502

A.4 Hyperbolic Functions 502

A.5 Fundamental Constants 503

A.6 Conversion Factors 503

A.7 Useful Formulas 503

A.8 Properties of Water 504

A.9 Properties of Materials 504

Answers to Problems 505

Index 509