Thermodynamics and Statistical Mechanics: An Integrated Approach

Thermodynamics and Statistical Mechanics 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.

Table of Contents:
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

About the Author :
ROBERT J. HARDY and CHRISTIAN BINEK Department of Physics, University of Nebraska-Lincoln, USA