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Contents Preface 23 About the Author 29 1. Introduction 31 1.1. The Statistical Foundation of Classical Thermodynamics 32 1.2. A Classification Scheme for Statistical Thermodynamics 36 1.3. Why Statistical Thermodynamics? 37 PART ONE FUNDAMENTALS OF STATISTICAL THERMODYNAMICS 43 2. Probability and Statistics 44 2.1. Probability: Definitions and Basic Concepts 44 2.2. Permutations and Combinations 50 2.3. Probability Distributions: Discrete and Continuous 53 2.4. The Binomial Distribution 56 2.5. The Poisson Distribution 59 2.6. The Gaussian Distribution 61 2.7. Combinatorial Analysis for Statistical Thermodynamics 66 2.7.1. Distinguishable Objects 68 2.1.2. Indistinguishable Objects 70 Problem Set I: Probability Theory and Statistical Mathematics (Chapter 2) 81 The Statistics of Independent Particles 91 3.1. Essential Concepts from Quantum Mechanics 93 3.2. The Ensemble Method of Statistical Thermodynamics 95 3.3. The Two Basic Postulates of Statistical Thermodynamics 99 3.3.1. The M-B Method: System Constraints and Particle Distribution 101 3.3.2. The M-B Method: Microstates and Macrostates 102 3.4. The Most Probable Macrostate 105 3.5. Bose-Einstein and Fermi-Dirac Statistics 110 3.5.1. Bose-Einstein Statistics 111 3.5.2. Fermi-Dirac Statistics 112 3.5.3. The Most Probable Particle Distribution 115 3.6. Entropy and the Equilibrium Particle Distribution 118 3.6.1. The Boltzmann Relation for Entropy 118 3.6.2. Identification of Lagrange Multipliers 120 3.6.3. The Equilibrium Particle Distribution 122 4. Thermodynamic Properties in the Dilute Limit 129 4.1. The Dilute Limit 130 4.2. Corrected Maxwell-Boltzmann Statistics 131 4.3. The Molecular Partition Function 134 4.3.1. The Influence of Temperature 137 4.3.2. Criterion for Dilute Limit 139 4.4. Internal Energy and Entropy in the Dilute Limit 141 4.5. Additional Thermodynamic Properties in the Dilute Limit 146 4.6. The Zero of Energy and Thermodynamic Properties 151 4.7. Intensive Thermodynamic Properties for the Ideal Gas 153 Problem Set II: Statistical Modeling for Thermodynamics (Chapters 3--4) 158 PART TWO QUANTUM MECHANICS AND SPECTROSCOPY 173 5. Basics of Quantum Mechanics 174 5.1. Historical Survey of Quantum Mechanics 175 5.2. The Bohr Model for the Spectrum of Atomic Hydrogen 182 5.3. The De Broglie Hypothesis 191 5.4. A Heuristic Introduction to the Schr¿odinger Equation 195 5.5. The Postulates of Quantum Mechanics 200 5.6. The Steady-State Schr¿odinger Equation 208 5.6.1. Single-Particle Analysis 209 5.6.2. Multi-Particle Analysis 211 5.7. The Particle in a Box 213 5.8. The Uncertainty Principle 222 5.9. Indistinguishability and Symmetry 227 5.10. The Pauli Exclusion Principle 230 5.11. The Correspondence Principle 233 6. Quantum Analysis of Internal Energy Modes 241 6.1. Schr¿odinger Wave Equation for Two-Particle System 242 6.1.1. Conversion to Center-of-Mass Coordinates 243 6.1.2. Separation of External from Internal Modes 244 6.2. The Internal Motion for a Two-Particle System 246 6.3. The Rotational Energy Mode for a Diatomic Molecule 248 6.4. The Vibrational Energy Mode for a Diatomic Molecule 255 6.5. The Electronic Energy Mode for Atomic Hydrogen 264 6.6. The Electronic Energy Mode for Multi-Electron Species 276 6.6.1. Electron Configuration for Multi-Electron Atoms 278 6.6.2. Spectroscopic Term Symbols for Multi-Electron Atoms 280 6.6.3. Electronic Energy Levels and Degeneracies for Atoms 284 6.6.4. Electronic Energy Levels and Degeneracies for Diatomic Molecules 286 6.7. Combined Energy Modes for Atoms and Diatomic Molecules 290 6.8. Selection Rules for Atoms and Molecules 292 7. The Spectroscopy of Diatomic Molecules 323 7.1. Rotational Spectroscopy using the Rigid-Rotor Model 325 7.2. Vibrational Spectroscopy using the Harmonic-Oscillator Model 328 7.3. Rovibrational Spectroscopy: The Simplex Model 330 7.4. The Complex Model for Combined Rotation and Vibration 337 7.5. Rovibrational Spectroscopy: The Complex Model 342 7.6. Electronic Spectroscopy 349 7.7. Energy-Mode Parameters for Diatomic Molecules 357 Problem Set III: Quantum Mechanics and Spectroscopy (Chapters 5--7) 370 PART THREE STATISTICAL THERMODYNAMICS IN THE DILUTE LIMIT 389 8. Interlude: From Particle to Assembly 390 8.1. Energy and Degeneracy 391 8.2. Separation of Energy Modes 394 8.3. The Molecular Internal Energy 396 8.4. The Partition Function and Thermodynamic Properties 398 8.5. Energy-Mode Contributions in Classical Mechanics 402 8.5.1. The Phase Integral 403 8.5.2. The Equipartition Principle 408 8.5.3. Mode Contributions 410 9. Thermodynamic Properties of the Ideal Gas 421 9.1. The Monatomic Gas 422 9.1.1. Translational Mode 423 9.1.2. Electronic Mode 430 9.2. The Diatomic Gas 434 9.2.1. Translational and Electronic Modes 435 9.2.2. The Zero of Energy 436 9.2.3. Rotational Mode 438 9.2.4. Quantum Origin of Rotational Symmetry Factor 446 9.2.5. Vibrational Mode 452 9.3. Rigorous and Semi-rigorous Models for the Diatomic Gas 458 9.4. The Polyatomic Gas 469 9.4.1. Rotational Contribution 472 9.4.2. Vibrational Contribution 477 9.4.3. Property Calculations for Polyatomic Molecules 480 Problem Set IV: Thermodynamic Properties of the Ideal Gas (Chapters 8--9) 496 10. Statistical Thermodynamics for Ideal Gas Mixtures 503 10.1. Equilibrium Particle Distribution for the Ideal Gas Mixture 504 10.2. Thermodynamic Properties of the Ideal Gas Mixture 509 10.3. The Reacting Ideal Gas Mixture 515 10.3.1. Equilibrium Particle Distribution for Reactive Ideal Gas Mixture 516 10.3.2. Equilibrium Constant: Introduction and Development 520 10.4. Equilibrium Constant: General Expression and Specific Examples 523 10.4.1. Dissociation of a Homonuclear Diatomic 528 10.4.2. The Homonuclear-Heteronuclear Conversion Reaction 533 10.4.3. The Ionization Reaction 534 11. Concentration and Temperature Measurements 538 11.1. Mode Temperatures 540 11.2. Radiative Transitions 542 11.2.1. Spectral Transfer of Radiation 546 11.2.2. The Einstein Coefficients 548 11.2.3. Line Broadening 550 11.3. Absorption Spectroscopy 553 11.4. Emission Spectroscopy 560 11.4.1. Emissive Diagnostics 561 11.4.2. The Problem of Self-Absorption 563 11.5. Fluorescence Spectroscopy 566 11.6. Sodium D-Line Reversal 573 11.7. Advanced Diagnostic Techniques 575 Problem Set V: Chemical Equilibrium and Diagnostics (Chapters 10--11) 588 PART FOUR STATISTICAL THERMODYNAMICS BEYOND THE DILUTE LIMIT 601 12. Thermodynamics and Information 602 12.1. Reversible Work and Heat 603 12.2. The Second Law of Thermodynamics 604 12.3. The Boltzmann Definition of Entropy 607 12.4. Information Theory 608 12.5. Spray Size Distribution from Information Theory 613 13. Elements of the Solid State 619 13.1. Statistical Thermodynamics of the Crystalline Solid 620 13.2. Einstein Theory for the Crystalline Solid 625 13.3. Debye Theory for the Crystalline Solid 628 13.4. Critical Evaluation of the Debye Formulation 634 13.5. The Band Theory of Metallic Solids 638 13.6. Thermodynamic Properties of the Electron Gas 642 13.7. The Metallic Crystal near Absolute Zero 648 14. Equilibrium Radiation 660 14.1. Bose-Einstein Statistics for the Photon Gas 660 14.2. Photon Quantum States 662 14.3. The Planck Distribution Law 663 14.4. Thermodynamics of Black-Body Radiation 667 14.5. The Influence of Wavelength for the Planck Distribution 672 Problem Set VI: The Solid State and Radiation (Chapters 13--14) 676 PART FIVE NON-EQUILIBRIUM STATISTICAL THERMODYNAMICS 683 15. Elementary Kinetic Theory 684 15.1. The Maxwell-Boltzmann Velocity Distribution 685 15.2. The Maxwell-Boltzmann Speed Distribution 689 15.3. The Maxwell-Boltzmann Energy Distribution 694 15.4. Molecular Effusion 696 15.5. The Ideal Gas Pressure 702 16. Kinetics of Molecular Transport 711 16.1. Binary Collision Theory 711 16.2. Fundamentals of Molecular Transport 720 16.2.1. The Mean Free Path 721 16.2.2. The Molecular Flux 724 16.2.3. Transport Properties 729 16.3. Rigorous Transport Theory 734 16.3.1. Dimensionless Transport Parameters 735 16.3.2. Collision Integrals 737 16.3.3. The Lennard-Jones Potential 740 16.3.4. Rigorous Expressions for Transport Properties 742 17. Chemical Kinetics 756 17.1. The Bimolecular Reaction 756 17.2. The Rate of Bimolecular Reactions 758 17.3. Chemical Kinetics from Collision Theory 761 17.4. The Significance of Internal Energy Modes 768 17.5. Chemical Kinetics from Transition State Theory 769 Problem Set VII: Kinetic Theory and Molecular Transport (Chapters 15--17) 783 PART SIX THE ENSEMBLE METHOD OF STATISTICAL THERMODYNAMICS 797 18. The Canonical and Grand Canonical Ensembles 798 18.1. The Ensemble Method 799 18.2. The Canonical Ensemble 800 18.2.1. The Equilibrium Distribution for the Canonical Ensemble 802 18.2.2. Equilibrium Properties for the Canonical Ensemble 806 18.2.3. Independent Particles in the Dilute Limit 812 18.2.4. Fluctuations in Internal Energy 815 18.3. Grand Canonical Ensemble 821 18.3.1. The Equilibrium Distribution for the Grand Canonical Ensemble 824 18.3.2. Equilibrium Properties for the Grand Canonical Ensemble 827 18.3.3. Independent Particles in the Dilute Limit Revisited 832 19. Applications of Ensemble Theory to Real Gases 840 19.1. The Behavior of Real Gases 841 19.2. Equation of State for Real Gases 843 19.2.1. Canonical Partition Function for Real Gases 844 19.2.2. The Virial Equation of State 846 19.3. The Second Virial Coefficient 852 19.3.1. Rigid-Sphere and Square-Well Potentials 855 19.3.2. Implementation of Lennard-Jones Potential 858 19.4. The Third Virial Coefficient 861 19.5. Properties for Real Gases 864 Problem Set VIII: Ensemble Theory and the Non-ideal Gas (Chapters 18--19) 872 20. Whence and Whither 882 20.1. Reprising the Journey 883 20.2. Preparing for New Journeys 892 20.3. The Continuing Challenge of Thermodynamics 898 PART SEVEN APPENDICES 901 A. Physical Constants and Conversion Factors 902 B. Series and Integrals 903 C. Periodic Table 905 D. Mathematical Procedures 907 E. Thermochemical Data for Ideal Gases 912 F. Summary of Classical Thermodynamics 926 G. Review of Classical Mechanics 938 H. Review of Operator Theory 943 I. The Spherical Coordinate System 949 J. Electronic Energy Levels 955 K. Energy-Mode Parameters for Molecules 962 L. Normal Mode Analysis 966 M. Tabulation of Debye Function 975 N. Maxwell-Boltzmann Energy Distribution 977 O. Force Constants for Lennard-Jones Potential 980 P. Collision Integrals for Calculating Transport Properties 981 Q. Reduced Second Virial Coefficient from Lennard-Jones Potential 982 R. References and Acknowledgements 983
Library of Congress Subject Headings for this publication:
Statistical thermodynamics.