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STATISTICAL THERMODYNAMICS: FUNDAMENTALS AND APPLICATIONS

Statistical Thermodynamics: Fundamentals and Applications discusses the fundamentals and applications of statistical thermodynamics for beginning graduate students in the engineering sciences. Building on the prototypical Maxwell–Boltzmann method and maintaining a step-by-step development of the subject, this book makes few presumptions concerning students’ previous exposure to statistics, quantum mechanics, or spectroscopy. The book begins with the essentials of statistical thermodynamics, pauses to recover needed knowledge from quantum mechanics and spectroscopy, and then moves on to applications involving ideal gases, the solid state, and radiation. A full intro- duction to kinetic theory is provided, including its applications to transport phenomena and chemical kinetics. A highlight of the textbook is its discussion of modern applications, such as laser-based diagnostics. The book concludes with a thorough presentation of the ensemble method, featuring its use for real gases. Each chapter is carefully written to address student difficulties in learn- ing this challenging subject, which is fundamental to combustion, propulsion, transport phenomena, spectroscopic measurements, and nanotechnology. Stu- dents are made comfortable with their new knowledge by the inclusion of both example and prompted homework problems.

Normand M. Laurendeau is the Ralph and Bettye Bailey Professor of Combus- tion at Purdue University. He teaches at both the undergraduate and graduate levels in the areas of thermodynamics, combustion, and engineering ethics. He conducts research in the combustion sciences, with particular emphasis on laser diagnostics, pollutant formation, and flame structure. Dr. Laurendeau is well known for his pioneering research on the development and application of both nanosecond and picosecond laser-induced fluorescence strategies to quantita- tive species concentration measurements in laminar and turbulent flames. He has authored or coauthored over 150 publications in the archival scientific and engineering literature. Professor Laurendeau is a Fellow of the American Soci- ety of Mechanical Engineers and a member of the Editorial Advisory Board for the peer-reviewed journal Combustion Science and Technology.

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Statistical Thermodynamics

Fundamentals and Applications

NORMAND M. LAURENDEAU Purdue University

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camʙʀɪdɢe uɴɪveʀsɪtʏ pʀess Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press The Edinburgh Building, Cambridge cʙ2 2ʀu, UK

First published in print format

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© Cambridge University Press 2005

2005

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I dedicate this book to my parents,

Maurice and Lydia Roy Laurendeau.

Their gift of bountiful love and support . . .

Continues to fill me with the joy of discovery.

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Contents

Preface page xv

1 Introduction 1

1.1 The Statistical Foundation of Classical Thermodynamics 1 1.2 A Classification Scheme for Statistical Thermodynamics 3 1.3 Why Statistical Thermodynamics? 3

PART ONE. FUNDAMENTALS OF STATISTICAL THERMODYNAMICS

2 Probability and Statistics 7

2.1 Probability: Definitions and Basic Concepts 7 2.2 Permutations and Combinations 10 2.3 Probability Distributions: Discrete and Continuous 11 2.4 The Binomial Distribution 13 2.5 The Poisson Distribution 15 2.6 The Gaussian Distribution 16 2.7 Combinatorial Analysis for Statistical Thermodynamics 18

2.7.1 Distinguishable Objects 19 2.7.2 Indistinguishable Objects 20

Problem Set I. Probability Theory and Statistical Mathematics (Chapter 2) 23

3 The Statistics of Independent Particles 29

3.1 Essential Concepts from Quantum Mechanics 30 3.2 The Ensemble Method of Statistical Thermodynamics 31 3.3 The Two Basic Postulates of Statistical Thermodynamics 32

3.3.1 The M–B Method: System Constraints and Particle Distribution 33

3.3.2 The M–B Method: Microstates and Macrostates 33 3.4 The Most Probable Macrostate 35

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3.5 Bose–Einstein and Fermi–Dirac Statistics 37 3.5.1 Bose–Einstein Statistics 37 3.5.2 Fermi–Dirac Statistics 38 3.5.3 The Most Probable Particle Distribution 39

3.6 Entropy and the Equilibrium Particle Distribution 40 3.6.1 The Boltzmann Relation for Entropy 40 3.6.2 Identification of Lagrange Multipliers 41 3.6.3 The Equilibrium Particle Distribution 42

4 Thermodynamic Properties in the Dilute Limit 45

4.1 The Dilute Limit 45 4.2 Corrected Maxwell–Boltzmann Statistics 46 4.3 The Molecular Partition Function 47

4.3.1 The Influence of Temperature 49 4.3.2 Criterion for Dilute Limit 50

4.4 Internal Energy and Entropy in the Dilute Limit 51 4.5 Additional Thermodynamic Properties in the Dilute

Limit 53 4.6 The Zero of Energy and Thermodynamic Properties 55 4.7 Intensive Thermodynamic Properties for the Ideal Gas 56

Problem Set II. Statistical Modeling for Thermodynamics (Chapters 3–4) 59

PART TWO. QUANTUM MECHANICS AND SPECTROSCOPY

5 Basics of Quantum Mechanics 69

5.1 Historical Survey of Quantum Mechanics 69 5.2 The Bohr Model for the Spectrum of Atomic Hydrogen 72 5.3 The de Broglie Hypothesis 76 5.4 A Heuristic Introduction to the Schrödinger Equation 78 5.5 The Postulates of Quantum Mechanics 80 5.6 The Steady-State Schrödinger Equation 83

5.6.1 Single-Particle Analysis 84 5.6.2 Multiparticle Analysis 85

5.7 The Particle in a Box 86 5.8 The Uncertainty Principle 90 5.9 Indistinguishability and Symmetry 92 5.10 The Pauli Exclusion Principle 94 5.11 The Correspondence Principle 95

6 Quantum Analysis of Internal Energy Modes 97

6.1 Schrödinger Wave Equation for Two-Particle System 97 6.1.1 Conversion to Center-of-Mass Coordinates 98 6.1.2 Separation of External from Internal Modes 99

6.2 The Internal Motion for a Two-Particle System 99 6.3 The Rotational Energy Mode for a Diatomic Molecule 100 6.4 The Vibrational Energy Mode for a Diatomic Molecule 104

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Contents � ix

6.5 The Electronic Energy Mode for Atomic Hydrogen 108 6.6 The Electronic Energy Mode for Multielectron Species 115

6.6.1 Electron Configuration for Multielectron Atoms 116 6.6.2 Spectroscopic Term Symbols for Multielectron

Atoms 118 6.6.3 Electronic Energy Levels and Degeneracies for

Atoms 119 6.6.4 Electronic Energy Levels and Degeneracies for

Diatomic Molecules 121 6.7 Combined Energy Modes for Atoms and Diatomic Molecules 123 6.8 Selection Rules for Atoms and Molecules 124

7 The Spectroscopy of Diatomic Molecules 129

7.1 Rotational Spectroscopy Using the Rigid-Rotor Model 130 7.2 Vibrational Spectroscopy Using the Harmonic-Oscillator

Model 131 7.3 Rovibrational Spectroscopy: The Simplex Model 132 7.4 The Complex Model for Combined Rotation and Vibration 136 7.5 Rovibrational Spectroscopy: The Complex Model 138 7.6 Electronic Spectroscopy 141 7.7 Energy-Mode Parameters for Diatomic Molecules 144

Problem Set III. Quantum Mechanics and Spectroscopy (Chapters 5–7) 147

PART THREE. STATISTICAL THERMODYNAMICS IN THE DILUTE LIMIT

8 Interlude: From Particle to Assembly 157

8.1 Energy and Degeneracy 157 8.2 Separation of Energy Modes 159 8.3 The Molecular Internal Energy 160 8.4 The Partition Function and Thermodynamic Properties 161 8.5 Energy-Mode Contributions in Classical Mechanics 163

8.5.1 The Phase Integral 164 8.5.2 The Equipartition Principle 166 8.5.3 Mode Contributions 167

9 Thermodynamic Properties of the Ideal Gas 169

9.1 The Monatomic Gas 169 9.1.1 Translational Mode 169 9.1.2 Electronic Mode 173

9.2 The Diatomic Gas 175 9.2.1 Translational and Electronic Modes 176 9.2.2 The Zero of Energy 176 9.2.3 Rotational Mode 178 9.2.4 Quantum Origin of Rotational Symmetry Factor 182 9.2.5 Vibrational Mode 184

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9.3 Rigorous and Semirigorous Models for the Diatomic Gas 187 9.4 The Polyatomic Gas 192

9.4.1 Rotational Contribution 194 9.4.2 Vibrational Contribution 196 9.4.3 Property Calculations for Polyatomic Molecules 198

Problem Set IV. Thermodynamic Properties of the Ideal Gas (Chapters 8–9) 201

10 Statistical Thermodynamics for Ideal Gas Mixtures 205

10.1 Equilibrium Particle Distribution for the Ideal Gas Mixture 205 10.2 Thermodynamic Properties of the Ideal Gas Mixture 208 10.3 The Reacting Ideal Gas Mixture 211

10.3.1 Equilibrium Particle Distribution for Reactive Ideal Gas Mixture 211

10.3.2 Equilibrium Constant: Introduction and Development 213 10.4 Equilibrium Constant: General Expression and Specific

Examples 214 10.4.1 Dissociation of a Homonuclear Diatomic 217 10.4.2 The Homonuclear–Heteronuclear Conversion Reaction 219 10.4.3 The Ionization Reaction 220

11 Concentration and Temperature Measurements 223

11.1 Mode Temperatures 224 11.2 Radiative Transitions 225

11.2.1 Spectral Transfer of Radiation 227 11.2.2 The Einstein Coefficients 228 11.2.3 Line Broadening 229

11.3 Absorption Spectroscopy 230 11.4 Emission Spectroscopy 234

11.4.1 Emissive Diagnostics 234 11.4.2 The Problem of Self-Absorption 235

11.5 Fluorescence Spectroscopy 237 11.6 Sodium D-Line Reversal 240 11.7 Advanced Diagnostic Techniques 241

Problem Set V. Chemical Equilibrium and Diagnostics (Chapters 10–11) 243

PART FOUR. STATISTICAL THERMODYNAMICS BEYOND THE DILUTE LIMIT

12 Thermodynamics and Information 251

12.1 Reversible Work and Heat 251 12.2 The Second Law of Thermodynamics 252 12.3 The Boltzmann Definition of Entropy 253 12.4 Information Theory 254 12.5 Spray Size Distribution from Information Theory 256

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Contents � xi

13 Elements of the Solid State 259

13.1 Statistical Thermodynamics of the Crystalline Solid 259 13.2 Einstein Theory for the Crystalline Solid 262 13.3 Debye Theory for the Crystalline Solid 263 13.4 Critical Evaluation of the Debye Formulation 266 13.5 The Band Theory of Metallic Solids 268 13.6 Thermodynamic Properties of the Electron Gas 270 13.7 The Metallic Crystal near Absolute Zero 273

14 Equilibrium Radiation 275

14.1 Bose–Einstein Statistics for the Photon Gas 275 14.2 Photon Quantum States 276 14.3 The Planck Distribution Law 276 14.4 Thermodynamics of Blackbody Radiation 278 14.5 The Influence of Wavelength for the Planck Distribution 280

Problem Set VI. The Solid State and Radiation (Chapters 13–14) 283

PART FIVE. NONEQUILIBRIUM STATISTICAL THERMODYNAMICS

15 Elementary Kinetic Theory 289

15.1 The Maxwell–Boltzmann Velocity Distribution 289 15.2 The Maxwell–Boltzmann Speed Distribution 291 15.3 The Maxwell–Boltzmann Energy Distribution 294 15.4 Molecular Effusion 295 15.5 The Ideal Gas Pressure 298

16 Kinetics of Molecular Transport 301

16.1 Binary Collision Theory 301 16.2 Fundamentals of Molecular Transport 305

16.2.1 The Mean Free Path 305 16.2.2 The Molecular Flux 307 16.2.3 Transport Properties 309

16.3 Rigorous Transport Theory 311 16.3.1 Dimensionless Transport Parameters 312 16.3.2 Collision Integrals 313 16.3.3 The Lennard–Jones Potential 314 16.3.4 Rigorous Expressions for Transport Properties 316

17 Chemical Kinetics 319

17.1 The Bimolecular Reaction 319 17.2 The Rate of Bimolecular Reactions 320 17.3 Chemical Kinetics from Collision Theory 321 17.4 The Significance of Internal Energy Modes 324 17.5 Chemical Kinetics from Transition State Theory 325

Problem Set VII. Kinetic Theory and Molecular Transport (Chapters 15–17) 331

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PART SIX. THE ENSEMBLE METHOD OF STATISTICAL THERMODYNAMICS

18 The Canonical and Grand Canonical Ensembles 339

18.1 The Ensemble Method 339 18.2 The Canonical Ensemble 340

18.2.1 The Equilibrium Distribution for the Canonical Ensemble 341

18.2.2 Equilibrium Properties for the Canonical Ensemble 342 18.2.3 Independent Particles in the Dilute Limit 345 18.2.4 Fluctuations in Internal Energy 347

18.3 Grand Canonical Ensemble 349 18.3.1 The Equilibrium Distribution for the Grand Canonical

Ensemble 351 18.3.2 Equilibrium Properties for the Grand Canonical

Ensemble 352 18.3.3 Independent Particles in the Dilute Limit Revisited 355

19 Applications of Ensemble Theory to Real Gases 359

19.1 The Behavior of Real Gases 359 19.2 Equation of State for Real Gases 360

19.2.1 Canonical Partition Function for Real Gases 361 19.2.2 The Virial Equation of State 362

19.3 The Second Virial Coefficient 364 19.3.1 Rigid-Sphere and Square-Well Potentials 366 19.3.2 Implementation of Lennard–Jones Potential 367

19.4 The Third Virial Coefficient 369 19.5 Properties for Real Gases 371

Problem Set VIII. Ensemble Theory and the Nonideal Gas (Chapters 18–19) 375

20 Whence and Whither 379

20.1 Reprising the Journey 379 20.2 Preparing for New Journeys 383 20.3 The Continuing Challenge of Thermodynamics 385

PART SEVEN. APPENDICES

A. Physical Constants and Conversion Factors 389

B. Series and Integrals 390

C. Periodic Table 391

D. Mathematical Procedures 393

E. Thermochemical Data for Ideal Gases 396

F. Summary of Classical Thermodynamics 409

G. Review of Classical Mechanics 415

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Contents � xiii

H. Review of Operator Theory 418

I. The Spherical Coordinate System 421

J. Electronic Energy Levels 424

K. Energy-Mode Parameters for Molecules 427

L. Normal Mode Analysis 430

M. Tabulation of Debye Function 433

N. Maxwell–Boltzmann Energy Distribution 434

O. Force Constants for the Lennard–Jones Potential 436

P. Collision Integrals for Calculating Transport Properties from the Lennard–Jones Potential 437

Q. Reduced Second Virial Coefficient from the Lennard–Jones Potential 438

R. References and Acknowledgments 439

Index 445

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Preface

My intention in this textbook is to provide a self-contained exposition of the fundamentals and applications of statistical thermodynamics for beginning graduate students in the engi- neering sciences. Especially within engineering, most students enter a course in statistical thermodynamics with limited exposure to statistics, quantum mechanics, and