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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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Published in the United States of America by Cambridge University Press, New York

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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 spectroscopy. Hence, I have found it necessary over the years to “start from the beginning,” not leaving out intermediary steps and presuming little knowledge in the discrete, as compared to the continuum, domain of physics. Once these things are done carefully, I find that good graduate students can follow the ideas, and that they leave the course excited and satisfied with their newfound understanding of both statistical and classical thermodynamics.

Nevertheless, a first course in statistical thermodynamics remains challenging and sometimes threatening to many graduate students. Typically, all their previous experience is with the equations of continuum mechanics, whether applied to thermodynamics, fluid mechanics, or heat transfer. For most students, therefore, the mathematics of probability theory, the novelty of quantum mechanics, the confrontation with entropy, and indeed the whole new way of thinking that surrounds statistical thermodynamics are all built-in hills that must be climbed to develop competence and confidence in the subject. For this reason, although I introduce the ensemble method at the beginning of the book, I have found it preferable to build on the related Maxwell–Boltzmann method so that novices are not confronted immediately with the conceptual difficulties of ensemble theory. In this way, students tend to become more comfortable with their new knowledge earlier in the course. Moreover, they are prepared relatively quickly for applications, which is very important to maintaining an active interest in the subject for most engineering students. Using this pedagogy, I find that the ensemble approach then becomes very easy to teach later in the semester, thus effectively preparing the students for more advanced courses that apply statistical mechanics to liquids, polymers, and semiconductors.

To hold the students’ attention, I begin the book with the fundamentals of statisti- cal thermodynamics, pause to recover needed knowledge from quantum mechanics and spectroscopy, and then move on to applications involving ideal gases, the solid state, and radiation. An important distinction between this book and previous textbooks is the inclu- sion of an entire chapter devoted to laser-based diagnostics, as applied to the thermal sciences. Here, I cover the essentials of absorption, emission, and laser-induced fluores- cence techniques for the measurement of species concentrations and temperature. A full

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introduction to kinetic theory is also provided, including its applications to transport phe- nomena and chemical kinetics.

During the past two decades, I have developed many problems for this textbook that are quite different from the typical assignments found in other textbooks, which are often either too facile or too ambiguous. Typically, the students at Purdue complete eight problem sets during a semester, with 4–6 problems per set. Hence, there are enough problems included in the book for approximately three such course presentations. My approach has been to construct problems using integrally related subcomponents so that students can learn the subject in a more prompted fashion. Even so, I find that many students need helpful hints at times, and the instructor should indeed be prepared to do so. In fact, I trust that the instructor will find, as I have, that these interactions with students, showing you what they have done and where they are stuck, invariably end up being one of the most rewarding parts of conducting the course. The reason is obvious. One-on-one discussions give the instructor the opportunity to get to know a person and to impart to each student his or her enthusiasm for the drama and subtleties of statistical thermodynamics.

As a guide to the instructor, the following table indicates the number of 50-minute lectures devoted to each chapter in a 42-lecture semester at Purdue University.

Chapter Number of Lectures Chapter

Number of Lectures

1 1 11 2 2 1 12 1 3 4 13 2 4 2 14 1 5 3 15 2 6 3 16 3 7 2 17 1 8 2 18 2 9 4 19 2

10 3 20 1

In conclusion, I would be remiss if I did not thank my spouse, Marlene, for her for- bearance and support during the writing of this book. Only she and I know firsthand the trials and tribulations confronting a partnership wedded to the long-distance writer. Pro- fessor Lawrence Caretto deserves my gratitude for graciously permitting the importation of embellished portions of his course notes to the text. I thank Professor Michael Renfro for his reading of the drafts and for his helpful suggestions. Many useful comments were also submitted by graduate students who put up with preliminary versions of the book at Purdue University and at the University of Connecticut. I appreciate Professor Robert Lucht, who aided me in maintaining several active research projects during the writing of the book. Finally, I thank the School of Mechanical Engineering at Purdue for providing me with the opportunity and the resources over these many years to blend my enthusiasm for statistical thermodynamics with that for my various research programs in combustion and optical diagnostics.

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1 Introduction

To this point in your career, you have probably dealt almost exclusively with the behav- ior of macroscopic systems, either from a scientific or engineering viewpoint. Examples of such systems might include a piston–cylinder assembly, a heat exchanger, or a battery. Typically, the analysis of macroscopic systems uses conservation or field equations related to classical mechanics, thermodynamics, or electromagnetics. In this book, our focus is on thermal devices, as usually described by thermodynamics, fluid mechanics, and heat transfer. For such devices, first-order calculations often employ a series of simple ther- modynamic analyses. Nevertheless, you should understand that classical thermodynamics is inherently limited in its ability to explain the behavior of even the simplest thermody- namic system. The reason for this deficiency rests with its inadequate treatment of the atomic behavior underlying the gaseous, liquid, or solid states of matter. Without proper consideration of constituent microscopic systems, such as a single atom or molecule, it is impossible for the practitioner to understand fully the evaluation of thermodynamic properties, the meaning of thermodynamic equilibrium, or the influence of temperature on transport properties such as the thermal conductivity or viscosity. Developing this ele- mentary viewpoint is the purpose of a course in statistical thermodynamics. As you will see, such fundamental understanding is also the basis for creative applications of classical thermodynamics to macroscopic devices.

1.1 The Statistical Foundation of Classical Thermodynamics

Since a typical thermodynamic system is composed of an assembly of atoms or molecules, we can surely presume that its macroscopic behavior can be expressed in terms of the microscopic properties of its constituent particles. This basic tenet provides the founda- tion for the subject of statistical thermodynamics. Clearly, statistical methods are manda- tory as even one cm3 of a perfect gas contains some 1019 atoms or molecules. In other words, the huge number of particles forces us to eschew any approach based on having an exact knowledge of the position and momentum of each particle within a macroscopic thermodynamic system.

The properties of individual particles can be obtained only through the methods of quantum mechanics. One of the most important results of quantum mechanics is that the energy of a single atom or molecule is not continuous, but discrete. Discreteness arises

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2 � Introduction

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