Tài liệu Tài liệu vật lý statistical thermodynamics fundermental and application

This page intentionally left blank 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 introduction 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 learning this challenging subject, which is fundamental to combustion, propulsion, transport phenomena, spectroscopic measurements, and nanotechnology. Students 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 Combustion 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 quantitative 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 Society of Mechanical Engineers and a member of the Editorial Advisory Board for the peer-reviewed journal Combustion Science and Technology. Statistical Thermodynamics Fundamentals and Applications NORMAND M. LAURENDEAU Purdue University camʙʀɪdɢe uɴɪveʀsɪtʏ pʀess Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cʙ2 2ʀu, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521846356 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 ɪsʙɴ-13 ɪsʙɴ-10 978-0-511-14062-4 eBook (NetLibrary) 0-511-14062-2 eBook (NetLibrary) ɪsʙɴ-13 ɪsʙɴ-10 978-0-521-84635-6 hardback 0-521-84635-8 hardback Cambridge University Press has no responsibility for the persistence or accuracy of uʀʟs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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. Contents Preface page xv 1 Introduction 1.1 The Statistical Foundation of Classical Thermodynamics 1.2 A Classification Scheme for Statistical Thermodynamics 1.3 Why Statistical Thermodynamics? 1 1 3 3 PART ONE. FUNDAMENTALS OF STATISTICAL THERMODYNAMICS 2 Probability and Statistics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Probability: Definitions and Basic Concepts Permutations and Combinations Probability Distributions: Discrete and Continuous The Binomial Distribution The Poisson Distribution The Gaussian Distribution Combinatorial Analysis for Statistical Thermodynamics 2.7.1 Distinguishable Objects 2.7.2 Indistinguishable Objects Problem Set I. Probability Theory and Statistical Mathematics (Chapter 2) 3 The Statistics of Independent Particles 3.1 Essential Concepts from Quantum Mechanics 3.2 The Ensemble Method of Statistical Thermodynamics 3.3 The Two Basic Postulates of Statistical Thermodynamics 3.3.1 The M–B Method: System Constraints and Particle Distribution 3.3.2 The M–B Method: Microstates and Macrostates 3.4 The Most Probable Macrostate 7 7 10 11 13 15 16 18 19 20 23 29 30 31 32 33 33 35 vii viii r Contents 3.5 Bose–Einstein and Fermi–Dirac Statistics 3.5.1 Bose–Einstein Statistics 3.5.2 Fermi–Dirac Statistics 3.5.3 The Most Probable Particle Distribution 3.6 Entropy and the Equilibrium Particle Distribution 3.6.1 The Boltzmann Relation for Entropy 3.6.2 Identification of Lagrange Multipliers 3.6.3 The Equilibrium Particle Distribution 4 Thermodynamic Properties in the Dilute Limit 4.1 The Dilute Limit 4.2 Corrected Maxwell–Boltzmann Statistics 4.3 The Molecular Partition Function 4.3.1 The Influence of Temperature 4.3.2 Criterion for Dilute Limit 4.4 Internal Energy and Entropy in the Dilute Limit 4.5 Additional Thermodynamic Properties in the Dilute Limit 4.6 The Zero of Energy and Thermodynamic Properties 4.7 Intensive Thermodynamic Properties for the Ideal Gas Problem Set II. Statistical Modeling for Thermodynamics (Chapters 3–4) 37 37 38 39 40 40 41 42 45 45 46 47 49 50 51 53 55 56 59 PART TWO. QUANTUM MECHANICS AND SPECTROSCOPY 5 Basics of Quantum Mechanics 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 Historical Survey of Quantum Mechanics The Bohr Model for the Spectrum of Atomic Hydrogen The de Broglie Hypothesis A Heuristic Introduction to the Schrödinger Equation The Postulates of Quantum Mechanics The Steady-State Schrödinger Equation 5.6.1 Single-Particle Analysis 5.6.2 Multiparticle Analysis The Particle in a Box The Uncertainty Principle Indistinguishability and Symmetry The Pauli Exclusion Principle The Correspondence Principle 6 Quantum Analysis of Internal Energy Modes 6.1 Schrödinger Wave Equation for Two-Particle System 6.1.1 Conversion to Center-of-Mass Coordinates 6.1.2 Separation of External from Internal Modes 6.2 The Internal Motion for a Two-Particle System 6.3 The Rotational Energy Mode for a Diatomic Molecule 6.4 The Vibrational Energy Mode for a Diatomic Molecule 69 69 72 76 78 80 83 84 85 86 90 92 94 95 97 97 98 99 99 100 104 Contents r ix 6.5 The Electronic Energy Mode for Atomic Hydrogen 6.6 The Electronic Energy Mode for Multielectron Species 6.6.1 Electron Configuration for Multielectron Atoms 6.6.2 Spectroscopic Term Symbols for Multielectron Atoms 6.6.3 Electronic Energy Levels and Degeneracies for Atoms 6.6.4 Electronic Energy Levels and Degeneracies for Diatomic Molecules 6.7 Combined Energy Modes for Atoms and Diatomic Molecules 6.8 Selection Rules for Atoms and Molecules 7 The Spectroscopy of Diatomic Molecules 7.1 Rotational Spectroscopy Using the Rigid-Rotor Model 7.2 Vibrational Spectroscopy Using the Harmonic-Oscillator Model 7.3 Rovibrational Spectroscopy: The Simplex Model 7.4 The Complex Model for Combined Rotation and Vibration 7.5 Rovibrational Spectroscopy: The Complex Model 7.6 Electronic Spectroscopy 7.7 Energy-Mode Parameters for Diatomic Molecules Problem Set III. Quantum Mechanics and Spectroscopy (Chapters 5–7) 108 115 116 118 119 121 123 124 129 130 131 132 136 138 141 144 147 PART THREE. STATISTICAL THERMODYNAMICS IN THE DILUTE LIMIT 8 Interlude: From Particle to Assembly 8.1 8.2 8.3 8.4 8.5 Energy and Degeneracy Separation of Energy Modes The Molecular Internal Energy The Partition Function and Thermodynamic Properties Energy-Mode Contributions in Classical Mechanics 8.5.1 The Phase Integral 8.5.2 The Equipartition Principle 8.5.3 Mode Contributions 9 Thermodynamic Properties of the Ideal Gas 9.1 The Monatomic Gas 9.1.1 Translational Mode 9.1.2 Electronic Mode 9.2 The Diatomic Gas 9.2.1 Translational and Electronic Modes 9.2.2 The Zero of Energy 9.2.3 Rotational Mode 9.2.4 Quantum Origin of Rotational Symmetry Factor 9.2.5 Vibrational Mode 157 157 159 160 161 163 164 166 167 169 169 169 173 175 176 176 178 182 184 x r Contents 9.3 Rigorous and Semirigorous Models for the Diatomic Gas 9.4 The Polyatomic Gas 9.4.1 Rotational Contribution 9.4.2 Vibrational Contribution 9.4.3 Property Calculations for Polyatomic Molecules 187 192 194 196 198 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 10.2 Thermodynamic Properties of the Ideal Gas Mixture 10.3 The Reacting Ideal Gas Mixture 10.3.1 Equilibrium Particle Distribution for Reactive Ideal Gas Mixture 10.3.2 Equilibrium Constant: Introduction and Development 10.4 Equilibrium Constant: General Expression and Specific Examples 10.4.1 Dissociation of a Homonuclear Diatomic 10.4.2 The Homonuclear–Heteronuclear Conversion Reaction 10.4.3 The Ionization Reaction 205 208 211 211 213 214 217 219 220 11 Concentration and Temperature Measurements 223 11.1 Mode Temperatures 11.2 Radiative Transitions 11.2.1 Spectral Transfer of Radiation 11.2.2 The Einstein Coefficients 11.2.3 Line Broadening 11.3 Absorption Spectroscopy 11.4 Emission Spectroscopy 11.4.1 Emissive Diagnostics 11.4.2 The Problem of Self-Absorption 11.5 Fluorescence Spectroscopy 11.6 Sodium D-Line Reversal 11.7 Advanced Diagnostic Techniques 224 225 227 228 229 230 234 234 235 237 240 241 Problem Set V. Chemical Equilibrium and Diagnostics (Chapters 10–11) 243 PART FOUR. STATISTICAL THERMODYNAMICS BEYOND THE DILUTE LIMIT 12 Thermodynamics and Information 12.1 12.2 12.3 12.4 12.5 Reversible Work and Heat The Second Law of Thermodynamics The Boltzmann Definition of Entropy Information Theory Spray Size Distribution from Information Theory 251 251 252 253 254 256 Contents r xi 13 Elements of the Solid State 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Statistical Thermodynamics of the Crystalline Solid Einstein Theory for the Crystalline Solid Debye Theory for the Crystalline Solid Critical Evaluation of the Debye Formulation The Band Theory of Metallic Solids Thermodynamic Properties of the Electron Gas The Metallic Crystal near Absolute Zero 14 Equilibrium Radiation 14.1 14.2 14.3 14.4 14.5 Bose–Einstein Statistics for the Photon Gas Photon Quantum States The Planck Distribution Law Thermodynamics of Blackbody Radiation The Influence of Wavelength for the Planck Distribution Problem Set VI. The Solid State and Radiation (Chapters 13–14) 259 259 262 263 266 268 270 273 275 275 276 276 278 280 283 PART FIVE. NONEQUILIBRIUM STATISTICAL THERMODYNAMICS 15 Elementary Kinetic Theory 15.1 15.2 15.3 15.4 15.5 The Maxwell–Boltzmann Velocity Distribution The Maxwell–Boltzmann Speed Distribution The Maxwell–Boltzmann Energy Distribution Molecular Effusion The Ideal Gas Pressure 16 Kinetics of Molecular Transport 16.1 Binary Collision Theory 16.2 Fundamentals of Molecular Transport 16.2.1 The Mean Free Path 16.2.2 The Molecular Flux 16.2.3 Transport Properties 16.3 Rigorous Transport Theory 16.3.1 Dimensionless Transport Parameters 16.3.2 Collision Integrals 16.3.3 The Lennard–Jones Potential 16.3.4 Rigorous Expressions for Transport Properties 17 Chemical Kinetics 17.1 17.2 17.3 17.4 17.5 The Bimolecular Reaction The Rate of Bimolecular Reactions Chemical Kinetics from Collision Theory The Significance of Internal Energy Modes Chemical Kinetics from Transition State Theory Problem Set VII. Kinetic Theory and Molecular Transport (Chapters 15–17) 289 289 291 294 295 298 301 301 305 305 307 309 311 312 313 314 316 319 319 320 321 324 325 331 xii r Contents PART SIX. THE ENSEMBLE METHOD OF STATISTICAL THERMODYNAMICS 18 The Canonical and Grand Canonical Ensembles 18.1 The Ensemble Method 18.2 The Canonical Ensemble 18.2.1 The Equilibrium Distribution for the Canonical Ensemble 18.2.2 Equilibrium Properties for the Canonical Ensemble 18.2.3 Independent Particles in the Dilute Limit 18.2.4 Fluctuations in Internal Energy 18.3 Grand Canonical Ensemble 18.3.1 The Equilibrium Distribution for the Grand Canonical Ensemble 18.3.2 Equilibrium Properties for the Grand Canonical Ensemble 18.3.3 Independent Particles in the Dilute Limit Revisited 19 Applications of Ensemble Theory to Real Gases 19.1 The Behavior of Real Gases 19.2 Equation of State for Real Gases 19.2.1 Canonical Partition Function for Real Gases 19.2.2 The Virial Equation of State 19.3 The Second Virial Coefficient 19.3.1 Rigid-Sphere and Square-Well Potentials 19.3.2 Implementation of Lennard–Jones Potential 19.4 The Third Virial Coefficient 19.5 Properties for Real Gases Problem Set VIII. Ensemble Theory and the Nonideal Gas (Chapters 18–19) 20 Whence and Whither 20.1 Reprising the Journey 20.2 Preparing for New Journeys 20.3 The Continuing Challenge of Thermodynamics 339 339 340 341 342 345 347 349 351 352 355 359 359 360 361 362 364 366 367 369 371 375 379 379 383 385 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 Contents r 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 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 engineering 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 statistical 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 inclusion 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 fluorescence techniques for the measurement of species concentrations and temperature. A full xv xvi r Preface introduction to kinetic theory is also provided, including its applications to transport phenomena 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 2 3 4 5 6 7 8 9 10 1 1 4 2 3 3 2 2 4 3 11 12 13 14 15 16 17 18 19 20 2 1 2 1 2 3 1 2 2 1 In conclusion, I would be remiss if I did not thank my spouse, Marlene, for her forbearance 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. Professor 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. 1 Introduction To this point in your career, you have probably dealt almost exclusively with the behavior 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 thermodynamic analyses. Nevertheless, you should understand that classical thermodynamics is inherently limited in its ability to explain the behavior of even the simplest thermodynamic 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 elementary 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 foundation for the subject of statistical thermodynamics. Clearly, statistical methods are mandatory 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 1 2 r Introduction (b) Emissive Signal Emissive Signal (a) Wavelength Wavelength Figure 1.1 Schematic of simplified (a) continuous spectrum and (b) discrete spectrum. from the distinct energy values permitted for either an atom or molecule. The best evidence for this quantized behavior comes from the field of spectroscopy. Consider, for example, the simplified emission spectra shown in Fig. 1.1. Spectrum (a) displays a continuous variation of emissive signal versus wavelength, while spectrum (b) displays individual “lines” at specific wavelengths. Spectrum (a) is typical of the radiation given off by a hot solid while spectrum (b) is typical of that from a hot gas. As we will see in Chapter 7, the individual lines of spectrum (b) reflect discrete changes in the energy stored by an atom or molecule. Moreover, the height of each line is related to the number of particles causing the emissive signal. From the point of view of statistical thermodynamics, the number of relevant particles (atoms or molecules) can only be determined by using probability theory, as introduced in Chapter 2. The total energy of a single molecule can be taken, for simplicity, as the sum of individual contributions from its translational, rotational, vibrational, and electronic energy modes. The external or translational mode specifies the kinetic energy of the molecule’s center of mass. In comparison, the internal energy modes reflect any molecular motion with respect to the center of mass. Hence, the rotational mode describes energy stored by molecular rotation, the vibrational mode energy stored by vibrating bonds, and the electronic mode energy stored by the motion of electrons within the molecule. By combining predictions from quantum mechanics with experimental data obtained via spectroscopy, it turns out that we can evaluate the contributions from each mode and thus determine the microscopic properties of individual molecules. Such properties include bond distances, rotational or vibrational frequencies, and translational or electronic energies. Employing statistical methods, we can then average over all particles to calculate the macroscopic properties associated with classical thermodynamics. Typical phenomenological properties include the temperature, the internal energy, and the entropy. Figure 1.2 summarizes the above discussion and also provides a convenient road map for our upcoming study of statistical thermodynamics. Notice that the primary subject of this book plays a central role in linking the microscopic and macroscopic worlds. Moreover, while statistical thermodynamics undoubtedly constitutes an impressive application of probability theory, we observe that the entire subject can be founded on only two major postulates. As for all scientific adventures, our acceptance of these basic postulates as