Preface ......................................................................................................... vii Definition of Symbols .................................................................................. ix Table of Constants and Conversion Factors................................................ xi

Introduction and Overview A. Classical to Quantum Physics ............................................................... xiii B. Systems with Many Particles ..................................................................xiv C. Statistics and Entropy..............................................................................xv D. Road Map for this Course .....................................................................xvii Chapter 1: Origins of Mechanical Energy A. Kinetic Energy and Work ......................................................................... 1 B. Extension to Many-Particle Systems........................................................ 3 C. Internal Energy ....................................................................................... 5 D. Potential Energy ...................................................................................... 7 E. Vibrational Energy—Kinetic plus Potential ............................................ 8 Chapter 2: Irreversibility and the Second Law of Thermodynamics A. Thermal Energy ..................................................................................... 13 B. Irreversibility of Many-Body Systems .................................................... 14 C. Entropy and the Approach to Equilibrium ............................................ 14 D. Entropy Maximization and the Calculus of Several Variables .............. 17 Chapter 3: Kinetic Theory of the Ideal Gas A. Common Particles .................................................................................. 21 B. Pressure and Kinetic Energy .................................................................. 22 C. Equipartition Theorem .......................................................................... 23 D. Equipartition Applied to a Solid ........................................................... 25 E. Ideal Gas Law ......................................................................................... 26 F. Distribution of Energies in a Gas ........................................................... 27 Chapter 4: Ideal-Gas Heat Engines A. The First Law of Thermodynamics ...................................................... 31 B. Quasi-static Processes and State Functions ........................................... 32 iii

Physics 213 Elements of Thermal Physics

C. Isothermal and Adiabatic Processes—Reversibility ............................... 32 D. Entropy of the Ideal Gas—a First Look ................................................ 36 E. Converting Heat into Work ................................................................... 37 F. Refrigerators and Heat Pumps ................................................................ 40

Chapter 5: Statistical Processes I: Two-State Systems A. Macrostates and Microstates .................................................................. 45 B. Multiple Spins ......................................................................................... 46 C. The Random Walk Problem—Diffusion of Particles ........................... 50 D. Heat Conduction.................................................................................... 54 Chapter 6: Statistical Processes II: Entropy and the Second Law A. Meaning of Equilibrium ......................................................................... 59 B. Objects in Multiple Bins ........................................................................ 60 C. Application to a Gas of Particles ............................................................ 62 D. Volume Exchange and Entropy ............................................................. 64 E. Indistinguishable Particles ..................................................................... 68 F. Maximum Entropy in Equilibrium ......................................................... 68 Chapter 7: Energy Exchange A. Model System for Exchanging Energy .................................................. 73 B. Thermal Equilibrium and Absolute Temperature.................................. 78 C. Equipartition Revisited .......................................................................... 79 D. Why Energy Flows from Hot to Cold .................................................. 81 E. Entropy of the Ideal Gas—Temperature Dependence .......................... 82 Chapter 8: Boltzmann Distribution A. Concept of a Thermal Reservoir ............................................................ 87 B. The Boltzmann Factor............................................................................ 88 C. Paramagnetism ....................................................................................... 91 D. Elasticity in Polymers............................................................................. 94 E. Harmonic Oscillator .............................................................................. 95 Chapter 9: Distributions of Molecules and Photons A. Applying the Boltzmann Factor ............................................................. 99 B. Particle States in a Classical Gas .......................................................... 100 C. Maxwell-Boltzmann Distribution ........................................................ 102 D. Photons ................................................................................................. 103 E. Thermal Radiation................................................................................ 105 F. Global Warming .................................................................................... 107 Chapter 10: Work and Free Energy A. Heat Flow and Entropy ........................................................................ 111 B. Ideal Heat Engines ............................................................................... 112 C. Free Energy and Available Work ......................................................... 114 D. Free Energy Minimum in Equilibrium ............................................... 115 iv

Elements of Thermal Physics Physics 213

E. Principle of Minimum Free Energy .................................................. 116 F. Equipartition of Energy ........................................................................ 117 G. Paramagnetism—the Free Energy Approach ...................................... 119

Chapter 11: Equilibrium between Particles I A. Free Energy and Chemical Potential ................................................... 123 B. Absolute Entropy of an Ideal Gas......................................................... 125 C. Chemical Potential of an Ideal Gas ..................................................... 128 D. Law of Atmospheres ............................................................................. 129 E. Physical Interpretations of Chemical Potential ................................... 130 Chapter 12: Equilibrium between Particles II A. Ionization of Atoms .............................................................................. 135 B. Chemical Equilibrium in Gases............................................................ 137 C. Carrier Densities in a Semiconductor ................................................. 139 D. Law of Mass Action: Doped Semiconductors .................................... 142 Chapter 13: Adsorption of Atoms and Phase Transitions A. Adsorption of Atoms on a Solid Surface .............................................. 145 B. Oxygen in Myoglobin ........................................................................... 147 C. Why Gases Condense .......................................................................... 148 D. Vapor Pressure of a Solid ..................................................................... 148 E. Solid/Liquid/Gas Phase Transitions .................................................... 151 F. Model of Liquid–Gas Condensation .................................................... 154 Chapter 14: Processes at Constant Pressure A. Gibbs Free Energy................................................................................ 157 B. Vapor Pressures of Liquids—General Aspects ..................................... 160 C. Chemical Reactions at Constant Pressure ........................................... 161 Appendices Appendix 1: Vibrations in Molecules and Solids—Normal Modes ......... 165 Appendix 2: The Stirling Cycle ................................................................ 169 Appendix 3: Statistical Tools ..................................................................... 173 Appendix 4: Table of Integrals .................................................................. 179 Appendix 5: Exclusion Principle and Identical Particles .......................... 181 Appendix 6: Sum over States and Average Energy ................................... 185 Appendix 7: Debye Specific Heat of a Solid ............................................. 189 Appendix 8: Absolute Entropy of an Ideal Gas ........................................ 191 Appendix 9: Entropy and Diatomic Molecules ........................................ 195 Appendix 10: Vapor Pressure of a Vibrating Solid ................................... 199 Solutions to Exercises ............................................................................... 201 Index .......................................................................................................... 215 *May not be covered in Physics 213 v

Physics 213 Elements of Thermal Physics

The central ideas in this course have a wide range of applications. For example:

← fabrication of materials chemical reactions →

← biological processes phase transitions →

← magnetism electrons and holes in semiconductors →

← converting energy into work thermal radiation (global warming) →

← thin films and surface chemistry and much more...

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Preface

The unifying concepts of entropy and free energy are essential to the understanding of physical, chemical, and biological systems. Recognizing that these concepts permeate the undergraduate science and engineering curricula, the Physics Department has created this sophomore-level course dealing with thermodynamics and statistical mechanics. Starting with a few basic principles, we introduce practical tools for solving a variety of problems in the areas of materials science, electrical engineering, chemistry, and biology. These introductory notes on Thermal Physics are designed to be used in concert with the Physics 213 Lectures, Discussion Exercises, Homework Problems, and Laboratory Exercises. The Lectures summarize the principal ideas of the course with live demonstrations and active-learning exercises. Discussion problems are solved cooperatively by students. The lab experiments lend reality to the basic principles. Exercises at the end of each chapter are designed to complement discussion and homework problems. Solutions to most Exercises are provided in the back pages. Appendices (and Chapter 14) are included for students who want to dig a little deeper into the subjects of this course and gain additional links to advanced courses.

Acknowledgements The precursor to this course was first taught in Fall 1997 and Spring 1998 by Michael Weissman and Dale Van Harlingen. Subsequent versions of the course were developed by Doug Beck, Michael Weissman, Jon Thaler, Michael Stone, Paul Debevec, Lance Cooper, Yoshi Oono, Paul Kwiat, and myself. I particularly wish to thank Mike Weissman, Lance Cooper, Yoshi Oono, Inga Karliner, and Paul Kwiat for insightful suggestions and corrections to this book.

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Physics 213 Elements of Thermal Physics

Reference Texts In developing material for this course, I have drawn heavily from three excellent books. For students who wish to extend their knowledge in this area, I highly recommend them: C. Kittel and H. Kroemer, Thermal Physics, Second Edition (W. H. Freeman, 1980) D.V. Schroeder, An Introduction to Thermal Physics (Addison-Wesley, 1999) F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) The following books also provide very useful perspectives: Thomas A. Moore, Six Ideas That Shaped Physics, Unit T (McGraw-Hill, 1998) F. Reif, Statistical Physics, Berkeley Physics Course—Vol. 5 (McGraw-Hill, 1965) Steven S. Zumdahl, Chemical Principles, 5th Edition, (Houghton Mifflin, 2005) Professor Gino Segre has written a fascinating historical perspective of the world from the viewpoint of thermodynamics. It’s a “must read” for science and engineering majors: Gino Segre, A Matter of Degrees: What Temperature Reveals about the Past and Future of Our Species, Planet, and Universe (Penguin, USA, 2003).

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Elements of Thermal Physics Physics 213

Deﬁnition of Symbols (Alphabetically arranged) A = U/pV = U /NkT