Thermodynamics and Statistical Mechanics

Lectures on Theoretical Physics, Volume V: Thermodynamics and Statistical Mechanics discusses the significant developments and problems in the study of thermodynamics and statistical mechanics. This volume contains five chapters. The first two chapters provide an overview of the various aspects and applications of thermodynamics. Chapter III contains a preliminary introduction to statistical mechanics, with an emphasis on the Brownian motion, which is the most important example of statistical fluctuations. Chapter IV describes the Boltzmann’s original form of combinatorial method, in which the molecules of a gas are endowed with a physically real existence. This chapter also considers the various numerical combinations that govern the way in which the mutually indistinguishable particles are distributed over the states constituting the substance of the statistics. Chapter V explores the behavior of molecules in perfect gases following the course of historical development. This chapter covers an exact formulation of the kinetic theory of gases. Physics teachers and students will find this book invaluable.

Author's Preface

Editor's Preface

Translator's Preface

Chapter I. Thermodynamics. General Considerations

1. Temperature as a Property of a System

2. Work and Heat

3. The Perfect Gas

A. Boyle's Law (The Law of Boyle and Mariotte)

B. Charles' Law (The Law of Gay-Lussac)

C. Avogadro's Law and the Universal Gas Constant

4. The First Law. Energy and Enthalpy as Properties

A. Equivalence of Heat and Work

B. The Enthalpy as a Property

C. Digression on the Ratio of Specific Heats cp and cv

5. The Reversible and the Irreversible Adiabatic Process

A. The Reversible Adiabatic Process

B. The Irreversible Adiabatic Process

C. The Joule-Kelvin Porous Plug Experiment

D. A Conclusion of Great Consequence

6. The Second Law

A. The Carnot Cycle and Its Efficiency

B. The First Part of the Second Law

C. The Second Part of the Second Law

D. Simplest Numerical Examples

E. Remarks on the Literature of the Second Law

F. On the Relative Rank of Energy and Entropy

7. The Thermodynamic Potentials and the Reciprocity Relations

8. Thermodynamic Equilibria

A. Unconstrained Thermodynamic Equilibrium and Maximum of Entropy

B. An Isothermal and Isobaric System in Unconstrained Thermodynamic Equilibrium

C. Additional Degrees of Freedom in Retarded Equilibrium

D. Extremum Properties of the Thermodynamic Potentials

E. The Theorem on Maximum Work

9. The van der Waals Equation

A. Course of Isotherms

B. Entropy and the Caloric Behavior of the van der Waals Gas

10. Remarks on the Liquefaction of Gases According to van der Waals

A. The Integral and the Differential Joule-Thomson Effect

B. The Inversion Curve and Its Practical Utilization

C. The Boundary of the Region of Co-existing Liquid-Vapor Phases in the p, v Plane

11. The Kelvin Temperature Scale

12. Nernst's Third Law of Thermodynamics

Chapter II. The Application of Thermodynamics To Special Systems

13. Gaseous Mixtures. Gibbs' Paradox. The Law Due to Guldberg and Waage

A. Reversible Separation of Gases

B. The Increase in Entropy During Diffusion and Gibbs' Paradox

C. The Law of Mass Action Due to Guldberg and Waage

14. Chemical Potentials and Chemical Constants

A. The Chemical Potentials μi

B. Relation Between the μi's and the gi's for Ideal Mixtures

C. The Chemical Constant of a Perfect Gas

15. Dilute Solutions

A. General and Historical Remarks

B. Van 't Hoff's Equation of State for Dilute Solutions

16. The Different Phases of Water. Remarks on the Theory of the Steam Engine

A. The Vapor-Pressure Curve and Clapeyron's Equation

B. Phase Equilibrium Between Ice and Water

C. The Specific Heat of Saturated Steam

17. General Remarks on the Theory of Phase Equilibria

A. The Triple Point of Water

B. Gibbs' Phase Rule

C. Raoult's Laws for Dilute Solutions

D. Henry's Law of Absorption (1803)

18. The Electromotive Force of Galvanic Cells

A. Electrochemical Potentials

B. The Daniell Cell, 1836

C. Contraction of Individual Reactions into a Simplified Overall Reaction

D. The Gibbs-Helmholtz Fundamental Equation

E. Numerical Example

F. Remarks on the Integration of the Fundamental Equation

19. Ferro- and Paramagnetism

A. Work of Magnetization and Magnetic Equation of State

B. Langevin's Equation for Paramagnetic Substances

C. The Theory of Ferromagnetic Phenomena Due to Weiss

D. The Specific Heats cH and cM

E. The Magneto-Caloric Effect

20. Black Body Radiation

A. Kirchhoff 's Law

B. The Stefan-Boltzmann Law

C. Wien's Law

D. Planck's Law of Radiation

21. Irreversible Processes. Thermodynamics of Near-Equilibrium Processes

A. Conduction of Heat and Local Entropy Generation

B. The Conduction of Heat in an Anisotropie Body and Onsager's Reciprocal Relations

C. Thermoelectric Phenomena

D. Internal Transformations

E. General Relations

F. Limitations of the Thermodynamic Theory of Irreversible Processes

Chapter III. The Elementary Kinetic Theory of Gases

22. The Equation of State of a Perfect Gas

23. The Maxwellian Velocity Distribution

A. The Maxwellian Distribution for a Monatomic Gas. Proof of 1860

B. Numerical Values and Experimental Results

C. General Remarks on the Energy Distribution. The Boltzmann Factor

24. Brownian Motion

25. Statistical Considerations on Paramagnetic Substances

A. The Classical Langevin Function

B. Modification of Langevin's Function with the Aid of Quantum Mechanics

26. The Statistical Significance of the Constants in van der Waals' Equation

A. The Volume of a Molecule and the Constant b

B. The van der Waals Cohesion Forces and the Constant a

27. The Problem of the Mean Free Path

A. Calculation of the Mean Free Path in One Special Case

B. Viscosity

C. Thermal Conductivity

D. Some General Remarks on the Problems Associated with the Concept of the Mean Free Path

Chapter IV. General Statistical Mechanics: Combinatorial Method

28. Liouville's Theorem, Γ-space and μ-space

A. The Multidimensional Γ-space (Phase Space)

B. Liouville's Theorem

C. Equality of Probability for the Perfect Gas

29. Boltzmann's Principle

A. Permutability as a Measure of the Probability of a State

B. The Maximum of Probability as a Measure of Entropy

C. The Combining of Elementary Cells

30. Comparison with Thermodynamics

A. Constant Volume Process

B. General Process Performed by a Gas in the Absence of External Forces

C. A Gas in a Field of Forces; the Boltzmann Factor

D. The Maxwell-Boltzmann Velocity Distribution Law

E. Gaseous Mixtures

31. Specific Heat and Energy of Rigid Molecules

A. The Monatomic Gas

B. Gas Composed of Diatomic Molecules

C. The Polyatomic Gas and Kelvin's Clouds

32. The Specific Heat of Vibrating Molecules and of Solid Bodies

A. The Diatomic Molecule

B. Polyatomic Gases

C. The Solid Body and the Dulong-Petit Rule

33. The Quantization of Vibrational Energy

A. The Linear Oscillator

B. The Solid Body

C. Generalization to Arbitrary Quantum States

34. The Quantization of Rotational Energy

35. Supplement to the Theory of Radiation and to that of Solid Bodies

A. Method of Natural Vibrations

B. Debye's Theory of the Specific Heat of a Solid

36. Partition Function in the Γ-space

A. The Gibbs Condition

B. Connection with Boltzmann's Method

C. Correction for Quantum Effects

D. Analysis of Gibbs' Hypothesis

37. Fundamentals of Quantum Statistics

A. Quantum Statistics of Identical Particles

B. The Method Due to Darwin and Fowler

C. Bose-Einstein and Fermi-Dirac Statistics

D. The Saddle-Point Method

38. Degenerate Gases

A. Bose-Einstein and Fermi-Dirac Distribution

B. Degree of Gas Degeneration

C. Highly Degenerate Bose-Einstein Gas

39. Electron Gas in Metals

A. Introductory Remark to Drude's Method

B. The Completely Degenerate Fermi-Dirac Gas

C. Almost Complete Degeneracy

D. Special Problems

40. The Mean Square of Fluctuations

Chapter V. Outline of an Exact Kinetic Theory of Gases

41. The Maxwell-Boltzmann Collision Equation

A. Description of a State in the Kinetic Theory of Gases

B. The Variation of / with Time

C. The Laws of Elastic Collision

D. Boltzmann's Collision Integral

E. Boltzmann's Hypothesis About Molecular Chaos

42. The H-theorem and Maxwellian Distribution

A. The H-theorem

B. Maxwellian Distribution

C. Equilibrium Distributions

43. Fundamental Equations of Fluid Dynamics

A. Series Expansion for the Distribution Function

B. Maxwell's Transport Equation

C. Conservation of Mass

D. Conservation of Momentum

E. Conservation of Energy

F. Entropy Theorem

44. On the Integration of the Collision Equation

A. Integration with the Aid of Moment Equations

B. Transformation of the Equations for Moments

C. Evaluation of Collision Moments

D. Viscosity and Thermal Conductivity

45. Conductivity and the Wiedemann-Franz Law

A. The Collision and Transfer Equations for Electrons in Metals

B. Approximate Solution of the Collision Equation

C. Flux of Current and Energy

D. Ohm's Law

E. Thermal Conductivity and Absolute Thermal Electromotive Force

F. The Wiedemann-Franz Law

Problems for Chapter I

Problems for Chapter II

Problems for Chapter III

Problems for Chapter IV

Problems for Chapter V

Hints for the Solution of Problems

Index

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