In this clear and concise introduction to thermodynamics and statistical mechanics the reader, who will have some previous exposure to thermodynamics, will be guided through each of the two disciplines separately initially to provide an in-depth understanding of the area and thereafter the connection between the two is presented and discussed.
In addition, mathematical techniques are introduced at appropriate times, highlighting such use as: exact and inexact differentials, partial derivatives, Caratheodory's theorem, Legendre transformation, and combinatory analysis.
* Emphasis is placed equally on fundamentals and applications
* Several problems are included

"…a thorough treatment of thermodynamics at a level somewhat higher level than that in a typical undergraduate physical chemistry work." (CHOICE, February 2005)

BRUNO LINDER is Professor Emeritus of Chemistry at Florida State University. Founder of the Southeastern Theoretical Chemistry Association, he was formerly a John Simon Guggenheim Fellow at the Theoretical Physics Institute of the University of Amsterdam.

Thermodynamics, as developed in this course, deals with the macroscopic properties of matter or, more precisely, with processes on a macroscopic level. Mechanics (especially quantum mechanics) is concerned with molecular behavior. In principle, and in some limited cases, the molecular properties can be calculated directly from quantum mechanics. In the majority of cases, however, such properties are obtained from experimental studies such as spectral behavior or other devices, but the interpretation is based on quantum mechanics. Statistical mechanics is the branch of science that interconnects these seemingly unrelated disciplines: statistical mechanics interprets and, as far as possible, predicts the macroscopic properties in terms of the microscopic constituents.

For the purposes of the course presented in this book, thermodynamics and statistical mechanics are developed as separate disciplines. Only after the introduction of the fundamentals of statistical mechanics will the connection be made between statistical mechanics and thermodynamics. As noted, the laws of (macroscopic) thermodynamics deal with processes not structures. Therefore, no theory of matter is contained in theselaws. Traditional thermodynamics is based on common everyday experiences. For example, if two objects are brought in contact with each other, and one feels hotter than the other, the hotter object will cool while the colder one will heat up. Because thermodynamics is based on the common experience of macroscopic observations it has a generality unequaled in science. "Classical Thermodynamics," Einstein remarked, "... is the only physical theory of universal content ... which ... will never be overthrown" (Schilpp, 1949).

1.1 SCOPE AND OBJECTIVES

Class make-up varies greatly. Some students take this course as part of one-year course, in preparation for a comprehensive or preliminary exam, required for a Master's or Ph.D. degree. Others sign up because they heard it was a "snap" course. Still others take it because they think, or their major professor thinks, that it may help them in their research. A course designed to satisfy all students' aspirations is difficult, if not impossible. A suitable compromise is one, which provides a reasonable balance between fundamentals and applications, which is the aim of this book.

1.2 LEVEL OF COURSE

Most students are likely to have had previous exposure to thermodynamics in some undergraduate course, such as physical chemistry, physics, or engineering. The present course is intended to be more advanced from the standpoints of both principles and applications. The emphasis is on the logical structure and generality of the subject. All topics of interest cannot possibly be covered in a semester course; therefore, topics that are likely to have been adequately treated in undergraduate courses are skipped.

1.3 COURSE OUTLINE

The idea is to proceed from the general to the particular. The following outline suggests itself.

Part I: Thermodynamics

A. Fundamentals

1. Basic concepts and definitions

2. The laws of thermodynamics

2.1 Traditional approach 2.2 Axiomatic approach

3. General conditions for equilibrium and stability

B. Applications

1. Thermodynamics of (Real) gases, condensed systems

2. Chemical equilibrium

2.1 Homogeneous and heterogeneous systems 2.2 Chemical reactions

3. Phase transitions and critical phenomena

4. Thermodynamics of one- and two-dimensional systems

In addition, mathematical techniques are introduced at appropriate times, highlighting such use as:

1) Exact and inexact differentials (Section 3.3)

2) Partial Derivatives (Section 3.6)

3) Pfaffian Differential Forms (Section 4.6)

4) Legendre Transformation (Section 5.1)

5) Euler's Theorem (Section 5.7)

6) Combinatory Analysis (Section 13.5)

1.4 BOOKS

Because of the universality of the subject, books on Thermodynamics run into the thousands. Not all are textbooks, and not all are aimed at a particular discipline, such as chemistry, physics, or engineering. Most elementary chemical texts rely heavily on applications but treat the fundamentals lightly. Real systems (real gases, condensed systems, etc) are often not treated in any detail. Some books are strong on fundamentals but ignore applications. Other books are authoritative but highly opinionated, pressing for a particular point of view.

Two chemical thermodynamics books, which discuss the fundamentals in depth, are listed below.

1. J. de Heer, Phenomenological Thermodynamics, Prentice-Hall, 1986.

2. J. G. Kirkwood and I. Oppenheim, Chemical Thermodynamics, McGraw-Hill, 1961.

Other books that may provide additional insight into various topics are listed in the Annotated Bibliography on page....

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## More About This Textbook

## Overview

In this clear and concise introduction to thermodynamics and statistical mechanics the reader, who will have some previous exposure to thermodynamics, will be guided through each of the two disciplines separately initially to provide an in-depth understanding of the area and thereafter the connection between the two is presented and discussed.

In addition, mathematical techniques are introduced at appropriate times, highlighting such use as: exact and inexact differentials, partial derivatives, Caratheodory's theorem, Legendre transformation, and combinatory analysis.

* Emphasis is placed equally on fundamentals and applications

* Several problems are included

## Editorial Reviews

## From the Publisher

"…a thorough treatment of thermodynamics at a level somewhat higher level than that in a typical undergraduate physical chemistry work." (CHOICE, February 2005)## Product Details

## Related Subjects

## Meet the Author

BRUNO LINDER is Professor Emeritus of Chemistry at Florida State University. Founder of the Southeastern Theoretical Chemistry Association, he was formerly a John Simon Guggenheim Fellow at the Theoretical Physics Institute of the University of Amsterdam.

## Table of Contents

PREFACE.

1 INTRODUCTORY REMARKS.

1.1 Scope and Objectives.

1.2 Level of Course.

1.3 Course Outline.

1.4 Books.

PART I: THERMODYNAMICS.

2 BASIC CONCEPTS AND DEFINITIONS.

2.1 Systems and Surroundings.

2.2 State Variables and Thermodynamic Properties.

2.3 Intensive and Extensive Variables.

2.4 Homogeneous and Heterogeneous Systems, Phases.

2.5 Work.

2.6 Reversible and Quasi-Static Processes.

2.6.1 Quasi-Static Process.

2.6.2 Reversible Process.

2.7 Adiabatic and Diathermal Walls.

2.8 Thermal Contact and Thermal Equilibrium.

3 THE LAWS OF THERMODYNAMICS I.

3.1 The Zeroth Law—Temperature.

3.2 The First Law—Traditional Approach.

3.3 Mathematical Interlude I: Exact and InexactDifferentials.

3.4 The First Law—Axiomatic Approach.

3.5 Some Applications of the First Law.

3.5.1 Heat Capacity.

3.5.2 Heat and Internal Energy.

3.5.3 Heat and Enthalpy.

3.6 Mathematical Interlude II: Partial Derivatives.

3.6.1 Relations Between Partials of Dependent Variables.

3.6.2 Relations Between Partials with Different Subscripts.

3.7 Other Applications of the First Law.

3.7.1 C

_{P}—C_{V}.3.7.2 Isothermal Change, Ideal Gas.

3.7.3 Adiabatic Change, Ideal Gas.

3.7.4 The Joule and the Joule-Thomson Coefficients.

4 THE LAWS OF THERMODYNAMICS II.

4.1 The Second Law—Traditional Approach.

4.2 Engine Efficiency: Absolute Temperature.

4.2.1 Ideal Gas.

4.2.2 Coupled Cycles.

4.3 Generalization: Arbitrary Cycle.

4.4 The Clausius Inequality.

4.5 The Second Law—Axiomatic Approach(Caratheòodory).

4.6 Mathematical Interlude III: Pfaffian Differential Forms.

4.7 Pfaffian Expressions in Two Variables.

4.8 Pfaffian Expressions in More Than Two Dimensions.

4.9 Caratheòodory’s Theorem.

4.10 Entropy—Axiomatic Approach.

4.11 Entropy Changes for Nonisolated Systems.

4.12 Summary.

4.13 Some Applications of the Second Law.

4.13.1 Reversible Processes (PV Work Only).

4.13.2 Irreversible Processes.

5 USEFUL FUNCTIONS: THE FREE ENERGY FUNCTIONS.

5.1 Mathematical Interlude IV: Legendre Transformations.

5.1.1 Application of the Legendre Transformation.

5.2 Maxwell Relations.

5.3 The Gibbs-Helmholtz Equations.

5.4 Relation of ΔA and ΔG to Work: Criteriafor Spontaneity.

5.4.1 Expansion and Other Types of Work.

5.4.2 Comments.

5.5 Generalization to Open Systems and Systems of VariableComposition.

5.5.1 Single Component System.

5.5.2 Multicomponent Systems.

5.6 The Chemical Potential.

5.7 Mathematical Interlude V: Euler’s Theorem.

5.8 Thermodynamic Potentials.

6 THE THIRD LAW OF THERMODYNAMICS.

6.1 Statements of the Third Law.

6.2 Additional Comments and Conclusions.

7 GENERAL CONDITIONS FOR EQUILIBRIUM AND STABILITY.

7.1 Virtual Variations.

7.2 Thermodynamic Potentials—Inequalities.

7.3 Equilibrium Condition From Energy.

7.3.1 Boundary Fully Heat Conducting, Deformable, Permeable(Normal System).

7.3.2 Special Cases: Boundary Semi-Heat Conducting,Semi-Deformable, or Semi-Permeable.

7.4 Equilibrium Conditions From Other Potentials.

7.5 General Conditions for Stability.

7.6 Stability Conditions From E.

7.7 Stability Conditions From Cross Terms.

7.8 Stability Conditions From Other Potentials.

7.9 Derivatives of Thermodynamic Potentials With Respect toIntensive Variables.

8 APPLICATION OF THERMODYNAMICS TO GASES, LIQUIDS, ANDSOLIDS.

8.1 Gases.

8.2 Enthalpy, Entropy, Chemical Potential, Fugacity.

8.2.1 Enthalpy.

8.2.2 Entropy.

8.2.3 Chemical Potential.

8.2.4 Fugacity.

8.3 Standard States of Gases.

8.4 Mixtures of Gases.

8.4.1 Partial Fugacity.

8.4.2 Free Energy, Entropy, Enthalpy, and Volume of Mixing ofGases.

8.5 Thermodynamics of Condensed Systems.

8.5.1 The Chemical Potential.

8.5.2 Entropy.

8.5.3 Enthalpy.

9 PHASE AND CHEMICAL EQUILIBRIA.

9.1 The Phase Rule.

9.2 The Clapeyron Equation.

9.3 The Clausius-Clapeyron Equation.

9.4 The Generalized Clapeyron Equation.

9.5 Chemical Equilibrium.

9.6 The Equilibrium Constant.

10 SOLUTIONS—NONELECTROLYTES.

10.1 Activities and Standard State Conventions.

10.1.1 Gases.

10.1.2 Pure Liquids and Solids.

10.1.3 Mixtures.

10.1.3.1 Liquid–Liquid Solutions—Convention I (ConI).

10.1.3.2 Solid–Liquid Solutions—Convention II (ConII).

10.2 Ideal and Ideally Dilute Solutions; Raoult’s andHenry’s Laws.

10.2.1 Ideal Solutions.

10.2.2 Ideally Dilute Solutions.

10.3 Thermodynamic Functions of Mixing.

10.3.1 For Ideal Solutions.

10.3.2 For Nonideal Solutions.

10.4 Colligative Properties.

10.4.1 Lowering of Solvent Vapor Pressure.

10.4.2 Freezing Point Depression.

10.4.3 Boiling Point Elevation.

10.4.4 Osmotic Pressure.

11 PROCESSES INVOLVING WORK OTHER THAN PRESSURE-VOLUME WORK.

11.1 P-V Work and One Other Type of Work.

11.2 P-V, σA, and fL Work.

12 PHASE TRANSITIONS AND CRITICAL PHENOMENA.

12.1 Stable, Metastable, and Unstable Isotherms.

12.2 The Critical Region.

PART II: INTRODUCTORY STATISTICAL MECHANICS.

13 PRINCIPLES OF STATISTICAL MECHANICS.

13.1 Introduction.

13.2 Preliminary Discussion—Simple Problem.

13.3 Time and Ensemble Averages.

13.4 Number of Microstates, Ω

_{D},Distributions D_{I}.13.5 Mathematical Interlude VI: Combinatory Analysis.

13.6 Fundamental Problem in Statistical Mechanics.

13.7 Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein Statistics‘‘Corrected’’ Maxwell-BoltzmannStatistics.

13.7.1 Maxwell-Boltzmann Statistics.

13.7.2 Fermi-Dirac Statistics.

13.7.3 Bose-Einstein Statistics

13.7.4 ‘‘Corrected’’ Maxwell-BoltzmannStatistics.

13.8 Systems of Distinguishable (Localized) andIndistinguishable (Nonlocalized) Particles.

13.9 Maximizing Ω

_{D}13.10 Probability of a Quantum State: The PartitionFunction.

13.10.1 Maxwell-Boltzmann Statistics.

13.10.2 Corrected Maxwell-Boltzmann Statistics.

14 THERMODYNAMIC CONNECTION.

14.1 Energy, Heat, and Work.

14.2 Entropy.

14.2.1 Entropy of Nonlocalized Systems (Gases).

14.2.2 Entropy of Localized Systems (Crystalline Solids).

14.3 Identification of β with 1/kT.

14.4 Pressure.

14.5 The Functions E, H, S, A, G, and μ.

15 MOLECULAR PARTITION FUNCTION.

15.1 Translational Partition Function.

15.2 Vibrational Partition Function: Diatomics.

15.3 Rotational Partition Function: Diatomics.

15.4 Electronic Partition Function.

15.5 Nuclear Spin States.

15.6 The ‘‘Zero’’ of Energy.

16 STATISTICAL MECHANICAL APPLICATIONS.

16.1 Population Ratios.

16.2 Thermodynamic Functions of Gases.

16.3 Equilibrium Constants.

16.4 Systems of Localized Particles: The Einstein Solid.

16.4.1 Energy.

16.4.2 Heat Capacity.

16.4.3 Entropy.

16.5 Summary.

ANNOTATED BIBLIOGRAPHY.

APPENDIX I: HOMEWORK PROBLEM SETS.

Problem Set I.

Problem Set II.

Problem Set III.

Problem Set IV.

Problem Set V.

Problem Set VI.

Problem Set VII.

Problem Set VIII.

Problem Set IX.

Problem Set X.

APPENDIX II: SOLUTIONS TO PROBLEMS.

Solution to Set I.

Solution to Set II.

Solution to Set III.

Solution to Set IV.

Solution to Set V.

Solution to Set VI.

Solution to Set VII.

Solution to Set VIII.

Solution to Set IX.

Solution to Set X.

INDEX.

## First Chapter

## Thermodynamics and Introductory Statistical Mechanics

By Bruno LinderJohn Wiley & SonsCopyright © 2004John Wiley & Sons, Inc.All right reserved.

ISBN: 0-471-47459-2## Chapter One

INTRODUCTORY REMARKSThermodynamics, as developed in this course, deals with the macroscopic properties of matter or, more precisely, with processes on a macroscopic level. Mechanics (especially quantum mechanics) is concerned with molecular behavior. In principle, and in some limited cases, the molecular properties can be calculated directly from quantum mechanics. In the majority of cases, however, such properties are obtained from experimental studies such as spectral behavior or other devices, but the interpretation is based on quantum mechanics. Statistical mechanics is the branch of science that interconnects these seemingly unrelated disciplines: statistical mechanics interprets and, as far as possible, predicts the macroscopic properties in terms of the microscopic constituents.

For the purposes of the course presented in this book, thermodynamics and statistical mechanics are developed as separate disciplines. Only after the introduction of the fundamentals of statistical mechanics will the connection be made between statistical mechanics and thermodynamics. As noted, the laws of (macroscopic) thermodynamics deal with

processesnotstructures. Therefore, no theory of matter is contained in theselaws. Traditional thermodynamics is based on common everyday experiences. For example, if two objects are brought in contact with each other, and one feels hotter than the other, the hotter object will cool while the colder one will heat up. Because thermodynamics is based on the common experience of macroscopic observations it has a generality unequaled in science. "Classical Thermodynamics," Einstein remarked, "... is the only physical theory of universal content ... which ... will never be overthrown" (Schilpp, 1949).1.1 SCOPE AND OBJECTIVESClass make-up varies greatly. Some students take this course as part of one-year course, in preparation for a comprehensive or preliminary exam, required for a Master's or Ph.D. degree. Others sign up because they heard it was a "snap" course. Still others take it because they think, or their major professor thinks, that it may help them in their research. A course designed to satisfy all students' aspirations is difficult, if not impossible. A suitable compromise is one, which provides a reasonable balance between

fundamentalsandapplications, which is the aim of this book.1.2 LEVEL OF COURSEMost students are likely to have had previous exposure to thermodynamics in some undergraduate course, such as physical chemistry, physics, or engineering. The present course is intended to be more advanced from the standpoints of both

principlesandapplications. The emphasis is on the logical structure and generality of the subject. All topics of interest cannot possibly be covered in a semester course; therefore, topics that are likely to have been adequately treated in undergraduate courses are skipped.1.3 COURSE OUTLINEThe idea is to proceed from the

generalto theparticular.The following outline suggests itself.Part I: Thermodynamics

A.

Fundamentals1. Basic concepts and definitions

2. The laws of thermodynamics

2.1 Traditional approach 2.2 Axiomatic approach3. General conditions for equilibrium and stability

B. Applications1. Thermodynamics of (Real) gases, condensed systems

2. Chemical equilibrium

2.1 Homogeneous and heterogeneous systems 2.2 Chemical reactions3. Phase transitions and critical phenomena

4. Thermodynamics of one- and two-dimensional systems

4.1 Film enlarging 4.2 Rubber stretchingPart II: Introductory Statistical Mechanics

A. Fundamentals1. Preliminary discussion 2. Maxwell-Boltzmann, Corrected Maxwell-Boltzmann Statistics

3. Partition Functions

4. Thermodynamic connection

B. Applications1. Ideal gases2. Ideal solids 3. Equilibrium constant

4. The bases of chemical thermodynamics

In addition, mathematical techniques are introduced at appropriate times, highlighting such use as:

1) Exact and inexact differentials (Section 3.3)

2) Partial Derivatives (Section 3.6)

3) Pfaffian Differential Forms (Section 4.6)

4) Legendre Transformation (Section 5.1)

5) Euler's Theorem (Section 5.7)

6) Combinatory Analysis (Section 13.5)

1.4 BOOKSBecause of the universality of the subject, books on Thermodynamics run into the thousands. Not all are textbooks, and not all are aimed at a particular discipline, such as chemistry, physics, or engineering. Most elementary chemical texts rely heavily on applications but treat the fundamentals lightly. Real systems (real gases, condensed systems, etc) are often not treated in any detail. Some books are strong on fundamentals but ignore applications. Other books are authoritative but highly opinionated, pressing for a particular point of view.

Two chemical thermodynamics books, which discuss the fundamentals in depth, are listed below.

1. J. de Heer,

Phenomenological Thermodynamics, Prentice-Hall, 1986.2. J. G. Kirkwood and I. Oppenheim,

Chemical Thermodynamics,McGraw-Hill, 1961.Other books that may provide additional insight into various topics are listed in the Annotated Bibliography on page....

(Continues...)