Introduction to Applied Statistical Thermodynamics

STANLEY I. SANDLER is the H. B. du Pont Professor of Chemical Engineering at the University of Delaware as well as professor of chemistry and biochemistry. He is also the founding director of its Center for Molecular and Engineering Thermodynamics. In addition to this book, Sandler is the author of 235 research papers and a monograph, and is the editor of a book on thermodynamic modeling and five conference proceedings. He earned his B.Ch.E. degree in 1962 from the City College of New York, and his Ph.D. in chemical engineering from the University of Minnesota in 1966.

• PREFACE FOR INSTRUCTORS vPREFACE FOR STUDENTS ixCHAPTER 1 INTRODUCTION TO STATISTICAL THERMODYNAMICS 11.1 Probabilistic Description 11.2 Macroscopic States and Microscopic States 21.3 Quantum Mechanical Description of Microstates 31.4 The Postulates of Statistical Mechanics 51.5 The Boltzmann Energy Distribution 6CHAPTER 2 THE CANONICAL PARTITION FUNCTION 92.1 Some Properties of the Canonical Partition Function 92.2 Relationship of the Canonical Partition Function to Thermodynamic Properties 112.3 Canonical Partition Function for a Molecule with Several Independent Energy Modes 122.4 Canonical Partition Function for a Collection of Noninteracting Identical Atoms 13Chapter 2 Problems 15CHAPTER 3 THE IDEAL MONATOMIC GAS 163.1 Canonical Partition Function for the Ideal Monatomic Gas 163.2 Identification of β as 1/kT 183.3 General Relationships of the Canonical Partition Function to Other Thermodynamic Quantities 193.4 The Thermodynamic Properties of the Ideal Monatomic Gas 223.5 Energy Fluctuations in the Canonical Ensemble 293.6 The Gibbs Entropy Equation 333.7 Translational State Degeneracy 353.8 Distinguishability, Indistinguishability, and the Gibbs’ Paradox 373.9 A Classical Mechanics–Quantum Mechanics Comparison: The Maxwell-Boltzmann Distribution of Velocities 39Chapter 3 Problems 42CHAPTER 4 THE IDEAL DIATOMIC AND POLYATOMIC GASES 444.1 The Partition Function for an Ideal Diatomic Gas 444.1a The Translational and Nuclear Partition Functions 454.1b The Rotational Partition Function 454.1c The Vibrational Partition Function 474.1d The Electronic Partition Function 484.2 The Thermodynamic Properties of the Ideal Diatomic Gas 494.3 The Partition Function for an Ideal Polyatomic Gas 534.4 The Thermodynamic Properties of an Ideal Polyatomic Gas 554.5 The Heat Capacities of Ideal Gases 584.6 Normal Mode Analysis: The Vibrations of a Linear Triatomic Molecule 59Chapter 4 Problems 62CHAPTER 5 CHEMICAL REACTIONS IN IDEAL GASES 645.1 The Nonreacting Ideal Gas Mixture 645.2 Partition Function of a Reacting Ideal Chemical Mixture 655.3 Three Different Derivations of the Chemical Equilibrium Constant in an Ideal Gas Mixture 675.4 Fluctuations in a Chemically Reacting System 705.5 The Chemically Reacting Gas Mixture: The General Case 735.6 Two Illustrations 80Appendix: The Binomial Expansion 83Chapter 5 Problems 85CHAPTER 6 OTHER PARTITION FUNCTIONS 876.1 The Microcanonical Ensemble for a Pure Fluid 876.2 The Grand Canonical Ensemble for a Pure Fluid 896.3 The Isobaric-Isothermal Ensemble 926.4 The Restricted Grand or Semi-Grand Canonical Ensemble 936.5 Comments on the Use of Different Ensembles 94Chapter 6 Problems 96CHAPTER 7 INTERACTING MOLECULES IN A GAS 987.1 The Configuration Integral 987.2 Thermodynamic Properties from the Configuration Integral 1007.3 The Pairwise Additivity Assumption 1017.4 Mayer Cluster Function and Irreducible Integrals 1027.5 The Virial Equation of State 1097.6 Virial Equation of State for Polyatomic Molecules 1147.7 Thermodynamic Properties from the Virial Equation of State 1167.8 Derivation of Virial Coefficient Formulae from the Grand Canonical Ensemble 1187.9 Range of Applicability of the Virial Equation 123Chapter 7 Problems 124CHAPTER 8 INTERMOLECULAR POTENTIALS AND THE EVALUATION OF THE SECOND VIRIAL COEFFICIENT 1258.1 Interaction Potentials for Spherical Molecules 1258.2 The Second Virial Coefficient in a Mixture: Interaction Potentials Between Unlike Atoms 1368.3 Interaction Potentials for Multiatom, Nonspherical Molecules, Proteins, and Colloids 1378.4 Engineering Applications and Implications of the Virial Equation of State 140Chapter 8 Problems 144CHAPTER 9 MONATOMIC CRYSTALS 1479.1 The Einstein Model of a Crystal 1479.2 The Debye Model of a Crystal 1509.3 Test of the Einstein and Debye Heat Capacity Models for a Crystal 1579.4 Sublimation Pressure and Enthalpy of Crystals 1599.5 A Comment on the Third Law of Thermodynamics 161Chapter 9 Problems 161CHAPTER 10 SIMPLE LATTICE MODELS FOR FLUIDS 16310.1 Introduction 16410.2 Development of Equations of State from Lattice Theory 16510.3 Activity Coefficient Models for Similar-Size Molecules from Lattice Theory 16810.4 The Flory-Huggins and Other Models for Polymer Systems 17210.5 The Ising Model 178Chapter 10 Problems 184CHAPTER 11 INTERACTING MOLECULES IN A DENSE FLUID. CONFIGURATIONAL DISTRIBUTION FUNCTIONS 18511.1 Reduced Spatial Probability Density Functions 18511.2 Thermodynamic Properties from the Pair Correlation Function 19011.3 The Pair Correlation Function (Radial Distribution Function) at Low Density 19411.4 Methods of Determination of the Pair Correlation Function at High Density 19711.5 Fluctuations in the Number of Particles and the Compressibility Equation 19911.6 Determination of the Radial Distribution Function of Fluids using Coherent X-ray or Neutron Diffraction 20211.7 Determination of the Radial Distribution Functions of Molecular Liquids 21011.8 Determination of the Coordination Number from the Radial Distribution Function 21111.9 Determination of the Radial Distribution Function of Colloids and Proteins 213Chapter 11 Problems 214CHAPTER 12 INTEGRAL EQUATION THEORIES FOR THE RADIAL DISTRIBUTION FUNCTION 21612.1 The Yvon-Born-Green (YBG) Equation 21612.2 The Kirkwood Superposition Approximation 21912.3 The Ornstein-Zernike Equation 22012.4 Closures for the Ornstein-Zernike Equation 22212.5 The Percus-Yevick Hard-Sphere Equation of State 22712.6 The Radial Distribution Functions and Thermodynamic Properties of Mixtures 22812.7 The Potential of Mean Force 23012.8 Osmotic Pressure and the Potential of Mean Force for Protein and Colloidal Solutions 237Chapter 12 Problems 239CHAPTER 13 DETERMINATION OF THE RADIAL DISTRIBUTION FUNCTION AND FLUID PROPERTIES BY COMPUTER SIMULATION 24113.1 Introduction to Molecular Level Computer Simulation 24213.2 Thermodynamic Properties from Molecular Simulation 24513.3 Monte Carlo Simulation 24913.4 Molecular-Dynamics Simulation 253Chapter 13 Problems 255CHAPTER 14 PERTURBATION THEORY 25714.1 Perturbation Theory for the Square-Well Potential 25714.2 First Order Barker-Henderson Perturbation Theory 26214.3 Second-Order Perturbation Theory 26514.4 Perturbation Theory Using Other Reference Potentials 26914.5 Engineering Applications of Perturbation Theory 272Chapter 14 Problems 274CHAPTER 15 A THEORY OF DILUTE ELECTROLYTE SOLUTIONS AND IONIZED GASES 27615.1 Solutions Containing Ions (and Electrons) 27615.2 Debye-Huckel Theory 28015.3 The Mean Ionic Activity Coefficient 291Chapter 15 Problems 296CHAPTER 16 THE DERIVATION OF THERMODYNAMIC MODELS FROM THE GENERALIZED VAN DER WAALS PARTITION FUNCTION 29716.1 The Statistical-Mechanical Background 29816.2 Application of the Generalized van der Waals Partition Function to Pure Fluids 30116.3 Equation of State for Mixtures from the Generalized van der Waals Partition Function 31016.4 Activity Coefficient Models from the Generalized van der Waals Partition Function 31816.5 Chain Molecules and Polymers 32916.6 Hydrogen-Bonding and Associating Fluids 332Chapter 16 Problems 334INDEX 335