With the present emphasis on nano and bio technologies, molecular level descriptions and understandings offered by statistical mechanics are of increasing interest and importance. This text emphasizes how statistical thermodynamics is and can be used by chemical engineers and physical chemists. The text shows readers the path from molecular level approximations to the applied, macroscopic thermodynamic models engineers use, and introduces them to molecular-level computer simulation. Readers of this book will develop an appreciation for the beauty and utility of statistical mechanics.
Table of Contents:
PREFACE FOR INSTRUCTORS v
PREFACE FOR STUDENTS ix
CHAPTER 1 INTRODUCTION TO STATISTICAL THERMODYNAMICS 1
1.1 Probabilistic Description 1
1.2 Macroscopic States and Microscopic States 2
1.3 Quantum Mechanical Description of Microstates 3
1.4 The Postulates of Statistical Mechanics 5
1.5 The Boltzmann Energy Distribution 6
CHAPTER 2 THE CANONICAL PARTITION FUNCTION 9
2.1 Some Properties of the Canonical Partition Function 9
2.2 Relationship of the Canonical Partition Function to Thermodynamic Properties 11
2.3 Canonical Partition Function for a Molecule with Several Independent Energy Modes 12
2.4 Canonical Partition Function for a Collection of Noninteracting Identical Atoms 13
Chapter 2 Problems 15
CHAPTER 3 THE IDEAL MONATOMIC GAS 16
3.1 Canonical Partition Function for the Ideal Monatomic Gas 16
3.2 Identification of β as 1/kT 18
3.3 General Relationships of the Canonical Partition Function to Other Thermodynamic Quantities 19
3.4 The Thermodynamic Properties of the Ideal Monatomic Gas 22
3.5 Energy Fluctuations in the Canonical Ensemble 29
3.6 The Gibbs Entropy Equation 33
3.7 Translational State Degeneracy 35
3.8 Distinguishability, Indistinguishability, and the Gibbs’ Paradox 37
3.9 A Classical Mechanics–Quantum Mechanics Comparison: The Maxwell-Boltzmann Distribution of Velocities 39
Chapter 3 Problems 42
CHAPTER 4 THE IDEAL DIATOMIC AND POLYATOMIC GASES 44
4.1 The Partition Function for an Ideal Diatomic Gas 44
4.1a The Translational and Nuclear Partition Functions 45
4.1b The Rotational Partition Function 45
4.1c The Vibrational Partition Function 47
4.1d The Electronic Partition Function 48
4.2 The Thermodynamic Properties of the Ideal Diatomic Gas 49
4.3 The Partition Function for an Ideal Polyatomic Gas 53
4.4 The Thermodynamic Properties of an Ideal Polyatomic Gas 55
4.5 The Heat Capacities of Ideal Gases 58
4.6 Normal Mode Analysis: The Vibrations of a Linear Triatomic Molecule 59
Chapter 4 Problems 62
CHAPTER 5 CHEMICAL REACTIONS IN IDEAL GASES 64
5.1 The Nonreacting Ideal Gas Mixture 64
5.2 Partition Function of a Reacting Ideal Chemical Mixture 65
5.3 Three Different Derivations of the Chemical Equilibrium Constant in an Ideal Gas Mixture 67
5.4 Fluctuations in a Chemically Reacting System 70
5.5 The Chemically Reacting Gas Mixture: The General Case 73
5.6 Two Illustrations 80
Appendix: The Binomial Expansion 83
Chapter 5 Problems 85
CHAPTER 6 OTHER PARTITION FUNCTIONS 87
6.1 The Microcanonical Ensemble for a Pure Fluid 87
6.2 The Grand Canonical Ensemble for a Pure Fluid 89
6.3 The Isobaric-Isothermal Ensemble 92
6.4 The Restricted Grand or Semi-Grand Canonical Ensemble 93
6.5 Comments on the Use of Different Ensembles 94
Chapter 6 Problems 96
CHAPTER 7 INTERACTING MOLECULES IN A GAS 98
7.1 The Configuration Integral 98
7.2 Thermodynamic Properties from the Configuration Integral 100
7.3 The Pairwise Additivity Assumption 101
7.4 Mayer Cluster Function and Irreducible Integrals 102
7.5 The Virial Equation of State 109
7.6 Virial Equation of State for Polyatomic Molecules 114
7.7 Thermodynamic Properties from the Virial Equation of State 116
7.8 Derivation of Virial Coefficient Formulae from the Grand Canonical Ensemble 118
7.9 Range of Applicability of the Virial Equation 123
Chapter 7 Problems 124
CHAPTER 8 INTERMOLECULAR POTENTIALS AND THE EVALUATION OF THE SECOND VIRIAL COEFFICIENT 125
8.1 Interaction Potentials for Spherical Molecules 125
8.2 The Second Virial Coefficient in a Mixture: Interaction Potentials Between Unlike Atoms 136
8.3 Interaction Potentials for Multiatom, Nonspherical Molecules, Proteins, and Colloids 137
8.4 Engineering Applications and Implications of the Virial Equation of State 140
Chapter 8 Problems 144
CHAPTER 9 MONATOMIC CRYSTALS 147
9.1 The Einstein Model of a Crystal 147
9.2 The Debye Model of a Crystal 150
9.3 Test of the Einstein and Debye Heat Capacity Models for a Crystal 157
9.4 Sublimation Pressure and Enthalpy of Crystals 159
9.5 A Comment on the Third Law of Thermodynamics 161
Chapter 9 Problems 161
CHAPTER 10 SIMPLE LATTICE MODELS FOR FLUIDS 163
10.1 Introduction 164
10.2 Development of Equations of State from Lattice Theory 165
10.3 Activity Coefficient Models for Similar-Size Molecules from Lattice Theory 168
10.4 The Flory-Huggins and Other Models for Polymer Systems 172
10.5 The Ising Model 178
Chapter 10 Problems 184
CHAPTER 11 INTERACTING MOLECULES IN A DENSE FLUID. CONFIGURATIONAL DISTRIBUTION FUNCTIONS 185
11.1 Reduced Spatial Probability Density Functions 185
11.2 Thermodynamic Properties from the Pair Correlation Function 190
11.3 The Pair Correlation Function (Radial Distribution Function) at Low Density 194
11.4 Methods of Determination of the Pair Correlation Function at High Density 197
11.5 Fluctuations in the Number of Particles and the Compressibility Equation 199
11.6 Determination of the Radial Distribution Function of Fluids using Coherent X-ray or Neutron Diffraction 202
11.7 Determination of the Radial Distribution Functions of Molecular Liquids 210
11.8 Determination of the Coordination Number from the Radial Distribution Function 211
11.9 Determination of the Radial Distribution Function of Colloids and Proteins 213
Chapter 11 Problems 214
CHAPTER 12 INTEGRAL EQUATION THEORIES FOR THE RADIAL DISTRIBUTION FUNCTION 216
12.1 The Yvon-Born-Green (YBG) Equation 216
12.2 The Kirkwood Superposition Approximation 219
12.3 The Ornstein-Zernike Equation 220
12.4 Closures for the Ornstein-Zernike Equation 222
12.5 The Percus-Yevick Hard-Sphere Equation of State 227
12.6 The Radial Distribution Functions and Thermodynamic Properties of Mixtures 228
12.7 The Potential of Mean Force 230
12.8 Osmotic Pressure and the Potential of Mean Force for Protein and Colloidal Solutions 237
Chapter 12 Problems 239
CHAPTER 13 DETERMINATION OF THE RADIAL DISTRIBUTION FUNCTION AND FLUID PROPERTIES BY COMPUTER SIMULATION 241
13.1 Introduction to Molecular Level Computer Simulation 242
13.2 Thermodynamic Properties from Molecular Simulation 245
13.3 Monte Carlo Simulation 249
13.4 Molecular-Dynamics Simulation 253
Chapter 13 Problems 255
CHAPTER 14 PERTURBATION THEORY 257
14.1 Perturbation Theory for the Square-Well Potential 257
14.2 First Order Barker-Henderson Perturbation Theory 262
14.3 Second-Order Perturbation Theory 265
14.4 Perturbation Theory Using Other Reference Potentials 269
14.5 Engineering Applications of Perturbation Theory 272
Chapter 14 Problems 274
CHAPTER 15 A THEORY OF DILUTE ELECTROLYTE SOLUTIONS AND IONIZED GASES 276
15.1 Solutions Containing Ions (and Electrons) 276
15.2 Debye-Huckel Theory 280
15.3 The Mean Ionic Activity Coefficient 291
Chapter 15 Problems 296
CHAPTER 16 THE DERIVATION OF THERMODYNAMIC MODELS FROM THE GENERALIZED VAN DER WAALS PARTITION FUNCTION 297
16.1 The Statistical-Mechanical Background 298
16.2 Application of the Generalized van der Waals Partition Function to Pure Fluids 301
16.3 Equation of State for Mixtures from the Generalized van der Waals Partition Function 310
16.4 Activity Coefficient Models from the Generalized van der Waals Partition Function 318
16.5 Chain Molecules and Polymers 329
16.6 Hydrogen-Bonding and Associating Fluids 332
Chapter 16 Problems 334
INDEX 335
About the Author :
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.
