Statistical Mechanics of Liquids and Solutions

Roland Kjellander acquired a masters degree in chemical engineering, a Ph.D. in physical chemistry, and the title of docent in physical chemistry from the Royal Institute of Technology, Stockholm, Sweden. He is currently a professor emeritus of physical chemistry in the Department of Chemistry and Molecular Biology at the University of Gothenburg, Sweden. His previous appointments include roles in various academic and research capacities at the University of Gothenburg, Sweden; Australian National University, Canberra; Royal Institute of Technology, Stockholm, Sweden; Massachusetts Institute of Technology, Cambridge, USA; and Harvard Medical School, Boston, USA. He was awarded the 2004 Pedagogical Prize from the University of Gothenburg, Sweden, and the 2007 Norblad-Ekstrand Medal from the Swedish Chemical Society. Professor Kjellanders field of research is statistical mechanics, in particular liquid state theory.

Contents Preface, xi Overview of Contents, xv Author, xix PART I Basis of Equilibrium Statistical Mechanics CHAPTER 1 Introduction 3 1.1 The Microscopic Definitions of Entropy and Temperature 3 1.1.1 A Simple Illustrative Example 5 1.1.2 Microscopic Definition of Entropy and Temperature for Isolated Systems 12 1.2 Quantum vs Classical Mechanical Formulations of Statistical Mechanics: An Example 17 1.2.1 The Monatomic Ideal Gas: Quantum Treatment 18 1.2.2 The Monatomic Ideal Gas: Classical Treatment 27 Appendix 1A: Alternative Expressions for the Entropy of an Isolated System 31 CHAPTER 2 Statistical Mechanics from a Quantum Perspective 33 2.1 Postulates and Some Basic Definitions 33 2.2 Isolated Systems: The Microcanonical Ensemble 38 2.3 Thermal Equilibria and the Canonical Ensemble 52 2.3.1 The Canonical Ensemble and Boltzmanns Distribution Law 52 2.3.2 Calculations of Thermodynamical Quantities; the Connection with Partition Functions 56 2.3.2.1 The Helmholtz Free Energy 56 2.3.2.2 Thermodynamical Quantities as Averages 59 2.3.2.3 Entropy in the Canonical Ensemble 63 2.4 Constant Pressure: The Isobaric-Isothermal Ensemble 66 2.4.1 Probabilities and the Isobaric-Isothermal Partition Function 66 2.4.2 Thermodynamical Quantities in the Isobaric-Isothermal Ensemble 71 2.4.2.1 The Gibbs Free Energy 71 2.4.2.2 Probabilities and Thermodynamical Quantities 73 2.4.2.3 The Entropy in the Isobaric-Isothermal Ensemble 76 2.5 Open Systems: Chemical Potential and the Grand Canonical Ensemble 79 2.5.1 Probabilities and the Grand Canonical Partition Function 79 2.5.2 Thermodynamical Quantities in the Grand Canonical Ensemble 83 2.6 Fluctuations in Thermodynamical Variables 88 2.6.1 Fluctuations in Energy in the Canonical Ensemble 88 2.6.2 Fluctuations in Number of Particles in the Grand Canonical Ensemble 89 2.6.3 Fluctuations in the Isobaric-Isothermal Ensemble 90 2.7 Independent Subsystems 91 2.7.1 The Ideal Gas and Single-Particle Partition Functions 91 2.7.2 Translational Single-Particle Partition Function 95 Appendix 2A: The Volume Dependence of S and Quasistatic Work 99 Appendix 2B: Stricter Derivations of Probability Expressions 103 CHAPTER 3 Classical Statistical Mechanics 109 3.1 Systems with N Spherical Particles 110 3.2 The Canonical Ensemble 112 3.3 The Grand Canonical Ensemble 122 3.4 Real Gases 125 CHAPTER 4 Illustrative Examples from Some Classical Theories of Fluids 131 4.1 The Ising Model 131 4.2 The Ising Model Applied to Lattice Gases and Binary Liquid Mixtures 134 4.2.1 Ideal Lattice Gas 135 4.2.2 Ideal Liquid Mixture 136 4.2.3 The Bragg-William Approximation 138 4.2.3.1 Regular Solution Theory 138 4.2.3.2 Some Applications of Regular Solution Theory 142 4.2.3.3 Flory-Huggins Theory for Polymer Solutions 151 PART II Fluid Structure and Interparticle Interactions CHAPTER 5 Interaction Potentials and Distribution Functions 165 5.1 Bulk Fluids of Spherical Particles. The Radial Distribution Function 166 5.2 Number Density Distributions: Density Profiles 172 5.3 Force Balance and the Boltzmann Distribution for Density: Potential of Mean Force 175 5.4 The Relationship to Free Energy and Chemical Potential 181 5.5 Distribution Functions of Various Orders for Spherical Particles 184 5.5.1 Singlet Distribution Function 184 5.5.2 Pair Distribution Function 185 5.5.3 Distribution Functions in the Canonical Ensemble 188 5.6 The structure factor for homogeneous and inhomogeneous fluids 192 5.7 Thermodynamical Quantities from Distribution Functions 199 5.8 Microscopic density distributions and density-density correlations 212 5.9 Distribution Function Hierarchies and Closures, Preliminaries 215 5.10 Distribution Functions in the Grand Canonical Ensemble 218 5.11 The Born-Green-Yvon Equations 222 5.12 Mean Field Approximations for Bulk Systems 225 5.13 Computer Simulations and Distribution Functions 227 5.13.1 General Background 227 5.13.1.1 Basics