Statistical Thermodynamics

IWAO TERAOKA, PHD, is a professor of chemistry at Tandon School of Engineering of NYU. His current research areas include photonic sensors and liquid chromatography, but in early years he worked on physical chemistry of polymer solutions. He is a recipient of the 1994 NSF Young Investigator Award. Professor Teraoka is also the author of Polymer Solutions: An Introduction to Physical Properties and more than 100 original papers and 6 book chapters.

• Preface xiiiAcknowledgments xviiAbout the Companion Website xixSymbols and Constants xxi1 Introduction 11.1 Classical Thermodynamics and Statistical Thermodynamics 11.2 Examples of Results Obtained from Statistical Thermodynamics 21.2.1 Heat Capacity of Gas of Diatomic Molecules 21.2.2 Heat Capacity of a Solid 31.2.3 Blackbody Radiation 31.2.4 Adsorption 41.2.5 Helix–Coil Transition 51.2.6 Boltzmann Factor 61.3 Practices of Notation 62 Review of Probability Theory 92.1 Probability 92.2 Discrete Distributions 112.2.1 Binomial Distribution 122.2.2 Poisson Distribution 132.2.3 Multinomial Distribution 142.3 Continuous Distributions 152.3.1 Uniform Distribution 192.3.2 Exponential Distribution 192.3.3 Normal Distribution 212.3.4 Distribution of a Dihedral Angle 212.4 Means and Variances 222.4.1 Discrete Distributions 222.4.2 Continuous Distributions 262.4.3 Central Limit Theorem 272.5 Uncertainty 28Problems 313 Energy and Interactions 353.1 Kinetic Energy and Potential Energy of Atoms and Ions 353.1.1 Kinetic Energy 353.1.2 Gravitational Potential 363.1.3 Ion in an Electric Field 363.1.4 Total Energy of Atoms and Ions 373.2 Kinetic Energy and Potential Energy of Diatomic Molecules 373.2.1 Kinetic Energy (Translation, Rotation, Vibration) 373.2.2 Dipolar Potential 423.2.2.1 Potential of a Permanent Dipole 423.2.2.2 Potential of an Induced Dipole 443.3 Kinetic Energy of Polyatomic Molecules 463.3.1 Linear Polyatomic Molecule 463.3.2 Nonlinear Polyatomic Molecule 483.4 Interactions Between Molecules 503.4.1 Excluded-Volume Interaction 523.4.2 Coulomb Interaction 523.4.3 Dipole–Dipole Interaction 533.4.4 van der Waals Interaction 543.4.5 Lennard-Jones Potential 553.5 Energy as an Extensive Property 573.6 Kinetic Energy of a Gas Molecule in Quantum Mechanics 583.6.1 Quantization of Translational Energy 583.6.2 Quantization of Rotational Energy 613.6.3 Quantization of Vibrational Energy 633.6.4 Electronic Energy Levels 653.6.5 Comparison of Energy Level Spacings 66Problems 674 Statistical Mechanics 694.1 Basic Assumptions, Microcanonical Ensembles, and Canonical Ensembles 694.1.1 Basic Assumptions 694.1.2 Microcanonical Ensembles 734.1.3 Canonical Ensembles 754.2 Probability Distribution in Canonical Ensembles and Partition Functions 774.2.1 Probability Distribution 774.2.2 Partition Function for a System with Discrete States 794.2.3 Partition Function for a System with Continuous States 814.2.4 Energy Levels and States 834.3 Internal Energy 884.4 Identification of 𝛽 894.5 Equipartition Law 914.6 Other Thermodynamic Functions 934.7 Another View of Entropy 974.8 Fluctuations of Energy 994.9 Grand Canonical Ensembles 1004.10 Cumulants of Energy 107Problems 1105 Canonical Ensemble of Gas Molecules 1135.1 Velocity of Gas Molecules 1135.2 Heat Capacity of a Classical Gas 1165.2.1 Point Mass 1175.2.2 Rigid Dumbbell 1175.2.3 Elastic Dumbbell 1185.3 Heat Capacity of a Quantum-Mechanical Gas 1205.3.1 General Formulas 1205.3.2 Translation 1225.3.3 Rotation 1245.3.4 Vibration 1275.3.5 Comparison with Classical Models 1285.4 Distribution of Rotational Energy Levels 1295.5 Conformations of a Molecule 130Problems 1326 Indistinguishable Particles 1356.1 Distinguishable Particles and Indistinguishable Particles 1356.2 Partition Function of Indistinguishable Particles 1376.2.1 System of Distinguishable Particles 1376.2.2 System of Indistinguishable Particles 1376.3 Condition of Nondegeneracy 1426.4 Significance of Division by N! 1446.4.1 Gas in a Two-Part Box 1446.4.2 Chemical Potential 1456.4.3 Mixture of Two Gases 1466.5 Indistinguishability and Center-of-Mass Movement 1476.6 Open System of Gas 147Problems 1497 Imperfect Gas 1537.1 Virial Expansion 1537.2 Molecular Expression of Interaction in the Canonical Ensemble 1577.3 Second Virial Coefficients in Different Models 1647.3.1 Hard-Core Repulsion Only 1647.3.2 Square-well Potential 1657.3.3 Lennard-Jones Potential 1677.4 Joule–Thomson Effect 167Problems 1718 Rubber Elasticity 1758.1 Rubber 1758.2 Polymer Chain in One Dimension 1768.3 Polymer Chain in Three Dimensions 1808.4 Network of Springs 184Problems 1859 Law of Mass Action 1899.1 Reaction of Two Monatomic Molecules 1909.2 Decomposition of Homonuclear Diatomic Molecules 1939.3 Isomerization 1959.4 Method of the Steepest Descent 197Problems 19810 Adsorption 20110.1 Adsorption Phenomena 20110.2 Langmuir Isotherm 20210.3 BET Isotherm 20610.4 Dissociative Adsorption 21110.5 Interaction Between Adsorbed Molecules 213Problems 21311 Ising Model 21711.1 Model 21711.2 Partition Function 22011.2.1 One-Dimensional Ising Model 22011.2.2 Calculating Statistical Averages 22111.2.2.1 Average Number of Up Spins 22211.2.2.2 Average of the Number of Spin Alterations (Number of Domains – 1) 22211.2.2.3 Domain Size 22311.2.2.4 Size of a Domain of Uniform Spins 22311.2.3 A Few Examples of 1D Ising Model 22311.2.3.1 Linear Ising Model, N = 3 22311.2.3.2 Ring Ising Model, N = 3 22511.2.3.3 Ring Ising Model, N = 4 22511.3 Mean-FieldTheories 22611.3.1 Bragg–Williams (B–W) Approximation 22711.3.2 Flory–Huggins (F–H) Approximation 23111.3.3 Approximation by a Mean-Field (MF) Theory 23511.4 Exact Solution of 1D Ising Model 23611.4.1 General Formula 23611.4.2 Large-N Approximation 23911.4.3 Exact Partition Function for Arbitrary N 24111.4.4 Ring Ising Model, Arbitrary N 24411.4.5 Comparison of the Exact Results with Those of Mean-Field Approximations 24511.5 Variations of the Ising Model 24711.5.1 System of Uniform Spins 24711.5.2 Random Local Fields of Opposite Directions 24911.5.3 Dilute Local Fields 252Problems 25412 Helical Polymer 26312.1 Helix-Forming Polymer 26312.2 Optical Rotation and Circular Dichroism 26612.3 Pristine Poly(n-hexyl isocyanate) 26712.4 Variations to the Helical Polymer 27112.4.1 Copolymer of Chiral and Achiral Isocyanate Monomers 27212.4.2 Copolymer of R- and S-Enantiomers of Isocyanate 274Problems 27413 Helix–Coil Transition 27713.1 Historical Background 27713.2 Polypeptides 28113.3 Zimm–Bragg Model 283Problems 28914 Regular Solutions 29114.1 Binary Mixture of Equal-Size Molecules 29114.1.1 Free Energy of Mixing 29114.1.2 Derivatives of the Free Energy of Mixing 29614.1.3 Phase Separation 30014.2 Binary Mixture of Molecules of Different Sizes 304Problems 312Appendix A Mathematics 315A.1 Hyperbolic Functions 315A.2 Series 317A.3 Binomial Theorem and Trinomial Theorem 317A.4 Stirling’s formula 318A.5 Integrals 318A.6 Error Functions 318A.7 Gamma Functions 319References 321Index 325