General and Statistical Thermodynamics

General and Statistical Thermodynamics
	Preface to the Second Edition
	Preface to the First Edition
	Contents
1 Definitions and the Zeroth Law
	1.1 Some Definitions
	1.2 Large Numbers
	1.3 The Zeroth Law
	1.4 Some Mathematical Procedures
		1.4.1 Exact Differential
	1.5 Cyclic Identity
	1.6 Jacobian
		1.6.1 A Simple Technique
		1.6.2 Jacobian Employed
	1.7 Additional Helpful Identities
		1.7.1 Cyclic Identity Rederived
		1.7.2 Simple Identity
		1.7.3 Mixed Identity
	References
2 Perfect Gas
	2.1 Model
		2.1.1 Pressure
		2.1.2 Temperature
	2.2 Statistical Techniques
		2.2.1 Internal Energy
		2.2.2 Equation of State
	2.3 Monatomic and Diatomic Perfect Gases
		2.3.1 Monatomic Perfect Gas
		2.3.2 Diatomic Perfect Gas
	2.4 Mixture of Perfect Gases
	2.5 Dalton\'s Law of Partial Pressure
	2.6 Perfect Gas Atmosphere
		2.6.1 Barometric Equation
		2.6.2 A Related Calculation
	2.7 Energy of Isothermal Atmosphere
		2.7.1 Atmosphere with Height-Dependent Temperature
	2.8 Perfect Gas of Extremely Relativistic Particles
		2.8.1 Problems 2.1–2.7
		2.8.2 Exercises for the Student
3 The First Law
	3.1 Heat, Work, and Internal Energy
		3.1.1 Heat
		3.1.2 Work
		3.1.3 Internal Energy
	3.2 Specific Heat
	3.3 Notation
	3.4 Some Applications of the First Law
	3.5 Independent t and v
		3.5.1 Problems 3.1–3.32
	3.6 Independent t and p
	3.7 Independent p and v
		3.7.1 The First Law: Another Version
	3.8 Enthalpy
		3.8.1 Enthalpy is a State Function
		3.8.2 Enthalpy and the First Law
	3.9 Hess\' Rules
		3.9.1 Chemothermal Reactions
	3.10 Oxidation, Heat of Vaporization
	3.11 Ideal Gas Adiabatics and Polytropics
		3.11.1 Ideal Gas Adiabatics
	3.12 Some Interrelationships
	3.13 Equation of State from Bulk and Elastic Moduli
	3.14 Newton\'s Law of Cooling
	3.15 Internal Energy in Noninteracting Monatomic Gases Equals 3/2PV
	3.16 Volume Dependence of Single Particle Energy Levels
	References
4 The Second Law
	4.1 Ideal Gas Heat Engines
		4.1.1 Nonexistence of Perpetual Machines of the Second Kind
	4.2 Perfect Carnot Engine
	4.3 Kelvin Description of Absolute Temperature
	4.4 Infinitesimal and Finite Carnot Cycles
	4.5 Entropy Calculation
	4.6 Perfect Carnot Engine with Arbitrary Working Substance
	4.7 Statements of the Second Law
		4.7.1 The Carnot Statement
		4.7.2 Clausius Statement
	4.8 Entropy Increase in Spontaneous Processes
	4.9 Energy Exchange Increases Total Entropy
	4.10 Kelvin–Planck Version of the Second Law
		4.10.1 Entropy Always Increases in Irreversible Adiabats
	4.11 Non-Carnot Heat Cycle and Clausius Inequality
		4.11.1 Integral Form of Clausius Inequality
		4.11.2 Differential Form of Clausius Inequality
	4.12 Solved Problems 4.2–4.24
	4.13 Carnot Refrigerator
	4.14 Idealized Version of Realistic Engine Cycles
	4.15 Negative Temperature: Cursory Remark
	References
5 Introduction and the Zeroth Law
	5.1 The First and Second Laws
		5.1.1 The First–Second Law: The Clausius Version
		5.1.2 The First td s Equation
	5.2 Two Chambers: Entropy Calculation
		5.2.1 Two Chambers: Mixing of Ideal Gases
	5.3 Velocity of Sound: Newton\'s Solution
	5.4 Van der Waal\'s Gas: Energy and Entropy Change
6 Van der Waals Theory of Imperfect Gases
	6.1 Interacting Molecules
		6.1.1 Hard Core Volume Reduction
	6.2 Van der Waals Virial Expansion
	6.3 Critical Point
		6.3.1 Critical Constants Pc,Vc,Tc
	6.4 Reduced Equation of State
		6.4.1 Critical Region
	6.5 Behavior Below Tc
		6.5.1 Maxwell Construction
	6.6 Molar Specific Volume and Density
		6.6.1 Temperature Just Below the Critical Point
	6.7 Lever Rule
	6.8 Smooth Transition from Liquid to Gas and Vice Versa
	6.9 Principle of Corresponding States
		6.9.1 Figure 6.5.a: X0 as a Function of p0 for Various Fluids
	6.10 Dieterici\'s Equation of State
		6.10.1 Figure 6.9: Dieterici Isotherms
	References
7 Joule and Kelvin: Internal Energy and Enthalpy
	7.1 Gay-Lussac–Joule Coefficient
		7.1.1 Measurement of η(t,v)
	7.2 Enthalpy: Description
	7.3 Enthalpy Remaining Unchanged
	7.4 Joule–Kelvin Effect: Derivation
		7.4.1 JK Coefficient: Van der Waals Gas
		7.4.2 JK Coefficient: Inversion Point
		7.4.3 JK Coefficient: Positive and Negative Regions
	7.5 Enthalpy Minimum: Gas with Three Virial Coefficients
	7.6 From Empirical to Thermodynamic Temperature
		7.6.1 Thermodynamic Temperature Scale: Via JGL Coefficient
		7.6.2 Thermodynamic Scale: Via JK Coefficient
		7.6.3 Temperature of Ice-Point: An Estimate
		7.6.4 Temperature: Ideal Gas Thermodynamic Scale
	7.7 Negative Temperature: Cursory Remark
	References
8 Euler Equation and Gibbs–Duhem Relation
	8.1 Euler Equation
		8.1.1 Chemical Potential
		8.1.2 Multiple-Component Systems
	8.2 Equations of State
		8.2.1 Callen\'s Remarks
		8.2.2 Equation of State: Energy Representation
		8.2.3 Equation of State: Entropy Representation
		8.2.4 Equations of State: Two Equations for an Ideal Gas
		8.2.5 Where Is the Third Equation of State?
	8.3 Gibbs–Duhem Relation: Energy Representation
		8.3.1 Gibbs–Duhem Relation: Entropy Representation
	8.4 Ideal Gas: Third Equation of State
	8.5 Ideal Gas: Fundamental Equation
		8.5.1 Ideal Gas: Entropy Representation
		8.5.2 Ideal Gas: Energy Representation
		8.5.3 Ideal Gas: Three Equations of State
	References
9 Le Châtelier Principle
	9.1 The Zeroth Law of Thermodynamics: Second Look
		9.1.1 The Zeroth Law of Thermodynamics: Reconfirmed
	9.2 Entropy Extremum: Maximum Possible
		9.2.1 Heat Energy Flow
		9.2.2 Molecular Flow
		9.2.3 Isothermal Compression
		9.2.4 Energy Extremum: Minimum Possible
	9.3 Motive Forces: Energy Formalism
		9.3.1 Isobaric Entropy Flow
		9.3.2 Isothermal–Isobaric Molecular Flow
		9.3.3 Isothermal Compression
	9.4 Physical Criteria for Thermodynamic Stability
		9.4.1 Le Châtelier\'s Principle
		9.4.2 Stable Self-Equilibrium
	9.5 The First and Second Requirement
		9.5.1 Implications of the First Requirement
		9.5.2 Implications of the Second Requirement
	9.6 Intrinsic Thermodynamic Stability: Chemical Potential
		9.6.1 Intrinsic Stability: CP and χS>0
10 Gibbs, Helmholtz, and Clausius–Clapeyron
	10.1 Energy and Entropy Extrema
	10.2 Single Variety Constituent Systems: Two Phases
	10.3 Minimum Energy in an Adiabatically Isolated System
	10.4 Relative Size of Phases and Energy Minimum
		10.4.1 Specific Internal Energy of Two Phases Is Equal
	10.5 Maximum Entropy in an Adiabatically Isolated System
	10.6 Relative Size of Phases and Entropy Maximum
		10.6.1 Specific Entropy of Two Phases Is Equal
	10.7 Legendre Transformations
	10.8 Helmholtz Free Energy
		10.8.1 Helmholtz Thermodynamic Potential
		10.8.2 Clausius Inequality in Differential Form
		10.8.3 Maximum Possible Work
		10.8.4 Helmholtz Free Energy Is Decreased
		10.8.5 Helmholtz Free Energy: Extremum Principle
		10.8.6 Helmholtz Free Energy: Relative Size of Phases
		10.8.7 Specific Helmholtz Free Energy Is Equal for Different Phases
	10.9 Gibbs Free Energy
		10.9.1 Maximum Available Work: Constant t and p
	10.10 Decrease in Gibbs Free Energy
		10.10.1 The Pd V Work
		10.10.2 Gibbs Free Energy: Extremum Principle
		10.10.3 Gibbs Potential Minimum: Relative Size of Phases
		10.10.4 Specific Gibbs Free Energy: Equality for Different Phases
	10.11 Enthalpy: Remark
	10.12 Heat of Transformation
	10.13 Thermodynamic Potentials: s,f,g, and h
	10.14 Characteristic Equations
		10.14.1 Helmholtz Potential to Internal Energy
		10.14.2 Gibbs Potential to Enthalpy
	10.15 Maxwell Relations
	10.16 Metastable Equilibrium
	10.17 The Clausius–Clapeyron Differential Equation
	10.18 Gibbs Phase Rule
		10.18.1 MultiPhase, Multiconstituent Systems
	10.19 Phase Equilibrium Relationships
		10.19.1 Phase Rule
	10.20 The Variance
		10.20.1 Invariant Systems
		10.20.2 Monovariant Systems
	10.21 Phase Rule for Systems with Chemical Reactions
	References
11 Statistical Thermodynamics, the Third Law
	11.1 Boltzmann–Maxwell–Gibbs Ideas and Helmholtz Free Energy
	11.2 Noninteracting Classical Systems: Monatomic Perfect Gas in Three Dimensions
		11.2.1 Partition Function: Classical Monatomic Perfect Gas in Three Dimensions
		11.2.2 Monatomic Perfect Gas: Thermodynamic Potentials
	11.3 Same Monatomic Perfect Gas at Different Pressure: Changes Due to Isothermal Mixing
		11.3.1 Change in the Entropy Due to Mixing
	11.4 Different Monatomic Ideal Gases: Mixed at Same Temperature
		11.4.1 Thermodynamic Potentials
		11.4.2 Gibbs Paradox
	11.5 Perfect Gas of Classical Diatoms
		11.5.1 Noninteracting Free Diatoms
		11.5.2 Experimental Observation of Specific Heat
		11.5.3 Center of Mass: Motion of and Around
		11.5.4 Center of Mass: Translational Motion
	11.6 Transformation to Spherical Coordinates: Classical Diatom with Stationary Center of Mass
	11.7 Diatom with Stiff Bond: Rotational Kinetic Energy
		11.7.1 Diatoms with Free Bonds
	11.8 Classical Diatoms: High Temperature
	11.9 Simple Oscillators: Anharmonic
	11.10 Classical Dipole Pairs: Average Energy and Force
		11.10.1 Distribution Factor and Thermal Average
		11.10.2 Average Force Between a Pair
	11.11 Langevin Paramagnetism: Classical Picture
		11.11.1 Langevin Paramagnetism: Statistical Average
		11.11.2 Langevin Paramagnetism: High Temperature
		11.11.3 Langevin Susceptibility
		11.11.4 Langevin Paramagnetism: Low Temperature
	11.12 Extremely Relativistic Monatomic Ideal Gas
	11.13 Hamiltonian: Gas with Interaction
	11.14 Mayer\'s Cluster Expansion: Partition Function
	11.15 Hard-Core Interaction
	11.16 Lennard-Jones Potential
		11.16.1 Attractive Potential
	11.17 Quantum Mechanics: Cursory Remark
	11.18 Canonical Partition Function
	11.19 Quantum Particles
		11.19.1 Quantum Particles: Motion in One Dimension
	11.20 Classical Coquantum Gas
	11.21 Noninteracting Particles: Classical Coquantum Versus Quantum Statistics
	11.22 Quasiclassical Statistical Thermodynamics of Rigid Quantum Diatoms
	11.23 Heteronuclear Diatoms: Rotational Motion
	11.24 Partition Function: Quantum
	11.25 Partition Function: Analytical Treatment
	11.26 Thermodynamic Potential: Low Temperature
		11.26.1 Thermodynamic Potential: High Temperature
	11.27 Homonuclear Diatoms: Rotational Motion
		11.27.1 Homonuclear Diatoms: Very High Temperature
		11.27.2 Homnuclear Diatoms: Very Low and Intermediate Temperature
	11.28 Diatoms with Vibrational Motion
	11.29 Quantum-Statistical Treatment: Quasiclassical
	11.30 Langevin Paramagnet: Quantum-Statistical Picture
	11.31 Helmholtz Potential and Partition Function
	11.32 System Entropy: High Temperature
	11.33 Internal Energy
	11.34 Specific Heat: General Temperature
		11.34.1 Specific Heat: High Temperature
		11.34.2 Specific Heat: Low Temperature
		11.34.3 Langevin Paramagnet: Classical Statistical Picture
	11.35 Adiabatic Demagnetization: Very Low Temperatures
	11.36 Third Law: Nernst\'s Heat Theorem
	11.37 Negative Temperatures
	11.38 Grand Canonical Ensemble: Classical Systems
		11.38.1 Grand Canonical Ensemble: Partition Function
	11.39 Quantum States: Statistics
	11.40 Fermi–Dirac: Noninteracting System
		11.40.1 Fermi–Dirac: Single-State Partition Function
	11.41 Bose–Einstein: Noninteracting System
		11.41.1 Bose–Einstein: Chemical Potential Always Negative
		11.41.2 Bose–Einstein: Partition Function
	11.42 Fermi–Dirac and Bose–Einstein Systems
		11.42.1 Pressure, Internal Energy, and Chemical Potential
	11.43 Perfect Fermi–Dirac System
		11.43.1 Weakly-Degenerate Fermi–Dirac System
	11.44 Virial Expansion for a Perfect Fermi–Dirac Gas
	11.45 Highly or Partially-Degenerate Fermi–Dirac
		11.45.1 Complete Degeneracy: Zero Temperature
		11.45.2 Partial Degeneracy: Finite but Low Temperature
	11.46 Thermodynamic Potentials
	11.47 Pauli Paramagnetism
		11.47.1 Pauli Paramagnetism: Zero Temperature
		11.47.2 Pauli Paramagnetism: Finite Temperature
		11.47.3 Pauli Paramagnetism: Very High Temperature
	11.48 Hand-Waving Argument: Specific Heat
		11.48.1 Hand-Waving Argument: Zero Temperature Pauli Paramagnetism
		11.48.2 Hand-Waving Argument: Pauli Paramagnetism at Finite Temperature
	11.49 Landau Diamagnetism
	11.50 Richardson Effect: Thermionic Emission
		11.50.1 Richardson Effect: Quasiclassical Statistics
	11.51 Bose–Einstein Gas: Low Density, High Temperature
		11.51.1 Bose–Einstein Gas: Grand Potential
		11.51.2 Bose–Einstein Gas: Three Dimensions
		11.51.3 Bose–Einstein Gas: Condensation Temperature T<=Tc
		11.51.4 Bose–Einstein Gas: Pressure and Internal Energy
		11.51.5 Bose–Einstein Gas: Degenerate Ideal Gas
	11.52 Degenerate Regime: Specific Heat
		11.52.1 Degenerate Regime: State Functions
	11.53 Bose–Einstein Gas: Nondegenerate Regime Specific Heat
	11.54 Bose–Einstein Condensation: In δ-Dimensions
	11.55 Critical Temperature: Bose–Einstein Gas
		11.55.1 Temperature Dependence of Condensate: Bose–Einstein Gas
		11.55.2 Pressure and Internal Energy of Condensate: Bose–Einstein Gas
	11.56 Black Body Radiation: Thermodynamic Consideration
		11.56.1 Black Body Radiation: Chemical Potential Zero
		11.56.2 Black Body Radiation: Energy
		11.56.3 Black Body Radiation: Pressure
		11.56.4 Black Body Radiation: Internal Energy
		11.56.5 Black Body Radiation: Other Thermodynamic Potentials
	11.57 Phonons: In a Continuum
		11.57.1 Phonons in Lattices
		11.57.2 Phonons: Einstein Approximation
		11.57.3 Phonons: Debye Approximation
	References
12 Second-Order Phase Transitions
	12.1 Landau Theory
		12.1.1 Ginzburg Contribution
	References
13 Cooper Pair
	13.1 Fermi Sea
		13.1.1 Classical Hamiltonian
		13.1.2 Quantum Hamiltonian
		13.1.3 Center-of-Mass Coordinates
	13.2 Schrödinger Equation
	13.3 Solution of Schrödinger Equation
		13.3.1 No Center-of-Mass Motion
	13.4 Binding Energy
	References
14 Bogolyubov Representation
	14.1 Hamiltonian and Solution
	14.2 Quasiparticles
	References
15 London and London Theory
	15.1 Electricity and Magnetism
	15.2 Perfect Conductivity
	15.3 One-Dimensional Theory
	15.4 Arbitrary Change of Theory: Meissner Effect
	References
16 Superconductivity
	16.1 Theory
	16.2 [HBCS]MFA Hamiltonian
	16.3 Fermionic Quasiparticles
	16.4 Procedure to Invert
	16.5 Nondiagonal Hamiltonian: Elimination of Terms
	16.6 Transformation of Hnondiagonal
	16.7 Evaluation of Diagonal Terms
	16.8 Fermi Surface
	16.9 Fermi Sea: Above and Within
	16.10 Gap Function and Solutions
	16.11 Critical Temperature Tc
	16.12 Debye Potential hΩD
	16.13 Isotope Effect
	16.14 Specific Heat Near Tc
	16.15 Specific Heat: Calculation of Jump in
	16.16 A Universal Ratio
	16.17 Another Universal Ratio and Result at Zero Temperature
	References
A Large Numbers. The Most Probable State
	A.1 Random Number Generator
	A.2 Binomial Expansion
	A.3 Large Number of Calls
		A.3.1 Remark: A Gaussian Distribution
B Moments of the Distribution Function: Remark
	B.1 Moments of the Gaussian Distribution Function
		B.1.1 Unnormalized Moments: Gaussian Distribution
		B.1.2 Normalized Moments of the Gaussian Distribution Function
C Normalized Moments of the Exact Distribution Function for General Occupancy
	C.1 Exact Second Normalized Moment
	C.2 Exact Third–Sixth Normalized Moments
		C.2.1 The Third Moment
		C.2.2 The Fourth Moment
		C.2.3 Calculation of the Fifth and Sixth Normalized Moments
	C.3 Concluding Remark
	C.4 Summary
D Perfect Gas Revisited
	D.1 Monatomic Perfect Gas
		D.1.1 Monatomic Perfect Gas: Pressure
	D.2 Classical Statistics: Boltzmann–Maxwell–Gibbs Distribution
		D.2.1 Energy in a Monatomic Perfect Gas
E Second Law: Carnot Version Leads to Clausius Version
	E.1 A Carnot and an Ordinary Engine in Tandem
F Positivity of the Entropy Increase: Equation (4.71)
G Mixture of Van der Waals Gases
H Positive-Definite Homogeneous Quadratic Form
	H.1 Positive Definiteness of 3x3 Quadratic Form
	H.2 Helpful Surprise
I Thermodynamic Stability: Three Extensive Variables
	I.1 Energy Minimum Procedure: Intrinsic Stability
	I.2 The First Requirement for Intrinsic Stability
	I.3 The Second Requirement for Intrinsic Stability
	I.4 The Third Requirement For Intrinsic Stability
	I.5 Solved Problems
J Massieu Transforms: The Entropy Representation
	J.1 Massieu Potential, M{v,u}
		J.1.1 Massieu Potential, M {v,1/t }
		J.1.2 Massieu Potential, M {p/t,u }
		J.1.3 Massieu Potential, M {p/t,1/t }
K Integral (11.83)
	K.1 Average Force Between a Pair
L Indistinguishable, Noninteracting Quantum Particles
	L.1 Quantum Statistics: Grand Canonical Partition Function
M Landau Diamagnetism
	M.1 Multiplicity Factor: Landau Diamagnetism
	M.2 High Temperature: Landau Diamagnetism
N Specific Heat for the B–E Gas
O Bogolyubov Transformation: Inversion Procedure
List of Problems and Proofs
Scientists
Index