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