| Modern Thermodynamics |
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| John Denker
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- 0 Introduction
- 0.1 Overview
- 0.2 Availability
- 0.3 Prerequisites, Goals, and Non-Goals
- 1 Energy
- 1.1 Preliminary Remarks
- 1.2 Conservation of Energy
- 1.3 Examples of Energy
- 1.4 Remark: Recursion
- 1.5 Energy is Completely Abstract
- 1.6 Additional Remarks
- 1.7 Energy versus “Capacity to do Work” or “Available Energy”
- 1.7.1 Best Case : Non-Thermal Situation
- 1.7.2 Equation versus Definition
- 1.7.3 General Case : Some Energy Not Available
- 1.8 Conflict with the Vernacular
- 1.8.1 Energy
- 1.8.2 Conservation
- 1.8.3 Energy Conservation
- 1.9 Range of Validity
- 1.10 Internal Energy
- 2 Entropy
- 2.1 Paraconservation
- 2.2 Scenario: Cup Game
- 2.3 Scenario: Card Game
- 2.4 Peeking
- 2.5 Discussion
- 2.5.1 States and Probabilities
- 2.5.2 Entropy is Not Knowing
- 2.5.3 Entropy versus Energy
- 2.5.4 Entropy versus Disorder
- 2.5.5 False Dichotomy, or Not
- 2.5.6 dQ, or Not
- 2.6 Quantifying Entropy
- 2.7 Microstate versus Macrostate
- 2.7.1 Surprisal
- 2.7.2 Contrasts and Consequences
- 2.8 Entropy of Independent Subsystems
- 3 Basic Concepts (Zeroth Law)
- 4 Low-Temperature Entropy (Alleged Third Law)
- 5 The Rest of Physics, Chemistry, etc.
- 6 Classical Thermodynamics
- 6.1 Overview
- 6.2 Stirling Engine
- 6.2.1 Basic Structure and Operations
- Cooling
- Warming
- Expansion
- Compression
- 6.2.2 Energy, Entropy, and Efficiency
- 6.2.3 Practical Considerations
- 6.2.4 Discussion: Reversibility
- 6.3 All Reversible Heat Engines are Equally Efficient
- 6.4 Not Everything is a Heat Engine
- 6.5 Carnot Efficiency Formula
- 6.5.1 Definition of Heat Engine
- 6.5.2 Analysis
- 6.5.3 Discussion
- 7 Functions of State
- 7.1 Functions of State : Basic Notions
- 7.2 Path Independence
- 7.3 Hess’s Law, Or Not
- 7.4 Partial Derivatives
- 7.5 Heat Capacities, Energy Capacity, and Enthalpy Capacity
- 7.6 Yet More Partial Derivatives
- 7.7 Integration
- 7.8 Advection
- 7.9 Deciding What’s True
- 7.10 Deciding What’s Fundamental
- 8 Thermodynamic Paths and Cycles
- 8.1 A Path Projected Onto State Space
- 8.1.1 State Functions
- 8.1.2 Out-of-State Functions
- 8.1.3 Converting Out-of-State Functions to State Functions
- 8.1.4 Reversibility and/or Uniqueness
- 8.1.5 The Importance of Out-of-State Functions
- 8.1.6 Heat Content, or Not
- 8.1.7 Some Mathematical Remarks
- 8.2 Grady and Ungrady One-Forms
- 8.3 Abuse of the Notation
- 8.4 Procedure for Extirpating dW and dQ
- 8.5 Some Reasons Why dW and dQ Might Be Tempting
- 8.6 Boundary versus Interior
- 8.7 The Carnot Cycle
- 9 Connecting Entropy with Energy
- 9.1 The Boltzmann Distribution
- 9.2 Systems with Subsystems
- 9.3 Remarks
- 9.3.1 Predictable Energy is Freely Convertible; Random Energy is Not
- 9.3.2 Thermodynamic Laws without Temperature
- 9.3.3 Kinetic and Potential Microscopic Energy
- 9.3.4 Ideal Gas : Potential Energy as well as Kinetic Energy
- 9.3.5 Relative Motion versus “Thermal” Energy
- 9.4 Entropy Without Constant Re-Shuffling
- 9.5 Units of Entropy
- 9.6 Probability versus Multiplicity
- 9.6.1 Exactly Equiprobable
- 9.6.2 Approximately Equiprobable
- 9.6.3 Not At All Equiprobable
- 9.7 Discussion
- 9.8 Misconceptions about Spreading
- 9.9 Spreading in Probability Space
- 10 Additional Fundamental Notions
- 10.1 Equilibrium
- 10.2 Non-Equilibrium; Timescales
- 10.3 Efficiency; Timescales
- 10.4 Spontaneity and Irreversibility
- 10.5 Stability
- 10.6 Relationship between Static Stability and Damping
- 10.7 Finite Size Effects
- 10.8 Words to Live By
- 11 Experimental Basis
- 11.1 Basic Notions of Temperature and Equilibrium
- 11.2 Exponential Dependence on Energy
- 11.3 Metastable Systems with a Temperature
- 11.4 Metastable Systems without a Temperature
- 11.5 Dissipative Systems
- 11.5.1 Sudden Piston : Sound
- 11.5.2 Sudden Piston : State Transitions
- 11.5.3 Rumford’s Experiment
- 11.5.4 Simple Example: Decaying Current
- 11.5.5 Simple Example: Oil Bearing
- 11.5.6 Misconceptions : Heat
- 11.5.7 Misconceptions : Work
- 11.5.8 Remarks
- 11.6 The Gibbs Gedankenexperiment
- 11.7 Spin Echo Experiment
- 11.8 Melting
- 11.9 Isentropic Expansion and Compression
- 11.10 Demagnetization Refrigerator
- 11.11 Thermal Insulation
- 12 More About Entropy
- 12.1 Terminology: Microstate versus Macrostate
- 12.2 What the Second Law Doesn’t Tell You
- 12.3 Phase Space
- 12.4 Entropy in a Crystal; Phonons, Electrons, and Spins
- 12.5 Entropy is Entropy
- 12.6 Spectator Entropy
- 12.7 No Secret Entropy, No Hidden Variables
- 12.8 Entropy is Context Dependent
- 12.9 Slice Entropy and Conditional Entropy
- 12.10 Extreme Mixtures
- 12.10.1 Simple Model System
- 12.10.2 Two-Sample Model System
- 12.10.3 Helium versus Snow
- 12.10.4 Partial Information aka Weak Peek
- 12.11 Entropy is Not Necessarily Extensive
- 12.12 Mathematical Properties of the Entropy
- 12.12.1 Entropy Can Be Infinite
- 13 Temperature : Definition and Fundamental Properties
- 13.1 Example Scenario: Two Subsystems, Same Stuff
- 13.2 Remarks about the Simple Special Case
- 13.3 Two Subsystems, Different Stuff
- 13.4 Discussion: Constants Drop Out
- 13.5 Calculations
- 13.6 Chemical Potential
- 14 Spontaneity, Reversibility, Equilibrium, Stability, Solubility, etc.
- 14.1 Fundamental Notions
- 14.1.1 Equilibrium
- 14.1.2 Stability
- 14.1.3 A First Example: Heat Transfer
- 14.1.4 Graphical Analysis – One Dimension
- 14.1.5 Graphical Analysis – Multiple Dimensions
- 14.1.6 Reduced Dimensionality
- 14.1.7 General Analysis
- 14.1.8 What’s Fundamental and What’s Not
- 14.2 Useful Proxies for Predicting Spontaneity, Reversibility, Equilibrium, etc.
- 14.2.1 Isolated System; Proxy = Entropy
- 14.2.2 External Damping; Proxy = Energy
- 14.2.3 Constant V and T; Proxy = Helmholtz Free Energy
- 14.2.4 Constant P and T; Proxy = Gibbs Free Enthalpy
- 14.3 Discussion: Some Fine Points
- 14.3.1 Local Conservation
- 14.3.2 Lemma: Conservation of Enthalpy, Maybe
- 14.3.3 Energy and Entropy (as opposed to «Heat»
- 14.3.4 Spontaneity
- 14.3.5 Conditionally Allowed and Unconditionally Disallowed
- 14.3.6 Irreversible by State or by Rate
- 14.4 Temperature and Chemical Potential in the Equilibrium State
- 14.5 The Approach to Equilibrium
- 14.5.1 Non-Monotonic Case
- 14.5.2 Monotonic Case
- 14.5.3 Approximations and Misconceptions
- 14.6 Natural Variables, or Not
- 14.6.1 The “Big Four” Thermodynamic Potentials
- 14.6.2 A Counterexample: Heat Capacity
- 14.7 Going to Completion
- 14.8 Example: Shift of Equilibrium
- 14.9 Le Châtelier’s «Principle», Or Not
- 14.10 Appendix: The Cyclic Triple Derivative Rule
- 14.10.1 Graphical Derivation
- 14.10.2 Validity is Based on Topology
- 14.10.3 Analytic Derivation
- 14.10.4 Independent and Dependent Variables, or Not
- 14.10.5 Axes, or Not
- 14.11 Entropy versus “Irreversibility” in Chemistry
- 15 The “Big Four” Energy-Like State Functions
- 15.1 Energy
- 15.2 Enthalpy
- 15.2.1 Integration by Parts; PV and its Derivatives
- 15.2.2 More About PdV versus VdP
- 15.2.3 Definition of Enthalpy
- 15.2.4 Enthalpy is a Function of State
- 15.2.5 Derivatives of the Enthalpy
- 15.3 Free Energy
- 15.4 Free Enthalpy
- 15.5 Thermodynamically Available Energy – Or Not
- 15.5.1 Overview
- 15.5.2 A Calculation of “Available” Energy
- 15.6 Relationships among E, F, G, and H
- 15.7 Yet More Transformations
- 15.8 Example: Hydrogen/Oxygen Fuel Cell
- 15.8.1 Basic Scenario
- 15.8.2 Enthalpy
- 15.8.3 Gibbs Free Enthalpy
- 15.8.4 Discussion: Assumptions
- 15.8.5 Plain Combustion ⇒ Dissipation
- 15.8.6 Underdetermined
- 15.8.7 H Stands For Enthalpy – Not «Heat»
- 16 Adiabatic Processes
- 16.1 Multiple Definitions of “Adiabatic”
- 16.2 Adiabatic versus Isothermal Expansion
- 17 Heat
- 17.1 Definitions
- 17.2 Idiomatic Expressions
- 17.3 Resolving or Avoiding the Ambiguities
- 18 Work
- 18.1 Definitions
- 18.1.1 Integral versus Differential
- 18.1.2 Coarse Graining
- 18.1.3 Local versus Overall
- 18.2 Energy Flow versus Work
- 18.3 Remarks
- 18.4 Hidden Energy
- 18.5 Pseudowork
- 19 Cramped versus Uncramped Thermodynamics
- 19.1 Overview
- 19.2 A Closer Look
- 19.3 Real-World Compound Cramped Systems
- 19.4 Heat Content, or Not
- 19.5 No Unique Reversible Path
- 19.6 Vectors: Direction and Magnitude
- 19.7 Reversibility
- 20 Ambiguous Terminology
- 20.1 Background
- 20.2 Overview
- 20.3 Energy
- 20.4 Conservation
- 21 Thermodynamics, Restricted or Not
- 22 The Relevance of Entropy
- 23 Equilibrium, Equiprobability, Boltzmann Factors, and Temperature
- 23.1 Background and Preview
- 23.2 Example: N=1001
- 23.3 Example: N=1002
- 23.4 Example: N=4
- 23.5 Role Reversal: N=1002; TM versus Tµ
- 23.6 Example: Light Blue
- 23.7 Discussion
- 23.8 Relevance
- 24 Partition Function
- 24.1 Basic Properties
- 24.2 Calculations Using the Partition Function
- 24.3 Example: Harmonic Oscillator
- 24.4 Example: Two-State System
- 24.5 Rescaling the Partition Function
- 25 Equipartition
- 25.1 Generalized Equipartition Theorem
- 25.2 Corollaries: Power-Law Equipartition
- 25.3 Interpolating Harmonic Oscillator ↔ Particle in a Box
- 25.4 Remarks
- 26 Partition Function: Some Examples
- 26.1 Preview: Single Particle in a Box
- 26.2 Ideal Gas of Point Particles
- 26.2.1 Distinguishable Particles
- 26.2.2 Indistinguishable Particles; Delabeling
- 26.2.3 Mixtures
- 26.2.4 Energy, Heat Capacity, and Entropy for a Pure Gas
- 26.2.5 Entropy of a Mixture
- 26.2.6 Extreme Mixtures
- 26.2.7 Entropy of the Deal
- 26.3 Rigid Rotor
- 26.4 Isentropic Processes
- 26.5 Polytropic Processes ⋯ Gamma etc.
- 26.6 Low Temperature
- 26.7 Degrees of Freedom, or Not
- 26.8 Discussion
- 26.9 Derivation: Particle in a Box
- 26.10 Area per State in Phase Space
- 26.10.1 Particle in a Box
- 26.10.2 Periodic Boundary Conditions
- 26.10.3 Harmonic Oscillator
- 26.10.4 Non-Basis States
- 27 Density Matrices
- 28 Summary
- 29 About the Book
- 30 References