| Modern Thermodynamics |
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| John Denker |
- 0 Introduction
- 0.1 Overview
- 0.2 Availability
- 0.3 Prerequisites, Goals, and Non-Goals
- 1 Energy
- 1.1 Preliminary Remarks
- 1.2 Definition of Energy
- 1.3 More Remarks
- 1.4 Conservation of Energy
- 1.5 Energy versus “Capacity to do Work” or “Available Energy”
- 1.5.1 Best Case : Non-Thermal Situation
- 1.5.2 Equation versus Definition
- 1.5.3 General Case : Some Energy Not Available
- 1.6 Conflict with the Vernacular
- 1.7 Range of Validity
- 1.8 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
- 2.6 Quantifying Entropy
- 2.7 Surprise Value
- 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 Functions of State
- 6.1 Functions of State : Basic Notions
- 6.2 Path Independence
- 6.3 Hess’s Law, Or Not
- 6.4 Partial Derivatives
- 6.5 Heat Capacities, Energy Capacity, and Enthalpy Capacity
- 6.6 Yet More Partial Derivatives
- 6.7 Integration
- 6.8 Advection
- 6.9 Deciding What’s True
- 6.10 Deciding What’s Fundamental
- 7 Thermodynamic Paths and Cycles
- 7.1 A Path Projected Onto State Space
- 7.1.1 State Functions
- 7.1.2 Out-of-State Functions
- 7.1.3 Converting Out-of-State Functions to State Functions
- 7.1.4 Reversibility and/or Uniqueness
- 7.1.5 The Importance of Out-of-State Functions
- 7.1.6 Heat Content, or Not
- 7.1.7 Some Mathematical Remarks
- 7.2 Grady and Ungrady One-Forms
- 7.3 Abuse of the Notation
- 7.4 Procedure for Extirpating dW and dQ
- 7.5 Some Reasons Why dW and dQ Might Be Tempting
- 7.6 Boundary versus Interior
- 7.7 The Carnot Cycle
- 8 Connecting Entropy with Energy
- 8.1 The Boltzmann Distribution
- 8.2 Locrian and Non-Locrian
- 8.3 An Illustration : Flywheels, Springs, and Batteries
- 8.4 Remarks
- 8.4.1 Predictable Energy is Freely Convertible; Random Energy
is Not
- 8.4.2 Thermodynamic Laws without Temperature
- 8.4.3 Kinetic and Potential Microscopic Energy
- 8.4.4 Ideal Gas : Potential Energy as well as Kinetic Energy
- 8.4.5 Relative Motion versus “Thermal” Energy
- 8.5 Entropy Without Constant Re-Shuffling
- 8.6 Units of Entropy
- 8.7 Probability versus Multiplicity
- 8.7.1 Exactly Equiprobable
- 8.7.2 Approximately Equiprobable
- 8.7.3 Not At All Equiprobable
- 8.8 Discussion
- 8.9 Misconceptions about Spreading
- 8.10 Spreading in Probability Space
- 9 Additional Fundamental Notions
- 9.1 Equilibrium
- 9.2 Non-Equilibrium; Timescales
- 9.3 Efficiency; Timescales
- 9.4 Spontaneity and Irreversibility
- 9.5 Stability
- 9.6 Relationship between Static Stability and Damping
- 9.7 Finite Size Effects
- 10 Experimental Basis
- 10.1 Basic Notions of Temperature and Equilibrium
- 10.2 Exponential Dependence on Energy
- 10.3 Metastable Systems with a Temperature
- 10.4 Metastable Systems without a Temperature
- 10.5 Dissipative Systems
- 10.5.1 Sudden Piston : Sound
- 10.5.2 Sudden Piston : State Transitions
- 10.5.3 Rumford’s Experiment
- 10.5.4 Flywheels with Oil Bearing
- 10.5.5 Misconceptions : Heat
- 10.5.6 Misconceptions : Work
- 10.5.7 Remarks
- 10.6 The Gibbs Gedankenexperiment
- 10.7 Spin Echo Experiment
- 10.8 Melting
- 10.9 Isentropic Expansion and Compression
- 10.10 Demagnetization Refrigerator
- 10.11 Thermal Insulation
- 11 More About Entropy
- 11.1 Microstate versus Macrostate
- 11.2 What the Second Law Doesn’t Tell You
- 11.3 Phase Space
- 11.4 Entropy in a Crystal; Phonons, Electrons, and Spins
- 11.5 Entropy is Entropy
- 11.6 Spectator Entropy
- 11.7 No Secret Entropy, No Hidden Variables
- 11.8 Entropy is Context Dependent
- 11.9 Extreme Mixtures
- 11.9.1 Simple Model System
- 11.9.2 Two-Sample Model System
- 11.9.3 Helium versus Snow
- 11.9.4 Partial Information aka Weak Peek
- 11.10 Entropy is Not Necessarily Extensive
- 11.11 Mathematical Properties of the Entropy
- 11.11.1 Entropy Can Be Infinite
- 12 Spontaneity, Reversibility, and Equilibrium
- 12.1 Fundamental Notions
- 12.1.1 Equilibrium
- 12.1.2 Stability
- 12.1.3 Irreversible by State or by Rate
- 12.1.4 Transformations, One-Dimensional or Otherwise
- 12.1.5 Conditionally Allowed and Unconditionally Disallowed
- 12.1.6 General Analysis
- 12.2 Example: Heat Transfer
- 12.3 Carnot Efficiency Formula
- 12.3.1 Definition of Heat Engine
- 12.3.2 Analysis
- 12.3.3 Discussion
- 12.4 Properties of the Equilibrium State
- 12.5 The Approach to Equilibrium
- 12.5.1 Non-Monotonic Case
- 12.5.2 Monotonic Case
- 12.5.3 Approximations and Misconceptions
- 12.6 Useful Proxies for Predicting Spontaneity, Reversibility,
Equilibrium, etc.
- 12.6.1 Reduced Dimensionality
- 12.6.2 Constant V and T
- 12.6.3 Constant P and T
- 12.6.4 Externally Damped Oscillator: Constant S and Decoupled V
- 12.6.5 Lemma: Conservation of Enthalpy, Maybe
- 12.6.6 Local Conservation
- 12.7 Natural Variables, or Not
- 12.7.1 The “Big Four” Thermodynamic Potentials
- 12.7.2 A Counterexample: Heat Capacity
- 12.8 Going to Completion
- 12.9 Example: Shift of Equilibrium
- 12.10 Le Châtelier’s Principle, Or Not
- 12.11 Appendix: The Cyclic Triple Derivative Rule
- 12.11.1 Graphical Derivation
- 12.11.2 Validity is Based on Topology
- 12.11.3 Analytic Derivation
- 12.11.4 Independent and Dependent Variables, or Not
- 12.11.5 Axes, or Not
- 12.12 Entropy versus “Irreversibility” in Chemistry
- 13 The “Big Four” Energy-Like State Functions
- 13.1 Energy
- 13.2 Enthalpy
- 13.3 Free Energy
- 13.4 Free Enthalpy
- 13.5 Thermodynamically Available Energy – Or Not
- 13.5.1 overview
- 13.5.2 A Calculation
- 13.6 Relationships among E, F, G, and H
- 13.7 Yet More Transformations
- 14 Adiabatic Processes
- 15 Heat
- 15.1 Definitions
- 15.2 Idiomatic Expressions
- 15.3 Resolving or Avoiding the Ambiguities
- 16 Work
- 16.1 Definitions
- 16.1.1 Integral versus Differential
- 16.1.2 Coarse Graining
- 16.1.3 Local versus Overall
- 16.2 Energy Flow versus Work
- 16.3 Remarks
- 16.4 Hidden Energy
- 16.5 Pseudowork
- 17 Cramped versus Uncramped Thermodynamics
- 17.1 The Two Options
- 17.2 Vectors: Direction and Magnitude
- 17.3 Reversibility
- 17.4 Heat Content, or Not
- 18 Ambiguous Terminology
- 19 Thermodynamics, Restricted or Not
- 20 The Relevance of Entropy
- 21 Equilibrium, Equiprobability, Boltzmann Factors, and Temperature
- 21.1 Background and Preview
- 21.2 Example: N=1001
- 21.3 Example: N=1002
- 21.4 Example: N=4
- 21.5 Role Reversal: N=1002; TM versus Tµ
- 21.6 Example: Light Blue
- 21.7 Discussion
- 21.8 Relevance
- 22 Partition Function
- 22.1 Basic Properties
- 22.2 Calculations Using the Partition Function
- 22.3 Example: Harmonic Oscillator
- 22.4 Example: Two-State System
- 22.5 Rescaling the Partition Function
- 23 Equipartition
- 23.1 Generalized Equipartition Theorem
- 23.2 Corollaries: Power-Law Equipartition
- 23.3 Interpolating Harmonic Oscillator ↔ Particle in a Box
- 23.4 Remarks
- 24 Partition Function: Some Examples
- 24.1 Preview: Single Particle in a Box
- 24.2 Ideal Gas of Point Particles
- 24.2.1 Distinguishable Particles
- 24.2.2 Indistinguishable Particles; Delabeling
- 24.2.3 Mixtures
- 24.2.4 Energy, Heat Capacity, and Entropy for a Pure Gas
- 24.2.5 Entropy of a Mixture
- 24.2.6 Extreme Mixtures
- 24.2.7 Entropy of the Deal
- 24.3 Rigid Rotor
- 24.4 Isentropic Processes
- 24.5 Polytropic Processes ⋯ Gamma etc.
- 24.6 Low Temperature
- 24.7 Degrees of Freedom, or Not
- 24.8 Discussion
- 24.9 Derivation: Particle in a Box
- 24.10 Area per State in Phase Space
- 24.10.1 Particle in a Box
- 24.10.2 Periodic Boundary Conditions
- 24.10.3 Non-Basis States
- 25 Density Matrices
- 26 Summary
- 27 References