# Thermodynamics > ----- - Skill: `neuralblitz/thermodynamics` (Agent Skill) - Install (CLI): `npx skillmds@latest add neuralblitz/thermodynamics` - Raw SKILL.md: https://api.skillmd.com/api/skills/neuralblitz/thermodynamics/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - Author: NeuralBlitz (https://skillmd.com/u/neuralblitz) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/neuralblitz/thermodynamics --- ----- ## name: thermodynamics description: > Expert thermodynamics assistant for physicists and engineers. Use this skill whenever the user needs: help with thermodynamic laws, heat engines, entropy, phase transitions, or thermodynamic potentials. Includes both classical and statistical mechanics perspectives. trigger: Any physics or engineering problem involving heat, work, energy, or entropy. license: MIT compatibility: opencode metadata: audience: physicists category: physics # Thermodynamics — Energy, Heat, and Entropy Covers: **Laws of Thermodynamics · Thermodynamic Potentials · Heat Engines · Phase Transitions · Kinetic Theory · Statistical Mechanics** ----- ## The Four Laws ### Zeroth Law If system A is in equilibrium with B, and B with C, then A is in equilibrium with C. Establishes temperature as property of equilibrium. ### First Law Energy conservation: ``` ΔU = Q - W ``` Or differential: ``` dU = δQ - δW ``` U is state function. Forms of work: - PdV work: δW = P dV - Electrical: δW = μdq - Surface: δW = γ dA ### Second Law Entropy increases in isolated system: ``` ΔS ≥ 0 ``` Kelvin-Planck: No process converting all heat to work. Clausius: No process transferring heat from cold to hot spontaneously. ### Third Law (Nernst) As T → 0, S → S₀ (often 0 for perfect crystal). Cannot reach absolute zero in finite steps. ### Carnot Theorem Maximum efficiency for heat engine: ``` η_C = 1 - T_C/T_H ``` All reversible engines have same efficiency. ### Clausius Inequality For any cyclic process: ``` ∮ δQ/T ≤ 0 ``` Equality for reversible cycles. ### Entropy Definition For reversible process: ``` dS = δQ_rev/T ``` Statistical (Boltzmann): ``` S = k_B ln Ω ``` ----- ## Thermodynamic Potentials ### Internal Energy U = U(S, V, N) Fundamental equation. ### Enthalpy ``` H = U + PV ``` H = H(S, P, N) Useful for constant pressure processes. ### Helmholtz Free Energy ``` F = U - TS ``` F = F(T, V, N) Useful at constant T, V. ### Gibbs Free Energy ``` G = H - TS = μN ``` G = G(T, P, N) Useful at constant T, P. ### Gibbs Potential ``` Φ = μN ``` Grand canonical: J = G - μN. ### Maxwell Relations From fundamental equations: ``` (∂S/∂V)_T = (∂P/∂T)_V (∂S/∂P)_T = -(∂V/∂T)_P (∂V/∂S)_P = (∂T/∂P)_S (∂P/∂S)_V = -(∂T/∂V)_S ``` General pattern: switch variables, sign changes. ### Thermodynamic Identities ``` dU = T dS - P dV + μ dN dH = T dS + V dP + μ dN dF = -S dT - P dV + μ dN dG = -S dT + V dP + μ dN ``` ### Euler Relations Extensive variables homogeneous of degree 1: ``` U = TS - PV + μN H = TS + μN F = -TS + μN G = μN ``` ----- ## State Equations ### Ideal Gas ``` PV = nRT ``` Molecular: m v²/2 = 3kT/2 ### Van der Waals ``` (P + a n²/V²)(V - nb) = nRT ``` a: attractive b: excluded volume ### Virial Expansion ``` Z = PV/RT = 1 + B(T)/V + C(T)/V² + ... ``` ### Compressibility Isothermal: ``` κ_T = - (1/V) (∂V/∂P)_T ``` Adiabatic: ``` κ_S = - (1/V) (∂V/∂P)_S ``` ### Thermal Expansion ``` α = (1/V) (∂V/∂T)_P ``` ### Heat Capacities ``` C_V = (∂U/∂T)_V C_P = (∂H/∂T)_P ``` Relation: ``` C_P - C_V = TVα²/(κ_T) ``` For ideal gas: C_P - C_V = nR ### Gamma Ratio ``` γ = C_P/C_V ``` Monatomic ideal: γ = 5/3 Diatomic: γ = 7/5 ----- ## Heat Engines and Refrigerators ### Heat Engine Device converting heat to work: ``` η = W/Q_H = 1 - Q_C/Q_H ``` Carnot efficiency: ``` η_C = 1 - T_C/T_H ``` ### Otto Cycle Intake → compression → combustion → exhaust. Spark ignition engine. Efficiency: η = 1 - r^{γ-1} ### Diesel Cycle Compression → injection → combustion → exhaust. Compression ignition. Efficiency: η = 1 - 1/r^{γ-1}(T_3 - T_1)/(T_4 - T_2) ### Rankine Cycle Steam power cycle: - Boiler → turbine → condenser → pump → boiler Superheat, regeneration improve efficiency. ### Brayton Cycle Gas turbine: - Compressor → combustion → turbine → exhaust ### Refrigerator Heat pump from cold to hot: ``` COP = Q_C/W = T_C/(T_H - T_C) ``` ### Heat Pump ``` COP = Q_H/W = T_H/(T_H - T_C) ``` ### Carnot Refrigerator ``` COP_max = T_C/(T_H - T_C) ``` ### Coefficient of Performance Carnot is maximum, all real devices less efficient. ----- ## Kinetic Theory ### Pressure From molecular collisions: ``` P = (1/3) n m ⟨v²⟩ ``` For ideal gas: ``` P = n k T ``` ### Temperature Average kinetic energy: ``` ⟨KE⟩ = (3/2) k T ``` Translational only. ### Maxwell-Boltzmann Distribution Speed distribution: ``` f(v) = 4π (m/2πkT)^{3/2} v² e^{-mv²/2kT} ``` Most probable: v_mp = √(2kT/m) Mean: ⟨v⟩ = √(8kT/πm) RMS: √(3kT/m) ### Mean Free Path ``` λ = 1/(√2 n σ) ``` For hard spheres: σ = πd² ### Transport Coefficients Viscosity: η = (1/3) n m v̄ λ Thermal conductivity: κ = (1/3) n c_v v̄ λ Diffusion: D = (1/3) v̄ λ ### Chapman-Enskog Solve Boltzmann equation. Corrections to simple kinetic theory. ### Diffusion Fick's first law: ``` J = -D ∇n ``` Second law: ``` ∂n/∂t = D ∇²n ``` ### Brownian Motion Random walk: ⟨x²⟩ = 2Dt Einstein relation: ``` D = kT/ξ ``` ----- ## Statistical Mechanics ### Microstates and Macrostates - Microstate: exact specification - Macrostate: macroscopic variables Boltzmann: S = k_B ln Ω ### Partition Function ``` Z = Σ_i e^{-βE_i} ``` For canonical ensemble. ### Thermodynamic quantities U = -∂ ln Z/∂β S = k_B (ln Z + βU) F = -k_B T ln Z ### Maxwell-Boltzmann Statistics Distinguishable particles: ``` f(E) = e^{-βE} ``` Applies to classical ideal gas. ### Fermi-Dirac Statistics Indistinguishable, half-integer spin: ``` f(E) = 1/(e^{(E-μ)/kT} + 1) ``` Pauli principle. ### Bose-Einstein Statistics Indistinguishable, integer spin: ``` f(E) = 1/(e^{(E-μ)/kT} - 1) ``` Bosons can condense. ### Quantum Statistics At high T → MB limit. At low T: degeneracy important. ### Degeneracy Temperature Fermi: ``` T_F = E_F/k ``` Bose: ``` T_c = 2πℏ²/(mk)[n/ζ(3/2)]^{2/3} ``` ### Blackbody Radiation Photon gas: ``` u = a T⁴ ``` Energy density: u = σ/c T⁴ Stefan-Boltzmann: σ = π²k⁴/(60ℏ³c²) ### Debye Model Phonons in solid: - 3 acoustic branches - Cutoff frequency (Debye) Heat capacity: C ~ T³ at low T. ----- ## Phase Transitions ### Phase Equilibrium Gibbs phase rule: ``` F = C - P + 2 ``` C components, P phases. ### Clausius-Clapeyron For phase boundary: ``` dP/dT = ΔS/ΔV = ΔH/(TΔV) ``` ### Clapeyron Equation For liquid-vapor: ``` dP/dT = L/(TΔV) ``` ### Latent Heat Heat absorbed/released: ``` L = T ΔS = ΔH ``` ### Phase Diagrams Lines of equilibrium: - Triple point - Critical point ### Critical Point Properties diverge: - Heat capacity - Correlation length Critical exponents. ### Order Parameter Zero in disordered phase: - Magnetization - Density difference Breaks symmetry. ### First Order Discontinuous change: - Latent heat - Hysteresis ### Second Order Continuous change: - Heat capacity divergence - Critical opalescence ### Ehrenfest Classification First derivatives discontinuous: 1st order Second derivatives discontinuous: 2nd order ### Landau Theory Expand free energy in order parameter: F = F₀ + a(T-T₀)φ² + b φ⁴ + ... Predicts mean-field exponents. ### Scaling Near critical point: ``` X ~ |t|^{-γ}, t = (T-T_c)/T_c ``` Universal exponents. ### Renormalization Group Wilsonian: - Integrate out short wavelengths - Fixed points → critical points ### Ginzburg Criterion Mean-field valid when fluctuations small: ``` |(T-T_c)/T_c| >> Gi ``` ### Superfluidity Bose-Einstein of ⁴He. Critical velocity. Lambda point. ### Superconductivity Zero resistance. Meissner effect. Cooper pairs. ### Phase Transitions in Fields - Magnetic field for ferromagnets - Electric field for ferroelectrics ----- ## Mixtures and Solutions ### Chemical Potential For component i: ``` μ_i = (∂G/∂n_i)_{T,P,n_j≠i} ``` Equilibrium: μ_i equal in all phases. ### Partial Molar Properties Partial molar volume: ``` V̄_i = (∂V/∂n_i)_{T,P,n_j} ``` ### Gibbs-Duhem ``` Σ n_i dμ_i = 0 ``` ### Raoult's Law For ideal solution: ``` p_i = x_i p_i* ``` ### Henry's Law For dilute solution: ``` p_i = x_i k_i ``` ### Activity ``` a_i = γ_i x_i ``` Corrects non-ideality. ### Free Energy of Mixing ``` ΔF_mix = nRT Σ x_i ln x_i ``` Negative → mixing spontaneous. ### Colligative Properties - Boiling point elevation - Freezing point depression - Osmotic pressure ### Gibbs Phase Rule For non-reactive system: F = C - P + 2 ### Eutectic Complete liquid miscibility, solid immiscibility. ### Lever Rule Phase diagram analysis: Amounts = lever arm lengths. ----- ## Non-Equilibrium ### Local Equilibrium Near equilibrium: local variables well-defined. ### Entropy Production Rate of entropy increase: ``` σ = Σ J_i X_i ``` Fluxes × forces. ### Onsager Reciprocal Relations Linear response: ``` J_L = Σ L_{lk} X_k ``` L_{lk} = L_{kl}. ### Fourier's Law Heat conduction: ``` q = -κ ∇T ``` ### Newton's Law Viscosity: ``` τ = η γ̇ ``` ### Fick's Law Diffusion: ``` J = -D ∇c ``` ### Linear Response Transport coefficients related. ### Boltzmann Equation Distribution function evolution: ``` ∂f/∂t + v·∇_x f + F/m·∇_v f = (∂f/∂t)_c ``` Collision term. ### H-Theorem Entropy increases: ``` ∂S/∂t ≥ 0 ``` Approaches equilibrium. ### Green-Kubo Transport from equilibrium fluctuations: ``` D = ∫₀^∞ ⟨v(t)v(0)⟩ dt ``` ### Chapman-Enskog From Boltzmann to hydrodynamics: - Euler equations - Navier-Stokes ### Hydrodynamic Equations Continuity: ∂ρ/∂t + ∇·(ρv) = 0 Momentum: ρDv/Dt = -∇P + η∇²v Energy: ρC_v dT/dt = κ∇²T ### Irreversible Processes Thermoelectric effects: - Seebeck - Peltier - Thomson ----- ## Common Errors to Avoid - Confusing heat and temperature - Using wrong sign conventions for work - Applying reversible formulas to irreversible processes - Confusing internal energy with enthalpy - Forgetting that entropy is a state function - Using ideal gas for high-pressure conditions - Confusing partial molar with molar properties - Applying equilibrium formulas to non-equilibrium - Ignoring third law when computing entropy differences - Confusing spontaneous with reversible processes - Using mean-field exponents for low dimensions ----- ## Key References - **Thermodynamics** by Callen — Elegant introduction - **Statistical Physics** by Landau & Lifshitz — Comprehensive - **Physical Chemistry** by McQuarrie — Chemistry focus - **Heat and Thermodynamics** by Zemansky — Classical approach