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