thermodynamics

If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. Grounds the existence of…

Energy of a closed system changes only by heat added and work done: dU = δQ − δW. Specialization of conservation-of-energy to thermo.

Total entropy of an isolated system never decreases over time; equivalently, heat cannot spontaneously flow from a colder to a hotter body.

Entropy of a perfect crystal approaches a constant minimum as temperature approaches absolute zero.

State function quantifying unavailability of a system's energy for doing work; equivalently, the log of the number of accessible…

The statistical-mechanical definition: S = k_B ln W, where W is the number of microstates consistent with the macrostate.

Empirical observation (and theoretical bound) that no heat engine operating between reservoirs at temperatures T_h and T_c can exceed η_max…

Empirical fact that absolute zero cannot be reached in a finite number of thermodynamic operations: laser-cooling and adiabatic…

Equality of mixed second partials of thermodynamic potentials yields four classical identities among (T,S,p,V) partial derivatives; e.g.…

F = U − TS. Minimised at equilibrium in a system at fixed T and V; partition-function link: F = −k_B T ln Z.

G = U − TS + pV = H − TS. Minimised at equilibrium at fixed T and p; driver of chemical reactions and phase equilibria.

Partial derivative μ_i = (∂G/∂N_i)_{T,p,N_{j≠i}}. Equalised across phases and species at equilibrium; conjugate to particle number.

SdT − Vdp + Σ N_i dμ_i = 0: intensive variables are not all independent. Consequence of Euler's theorem on the extensive G.

Qualitative change in thermodynamic state at a critical point: first-order (latent heat, discontinuous first derivative of G) or continuous…

Heat flux through a material is proportional to the negative gradient of temperature: q = −k ∇T, where k is the thermal conductivity (a…

Molar flux of a species is proportional to the negative gradient of concentration: J = −D ∇c, where D is the diffusion coefficient…

Idealised reversible thermodynamic cycle composed of two isotherms (heat exchange Q_h at T_h, Q_c at T_c) and two adiabats. Operating…

Linear parabolic PDE for a conserved scalar density (concentration, temperature, probability density) under Fickian transport: ∂c/∂t = D…

S → const (0 for perfect crystal) as T → 0; implies impossibility of cooling to 0 K in finite steps; residual entropy in glasses.

Isenthalpic throttling: ΔT/ΔP = (1/C_p)(T(∂V/∂T)_P - V); inversion temperature above which real gas warms on expansion.

η_max = 1 - T_c/T_h for any reversible engine between two reservoirs; defines thermodynamic temperature.

dP/dT = L/(TΔV) along coexistence line; governs vapor pressure, humidity, melting-curve slope; integrable if ΔV≈V_g, ΔH=L const.

T defined via reversible Carnot engines; absolute zero = T at which η=1; SI kelvin redefined 2019 via k_B = 1.380649×10⁻²³ J/K.

(P + a/V²)(V - b) = nRT; predicts liquid-gas coexistence, critical point; reduced equation universal near T_c.

vdW near T_c in 3D Ising universality class: β ≈ 0.326, γ ≈ 1.237, ν ≈ 0.630. Deviations from mean field.

Heat evolution dQ/dx = -σJ ∂T/∂x in conductor with temperature gradient; Kelvin relations link to Seebeck & Peltier.

Entropy as ln multiplicity; additive for independent subsystems; Gibbs definition S = -k_B Σ p_i ln p_i reduces to it for equiprobable W…

T<0 for bounded-spectrum systems (e.g. population-inverted spins) with S(E) decreasing; hotter than any T>0; inaccessible to conservative-H…

Mixing entropy problem resolved by indistinguishability; ln N! factor in classical Z restores extensivity.

∂²U/∂S²>0, ∂²U/∂V²>0 imply C_V>0, (∂P/∂V)_T<0; violation triggers phase separation; spinodal where stability fails.