The Cluster's Maxwell Relations

A page about Maxwell relations — the symmetric relations between thermodynamic derivatives.

The Maxwell relations

Maxwell relations follow from the equality of mixed partial derivatives of thermodynamic potentials. For the internal energy U(S,V): dU = T dS - p dV, so dT/dV|_S = -dp/dS|_V. For the Helmholtz free energy F(T,V): dF = -S dT - p dV, so dS/dV|_T = dp/dT|_V. For the enthalpy H(S,p): dH = T dS + V dp, so dT/dp|_S = dV/dS|_p. For the Gibbs free energy G(T,p): dG = -S dT + V dp, so dS/dp|_T = -dV/dT|_p. In the cluster, the Maxwell relations follow from the symmetry of edit second derivatives.

The thermodynamic potentials

There are four thermodynamic potentials:

• Internal energy: U(S,V)
• Helmholtz free energy: F(T,V) = U - TS
• Enthalpy: H(S,p) = U + pV
• Gibbs free energy: G(T,p) = U - TS + pV

Each is a function of different natural variables. The Maxwell relations relate the derivatives of each potential. In the cluster, the four edit potentials are the edit internal energy, edit free energy, edit enthalpy, and edit Gibbs free energy.

The Legendre transform

The Legendre transform converts a function f(x) to g(p) = f(x) - px, where p = df/dx. The thermodynamic potentials are related by Legendre transforms. F = U - TS transforms U(S,V) to F(T,V). H = U + pV transforms U(S,V) to H(S,p). G = U - TS + pV transforms U(S,V) to G(T,p). In the cluster, the thermodynamic potentials are related by edit Legendre transforms.

The response functions

The response functions are:

• Heat capacity: C_V = T dS/dT|_V = dU/dT|_V
• Thermal expansion: alpha = (1/V) dV/dT|_p
• Compressibility: kappa_T = -(1/V) dV/dp|_T

These are related by: C_p - C_V = TV alpha^2 / kappa_T. In the cluster, the edit response functions are related by the same identity.

This relation

This page is about Maxwell relations. The four potentials are U, F, H, G. The Maxwell relations follow from mixed partial symmetry. The response functions are C_V, alpha, kappa_T. The identity C_p - C_V = TV alpha^2 / kappa_T holds. The relations are real.