Maxwell Relations

Definition and physical interpretation

A Maxwell relation is an identity obtained by equating mixed partial derivatives of a thermodynamic potential. The key input is that a thermodynamic potential is a state function with an exact differential, and (assuming sufficient smoothness, e.g. twice continuously differentiable ) mixed derivatives commute.

Physically, Maxwell relations let you replace derivatives involving hard-to-measure quantities such as the entropy by derivatives of more directly accessible observables such as pressure , volume , and temperature .

Standard set for a simple compressible system (fixed composition)

Holding $N$ fixed for clarity:

1. From the internal energy $U(S,V)$ with

   $$dU = T\,dS - P\,dV,$$

   one obtains

   $$\left(\frac{\partial T}{\partial V}\right)_{S} ={} -\left(\frac{\partial P}{\partial S}\right)_{V}.$$

2. From the Helmholtz free energy $F(T,V)$ with

   $$dF = -S\,dT - P\,dV,$$

   one obtains

   $$\left(\frac{\partial S}{\partial V}\right)_{T} ={} \left(\frac{\partial P}{\partial T}\right)_{V}.$$

3. From the enthalpy $H(S,P)$ with

   $$dH = T\,dS + V\,dP,$$

   one obtains

   $$\left(\frac{\partial T}{\partial P}\right)_{S} ={} \left(\frac{\partial V}{\partial S}\right)_{P}.$$

4. From the Gibbs free energy $G(T,P)$ with

   $$dG = -S\,dT + V\,dP,$$

   one obtains

   $$\left(\frac{\partial S}{\partial P}\right)_{T} ={} -\left(\frac{\partial V}{\partial T}\right)_{P}.$$

These identities are frequently combined with an equation of state and response functions to convert between different measurable derivatives.