Maxwell relations
From Thermal-FluidsPedia
The fundamental thermodynamic relation for a reversible process in a single-component system, where the only work term considered is pdV, is obtained from eq. 
from Thermodynamic property relations, i.e.,
which can also be rewritten in terms of enthalpy (H = E + pV), Helmholtz free energy (F = E − TS), and Gibbs free energy (G = H − TS) as
which all have the form of
Where
and dz is an exact differential, as thermodynamic properties like E,H,F, and G are path-independent functions.
Since eq. (5) is the total differential of function z,M and N are related by
Applying eq. (8) to eqs. (1) – (4), the following relationships are obtained:
which are referred to as Maxwell relations. The goal of Maxwell relations is to find equivalent partial derivatives containing p,T, and V that can be physically measured and therefore provide a means of determining the change of entropy, which cannot be measured directly.
References
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.