So far we have derived the differential forms of the four thermodynamic potentials in which we’re interested and have identified their natural variables. This is summarised in the following table:

INTERNAL ENERGY MAXWELL RELATION

Let’s start by again considering the differential form of the internal energy, given by dU=TdS−PdV. Consider now U as a function of entropy, S, and volume, V, such that U=U(S,V). Notice that these are the natural variables of internal energy. We can now write the total differential of U(S,V) as:

ENTHALPY MAXWELL RELATION