# 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 *p**d**V*, is obtained from eq. from Thermodynamic property relations, i.e.,

which can also be rewritten in terms of enthalpy (*H* = *E* + *p**V*), Helmholtz free energy (*F* = *E* − *T**S*), and Gibbs free energy (*G* = *H* − *T**S*) as

which all have the form of

Where

and *d**z* 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.