Thermodynamic potentials generate relations among measurable property derivatives. Equality of continuous mixed second derivatives then produces the Maxwell relations.

For Gibbs free energy,

$$dG=-S,dT+V,dp.$$

Therefore

$$S=-\left(\frac{\partial G}{\partial T}\right)_p,\qquad V=\left(\frac{\partial G}{\partial p}\right)_T.$$

Because both derivatives come from the same state function $G(T,p)$, differentiating once more in the opposite order gives

$$\left(\frac{\partial S}{\partial p}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_p.$$

The internal-energy, enthalpy and Helmholtz potentials generate three analogous Maxwell relations.

Their practical value is that a difficult-to-measure derivative involving entropy can be replaced by a derivative involving pressure, volume or temperature.

Maxwell relations add no new thermodynamic law. They follow mathematically from the state-function structure of thermodynamic potentials.