Four standard Maxwell relations are commonly derived from the four principal thermodynamic potentials. They connect otherwise difficult-to-measure entropy derivatives with measurable changes in temperature, pressure, and volume.
The four potentials are internal energy U(S,V), enthalpy H(S,p), Helmholtz free energy A(T,V), and Gibbs free energy G(T,p). Each has two natural variables, and equality of mixed second derivatives produces one commonly taught Maxwell relation. Together, these are the familiar four relations used in classical thermodynamics.
They are named after James Clerk Maxwell, who presented them in his 1872 book Theory of Heat. The number is not an absolute limit: adding particle number, chemical potential, magnetic variables, or other work coordinates generates additional relations. Thus “four” means the standard set for the four commonly used potentials, not every Maxwell relation that can exist.