Gibbs free energy, G, and Helmholtz free energy, A, are both extensive thermodynamic properties defined as:
where T is temperature, S entropy and U internal energy.
Differentiation of these equations and substitution of the fundamental equations
where P is pressure and V volume leads to:
indicating that T and P are the natural variables associated with G, and T and V those associated with A.
The main utility of the free energy functions is in the determination of equilibrium states. For a closed system maintained at a fixed pressure and temperature in which the only work done is that of expansion of the system against the surrounding pressure, the equilibrium state is the one in which the Gibbs Free Energy assumes its lowest value. If the system is maintained at fixed volume and temperature and no work is done, it is the Helmholz Free Energy which assumes its lowest value. These conditions apply whether the system is single or multicomponent, single or multiphase.