Thermodynamic potentials can be constructed so that their natural independent variables match the environmental constraints imposed on a system.

The Helmholtz free energy is

$$\boxed{F=U-TS}.$$

The Gibbs free energy has already been introduced as

$$G=H-TS=U+pV-TS.$$

These definitions are Legendre transforms of the energy representation: they replace entropy or volume by conjugate variables that are often easier to control experimentally.

For a simple compressible closed system of fixed composition, the fundamental thermodynamic relations give

$$\boxed{dF=-S,dT-p,dV},$$

$$\boxed{dG=-S,dT+V,dp}.$$

The coefficients reveal the natural variables:

$$F=F(T,V),\qquad G=G(T,p).$$

For example,

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

These derivative identities are the starting point for Maxwell relations and other property connections.

The potentials also provide equilibrium criteria under common constraints. At fixed temperature and volume, a closed system approaches a state that minimizes $F$ subject to its constraints. At fixed temperature and pressure, the corresponding potential is $G$.

For phase equilibrium in a pure substance at fixed $T$ and $p$, coexisting phases have equal molar or specific Gibbs free energy. A phase with lower Gibbs free energy is thermodynamically favored, although kinetic barriers may delay its appearance.

The role of Legendre transforms is therefore not to introduce a different set of physical laws. They reorganize the first and second laws into state functions whose variables and minimization principles are adapted to particular controlled conditions.