Gibbs Free Energy - Derivation

Derivation

The Gibbs free energy total differential natural variables may be derived via Legendre transforms of the internal energy.

.

Because S, V, and N_i are extensive variables, Euler's homogeneous function theorem allows easy integration of dU:

.

The definition of G from above is

.

Taking the total differential, we have

.

Replacing dU with the result from the first law gives

.

The natural variables of G are then p, T, and {N_i}. Because some of the natural variables are intensive, dG may not be integrated using Euler integrals as is the case with internal energy. However, simply substituting the result for U into the definition of G gives a standard expression for G:

.