Definition of F, G, H
Suppose that F = U - στ. Then when we take the differential, we need to remember
to use the product rule. We obtain:
dF = dU - σ dτ - τ dσ
Now, we can substitute in the Thermodynamic Identity to obtain:
dF = - σ dτ - p dV + μ dN
Notice that F is a function now of τ, V, and N. By adding the
term - στ, we were able to swap two of the variables, σ and τ. We
call F the Helmholtz Free Energy, and we will soon see why it is useful.
The quick mind will realize that we could define 6 such energies in total, by
successively swapping all of the variables. It turns out that we'll only be
interested in two more. The Enthalpy, H, swaps p and V. We write H = U + pV and
obtain dH = τ dσ + V dp + μ dN. We also define the Gibbs Free Energy by
utilizing both of these swaps.
Letting G = U + pV - τσ, we obtain dG = - σ dτ + V dp + μ dN.
We say that the energy of any of these types is a function of the variables
that appear as differentials. Remember that the terms that are not differentials
can be defined in relation to those that are.
The relationships between the energies are summarized in the following figure.