Suppose that F = U - στ. Then when we take the differential, we need to remember to use the product rule. We obtain:
Now, we can substitute in the Thermodynamic Identity to obtain:
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.