What is Helmholtz free energy, Gibbs free enthalpy?

Example 28.5 (The perfect gas)

For $n$ moles, $U = nc_{V}T + U_0$ and $S = nc_V\ln T + nR\ln V + S_0$ give $F(T,V) = -nRT\ln V + f(T)$: then $P = -\partial_VF = nRT/V$ — the equation of state comes out of $F$ — and $S = -\partial_TF$ gives back the entropy. In the variables $(T,P)$, $G(T,P) = nRT\ln P + g(T)$, so $V = \partial_PG = nRT/P$, and the molar free enthalpy $\mu(T,P) = \mu^\circ(T) + RT\ln(P/P^\circ)$ — the chemical potential of the gas, which grows with its pressure: gas flows from high to low $\mu$, here from high to low pressure.