# Basic Thermodynamics

The Helmholtz free energy $$A (N, V, T )$$ is a natural function of $$N, V$$ and $$T$$. The isothermal-isobaric ensemble is generated by transforming the volume $$V$$ in favor of the pressure $$P$$ so that the natural variables are $$N$$, $$P$$, and $$T$$ (which are conditions under which many experiments are performed, e.g., `standard temperature and pressure'.

Performing a Legendre transformation of the Helmholtz free energy

$\tilde{A}(N,P,T) = A(N,V(P),T) - V(P) \frac {\partial A}{\partial V}$

But

$\frac {\partial A}{\partial V} = -P$

Thus,

$\tilde{A}(N,P,T) = A(N,V(P),T) + PV \equiv G(N,P,T)$

where $$G (N, P, T )$$ is the Gibbs free energy.
The differential of $$G$$ is

$dG = \left(\frac {\partial G}{\partial P}\right)_{N,T} dP+ \left(\frac {\partial G}{\partial T}\right)_{N,P} dT+ \left(\frac {\partial G}{\partial N}\right)_{P,T} dN$

But from $$G = A + PV$$, we have

$dG = dA + PdV + VdP$

but $$dA = - SdT - PdV + \mu dN$$, thus

$dG = - SdT + VdP + \mu dN$

Equating the two expressions for $$dG$$, we see that

$V=\left(\frac {\partial G}{\partial P}\right)_{N,T}$

$S=-\left(\frac {\partial G}{\partial T}\right)_{N,P}$

$\mu=\left(\frac {\partial G}{\partial N}\right)_{P,T}$