PX284 - E1c - helmholtz free energy

F = U - T S

S = k_{B} (β U + \ln Z)

F = U - k_{B} T (β U + \ln Z)

∴ F = - k_{B} T \ln Z

Z = \exp (- β F)

d F = (T d S - p d V) - T d S - S d T = - S d - p d V

\begin{aligned} {(\frac{\partial F}{\partial T})}_{V} & = - S \\ {(\frac{\partial F}{\partial V})}_{T} & = - p \end{aligned}