The first thermodynamic law for a system of charged molecule

The first thermodynamic law for a system of charged molecules in electric field reads dU = TdS - pdV + mu dn + psi dq where q is the charge of the molecules. Hence U = U(S, V, n, q). Find an alternative thermodynamic function F that has natural variables T, V, n, and psi, F = F(T, V, n, psi).

Solution

Helmholtz function can be defined as

F = U-TS

OR

dF = dU-TdS-SdT

= TdS-pdV+dn+dq-TdS-SdT