Maxwell relations: be careful

You'll be tempted to start swapping around derivatives to get Maxwell relations, but remember not all derivatives can be transformed using this approach. For example,

(dS/dT)_(V,N) cannot be transformed using a Maxwell relation. It is not equal to (dP/dV)_(T,N), if that is what you were thinking.

Recall:
(dS/dT) = (d2A/dT2)
(dP/dV) = (d2A/dV2)

Therefore, these are not the same second derivative of a free energy.

If it had instead been (dS/dV)_(T,N) then we could have said that equals (dP/dT)_(V,N).