How to add in heat capacity term to Helmholtz Free Energy?

Question

The heat capacity is defined to be the amount of heat necessary to change the temperature of system/object, one divided by another:

$$dQ=C~dT \tag{1}$$

Usually we would like to input such a definition into the differential relations between a thermodynamic potential and its respective variables, such as the Helmholtz free energy:

$$dF=-P~dV-S~dT + \mu~dN \tag{2}$$

Where does the heat capacity come into the Helmholtz free energy? I'm confused because I can already read off $\partial F/\partial T$ from (2),

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

Answer 1

Material properties can often be expressed as the second derivative of a thermodynamic potential. For example, the thermal expansion coefficient is $\alpha=\frac{1}{V}\left(\frac{\partial^2 G}{\partial P \partial T}\right)$. The heat capacity is $C=-T\left(\frac{\partial^2 F}{\partial T^2}\right)$.