Maxwell relations

It is often useful to obtain relations between the partial derivatives of different thermodynamic quantities. For example, by writing

$$dE = \left( \frac{\partial E}{\partial S}\right )_V dS + \left( \frac{\partial E}{\partial V}\right )_S dV,$$ (3.100)

and cross differentiating the first and the second partial derivative w.r.t $S$ and $V$, respectively, we have

$$\left( \frac{\partial T}{\partial V}\right )_S = - \left( \frac{\partial P}{\partial S}\right )_V,$$ (3.101)

which is known as one of the four Maxwell relations. We can obtain the other three by similar manipulations of the other three thermodynamic potentials, $F$, $H$, and $G$, which give:

$$\left( \frac{\partial P}{\partial T}\right )_V = \left( \frac{\partial S}{\partial V}\right )_T,$$ (3.102)

$$\left( \frac{\partial V}{\partial T}\right )_P = -\left( \frac{\partial S}{\partial P}\right )_T,$$ (3.103)

$$\left( \frac{\partial T}{\partial P}\right )_S = \left( \frac{\partial V}{\partial S}\right )_P,$$ (3.104)