Write down the first law of thermodynamics in differential form for an infinitesimal reversible change in terms of the increments $d E, d S$ and $d V$, where $E, S$ and $V$ are to be defined. Briefly give an interpretation of each term and deduce that

$$P=-\left(\frac{\partial E}{\partial V}\right)_{S}, \quad T=\left(\frac{\partial E}{\partial S}\right)_{V}$$

Define the specific heat at constant volume $C_{V}$ and show that for an adiabatic change

$$C_{V} d T+\left(\left(\frac{\partial E}{\partial V}\right)_{T}+P\right) d V=0$$

Derive the Maxwell relation

$$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}$$

where $T$ is temperature and hence show that

$$\left(\frac{\partial E}{\partial V}\right)_{T}=-P+T\left(\frac{\partial P}{\partial T}\right)_{V}$$

An imperfect gas of volume $V$ obeys the van der Waals equation of state

$$\left(P+\frac{a}{V^{2}}\right)(V-b)=R T$$

where $a$ and $b$ are non-negative constants. Show that

$$\left(\frac{\partial C_{V}}{\partial V}\right)_{T}=0,$$

and deduce that $C_{V}$ is a function of $T$ only. It can further be shown that in this case $C_{V}$ is independent of $T$. Hence show that

$$T(V-b)^{R / C_{V}}$$

is constant on adiabatic curves.