The Van der Waals equation of state for a non-ideal gas is

$$\left(p+\frac{a N^{2}}{V^{2}}\right)(V-b N)=N k T,$$

where $a$ and $b$ are constants.

(i) Briefly explain the physical motivation for differences between the Van der Waals and ideal gas equations of state.

(ii) Find the volume dependence (at constant temperature) of the internal energy $E$ and the heat capacity $C_{V}$ of a Van der Waals gas.

(iii) A Van der Waals gas is initially at temperature $T_{1}$ in an insulated container with volume $V_{1}$. A small opening is then made so that the gas can expand freely into an empty container, occupying both the old and new containers. The final result is that the gas now occupies a volume $V_{2}>V_{1}$. Calculate the final temperature $T_{2}$ assuming $C_{V}$ is temperature independent. You may assume the process happens quasistatically.