$12$ moles of nitrogen undergo a polytropic process from an initial pressure of $1$ bar and an initial temperature of $800$ K to a final pressure of $0\mathrm{.}7$ bar and a final temperature of $400$ K$.$ Under these conditions, nitrogen has heat capacities of cP $=29\mathrm{.}3$ J mol$1$ K$1$ and cV $=21\mathrm{.}0$ J mol$1$ K$1.$ Calculate:

a$)$ The volume change over the process in m$3.$ Clearly state and justify any assumptions made.

$[10$ marks$]$

b$)$ The internal energy change over the process in kJ$.$ Clearly state and justify any assumptions made.