Example $5\mathrm{.}7$

$0\mathrm{.}5kg$ of nitrogren initially at a pressure of $5$MPa and a temperature of $145C$ is expanded to a volume of $0\mathrm{.}15m^3$ along the polytropic path $p{V}^{1\mathrm{.}35}=$ constant. If for nitrogen $R=297\frac{J}{k}g.K,$ calculate:

$($a$)$ the initial volume of the gas;

$($b$)$ the final pressure of the gas;

$($c$)$ the final temperature of the gas.