Q$5$: For the given gas mixture with a molecular weight of $33$ a$.$m$.$u$,$ contained in a gas$-$piston cylinder, it undergoes a polytropic process described by the pressure$-$volume relationship $P{V}^{1\mathrm{.}3}=$ const. The initial state of the gas mixture is characterized by a temperature of $300K,$ a pressure of $\frac{3}{b}ar,$ and a volume of $0\mathrm{.}1m^3.$ The specific heat constant $(C_v)$ of the gas varies with temperature according to the equation:

$C_V=0\mathrm{.}6+2\mathrm{.}5{10}^{-4}T$

Where $T$ in $K$ and $C_v$ in $k\frac{J}{k}g.K.$ Determine the following:

a$)$ The final temperature $(k)$

b$)$ The final pressure $($bar$)$

c$)$ The final volume $(m^3)$

d$)$ The word done $(kJ)$