Explanation

(a) Ratio of specific heat for a monoatomic gas is $\frac{5}{3}$ and for diatomic gas is $\frac{7}{5}$.
Given, $n_{1} = 1,\ n_{2} = 3,\ n = 4$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n}{\gamma - 1} = \frac{n_{1}}{\gamma_{1} - 1} + \frac{n_{2}}{\gamma_{2} - 1}$
$\frac{4}{\gamma - 1} = \frac{1}{\frac{5}{3} - 1} + \frac{3}{\frac{7}{5} - 1}$
$\Rightarrow$ $\frac{4}{\gamma - 1} = \frac{3}{2} + \frac{15}{2} = 9$
$\therefore$ $4 = 9\gamma - 9$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ 9\gamma = 13\ \ \ \ \ \ \Rightarrow \gamma = \frac{13}{9}$
Now, $C_{V}(\gamma - 1) = R$
or $C_{V} = \frac{R}{\gamma - 1} = \frac{8.3}{\frac{13}{9} - 1} = \frac{8.3 \times 9}{4}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ C_{V} = 18.7\ J\ mol^{- 1} - K^{- 1}$