Explanation

(b) $\left. \ \begin{array}{r}
v_{rms} = \sqrt{\frac{3RT}{M}} \\
\Rightarrow \frac{\left( v_{rms} \right)_{1}}{\left( v_{rms} \right)_{2}} = \sqrt{\frac{T_{1}}{T_{2}}}
\end{array} \right|\begin{matrix}
T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1} \\
\Rightarrow \frac{T_{1}}{T_{2}} = \left( \frac{V_{2}}{V_{1}} \right)^{\gamma - 1}
\end{matrix}$
$\Rightarrow \frac{\left( v_{rms} \right)_{1}}{\left( v_{rms} \right)_{2}} = \left( \frac{V_{2}}{V_{1}} \right)^{\frac{\gamma - 1}{2}}$
$\Rightarrow \frac{v}{\frac{v}{2}} = \left( \frac{V_{2}}{V_{1}} \right)^{\frac{\frac{7}{5} - 1}{2}}$
$\Rightarrow 2 = \left( \frac{V_{2}}{V_{1}} \right)^{\frac{2}{5} \times \frac{1}{2}} = \left( \frac{V_{2}}{V_{1}} \right)^{1/5}$
$\Rightarrow \left( \frac{V_{2}}{V_{1}} \right) = 2^{5} = 32$