An ideal gas \((\gamma=1.5)\) is expanded adiabatically. To reduce root mean square velocity of molecules two times, the gas should be expanded

Since r.m.s. velocity \(\mathrm{v} \propto \sqrt{\mathrm{T}}\),
\(\therefore \quad \frac{\mathrm{v}_2}{\mathrm{v}_1}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}}\)
Given the r.m.s. velocity is reduced two times.
\(\Rightarrow \mathrm{v}_2=\frac{\mathrm{v}_1}{2}\)
Substituting the above result in (i),
\(\Rightarrow \frac{1}{2}=\sqrt{\frac{T_2}{T_1}} \Rightarrow \frac{T_1}{T_2}=4...(ii)\)
Using Fraction of given heat energy utilised in doing external work is given by the formula, \(\left(\frac{\Delta W}{\Delta Q}\right)=\left(1-\frac{1}{\gamma}\right)\)
For adiabatic expansion, \(T_1 \mathrm{~V}_1^{\gamma-1}=T_2 V_2^{y-1}\)
\(\begin{array}{ll}
\therefore & \left(\frac{V_2}{V_1}\right)^{\gamma-1}=\frac{T_1}{T_2}=4 ...[from (ii)]\\
& \Rightarrow\left(\frac{V_2}{V_1}\right)^{1.5-1}=4 \\
\therefore & \left(\frac{V_2}{V_1}\right)^{0.5}=4 \quad \Rightarrow \quad \frac{V_2}{V_1}=16
\end{array}\)