An ideal gas ($\mathrm{\gamma } =$1.5) is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of molecules 2.0 times?

${\mathrm{v}}_{\mathrm{rms}} = \sqrt{\frac{3\mathrm{RT}}{\mathrm{M}}} \mathrm{or} {\mathrm{v}}_{\mathrm{rms}} \propto \sqrt{\mathrm{T}}$

${\mathrm{v}}_{\mathrm{rms}}$, is to reduce two times, i.e., temperature of the gas will have to reduce four times or

$\frac{{\mathrm{T}}^{\text{'}}}{\mathrm{T}} = \frac{1}{4}$

During adiabatic process,

${\mathrm{TV}}^{\mathrm{\gamma }-1} = {\mathrm{T}}^{\text{'}}{\mathrm{V}}^{\text{'}\mathrm{\gamma }-1}$

or   $\frac{{\mathrm{V}}^{\text{'}}}{\mathrm{V}} = {\left(\frac{\mathrm{T}}{{\mathrm{T}}^{\text{'}}}\right)}^{\frac{\mathrm{\gamma }}{\mathrm{\gamma }-1}}$

       = ${\left(4\right)}^{\frac{1}{1.5-1}} = 4^2 = 16$

$\therefore {\mathrm{V}}^{\text{'}} = 16 \mathrm{V}$