How many times a diatomic gas should be expanded adiabatically so as to reduce the RMS speed to half

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How many times a diatomic gas should be expanded adiabatically so as to reduce the RMS speed to half?

Accepted Answer

$\frac{U_{rms},2}{U_{rms},1} = \frac{1}{2} = \sqrt{\frac{T_{2}}{T_{1}}} \Rightarrow \frac{1}{4}$

Now $T.V^{\gamma - 1}$= constant

$\Rightarrow \frac{T_{2}}{T_{1}} = \left( \frac{V_{1}}{V_{2}} \right)^{\gamma - 1} \Rightarrow \frac{1}{4} = \left( \frac{V_{1}}{V_{2}} \right)^{7/5 - 1} $$$\therefore V_{2} = 32\text{ }\text{V}_{1}$$

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