Physics - Heat & Thermo Question with Solution | TestHub

The RMS speed is given by 

$v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$

The average speed is given by 

$v_{\mathrm{avg}}=\sqrt{\frac{8RT}{\pi M}}$

The ratio is 

$$\begin{gathered} \Rightarrow \frac{v_{\mathrm{rms}}}{v_{\text{avg }}}=\sqrt{\frac{{\displaystyle \frac{3RT}{M}}}{{\displaystyle \frac{8RT}{\pi M}}}=}\sqrt{\frac{3\pi }{8}}=\sqrt{\frac{3}{8}\times \frac{22}{7}} \\ \Rightarrow \frac{v_{\mathrm{rms}}}{v_{\text{avg }}}=\sqrt{\frac{33}{28}} \end{gathered}$$

Comparing it with the given value ${\left(1+\frac{5}{x}\right)}^{\frac{1}{2}}$,

$$\begin{gathered} \frac{v_{\mathrm{rms}}}{v_{\text{avg }}}=\sqrt{1+\frac{5}{28}} \\ \Rightarrow x=28 \end{gathered}$$