The speed of sound in an ideal gas at a given temperature \(T\) is \(v\). The rms speed of gas molecules at that temperature is \(v_{\mathrm{rms}}\). The ratio of the velocities \(v\) and \(v_{\text {rms }}\) for helium and oxygen gases are \(X\) and \(X^{\prime}\). respectively. Then, \(\frac{X}{X^{\prime}}\) is equal to

The speed of sound at a given temperature is
given by
\(v_{\text {sound }}=\sqrt{\frac{r R T}{M}} \Rightarrow v_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}\)
For oxygen molecules; \(\frac{v_0}{v_{\text {rms }}}=\frac{\sqrt{\frac{r_0 R T}{M}}}{\sqrt{\frac{3 R T}{M}}}=\sqrt{\frac{r_0}{3}}\)
For helium molecules; \(\frac{v_{\mathrm{He}}}{v_{\mathrm{rms}}}=\frac{\sqrt{r_{\mathrm{He}} \frac{R T}{M}}}{\sqrt{\frac{3 R T}{M}}}=\sqrt{\frac{r_{\mathrm{He}}}{3}}\) \(\left(r_{\mathrm{Be}}=\frac{5}{3}, r_{\mathrm{O}}=\frac{7}{5}\right)\)
\(\frac{x^{\prime}}{x}=\frac{\sqrt{\frac{5}{9}}}{\sqrt{\frac{7}{15}}}=\sqrt{\frac{25}{21}}=\frac{5}{\sqrt{21}}\)