Q153 / 180 · KCET 2023 · Physics
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
- A$\frac{21}{\sqrt{5}}$
- Correct answerB$\frac{5}{\sqrt{21}}$
- C$\sqrt{\frac{5}{21}}$
- D$\frac{21}{5}$
Solution
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; $\frac{v_{\mathrm{He}}}{v_{\mathrm{rms}}}=\frac{\sqrt{r_{\mathrm{He}} \frac{R T}{M}}}{\sqrt{\frac{3 R T}{M}}}=\sqrt{\frac{r_{\mathrm{He}}}{3}}$ $\left(r_{\mathrm{Be}}=\frac{5}{3}, r_{\mathrm{O}}=\frac{7}{5}\right)$
$\frac{x^{\prime}}{x}=\frac{\sqrt{\frac{5}{9}}}{\sqrt{\frac{7}{15}}}=\sqrt{\frac{25}{21}}=\frac{5}{\sqrt{21}}$
Hence, the correct answer is option: 'B'.