Question

Consider the sound wave travelling in ideal gases of $\mathrm{He}, \mathrm{CH}_{4}$, and $\mathrm{CO}_{2}$. All the gases have the same ratio $\frac{\mathrm{P}}{\rho}$, where P is the pressure and $\rho$ is the density. The ratio of the speed of sound through the gases $\mathrm{v}_{\mathrm{He}}: \mathrm{v}_{\mathrm{CH}_{4}}: \mathrm{v}_{\mathrm{CO}_{2}}$ is given by

• A. $\sqrt{\frac{7}{5}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}$
• B. $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}$
• C. $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$
• D. $\sqrt{\frac{4}{3}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{7}{5}}$

Hint:

The speed of sound in a gas depends on the ratio of specific heats ($\gamma$).

Answer 1

The Correct Option is C

1. Speed of sound formula: \[ v_{\text{sound}} = \sqrt{\frac{\gamma P}{\rho}} \]
2. Ratio of specific heats ($\gamma$): - $\gamma_{\mathrm{He}} = \frac{5}{3}$ - $\gamma_{\mathrm{CH}_{4}} \approx \frac{4}{3}$ - $\gamma_{\mathrm{CO}_{2}} \approx \frac{4}{3}$
3. Ratio of speeds of sound: \[ \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}} \] Therefore, the correct answer is (3) $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$.