JEE Physics 2025 Question with Solution

Given: Sound waves travel in ideal gases $\mathrm{He}$, $\mathrm{CH}_{4}$ and $\mathrm{CO}_{2}$, and all gases have the same ratio $\frac{P}{\rho}$.

Find: The ratio $v_{\mathrm{He}}:v_{\mathrm{CH}_{4}}:v_{\mathrm{CO}_{2}}$.

For an ideal gas, the speed of sound is

$$v_{\text{sound}} = \sqrt{\frac{\gamma P}{\rho}}$$

Since $\frac{P}{\rho}$ is the same for all three gases, the speed depends only on $\sqrt{\gamma}$.

The ratios of specific heats are:

• $\gamma_{\mathrm{He}} = \frac{5}{3}$
• $\gamma_{\mathrm{CH}_{4}} \approx \frac{4}{3}$
• $\gamma_{\mathrm{CO}_{2}} \approx \frac{4}{3}$

Therefore,

$$v_{\mathrm{He}}:v_{\mathrm{CH}_{4}}:v_{\mathrm{CO}_{2}} = \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$$

Hence, the correct option is C.

The expression

$$v = \sqrt{\frac{\gamma P}{\rho}}$$

can be written for each gas as:

$$v_{\mathrm{He}} = \sqrt{\gamma_{\mathrm{He}}\frac{P}{\rho}}, \quad v_{\mathrm{CH}_{4}} = \sqrt{\gamma_{\mathrm{CH}_{4}}\frac{P}{\rho}}, \quad v_{\mathrm{CO}_{2}} = \sqrt{\gamma_{\mathrm{CO}_{2}}\frac{P}{\rho}}$$

Because the factor $\frac{P}{\rho}$ is common, it cancels in the ratio. So,

$$v_{\mathrm{He}}:v_{\mathrm{CH}_{4}}:v_{\mathrm{CO}_{2}} = \sqrt{\gamma_{\mathrm{He}}}:\sqrt{\gamma_{\mathrm{CH}_{4}}}:\sqrt{\gamma_{\mathrm{CO}_{2}}}$$

Substituting the given values of $\gamma$ gives

$$\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$$

Therefore, the correct option is C.