Consider the sound wave travelling in ideal gases of $\mathrm{He}, \mathrm{CH}_4$, and $\mathrm{CO}_2$. All…

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 $\frac{P}{\rho}$, 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
  1. $\sqrt{\frac{7}{5}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}$
  2. $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}$
  3. $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$
  4. $\sqrt{\frac{4}{3}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{7}{5}}$

Solution

$\begin{aligned} & \mathrm{v}_{\text {sound }}=\sqrt{\frac{\gamma \mathrm{p}}{\rho}} \\ & \gamma=1+\frac{2}{\mathrm{f}} \\ & \gamma_{\mathrm{He}}=\frac{5}{3} \\ & \left.\gamma_{\mathrm{CH}_4}=\gamma_{\mathrm{CO}_2} \approx 1.33=\frac{4}{3} \quad \text { (Experimental data }\right)\end{aligned}$

Asked in: JEE Main 2025 (04 Apr Shift 1)

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