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

<p>The speed of sound in an ideal gas is given by</p> <p>$v = \sqrt{\gamma \frac{P}{\rho}},$</p> <p>where:</p> <p><p>$\gamma$ is the ratio of specific heats,</p></p> <p><p>$P$ is the pressure, and</p></p> <p><p>$\rho$ is the density.</p></p> <p>Since the problem states that $\frac{P}{\rho}$ is the same for all the gases, the speed of sound in each gas is determined solely by the factor $\sqrt{\gamma}$.</p> <p>Let's determine the appropriate $\gamma$ for each gas:</p> <p><strong>Helium (He):</strong></p> <p><p>Helium is a monatomic gas.</p></p> <p><p>For a monatomic gas, $\gamma = \frac{5}{3}$.</p></p> <p><p>Therefore, $v_{\mathrm{He}} \propto \sqrt{\frac{5}{3}}$.</p></p> <p><strong>Methane (CH$_4$):</strong></p> <p><p>Methane is a polyatomic gas (a tetrahedral molecule with three rotational degrees of freedom).</p></p> <p><p>For nonlinear polyatomic gases, $\gamma$ is typically taken as $\frac{4}{3}$.</p></p> <p><p>Thus, $v_{\mathrm{CH_4}} \propto \sqrt{\frac{4}{3}}$.</p></p> <p><strong>Carbon Dioxide (CO$_2$):</strong></p> <p><p>Carbon dioxide is a linear molecule.</p></p> <p><p>For linear molecules, $\gamma = \frac{7}{5}$.</p></p> <p><p>Hence, $v_{\mathrm{CO_2}} \propto \sqrt{\frac{7}{5}}$.</p></p> <p>Therefore, the ratio of the speeds of sound in the gases is:</p> <p>$$ v_{\mathrm{He}} : v_{\mathrm{CH_4}} : v_{\mathrm{CO_2}} = \sqrt{\frac{5}{3}} : \sqrt{\frac{4}{3}} : \sqrt{\frac{7}{5}}. $$</p> <p>Comparing this expression with the given options, we see that it matches <strong>Option C</strong>.</p>