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(B) The speed of sound in an ideal gas is given by the formula $V = \sqrt{\frac{\gamma P}{\rho}}$,where $\gamma$ is the adiabatic index,$P$ is the pre

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

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(B) The speed of sound in an ideal gas is given by the formula $V = \sqrt{\frac{\gamma P}{\rho}}$,where $\gamma$ is the adiabatic index,$P$ is the pressure,and $\rho$ is the density.Given that the ratio $\frac{P}{\rho}$ is constant for all gases,the speed of sound is proportional to the square root of the adiabatic index: $V \propto \sqrt{\gamma}$.For $He$ (monatomic gas),the degrees of freedom $f = 3$,so $\gamma_{He} = 1 + \frac{2}{3} = \frac{5}{3}$.For $CH_4$ (polyatomic gas),the degrees of freedom $f = 6$,so $\gamma_{CH_4} = 1 + \frac{2}{6} = 1 + \frac{1}{3} = \frac{4}{3}$.For $CO_2$ (polyatomic gas),the degrees of freedom $f = 6$,so $\gamma_{CO_2} = 1 + \frac{2}{6} = \frac{4}{3}$.Therefore,the ratio of the speeds is $V_{He} : V_{CH_4} : V_{CO_2} = \sqrt{\gamma_{He}} : \sqrt{\gamma_{CH_4}} : \sqrt{\gamma_{CO_2}} = \sqrt{\frac{5}{3}} : \sqrt{\frac{4}{3}} : \sqrt{\frac{4}{3}}$.

Consider the sound wave travelling in ideal gases of $He$,$CH_4$,and $CO_2$. 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 $V_{He}: V_{CH_4}: V_{CO_2}$ is given by

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