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 $v_{\mathrm{He}}: \mathrm{V}_{\mathrm{CH}_{4}}: \mathrm{V}_{\mathrm{CO}_{2}}$ is given by

(C) $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{4}{3}}$

(A) $\sqrt{\frac{7}{5}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}$

(B) $\sqrt{\frac{5}{3}}: \sqrt{\frac{4}{3}}: \sqrt{\frac{7}{5}}$

(D) $\sqrt{\frac{4}{3}}: \sqrt{\frac{5}{3}}: \sqrt{\frac{7}{5}}$

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Waves & Sound

Consider the sound wave travelling in ideal gases of

He

CH_{4}

, and

CO_{2}

. All the gases have the same ratio

\frac{P}{ρ}

, where

P

is the pressure and

ρ

is the density. The ratio of the speed of sound through the gases

v_{He} : V_{CH_{4}} : V_{CO_{2}}

is given by

Answer: (C)

The speed of sound in a gas is

v = γ P / ρ

. Since

P / ρ

is identical for the three gases, the speed depends only on the adiabatic index,

v \propto γ

. Helium is monatomic, so

γ = 5/3

. Methane and carbon dioxide behave as polyatomic gases, for which

γ = 4/3

. Hence the speeds are in the ratio

v_{H e} : v_{C H_{4}} : v_{C O_{2}} = \frac{5}{3} : \frac{4}{3} : \frac{4}{3} .

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Single correct option