Q117 / 150 · MHT CET 2026 (20 April Shift 1) · Physics
The ratio of the speed of sound in helium gas to that in nitrogen gas at $295$ K is ( Ratio of specific heats $\gamma$ for helium is $\dfrac{5}{3}$ and that for nitrogen is $\dfrac{7}{5}$. Molecular weight for helium and nitrogen are $4$ and $28$ respectively.)
- A$\sqrt{\dfrac{2}{7}}$
- B$\sqrt{\dfrac{3}{5}}$
- C$\dfrac{\sqrt{3}}{5}$
- Correct answerD$\dfrac{5}{\sqrt{3}}$
Solution
The speed of sound in a gas is given by $v = \sqrt{\dfrac{\gamma R T}{M}}$.
For helium, $\gamma_{He} = \dfrac{5}{3}$ and $M_{He} = 4$.
For nitrogen, $\gamma_{N_2} = \dfrac{7}{5}$ and $M_{N_2} = 28$.
The ratio of the speed of sound in helium to that in nitrogen at the same temperature $T$ is:
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{\gamma_{He}}{\gamma_{N_2}} \times \dfrac{M_{N_2}}{M_{He}}}$
Substituting the given values:
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{5/3}{7/5} \times \dfrac{28}{4}}$
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{25}{21} \times 7} = \sqrt{\dfrac{25}{3}} = \dfrac{5}{\sqrt{3}}$
Answer: $\dfrac{5}{\sqrt{3}}$
For helium, $\gamma_{He} = \dfrac{5}{3}$ and $M_{He} = 4$.
For nitrogen, $\gamma_{N_2} = \dfrac{7}{5}$ and $M_{N_2} = 28$.
The ratio of the speed of sound in helium to that in nitrogen at the same temperature $T$ is:
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{\gamma_{He}}{\gamma_{N_2}} \times \dfrac{M_{N_2}}{M_{He}}}$
Substituting the given values:
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{5/3}{7/5} \times \dfrac{28}{4}}$
$\dfrac{v_{He}}{v_{N_2}} = \sqrt{\dfrac{25}{21} \times 7} = \sqrt{\dfrac{25}{3}} = \dfrac{5}{\sqrt{3}}$
Answer: $\dfrac{5}{\sqrt{3}}$
Hence, the correct answer is option: 'D'.