Question

The ratio of the speed of sound in nitrogen gas to that in helium gas, at $300K$ is

• A. $\sqrt{\frac{2}{7}}$
• B. $\sqrt{\frac{1}{7}}$
• C. $\frac{\sqrt{3}}{5}$
• D. $\frac{\sqrt{6}}{5}$

Answer 1

Answer: (C)

Speed of sound in an ideal gas is given by
$v=\sqrt{\frac{\gamma RT}{M}}$
$\therefore v\propto \sqrt{\frac{\gamma }{M}}$ [ $T$ is same for both the gases]
${\gamma }_{{\text{N}}_2}=\frac{7}{5}$ (Diatomic)
${\gamma }_{\text{He}}=\frac{5}{3}$ (Monoatomic)
$\therefore \frac{v_{N_2}}{v_{He}}=\sqrt{\frac{{\gamma }_{N_2}}{{\gamma }_{He}}\cdot \frac{M_{He}}{M_{N_2}}}=\sqrt{\frac{7/5}{5/3}\left(\frac{4}{28}\right)}$
$=\frac{\sqrt{3}}{5}$