The ratio of the speed of sound in nitrogen gas to that in helium gas at $300\text{ K}$ is

The ratio of the speed of sound in nitrogen gas to that in helium gas at $300\text{ K}$ is
  1. $\sqrt{\frac{2}{7}}$
  2. $\sqrt{\frac{1}{7}}$
  3. $\frac{\sqrt{3}}{5}$
  4. $\frac{\sqrt{6}}{5}$

Solution

Velocity of sound in gas, $v = \sqrt{\frac{\gamma RT}{M}} \Rightarrow v \propto \sqrt{\frac{\gamma}{M}}$ $\Rightarrow \frac{v_{\text{N}_2}}{v_{\text{He}}} = \sqrt{\frac{\gamma_{\text{N}_2}}{\gamma_{\text{He}}} \times \frac{M_{\text{He}}}{M_{\text{N}_2}}} = \sqrt{\frac{\frac{7}{5} \times 4}{\frac{5}{3} \times 28}} = \frac{\sqrt{3}}{5}$

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