What is the ratio of the velocity of sound in hydrogen $\left(\gamma=\frac{7}{5}\right)$ to that in helium…
What is the ratio of the velocity of sound in hydrogen $\left(\gamma=\frac{7}{5}\right)$ to that in helium $\left(\gamma=\frac{5}{3}\right)$ at the same temperature? (Molecular weight of hydrogen and helium is 2 and 4 respectively)
$\frac{\sqrt{42}}{5}$
$\frac{5}{\sqrt{42}}$
$\frac{\sqrt{21}}{5}$
$\frac{5}{\sqrt{21}}$
Solution
Velocity of sound is given by
$\mathrm{V}=\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}} \therefore \frac{\mathrm{V}_{\mathrm{H}}}{\mathrm{V}_{\mathrm{He}}}=\sqrt{\frac{\gamma_{\mathrm{H}}}{\gamma_{\mathrm{He}}} \cdot \frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{H}}}}(\mathrm{T}$ is the same for both)
$=\sqrt{\frac{7}{5} \times \frac{3}{5} \times \frac{4}{2}}=\frac{\sqrt{42}}{5}$