The ratio of the speed of sound in helium gas to that in nitrogen gas at the same temperature is…
The ratio of the speed of sound in helium gas to that in nitrogen gas at the same temperature is $\left(\gamma_{\mathrm{He}}=\frac{5}{3}, \gamma_{\mathrm{N}_2}=\frac{7}{5}, M_{\mathrm{He}}=4, M_{\mathrm{N}_2}=28\right)$
$\frac{5}{\sqrt{3}}$
$\sqrt{\frac{7}{5}}$
$\sqrt{\frac{2}{7}}$
$\sqrt{\frac{5}{3}}$
Solution
The correct option is (A).
Concept: Newton-Laplace equation for the speed of sound in an ideal gas is given by, $c=\sqrt{\frac{\gamma P}{\rho}}$ where is the speed of sound, $\gamma$ is the adiabatic index, $P$ the pressure and $\rho$ the density of the gas.
On introducing pressure $P=\frac{\rho R T}{M}$ by using the ideal gas equation, the speed of sound can be written as $c=\sqrt{\frac{\gamma R T}{M}}$. The speed of sound is proportional to the square root of the ratio of adiabatic index $y$ and molecular weight $\mathrm{M}$, i.e., $c \propto \sqrt{\frac{\gamma}{M}}$. On taking the ratio for Helium
and Nitrogen: $\frac{c_{\mathrm{He}}}{c_{\mathrm{N}_2}}=\sqrt{\frac{\gamma_{\mathrm{He}}}{M_{\mathrm{He}}} \times \frac{M_{\mathrm{N}_2}}{\gamma_{\mathrm{N}_2}}}$
On plugging in the values: $\frac{c_{\mathrm{He}}}{c_{\mathrm{N}_2}}=\sqrt{\frac{\frac{5}{3}}{\frac{7}{5}} \times \frac{28}{4}}=\frac{5}{\sqrt{3}}$