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The ratio of the speed of sound in helium gas to that in nitrogen gas at the same temperature is $\left(\gamma_{\mathrm{He}}=\frac{5}{3}, \gamma_{\mathrm{N}_2}=\frac{7}{5}, M_{\mathrm{He}}=4, M_{\mathrm{N}_2}=28\right)$
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$\frac{5}{\sqrt{3}}$
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: $\frac{c_{\mathrm{He}}}{c_{\mathrm{N}_2}}=\sqrt{\frac{\frac{5}{3}}{\frac{7}{5}} \times \frac{28}{\sqrt{3}}}=\frac{5}{\sqrt{5}}$
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: $\frac{c_{\mathrm{He}}}{c_{\mathrm{N}_2}}=\sqrt{\frac{\frac{5}{3}}{\frac{7}{5}} \times \frac{28}{\sqrt{3}}}=\frac{5}{\sqrt{5}}$
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