The ratio of the speed of sound in helium gas | vedclass

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(D) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic index (ratio of speci

Vedclass · Accepted Answer

$\frac{5}{\sqrt{3}}$

Vedclass · Answer

(D) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic index (ratio of specific heats), $R$ is the universal gas constant, $T$ is the temperature, and $M$ is the molar mass.Given $T = 295 	ext{ K}$ for both gases, the ratio of the speed of sound in helium $(v_{He})$ to nitrogen $(v_{N_2})$ is:$\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{\gamma_{He} R T / M_{He}}{\gamma_{N_2} R T / M_{N_2}}} = \sqrt{\frac{\gamma_{He}}{\gamma_{N_2}} \cdot \frac{M_{N_2}}{M_{He}}}$Substituting the given values: $\gamma_{He} = 5/3$, $\gamma_{N_2} = 7/5$, $M_{He} = 4$, $M_{N_2} = 28$.$\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{5/3}{7/5} \cdot \frac{28}{4}} = \sqrt{\frac{5}{3} \cdot \frac{5}{7} \cdot 7} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}}$.

The ratio of the speed of sound in helium gas to that in nitrogen gas at $295 \text{ K}$ is (Ratio of specific heats for helium is $5/3$ and that for nitrogen is $7/5$. Molecular weight for helium and nitrogen are $4$ and $28$ respectively.)

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