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

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Question

(A) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$.Since the temperature $T$ is the same for both gases,t

Vedclass · Accepted Answer

$\sqrt{5/3}$

Vedclass · Answer

(A) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$.Since the temperature $T$ is the same for both gases,the ratio of the speed of sound in helium $(v_{He})$ to that in nitrogen $(v_{N_2})$ is given by:$\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{\gamma_{He}}{M_{He}} \cdot \frac{M_{N_2}}{\gamma_{N_2}}}$Substituting the given values $\gamma_{He} = 5/3$,$M_{He} = 4$,$\gamma_{N_2} = 7/5$,and $M_{N_2} = 28$:$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5/3}{4}ight) \cdot \left(\frac{28}{7/5}ight)}$$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5}{12}ight) \cdot \left(\frac{28 \cdot 5}{7}ight)}$$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5}{12}ight) \cdot (4 \cdot 5)} = \sqrt{\frac{5 \cdot 20}{12}} = \sqrt{\frac{100}{12}} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}}$Wait,re-evaluating the calculation: $\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{5/3}{4} 	imes \frac{28}{7/5}} = \sqrt{\frac{5}{12} 	imes \frac{140}{7}} = \sqrt{\frac{5}{12} 	imes 20} = \sqrt{\frac{100}{12}} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}}$.Given the options provided in the prompt,there appears to be a discrepancy. Based on the standard calculation,the result is $5/\sqrt{3}$. If we assume the question intended to test the ratio $\sqrt{\frac{\gamma_{He} M_{N_2}}{\gamma_{N_2} M_{He}}}$,the result is $\sqrt{\frac{5/3 	imes 28}{7/5 	imes 4}} = \sqrt{\frac{140/3}{28/5}} = \sqrt{\frac{140}{3} 	imes \frac{5}{28}} = \sqrt{5 	imes \frac{5}{3}} = \sqrt{25/3} = 5/\sqrt{3}$.

The ratio of the speed of sound in helium gas to that in nitrogen gas at the same temperature is $\left(\gamma_{He}=\frac{5}{3}, \gamma_{N_2}=\frac{7}{5}, M_{He}=4, M_{N_2}=28\right)$

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