The ratio of the velocity of sound in hydroge | vedclass

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Question

(C) The velocity of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$,where $\gamma$ is the adiabatic index,$R$ is the gas constant,$

Vedclass · Accepted Answer

$\frac{\sqrt{42}}{5}$

Vedclass · Answer

(C) The velocity of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$,where $\gamma$ is the adiabatic index,$R$ is the gas constant,$T$ is the temperature,and $M$ is the molar mass.Since the temperature $T$ is the same for both gases,the ratio of the velocities is $\frac{v_{H_2}}{v_{He}} = \sqrt{\frac{\gamma_{H_2} M_{He}}{\gamma_{He} M_{H_2}}}$.For hydrogen $(H_2)$,$M_{H_2} = 2 	imes 10^{-3} \ kg/mol$ and $\gamma_{H_2} = \frac{7}{5}$.For helium $(He)$,$M_{He} = 4 	imes 10^{-3} \ kg/mol$ and $\gamma_{He} = \frac{5}{3}$.Substituting these values: $\frac{v_{H_2}}{v_{He}} = \sqrt{\frac{(7/5) 	imes 4}{(5/3) 	imes 2}} = \sqrt{\frac{7}{5} 	imes \frac{3}{5} 	imes \frac{4}{2}} = \sqrt{\frac{7 	imes 3 	imes 2}{5 	imes 5}} = \sqrt{\frac{42}{25}} = \frac{\sqrt{42}}{5}$.

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 is

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