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 the formula $
u = \sqrt{\frac{\gamma R T}{M}}$.Since the temperature $T$ is the same for both g

Vedclass · Accepted Answer

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

Vedclass · Answer

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

The ratio of the velocity of sound in hydrogen $(\gamma = 7/5)$ to that in helium $(\gamma = 5/3)$ at the same temperature is

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