The ratio of the velocity of sound in hydroge | vedclass

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Question

(A) 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 universal gas

Vedclass · Accepted Answer

$\sqrt{42}/5$

Vedclass · Answer

(A) 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 universal gas constant,$T$ is the absolute temperature,and $M$ is the molar mass.Since the temperature $T$ is the same,the ratio of velocities is given by $\frac{v_{H_2}}{v_{He}} = \sqrt{\frac{\gamma_{H_2}}{\gamma_{He}} \cdot \frac{M_{He}}{M_{H_2}}}$.For hydrogen gas $(H_2)$,$\gamma_1 = 7/5$ and $M_1 = 2 	imes 10^{-3} \ kg/mol$.For helium gas $(He)$,$\gamma_2 = 5/3$ and $M_2 = 4 	imes 10^{-3} \ kg/mol$.Substituting the values:$\frac{v_{H_2}}{v_{He}} = \sqrt{\frac{7/5}{5/3} \cdot \frac{4}{2}} = \sqrt{\frac{7}{5} \cdot \frac{3}{5} \cdot 2} = \sqrt{\frac{42}{25}} = \frac{\sqrt{42}}{5}$.

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

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