The velocities of sound at the same temperature in two monoatomic gases of densities $p_{1}$ and $p_{2}$ are $v_{1}$ and $v_{2}$ respectively. If $p_{1} / p_{2} = 4$ , then the value of $v_{1} / v_{2}$ is

\frac{1}{4}

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2

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\frac{1}{2}

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4

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Solution

The correct options are
A $\frac{1}{4}$
C $\frac{1}{2}$
For velocity of sound in gas
$v = \sqrt{\frac{p γ}{p}}$
$[P$ is pressure and $p$ is density of gas, $γ$ is $C_{p} / C_{v}]$
Here, $v_{1} = \sqrt{\frac{γ P}{p_{1}}}$ and $v_{2} = \sqrt{\frac{γ P}{p_{2}}}$
$\frac{v_{1}}{v_{2}} = \sqrt{\frac{p_{2}}{p_{1}}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$

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