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A mixture of two diatomic gases exists in a closed cylinder. The volumes and velocities in the two gases are $V_{1}, V_{2}$ and $c_{1}, c_{2}$ respectively. Determine the velocities of the sound in the mixture.

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$C_{1} C_{2} \sqrt{\frac{V_{1} + V_{2}}{V_{1} c_{2}^{2} + V_{2} c_{1}^{2}}}$

$C_{1} C_{2} \sqrt{\frac{V_{1} + V_{2}}{V_{1} c_{1}^{2} + V_{2} c_{2}^{2}}}$

$C_{1} C_{2} \sqrt{\frac{V_{1} + V_{2}}{V_{1}^{2} c_{1} + V_{2} c_{2}^{2}}}$

$C_{1} C_{2} \sqrt{\frac{V_{1} + V_{2}}{V_{1}^{2} c_{1} + V_{2}^{2} c_{2}}}$

answer is A.

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Detailed Solution

$ρ_{m i x} = \frac{ρ_{1} V_{1} + ρ_{2} V_{2}}{V_{1} + V_{2}}$
Velocity of sound $\propto \frac{1}{\sqrt{ρ}} \Rightarrow ρ \propto \frac{1}{c^{2}}$

$\frac{ρ_{1}}{ρ_{2}} = \frac{c_{2}^{2}}{c_{1}^{2}}$

$ρ_{m i x} = \frac{ρ_{2} \frac{c_{2}^{2}}{c_{1}^{2}} V_{1} + ρ_{2} V_{2}}{V_{1} + V_{2}}$

$\frac{c_{\max}}{c_{2}} = \sqrt{\frac{ρ_{2}}{ρ_{\max}}}$

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