A mixture of two diatomic gases is obtained by mixing $m_{1} {and m}_{2}$ masses of two gases, with velocity of sound in them as $c_{1} {and c}_{2}$ respectively. Determine the velocity of sound in the mixture of gases. (Temperature of the gas remains constant)

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$c = \sqrt{\frac{m_{1} c_{2}^{2} - m_{2} c_{1}^{2}}{m_{2} {+m}_{1}}}$

$c = \sqrt{\frac{m_{1} c_{2}^{2} {+m}_{2} c_{1}^{2}}{m_{1} {+m}_{2}}}$

$c= \sqrt{\frac{m_{1} c_{1}^{2} {+m}_{2} c_{2}^{2}}{m_{1} {+m}_{2}}}$

$c = \sqrt{\frac{m_{1} c_{2}^{2} {+m}_{2} c_{1}^{2}}{m_{1} - m_{2}}}$

answer is C.

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Detailed Solution

$C = \sqrt{\frac{γ R T}{μ}} n = \frac{m}{μ} C = \sqrt{\frac{γ R T n}{m}} μ = \frac{m}{n} μ = \frac{m_{1} + m_{2}}{\frac{m_{1}}{μ_{1}} + \frac{m_{2}}{μ_{2}}} a n d μ_{1} = \frac{γ R T}{C_{1}^{2}}, μ_{2} = \frac{γ R T}{C_{2}^{2}} \frac{γ R T}{C^{2}} = \frac{m_{1} + m_{2}}{\frac{m_{1} C_{1}^{2}}{γ R T} + \frac{m_{2} C_{2}^{2}}{γ R J}} C = \sqrt{\frac{m_{1} C_{1}^{2} + m_{2} C_{2}^{2}}{m_{1} + m_{2}}}$