A mixture of two diatomic gases exist in a closed cylinder. The volumes and velocities in the two gases are $V_1, V_2$ and $c_1, c_2$ respectively ${\rho }_1 \mathrm{and} {\rho }_2$ are the densities of the two gases. Then

 

We can write $C_1=\sqrt{\frac{\gamma P}{{\rho }_1}} and C_2=\sqrt{\frac{\gamma P}{{\rho }_2}}$

$\therefore \frac{{\rho }_1}{{\rho }_2}=\frac{C_2^2}{C_1^2}$

If $\rho$ is the density of the mixture of gases, then

$$\begin{gathered} \rho =\frac{m_1+m_2}{V_1+V_2}=\frac{{\rho }_1{V}_1+{\rho }_2{V}_2}{V_1+V_2} \\ =\frac{{\rho }_1{V}_1\left(1+{\displaystyle \frac{{\rho }_2}{{\rho }_1}\frac{V_2}{V_1}}\right)}{V_1+V_2} \\ \mathrm{or} \frac{\rho }{{\rho }_1}=\frac{V_1\left(1+{\displaystyle \frac{C_1^2}{C_2^2}}{\displaystyle \frac{V_2}{V_1}}\right)}{V_1+V_2}=\frac{V_1\left(V_1{C}_2^2+V_2{C}_1^2\right)}{C_2^2\left(V_1+V_2\right)} \end{gathered}$$

The velocity of sound in a mixture is given by

$$\begin{gathered} c=\sqrt{\frac{\gamma P}{\rho }}. \\ \mathrm{Also} \frac{c}{c_1}=\sqrt{\frac{{\rho }_1}{\rho }} \therefore c=c_1{c}_2\sqrt{\frac{V_1+V_2}{V_1{c}_2^2+V_2{c}_1^2}} \end{gathered}$$