Question

Two gases with different densities and same ratio of specific heats $\left(\gamma \right)$ are mixed in proportions $V_1$ and $V_2$ by volume. The velocity $C$ of sound in mixture will be given by $(C_1,C_2$ are velocities in individual gases$)$

• A. $\frac{C_1+C_2}{2}$
• B. $\sqrt{C_1{C}_2}$
• C. $\frac{C_1{C}_2\sqrt{(V_1+V_2)}}{\sqrt{(V_1{C}_2^2+V_2{C}_1^2)}}$
• D. $\frac{C_1{C}_2\sqrt{(V_1+V-2)}}{\sqrt{(V_1{C}_1^2+V_2{C}_2^2)}}$

Answer 1

Answer: (C)

$\frac{C_1{C}_2\sqrt{(V_1+V_2)}}{\sqrt{(V_1{C}_2^2+V_2{C}_1^2)}}$

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

${\rho }_1=\frac{\gamma P}{{C_1}^2}$

${\rho }_2=\frac{\gamma P}{{C_2}^2}$

Mixture density,

$\rho =\frac{{\rho }_1\times V_1+{\rho }_2\times V_2}{V_1+V_2}$

$\rho =\frac{\frac{\gamma P}{{C_1}^2}\times V_1+\frac{\gamma P}{{C_2}^2}\times V_2}{V_1+V_2}$

$\rho =\gamma P\frac{\frac{1}{{C_1}^2}\times V_1+\frac{1}{{C_2}^2}\times V_2}{V_1+V_2}$

$C=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{1}{\frac{\frac{1}{{C_1}^2}\times V_1+\frac{1}{{C_2}^2}\times V_2}{V_1+V_2}}}$

$C=C_1{C}_2\sqrt{\frac{V_1+V_2}{V_1{C}_2^2+V_2{C}_1^2}}$