Question

The speed of sound in hydrogen gas at $\text{STP}$ is $v_o$. The speed of sound in a mixture containing $3$ parts of hydrogen gas and $2$ parts of oxygen gas at $\text{STP}$ will be

• A. $\frac{v_o}{\sqrt{3}}$
• B. $\frac{v_o}{\sqrt{5}}$
• C. $\frac{v_o}{\sqrt{6}}$
• D. $\frac{v_o}{\sqrt{7}}$

Answer 1

The correct option is D $\frac{v_o}{\sqrt{7}}$

At a given pressure and temperature, the density of gas is directly proportional to its molecular weight.
$PM=\rho RT$
$\Rightarrow \rho \propto M$
$\Rightarrow \frac{{\rho }_1}{{\rho }_2}=\frac{M_1}{M_2}=\frac{2}{32}=\frac{1}{16}$
$\Rightarrow {\rho }_2=16{\rho }_1$ ...............(1)
Here,
${\rho }_1\to$ density of hydrogen,
${\rho }_2\to$ density of oxygen

Now, density of mixture,
${\rho }_{mix}=\frac{{\rho }_1{V}_1+{\rho }_2{V}_2}{V_1+V_2}$
$\Rightarrow {\rho }_{mix}=\frac{{\rho }_1{V}_1+16{\rho }_1\times \frac{2V_1}{3}}{V_1+\frac{2V_1}{3}}=\frac{35{\rho }_1{V}_1}{5V_1}$
[ given $\frac{V_1}{V_2}=\frac{3}{2}$ and from (1)]
$\Rightarrow {\rho }_{mix}=7{\rho }_1$ ..........(2)
Speed of sound,
$v=\sqrt{\frac{\gamma P}{\rho }}$
At constant pressure,
$v\propto \frac{1}{\sqrt{\rho }}$
So,
$\frac{v_{H_2}}{v_{mix}}=\sqrt{\frac{{\rho }_{mix}}{{\rho }_{H_2}}}$
$\Rightarrow v_{mix}=v_{H_2}\times \sqrt{\frac{{\rho }_{H_2}}{{\rho }_{mix}}}$
$\Rightarrow v_{mix}=v_o\times \sqrt{\frac{{\rho }_1}{7{\rho }_1}}$
$\Rightarrow v_{mix}=\frac{v_o}{\sqrt{7}}$