Question

The velocities of sound at the same pressure in two monoatomic gases of densities ${\rho }_1$ and ${\rho }_2$ are $v_1$ and $v_2$ respectively. If $\frac{{\rho }_1}{{\rho }_2}=4$, then the value of $\frac{v_1}{v_2}$ is:

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. 2
• D. 4

Answer 1

The correct option is B $\frac{1}{2}$

Given that,
$\frac{{\rho }_1}{{\rho }_2}=4$
Velocity of sound is given by
$v=\sqrt{\frac{\gamma P}{\rho }}$

For the same monoatomic $\left(\gamma \right)$ gas and pressure $\left(P\right)$, $v\propto \sqrt{\frac{1}{\rho }}$
$v_1\propto \sqrt{\frac{1}{{\rho }_1}}$ ...(i)
$v_2\propto \sqrt{\frac{1}{{\rho }_2}}$ ...(ii)
From equation (i) and (ii),
$\frac{v_1}{v_2}=\sqrt{\frac{{\rho }_2}{{\rho }_1}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$