Question

The velocities of sound at same temperature in two monoatomic gases densities ${\rho }_1$ and ${\rho }_2$ are $v_1$ and $v_2$ respectively, if $\frac{{\rho }_1}{{\rho }_2}=4$, then, the value of $\frac{v_1}{v_2}$ will be:

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

Answer 1

Answer: (C)

Since both gases are monoatomic ratio of specific heats is same for both.
The velocity of sound is given by
$v=\sqrt{\frac{\gamma P}{\rho }},$
where $P$ is pressure, $\rho$ is density and $\gamma$ is ratio of specific heats.
$\therefore \frac{v_1}{v_2}=\sqrt{\frac{{\rho }_2}{{\rho }_1}}$ (since $\rho$ and $\gamma$ are same for both)
Given, $\frac{{\rho }_2}{{\rho }_1}=\frac{1}{4}$
$\therefore \frac{v_1}{v_2}=\sqrt{\frac{1}{4}}=\frac{1}{2}$