Question

If the speed of sound in a gas in beaker A is four times that of the speed of sound in another gas in beaker B, find the ratio of densities of the gases in the beakers A and B.

• A. $1:4$
• B. $1:16$
• C. $16:1$
• D. $4:1$

Answer 1

The correct option is B $1:16$

Laplace’s formula,
$V=\sqrt{\gamma P/\rho }$
$V-$Speed of gas
$\gamma -$Ratio of the specific heat of gas at constant temperature and pressure.
$P-$Pressure of the gas
$\rho -$Density of gas
Here,
$V\propto \sqrt{1/\rho }$
$V\sqrt{\rho }$ is a constant.
$i.e.,V_1{\sqrt{\rho }}_1=V_2{\sqrt{\rho }}_2$
Let the speed of sound in beaker B,
$V_2=V$
Therefore, the speed of sound in beaker A,
$V_1=4V$
We have,
$V_1{\sqrt{\rho }}_1=V_2{\sqrt{\rho }}_2$
$\sqrt{\frac{{\rho }_1}{{\rho }_2}}=\frac{V_2}{V_1}$
$\frac{{\rho }_1}{{\rho }_2}=\frac{{V_2}^2}{{V_1}^2}$
$\frac{{\rho }_1}{{\rho }_2}=\frac{V^2}{(4V{)}^2}$
$\frac{{\rho }_1}{{\rho }_2}=\frac{V^2}{16V^2}=\frac{1}{16}$

The ratio of densities of the gases in the beakers A and B is $1:16$