Question 143: Thermodynamics & KTG

Question

The ratio of vapour densities of two gases at the same temperature is $ \frac{4}{25} $, then the ratio of r.m.s. velocities will be :

Accepted Answer

We are given that the ratio of the vapour densities of the two gases is
$$\frac{4}{25}.$$
Since vapour density is proportional to the molecular mass, we can write
$$\frac{M_1}{M_2} = \frac{4}{25},$$
where $M_1$ and $M_2$ are the molecular masses of the gases.
The root mean square (r.m.s.) velocity of a gas is given by
$$v_{	ext{rms}} = \sqrt{\frac{3RT}{M}},$$
where:
$R$ is the universal gas constant,
$T$ is the temperature,
$M$ is the molecular mass of the gas.
Since both gases are at the same temperature, the ratio of their r.m.s. velocities is
$$\frac{v_{	ext{rms,1}}}{v_{	ext{rms,2}}} = \sqrt{\frac{M_2}{M_1}}.$$
Substitute the ratio of molecular masses:
$$\frac{v_{	ext{rms,1}}}{v_{	ext{rms,2}}} = \sqrt{\frac{25}{4}} = \frac{5}{2}.$$
Thus, the ratio of their r.m.s. velocities is
$$\frac{5}{2}.$$
The correct answer is Option A: $$\frac{5}{2}.$$

Answer

$ \frac{5}{2} $

Answer

$ \frac{25}{4} $

Answer

$ \frac{4}{25} $

Answer

$ \frac{2}{5} $

Thermodynamics & KTG

The ratio of vapour densities of two gases at the same temperature is \frac{4}{25} , then the ratio of r.m.s. velocities will be :

Choose the correct answer: