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 :

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