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:

• A. \( \frac{25}{4} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{5}{2} \)
• D. \( \frac{4}{25} \)

Hint:

The r.m.s. velocity of a gas is inversely proportional to the square root of its molecular mass. Vapour density is directly proportional to the molecular mass.

Answer 1

The Correct Option is C

Step 1: Given Vapour Density Ratio

We are given the ratio of the vapour densities as: \[ \frac{\rho_1}{\rho_2} = \frac{4}{25} \]

Step 2: Relate Vapour Density Ratio to r.m.s. Velocity Ratio

We know that the ratio of r.m.s. velocities \( v_1 \) and \( v_2 \) is related to the ratio of vapour densities by the formula: \[ \frac{v_1}{v_2} = \sqrt{\frac{\rho_2}{\rho_1}} \]

Step 3: Calculate the Ratio of r.m.s. Velocities

Substituting the given vapour density ratio: \[ \frac{v_1}{v_2} = \sqrt{\frac{25}{4}} = \frac{5}{2} \]

Final Answer: \[ \frac{v_1}{v_2} = \frac{5}{2} \]