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: Vapour Density Ratio Provided

The provided ratio of vapour densities is: \[ \frac{\rho_1}{\rho_2} = \frac{4}{25} \]

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

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

Step 3: r.m.s. Velocity Ratio Calculation

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

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