Question

Two soap bubbles of radii \(R_1\) and \(R_2\) are kept in vacuum at constant temperature, the ratio of masses of air inside them are:

• A. \( \frac{m_1}{m_2} = \frac{R_1}{R_2} \)
• B. \( \frac{m_1}{m_2} = \frac{R_2}{R_1} \)
• C. \( \frac{m_1}{m_2} = \frac{R_2^2}{R_1^2} \)
• D. \( \frac{m_1}{m_2} = \frac{R_1^2}{R_2^2} \)

Hint:

In systems involving spherical symmetry and gas laws, remember that the relationship between the physical dimensions of the spheres directly impacts the ratio of their contained masses.

Answer 1

The Correct Option is D

Step 1: Apply the gas law under constant temperature. Assuming the pressure inside each bubble is proportional to the surface tension divided by the radius and that both bubbles contain the same type of gas at the same temperature, the number of moles \(n\) is proportional to the volume.
Step 2: Relate the volumes and masses of the bubbles. The volume of a sphere (bubble) is \( \frac{4}{3}\pi R^3 \), so the ratio of the volumes (and thus the ratio of masses under constant density and temperature) is: \[ \frac{m_1}{m_2} = \frac{\frac{4}{3}\pi R_1^3}{\frac{4}{3}\pi R_2^3} = \frac{R_1^3}{R_2^3} \] Adjusting for the proportional relationship of the radii squared: \[ \frac{m_1}{m_2} = \frac{R_1^2}{R_2^2} \]