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: \[ \frac{m_1}{m_2} = \frac{R_1^2}{R_2^2} \]

Answer 2

To solve the problem, we need to find the relationship between the masses of air inside two soap bubbles of different radii kept at constant temperature in a vacuum.

1. Understanding the Problem:
We are given:

• The two soap bubbles have radii \( R_1 \) and \( R_2 \).
• The temperature is constant, and the bubbles are in a vacuum.
• We are asked to find the ratio of the masses of air inside them.

2. Relationship Between the Masses and Radii of the Bubbles:
For soap bubbles in a vacuum at constant temperature, the mass of air inside a bubble is directly related to the surface area of the bubble. The surface area \( A \) of a spherical bubble is proportional to \( R^2 \), where \( R \) is the radius of the bubble. Hence, the mass of air inside the bubble is proportional to \( R^3 \), since mass is related to volume (and volume is proportional to \( R^3 \)). So, the ratio of the masses \( m_1 \) and \( m_2 \) of air inside the bubbles is given by: \[ \frac{m_1}{m_2} = \frac{R_1^3}{R_2^3} \] This matches with the third option, which is the correct relationship.

Final Answer:
The correct option is (D) \( \frac{m_1}{m_2} = \frac{R_1^2}{R_2^2} \).