Question

Two spherical soap bubbles of radii \( r_1 \) and \( r_2 \) in vacuum coalesce under isothermal condition. The newly formed bubble has a radius (\( r \)) given by

• A. \( r_1 + r_2 \)
• B. \( \frac{r_1+r_2}{2} \)
• C. \( \frac{r_1 r_2}{r_1 + r_2} \)
• D. \( \sqrt{r_1^2 + r_2^2} \)

Hint:

Always distinguish whether the coalescence is happening in a vacuum or in the atmosphere.
- In vacuum: \( r = \sqrt{r_1^2 + r_2^2} \)
- In atmosphere (with pressure \( P_0 \)): \( P_0 (r^3 - r_1^3 - r_2^3) + 4T(r^2 - r_1^2 - r_2^2) = 0 \)
Paying attention to the word "vacuum" in the question text allows you to choose the correct relationship directly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Two soap bubbles of different radii coalesce to form a single larger soap bubble under isothermal conditions inside a vacuum. We need to find the radius of the resulting bubble.

Step 2: Key Formula or Approach:
Since the process is isothermal and the number of moles of air remains conserved, we can apply Boyle's Law:
\[ P_1 V_1 + P_2 V_2 = P_f V_f \]
where:
- For a soap bubble in vacuum, the internal pressure is equal to the excess pressure: \( P = \frac{4T}{R} \) (where \( T \) is the surface tension).
- Volume of a spherical bubble of radius \( R \) is: \( V = \frac{4}{3}\pi R^3 \)

Step 3: Detailed Explanation:
Let the initial bubbles have radii \( r_1 \) and \( r_2 \), and the final coalesced bubble have radius \( r \).
The internal pressures of the bubbles are:
\[ P_1 = \frac{4T}{r_1}, \quad P_2 = \frac{4T}{r_2}, \quad P_f = \frac{4T}{r} \]
The respective volumes are:
\[ V_1 = \frac{4}{3}\pi r_1^3, \quad V_2 = \frac{4}{3}\pi r_2^3, \quad V_f = \frac{4}{3}\pi r^3 \]
According to Boyle's law for isothermal mixture of ideal gases:
\[ P_1 V_1 + P_2 V_2 = P_f V_f \]
Substitute the pressure and volume terms:
\[ \left( \frac{4T}{r_1} \right) \left( \frac{4}{3}\pi r_1^3 \right) + \left( \frac{4T}{r_2} \right) \left( \frac{4}{3}\pi r_2^3 \right) = \left( \frac{4T}{r} \right) \left( \frac{4}{3}\pi r^3 \right) \]
Simplifying the terms:
\[ \frac{16}{3}\pi T r_1^2 + \frac{16}{3}\pi T r_2^2 = \frac{16}{3}\pi T r^2 \]
Divide both sides by the constant factor \( \frac{16}{3}\pi T \):
\[ r_1^2 + r_2^2 = r^2 \]
\[ r = \sqrt{r_1^2 + r_2^2} \]

Step 4: Final Answer:
The radius of the newly formed bubble is \( \sqrt{r_1^2 + r_2^2} \).