Question

Under isothermal conditions, two soap bubbles of radii $r_{1}$ and $r_{2}$ coalesce to form a big bubble. The radius of the big bubble is}

• A. $(r_{1}+r_{2})^{1/2}$
• B. $(r_{1}+r_{2})^{2}$
• C. $(r_{1}^{2}+r_{2}^{2})^{1/2}$
• D. $(r_{1}+r_{2})^{3}$

Hint:

Coalescence in vacuum: $R^2 = r_1^2 + r_2^2$. Coalescence under atmospheric pressure: $R^3 = r_1^3 + r_2^3$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Isothermal coalescence involves the conservation of total energy or total number of moles of gas.

Step 2: Key Formula or Approach:
For a soap bubble, excess pressure \( P = \frac{4T}{r} \).
Conservation of gas (Boyle's Law): \( P_1 V_1 + P_2 V_2 = P_R V_R \).

Step 3: Detailed Explanation:
Assume external pressure is zero (vacuum) or constant. The contribution from surface energy leads to:
\[ \left( \frac{4T}{r_1} \right) \left( \frac{4}{3}\pi r_1^3 \right) + \left( \frac{4T}{r_2} \right) \left( \frac{4}{3}\pi r_2^3 \right) = \left( \frac{4T}{R} \right) \left( \frac{4}{3}\pi R^3 \right) \]
Cancelling common terms \( \left( \frac{4T \cdot 4\pi}{3} \right) \):
\[ r_1^2 + r_2^2 = R^2 \implies R = \sqrt{r_1^2 + r_2^2} \]

Step 4: Final Answer:
The radius is \( (r_1^2 + r_2^2)^{1/2} \).