Step 1: Understanding the Concept:
In vacuum, the pressure inside a soap bubble is due solely to the surface tension.
For a soap bubble of radius \(r\) with surface tension \(S\), the excess pressure (and thus the internal pressure in vacuum) is \(P = \frac{4S}{r}\).
Under isothermal conditions, according to Boyle's Law (\(PV = \text{constant}\)), the total "PV" product is conserved during the coalescence of two bubbles.
Step 2: Key Formula or Approach:
Pressure: \(P = \frac{4S}{r}\).
Volume: \(V = \frac{4}{3}\pi r^3\).
Conservation of state: \(P_1 V_1 + P_2 V_2 = P_{final} V_{final}\).
Step 3: Detailed Explanation:
Step 1: Write terms for Bubble 1.
\[P_1 V_1 = \left(\frac{4S}{r_1}\right) \left(\frac{4}{3}\pi r_1^3\right) = \frac{16}{3}\pi S r_1^2\]
Step 2: Write terms for Bubble 2.
\[P_2 V_2 = \left(\frac{4S}{r_2}\right) \left(\frac{4}{3}\pi r_2^3\right) = \frac{16}{3}\pi S r_2^2\]
Step 3: Write terms for the resultant Bubble.
\[P V = \frac{16}{3}\pi S r^2\]
Step 4: Equating total products (Boyle's Law for moles conservation at const. T):
\[\frac{16}{3}\pi S r_1^2 + \frac{16}{3}\pi S r_2^2 = \frac{16}{3}\pi S r^2\]
Step 5: Simplify:
\[r_1^2 + r_2^2 = r^2\]
\[r = \sqrt{r_1^2 + r_2^2}\]
This result matches option (D).
Step 4: Final Answer:
The final radius is given by the square root of the sum of the squares of the initial radii, derived from the conservation of surface energy/molar content in vacuum.