The radius R of the soap bubble is doubled under isothermal condition. If T be the surface tension of soap bubble. The work done in doing so it given by

Options

A. `32pi R^(2)T`
B. `24pi R^(2)T`
C. `8pi R^(2)T`
D. `4pi R^(2)T`

Answer

To solve the problem of calculating the work done when the radius of a soap bubble is doubled under isothermal conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Radius and Surface Area**:
- Let the initial radius of the soap bubble be \( R \).
- The surface area \( A_1 \) of the soap bubble is given by the formula:
\[ A_1 = 4\pi R^2 \]

2. **Determine the Final Radius and Surface Area**:
- When the radius is doubled, the new radius \( R' \) is:
\[ R' = 2R \]
- The new surface area \( A_2 \) of the soap bubble is:
\[ A_2 = 4\pi (R')^2 = 4\pi (2R)^2 = 4\pi (4R^2) = 16\pi R^2 \]

3. **Calculate the Change in Surface Area**:
- The change in surface area \( \Delta A \) is:
\[ \Delta A = A_2 - A_1 = 16\pi R^2 - 4\pi R^2 = 12\pi R^2 \]

4. **Account for the Two Surfaces of the Bubble**:
- Since a soap bubble has two surfaces (inner and outer), the total change in surface area \( \Delta A_t \) is:
\[ \Delta A_t = 2 \times \Delta A = 2 \times 12\pi R^2 = 24\pi R^2 \]

5. **Calculate the Work Done**:
- The work done \( W \) in expanding the bubble is given by the product of the change in surface area and the surface tension \( T \):
\[ W = \Delta A_t \times T = 24\pi R^2 T \]

### Final Answer:
The work done in doubling the radius of the soap bubble is:
\[ W = 24\pi R^2 T \]