The gas inside a spherical bubble expands uniformly and slowly so that its radius increases from `R` to `2R`. Let the atmospheric pressure be `p_(0)` and surface tension be `S`. The work done by the gas in the process is

Options

A. `(28piR^3P_0)/3+24pisR^2`
B. `(25piR^3P_0)/3+24pisR^2`
C. `(25piR^3P_0)/3+(23pisR^2)/2`
D. `23piR^2`

Answer

To find the work done by the gas in expanding from a radius \( R \) to \( 2R \) in a spherical bubble, we need to consider both the work done against the atmospheric pressure and the work done against the surface tension of the bubble.

### Step-by-Step Solution:

1. **Identify the Work Done Against Atmospheric Pressure:**
The work done by the gas against the atmospheric pressure \( p_0 \) when the radius increases from \( R \) to \( 2R \) can be expressed as:
\[ W_{\text{pressure}} = \int_{R}^{2R} 4\pi r^2 p_0 \, dr \]
Here, \( 4\pi r^2 \) is the surface area of the bubble, and \( p_0 \) is the atmospheric pressure.

2. **Calculate the Work Done Against Surface Tension:**
The work done against the surface tension \( S \) is given by:
\[ W_{\text{surface}} = \int_{R}^{2R} 4S \, dr \]
The factor \( 4S \) comes from the fact that the surface area increases as the bubble expands.

3. **Combine the Work Done:**
The total work done by the gas during this process is the sum of the work done against pressure and surface tension:
\[ W_{\text{total}} = W_{\text{pressure}} + W_{\text{surface}} \]

4. **Perform the Integration:**
- For the work done against atmospheric pressure:
\[ W_{\text{pressure}} = 4\pi p_0 \int_{R}^{2R} r^2 \, dr = 4\pi p_0 \left[ \frac{r^3}{3} \right]_{R}^{2R} = 4\pi p_0 \left( \frac{(2R)^3}{3} - \frac{R^3}{3} \right) \]
\[ = 4\pi p_0 \left( \frac{8R^3 - R^3}{3} \right) = 4\pi p_0 \left( \frac{7R^3}{3} \right) = \frac{28\pi p_0 R^3}{3} \]

- For the work done against surface tension:
\[ W_{\text{surface}} = 4S \int_{R}^{2R} dr = 4S [r]_{R}^{2R} = 4S (2R - R) = 4S R \]

5. **Final Expression for Total Work Done:**
Combining both contributions:
\[ W_{\text{total}} = \frac{28\pi p_0 R^3}{3} + 4S R \]

### Final Answer:
The total work done by the gas in the process is:
\[ W = \frac{28\pi p_0 R^3}{3} + 4S R \]