Work `W` is required to form a bubble of volume `V` from a given solution. What amount of work is required to be done to form a bubble of volume `2V` ?

Options

A. `4^(2//3) W_(1)`
B. `4^(1//3) W_(1)`
C. `2^(1//2) W_(1)`
D. `2^(3//2) W_(1)`

Explanation

To solve the problem of determining the work required to form a bubble of volume \(2V\) when \(W\) is required for a bubble of volume \(V\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Work and Surface Area**:
The work done \(W\) to form a bubble is related to the surface tension \(\gamma\) and the change in surface area \(A\):
\[ W \propto \gamma \Delta A \]

2. **Surface Area of a Bubble**:
The surface area \(A\) of a bubble with radius \(r\) is given by:
\[ A = 4\pi r^2 \]

3. **Volume of a Bubble**:
The volume \(V\) of a bubble is given by:
\[ V = \frac{4}{3}\pi r^3 \]

4. **Relate Radius to Volume**:
From the volume equation, we can express the radius \(r\) in terms of volume \(V\):
\[ r = \left(\frac{3V}{4\pi}\right)^{1/3} \]

5. **Substituting Radius into Surface Area**:
Substitute \(r\) back into the surface area equation:
\[ A = 4\pi \left(\left(\frac{3V}{4\pi}\right)^{1/3}\right)^2 = 4\pi \cdot \left(\frac{3V}{4\pi}\right)^{2/3} \]

6. **Express Work in Terms of Volume**:
Since work is proportional to surface area, we can express the work done \(W\) in terms of volume:
\[ W \propto V^{2/3} \]

7. **Finding Work for Volume \(2V\)**:
Let \(W_2\) be the work required to form a bubble of volume \(2V\):
\[ W_2 \propto (2V)^{2/3} \]
Simplifying this gives:
\[ W_2 \propto 2^{2/3} V^{2/3} \]

8. **Relating \(W_2\) to \(W\)**:
Since \(W \propto V^{2/3}\), we can express \(W_2\) in terms of \(W\):
\[ W_2 = W \cdot 2^{2/3} \]

9. **Final Expression**:
Therefore, the final expression for the work required to form a bubble of volume \(2V\) is:
\[ W_2 = W \cdot 2^{2/3} \]

### Conclusion:
The amount of work required to form a bubble of volume \(2V\) is \(W \cdot 2^{2/3}\).