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` ?

Explanation

To solve the problem of determining the work required to form a bubble of volume `2V` given that the work `W` is required to form a bubble of volume `V`, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Done Formula**:
The work done `W` to form a bubble of volume `V` is given by the formula:
\[ W = T \cdot A \]
where \( T \) is the surface tension and \( A \) is the surface area of the bubble.

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

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

4. **Relate Volume to Radius**:
For the initial bubble of volume \( V \):
\[ V = \frac{4}{3} \pi r^3 \]
For the new bubble of volume \( 2V \):
\[ 2V = \frac{4}{3} \pi r'^3 \]
where \( r' \) is the radius of the new bubble.

5. **Set Up the Equation**:
Equating the two volumes, we have:
\[ \frac{4}{3} \pi r'^3 = 2 \cdot \frac{4}{3} \pi r^3 \]
Canceling \( \frac{4}{3} \pi \) from both sides gives:
\[ r'^3 = 2r^3 \]

6. **Find the Ratio of Radii**:
Taking the cube root of both sides:
\[ r' = r \cdot (2)^{1/3} \]

7. **Work Done Ratio**:
Since work is directly proportional to the square of the radius, we can express the ratio of work done for the two volumes:
\[ \frac{W'}{W} = \left(\frac{r'}{r}\right)^2 \]
Substituting \( r' \):
\[ \frac{W'}{W} = \left((2)^{1/3}\right)^2 = (2)^{2/3} \]

8. **Final Work Calculation**:
Therefore, the work required to form a bubble of volume \( 2V \) is:
\[ W' = (2)^{2/3} \cdot W \]

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