Calculate the work done in blowing out a soap bubble of diameter 1 cm. given that the surface tension of soap solution is `28 xx 10^(-3) Nm^(-1)`

Explanation

To calculate the work done in blowing out a soap bubble of diameter 1 cm, we can follow these steps:

### Step 1: Understand the Concept of Work Done
The work done (W) in blowing out a soap bubble is related to the surface tension (S) of the soap solution and the increase in surface area of the bubble. A soap bubble has two surfaces (inner and outer), so we need to account for both when calculating the work done.

### Step 2: Formula for Work Done
The formula for work done in creating a surface area is given by:
\[ W = S \times \Delta A \]
where \( \Delta A \) is the change in surface area.

### Step 3: Calculate the Surface Area of the Bubble
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
For a soap bubble, we need to consider the increase in surface area, which is:
\[ \Delta A = 4\pi r^2 \]
Since the bubble has two surfaces, we multiply by 2:
\[ \Delta A = 2 \times 4\pi r^2 = 8\pi r^2 \]

### Step 4: Convert Diameter to Radius
Given the diameter \( d = 1 \text{ cm} = 0.01 \text{ m} \), we can find the radius \( r \):
\[ r = \frac{d}{2} = \frac{0.01 \text{ m}}{2} = 0.005 \text{ m} \]

### Step 5: Substitute Radius into Surface Area Formula
Now, substituting the radius into the surface area formula:
\[ \Delta A = 8\pi (0.005)^2 \]
Calculating this gives:
\[ \Delta A = 8\pi (0.000025) = 8 \times 3.14 \times 0.000025 \approx 0.000628 \text{ m}^2 \]

### Step 6: Calculate Work Done Using Surface Tension
Now, we can substitute the values into the work done formula. The surface tension \( S \) is given as \( 28 \times 10^{-3} \text{ N/m} \):
\[ W = S \times \Delta A = (28 \times 10^{-3}) \times (0.000628) \]
Calculating this gives:
\[ W \approx 28 \times 10^{-3} \times 0.000628 \approx 1.76 \times 10^{-5} \text{ J} \]

### Final Answer
Thus, the work done in blowing out the soap bubble is approximately:
\[ W \approx 1.76 \times 10^{-5} \text{ J} \]

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