A rectangular film of liquid is extended from `(4cmxx2cm)` to `(5cmxx4cm)`. If the work done is `3xx10^-4J`, the value of the surface tension of the liquid is

Options

A. `0.2Nm^-1`
B. `8.0Nm^-1`
C. `0.250Nm^-1`
D. `0.125Nm^-1`

Explanation

To find the surface tension of the liquid given the dimensions of the film and the work done, we can follow these steps:

### Step 1: Understand the Problem
We have a rectangular film of liquid that is being stretched from dimensions \(4 \, \text{cm} \times 2 \, \text{cm}\) to \(5 \, \text{cm} \times 4 \, \text{cm}\). The work done in this process is \(3 \times 10^{-4} \, \text{J}\). We need to find the surface tension \(S\) of the liquid.

### Step 2: Calculate the Initial and Final Areas
- **Initial Area (A1)**:
\[ A_1 = 4 \, \text{cm} \times 2 \, \text{cm} = 8 \, \text{cm}^2 \]

- **Final Area (A2)**:
\[ A_2 = 5 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2 \]

### Step 3: Calculate the Change in Area
- **Change in Area (ΔA)**:
\[ \Delta A = A_2 - A_1 = 20 \, \text{cm}^2 - 8 \, \text{cm}^2 = 12 \, \text{cm}^2 \]

### Step 4: Convert Change in Area to Square Meters
Since surface tension is typically expressed in SI units, we need to convert the area from square centimeters to square meters:
\[ \Delta A = 12 \, \text{cm}^2 = 12 \times 10^{-4} \, \text{m}^2 = 1.2 \times 10^{-3} \, \text{m}^2 \]

### Step 5: Relate Work Done to Surface Tension
The work done (W) in stretching the film is related to the surface tension (S) and the change in area (ΔA) by the formula:
\[ W = 2S \Delta A \]
Here, the factor of 2 accounts for the two surfaces of the film.

### Step 6: Rearrange the Formula to Solve for Surface Tension
Rearranging the formula gives:
\[ S = \frac{W}{2 \Delta A} \]

### Step 7: Substitute the Known Values
Substituting the known values into the equation:
\[ S = \frac{3 \times 10^{-4} \, \text{J}}{2 \times 1.2 \times 10^{-3} \, \text{m}^2} \]

### Step 8: Calculate Surface Tension
Calculating the value:
\[ S = \frac{3 \times 10^{-4}}{2.4 \times 10^{-3}} = \frac{3}{2.4} \times 10^{-1} = 1.25 \times 10^{-1} \, \text{N/m} = 0.125 \, \text{N/m} \]

### Final Answer
The value of the surface tension of the liquid is:
\[ S = 0.125 \, \text{N/m} \]

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