Calculate the surface tension of a liquid if `5 xx 10^(-3)Nm^(-1)` of work is done to increase the surface area of a liquid film from `4 cm xx 3" cm to 8 cm "xx "5 cm."`

Answer

To calculate the surface tension of a liquid given the work done to change the surface area, we can follow these steps:

### Step 1: Identify the given values
- Work done (W) = \(5 \times 10^{-3} \, \text{Nm}\)
- Initial area (A_i) = \(4 \, \text{cm} \times 3 \, \text{cm}\)
- Final area (A_f) = \(8 \, \text{cm} \times 5 \, \text{cm}\)

### Step 2: Convert the areas from cm² to m²
- Initial area (A_i) = \(4 \, \text{cm} \times 3 \, \text{cm} = 12 \, \text{cm}^2 = 12 \times 10^{-4} \, \text{m}^2\)
- Final area (A_f) = \(8 \, \text{cm} \times 5 \, \text{cm} = 40 \, \text{cm}^2 = 40 \times 10^{-4} \, \text{m}^2\)

### Step 3: Calculate the change in surface area (ΔA)
\[ \Delta A = A_f - A_i = (40 \times 10^{-4} \, \text{m}^2) - (12 \times 10^{-4} \, \text{m}^2) = 28 \times 10^{-4} \, \text{m}^2 \]

### Step 4: Use the formula for work done in terms of surface tension (γ)
The work done (W) is related to surface tension (γ) and the change in area (ΔA) by the formula:
\[ W = \gamma \Delta A \]
Rearranging the formula to solve for surface tension gives:
\[ \gamma = \frac{W}{\Delta A} \]

### Step 5: Substitute the values into the formula
Substituting the known values:
\[ \gamma = \frac{5 \times 10^{-3} \, \text{Nm}}{28 \times 10^{-4} \, \text{m}^2} \]

### Step 6: Calculate the surface tension
\[ \gamma = \frac{5 \times 10^{-3}}{28 \times 10^{-4}} = \frac{5}{28} \times 10^{1} \approx 1.78 \, \text{N/m} \]

### Final Answer
The surface tension of the liquid is approximately \(1.78 \, \text{N/m}\).

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