The relation between surface tension and surface energy is

Options

A. W=A/T
B. W=T/A
C. T=W/A
D. T=W.A

Answer

To derive the relation between surface tension (T) and surface energy (U), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Surface Tension and Surface Energy**:
- Surface tension (T) is defined as the force per unit length acting along the surface of a liquid.
- Surface energy (U) is the energy required to increase the surface area of a liquid by a unit area.

2. **Consider a Liquid Surface**:
- Imagine a liquid surface with a length \( L \) and a surface tension \( T \). The surface has two sides (top and bottom), so the effective force acting on the surface is \( 2T \).

3. **Displacement of the Surface**:
- If we move a slider to displace the surface by a small distance \( dx \), the work done (W) in moving this slider can be calculated as:
\[ W = \text{Force} \times \text{Displacement} = (2T) \times L \times dx \]
- Therefore, the work done is:
\[ W = 2T \cdot L \cdot dx \]

4. **Change in Surface Area**:
- The change in area (\( \Delta A \)) due to the displacement \( dx \) is:
\[ \Delta A = 2L \cdot dx \]
- This is because there are two surfaces (top and bottom) contributing to the area change.

5. **Relating Work Done to Surface Energy**:
- The work done on the liquid surface is stored as surface energy. Thus, we can express surface energy (U) as:
\[ U = W = 2T \cdot L \cdot dx \]

6. **Surface Energy per Unit Area**:
- To find the surface energy per unit area, we divide the work done by the change in area:
\[ \text{Surface energy per unit area} = \frac{W}{\Delta A} = \frac{2T \cdot L \cdot dx}{2L \cdot dx} \]
- Simplifying this gives:
\[ \text{Surface energy per unit area} = T \]

7. **Final Relation**:
- Therefore, we conclude that the surface tension (T) is equal to the surface energy per unit area (U/A):
\[ T = \frac{U}{A} \]

### Conclusion:
The relation between surface tension and surface energy is given by:
\[ T = \frac{U}{A} \]
where \( T \) is the surface tension, \( U \) is the surface energy, and \( A \) is the area.