Question

Derive the relation between surface tension and surface energy per unit area.

Answer 1

Hint: Surface energy can be defined as excess energy on a material's surface in relation to bulk or as work required for constructing an area of a given surface. Another method of looking at the surface energy is to compare it to the work needed to cut a large sample, which produces two surfaces.

Complete answer:
Assuming that a soap film is distributed over the area surrounded by a frame ABCD which is U-shaped and a PQ crosspiece, which can move easily over the frame. Let T be the soap solution's surface tension and l, the wire PQ's length in contact with the soap film. Along with the wire, the film has 2 surfaces. The film exerts a force on the wire PQ and tends to contract. The total wire force is $2Tl$, since this surface exerts a $Tl$ force. Assume the PQ wire is pulled to P′Q' at a dx distance very slowly. The work was done by and external force against the force due to film is-
$ \Rightarrow $W = applied force $ \times $ displacement
Since the value of force is given as $F = 2Tl$.
Therefore-

$
   \Rightarrow W = Fdx \\
    \\
   \Rightarrow W = 2Tldx \\
$

The film's surface area increases due to the dx displacement. With two surfaces, the increase in its surface area will be-
$ \Rightarrow A = 2ldx$
Hence, the work done per unit area will be-
$ \Rightarrow \dfrac{W}{A} = \dfrac{{2Tldx}}{{2ldx}} = T$
This work is stored as potential energy in the unit surface. The surface energy is this potential energy.
The above relation shows that the surface energy is equal to its surface tension per unit area of a liquid.

Note:
Surface tension, the characteristic of a liquid surface, shown as being an elastic membrane stretched out. The almost spherical form of small drops of liquids and of soap bubbles shows this phenomenon. Some insects that stay on the surface of the water are there because of this property. Surface tension depends primarily on attractive forces between particles in the given liquid and also on the interaction between the gas, the solid or the liquid.