Define surface energy of the liquid. Obtain the relation between the surface energy and surface tension.

Answer: Surface energy is the extra (potential) energy possessed by the molecules in the surface layer of a liquid compared with those inside; surface energy per unit area is equal to the surface tension, \(T = \dfrac{W}{\Delta A}\).

Definition: Molecules in the surface film experience a net inward pull, so work must be done to bring molecules from the interior to the surface. This work is stored as extra potential energy of the surface molecules, called surface energy. Surface energy per unit area: \(E = \dfrac{W}{A}\).

Relation: [Diagram: a rectangular wire frame ABCD with a movable wire PQ of length \(l\) sliding on it, enclosing a soap film; force F applied on PQ, slider displaced by dx to P′Q′.]

1. A soap film is formed on the frame. The film has two free surfaces, so the force of surface tension on the slider PQ of length \(l\) is \(F = 2Tl\), pulling it inwards.
2. To pull the slider outwards slowly through a small distance \(dx\) (isothermally), an equal external force must be applied. Work done: \(dW = F\,dx = 2Tl\,dx\).
3. Increase in surface area of the film (both surfaces): \(dA = 2l\,dx\).
4. So \(dW = T\,dA\). This work is stored as the surface energy of the new surface: \(\text{surface energy per unit area} = \dfrac{dW}{dA} = T\).

Thus surface tension is numerically equal to the surface energy per unit area of the liquid surface.