Prove that surface tension is numerically equal to surface energy

Surface Tension (σ): It is measured as force acting per unit length of an imaginary line drawn on liquid surface.

Where,

F = Force that liquid exerts

l = length of imaginary line drawn on the liquid surface

Surface Energy: It is the work done in increasing the area of the surface film by a unit amount.

Consider a rectangular frame ABCD in which wire AB is movable. Dip the frame in soap solution. A film is formed which pulls the wire AB inwards due to surface tension with a force,

We have included a factor of 2 because the soap film has two free surfaces.

Suppose AB is moved out through a distance x to the position A’B’. Then,

Increase in surface area of the film is =

Hence surface energy is numerically equivalent to its surface tension.