Show that the surface tension is numerically equal to the surface energy of a liquid.

 

$a)$Let $ABCD$ be an open rectangular frame of wire on which a wire $PQ$ can slide without friction.

$\left(b\right)$The frame held in horizontal position is dipped into soap solution and taken out so that a soap film $APQB$ is formed.Due to surface tension of soap solution,a force $F$ will act on the wire $PQ$ which tends to pull it towards $AB$

$\left(c\right)$Magnitude of force due to surface tension is $F=2TI\because T=\frac{F}{I}$

(A factor of $2$ appears because soap film has two surfaces which are in contact with wire.)

$\left(d\right)$Let the wire $PQ$ be pulled outwards through a small distance $dx$ to the position $P^{\prime }{Q}^{\prime }$, by applying an external force $F^{\prime }$ equal and opposite to $F$.Soek done by this force, is

$\Delta W=F^{\prime }dx=2TIdx$

$\left(e\right)$But ,$2Idx=\Delta A=$increase in area of two surfaces of film.

$\therefore \Delta W=T\Delta A$

This work done is stored in the form of potential energy(surface energy)

$\therefore$Surface energy,$E=T\Delta A$

$\therefore \frac{E}{\Delta A}=T$

Hence, surface tension$=$Surface energy per unit area.

$\left(f\right)$Thus, surface tension is equal to the mechanical work done per unit surface area of the liquid, which is also called as surface energy.