Question

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

Answer 1

Suppose a soap film is isothermally stretched over the area enclosed by a U-shaped frame $ABCD$ and a cross-piece $PQ$ that can slide smoothly along the frame. Let $T$ be the surface tension of soap solution and $l$, the length of wire $PQ$ in contact with the soap film.
The film has two surfaces, both of which are in contact with the wire. The film tends to contract by exerting a force on wire $PQ$. As such surface exerts a force $Tl$, the total force on the wire is $2Tl$. Suppose the wire $PQ$ is pulled outward very slowly to $P^{\prime }{Q}^{\prime }$ through a distance $dx$. The work was done by and external force against the force due to film is
$W=$ applied force $\times$ displacement
$\therefore$ $W=Fdx=2Tldx$
$(\because F=2Tl)$
Due to the displacement $dx$, the surface area of the film increases. As the film has two surfaces, the increase in its surface area is
$A=2ldx$
Thus, the work done per unit surface area is
$\frac{W}{A}=\frac{2Tldx}{2ldx}=T$
This work is stored in the unit surface area in the form of potential energy. This potential energy is called the surface energy.
The above relation shows that the surface energy per unit area of a liquid is equal to its surface tension.