Derive the
relation between surface tension and
surface energy
Answers
Explanation:
During evaluation of the surface energy we found that the surface energy is equal to the surface tension in adiabatic process. In isothermal process surface energy (S) is equal to the sum of surface tension (T) and amount of heat absorbed by the surface per unit increment in area (H).
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
′
Q
′
through a distance dx. The work was done by and external force against the force due to film is
W= applied force × displacement
∴ W=Fdx=2Tldx
(∵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
A
W
=
2ldx
2Tldx
=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.