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Question

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

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Solution

(a)Let ABCD be an open rectangular frame of wire on which a wire PQ can slide without friction.
(b)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
(c)Magnitude of force due to surface tension is F=2TIT=FI
(A factor of 2 appears because soap film has two surfaces which are in contact with wire.)
(d)Let the wire PQ be pulled outwards through a small distance dx to the position PQ, by applying an external force F equal and opposite to F.Soek done by this force, is
ΔW=Fdx=2TIdx
(e)But ,2Idx=ΔA=increase in area of two surfaces of film.
ΔW=TΔA
This work done is stored in the form of potential energy(surface energy)
Surface energy,E=TΔA
EΔA=T
Hence, surface tension=Surface energy per unit area.
(f)Thus, surface tension is equal to the mechanical work done per unit surface area of the liquid, which is also called as surface energy.


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