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Questions
Derive the relation between surface tension and surface energy per unit area.
Derive the relation between surface energy and surface tension.
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Solution 1
Surface tension tries to decrease the surface area of a liquid. For increasing surface area, the work has to be done against the surface tension and it is stored in the surface molecules in the form of potential energy
Consider a rectangular frame PQRS having a movable wire CD. Let QR = CD = L. If a soap film is formed on the frame CQRD, then the surface tension will try to pull the wire inward by a force F.
Surface tension = `"Force"/"free Length"`
F = Surface tension × Free length
∴ F = T × (2L)
If the wire is pulled out to C'D' through distance ‘dx’.
∴ Work done = F. dx
∴ W = T (2Ldx)
But increase in area = dA = 2Ldx
Surface energy is defined as the work done per unit area to increase the free surface area, under isothermal conditions.
`therefore "Surface energy"="Work done"/"Free surface area"=W/"dA"=(T(2Ldx))/(2Ldx)=T`
∴ Surface tension is also equal to the surface energy per unit area.
Solution 2
Consider a C-shaped frame of wire P'PSS'. It is fitted with a movable arm QR, as shown in Fig. This frame is dipped in a soap solution and then taken out. A film of soap solution will be formed within the boundaries PQRS of the frame.
Each arm of the frame experiences an inward force due to the film. Under the action of this force, the movable arm QR moves towards side PS so as to decrease the area of the film. If the length of QR is L, then this inward force F acting on it is given by
F = (T) × (2L) ...(1)
Since the film has two surfaces, the upper surface and the lower surface, the total length over which surface tension acts on QR is 2L. Imagine an external force F' (equal and opposite to F) applied isothermally (gradually and at constant temperature) to the arm QR so that it pulls the arm away and tries to increase the surface area of the film. The arm QR moves to Q'R' through a distance dx. Therefore, the work done against F, the force due to surface tension, is given by
dw = F'dx
Using equation (1),
dw = T (2Ldx)
But, 2Ldx = dA, increase in area of the two surfaces of the film. Therefore, dw = T(dA).
This work done in stretching the film is stored in the area dA of the film as its potential energy. This energy is called surface energy.
∴ Surface energy = T (dA) ...(2)
Thus, surface tension is also equal to the surface energy per unit area.
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