Derive the relation between surface tension and surface energy per unit area.
Derive the relation between surface energy & surface tension.
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 x Free length
∴ F = T (2L)
If the wire is pulled out to C'D' through distance ‘dx’
∴ Work done = F. dx
∴ W = T (2Ldx)
∴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 condition.
`therefore "Surface energy"="Work done"/"Free surface area"=W/"dA"=(T(2Ldx))/(2Ldx)=T`
∴ Surface energy is equal to surface tension
The relation between surface tension and surface energy:
- Let ABCD be a rectangular frame of wire, fitted with a movable arm PQ.
- The frame held in a horizontal position is dipped into a soap solution and taken out so that a soap film APQB is formed. Due to the surface tension of the soap solution, a force ‘F’ will act on each arm of the frame. Under the action of this force, the movable arm PQ moves towards AB.
- The magnitude of force due to surface tension is, F = 2Tl .....[∵ T = F/l]
(A factor of 2 appears because soap film has two surfaces that are in contact with a wire.)
- Let the wire PQ be pulled outwards through a small distance ‘dx’ to the position P′Q′, by applying an external force F′ isothermally, which is equal and opposite to F. Work done by this force, dW = F′dx = 2Tldx.
- But, 2ldx = dA = increase in the area of two surfaces of the film.
∴ dW = T dA
- This work done in stretching the film is stored in the area dA in the form of potential energy (surface energy).
∴ Surface energy, E = T dA
∴ `E/(dA) = T`
Hence, surface tension = surface energy per unit area.
- Thus, surface tension is equal to the mechanical work done per unit surface area of the liquid, which is also called surface energy.
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