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Derive the Relation Between Surface Tension and Surface Energy per Unit Area. - Physics

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

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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 CD" role="presentation" style="position: relative;" data-mce-style="position: relative;">DD
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 energey="Work done"/"Free surface area"=W/"dA"=(T(2Ldx))/(2Ldx)=T`

∴ Surface energy is equal to surface tension

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