Question
Derive the relation between surface tension and surface energy per unit area.
✓
Answer
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 $Q R=C D=L$. If a soap film is formed on the frame CQRD, then the surface tension will try to pull the wire inward by a force $F$.
$\text { Surface tension }=\frac{\text { Force }}{\text { free Length }}$
$F=\text { Surface tension } \times \text { Free length }$
$\therefore F=T \times(2 L)$
If the wire is pulled out to C' $D$ ' through distance ' $d x$ '.
$\therefore$ Work done $= F . dx$
$\therefore W = T (2 Ldx )$
$\therefore W = T (2 Ldx )$
But increase in area $=d A=2 Ldx$
Surface energy is defined as the work done per unit area to increase the free surface area, under isothermal conditions.
$\therefore \text { Surface energy }=\frac{\text { Work done }}{\text { Free surface area }}=\frac{W}{ dA }=\frac{T(2 L d x)}{2 L d x}=T$
$\therefore$ Surface tension is also equal to the surface energy per unit area.
Consider a rectangular frame PQRS having a movable wire CD. Let $Q R=C D=L$. If a soap film is formed on the frame CQRD, then the surface tension will try to pull the wire inward by a force $F$.
$\text { Surface tension }=\frac{\text { Force }}{\text { free Length }}$
$F=\text { Surface tension } \times \text { Free length }$
$\therefore F=T \times(2 L)$
If the wire is pulled out to C' $D$ ' through distance ' $d x$ '.
$\therefore$ Work done $= F . dx$
$\therefore W = T (2 Ldx )$
$\therefore W = T (2 Ldx )$
But increase in area $=d A=2 Ldx$
Surface energy is defined as the work done per unit area to increase the free surface area, under isothermal conditions.
$\therefore \text { Surface energy }=\frac{\text { Work done }}{\text { Free surface area }}=\frac{W}{ dA }=\frac{T(2 L d x)}{2 L d x}=T$
$\therefore$ Surface tension is also equal to the surface energy per unit area.
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