Question

derive expression for capillary rise?

Answer 1

See the diagram in the attachment .

T is the upward component of the adhesive force F on the contact line of the fluid and the wall and supports the fluid column. The surface tension sigma is defined as the work done per unit increase in surface area when a surface under tension is stretched. \[ \sigma=\frac{ F \times \Delta x }{\Delta A } \rightarrow eq 1 \] F is the total force along the contact line and is producing the tension in the surface , delta x is the amount stretched and delta A is the increase in area due to F.

Now we use a technique called virtual work. If the force F is increased minutely the surface would stretch incrementally by say delta x and the surface area of the meniscus would increase by a ring around the meniscus by an amount\[\Delta A= 2 \pi R \Delta x\]

From the diagram \[F=T/ \cos \left( a \right)\]
T counteracts the weight of the column of fluid.\[weight - of- column = T= g \rho V=g \rho \pi R ^{2}h\] which is where V is the volume of fluid in the column. g is the acceleration due to gravity, rho is the density of the fluid. Substituting into eq. 1 we get


\[height -of-column=h=2\sigma \cos \left( a \right)/\rho g R\]

The angle a depends on the fluid and material the tube is made. for water and clean glass a=0 deg.

Any questions?