What is capillary action ? Derive the formula for rise of liquid in a capillary tube immersed vertically in liquid.

Explanation

The phenomenon , the surface tension of liquid plays an important role.

Capillary rise , (a) Schematic picture of a narrow tube immersed water .(b) Enlarged picture near interface.
The word Capilla means hair in Latin . If the tube were hair thin the rise would be very lrge for liquid of angle of contact is acute (This type tube is known as capillary tube)
In figure , consider a vertical capillary tube of circular cross action (radius a) inserted into an open vessel of water .
The contact angle between water and glass is acute .`(thetalt(90^(@))`. Thus the surface of water in the capillary is concave .This means that there is a pressure difference between in two sides of the top surface.
`P_(i)-P_(0)=(2S)/(r)` where r is radius of meaniscus ..(1)
From figure `costheta=(a)/(r)`
`thereforer=(a)/(costheta)` ....(2)
Putting value of equation(2) in (1),
`P_(i)-P_(0)=(2s)/(a)costheta`....(3)
Hence , the pressure `P_(i)` of the water inside the tube , just at the meniscus (air -water interface) is less then the atmospheric pressure `P_(0)`.
According to figure (a) , consider two points A and and B . they must be at the same pressure,
Hence , here `P_(B)=P_(A)` ....(4)
but `P_(B)=` pressure on convex surface of meniscus + pressure of liquid column of height h.
`thereforeP_(B)=P_(0)+hrhog`
and `P_(A)=P_(i)" " (because` From equ .(4) and (5)) Where `rho` is the density of liquid and h is the height of liquid column in capillary.
`thereforeP_(i) -P_(0)=hrhog`
From equation (3),(4),
`hrhog=(2Scostheta)/(a)`
`h=(2Scostheta)/(arhog)`
As the radius a is small , the height of column is more .
If meniscus is convex (for example mercury), means `costheta` is negative , liquid depressed in capillary.