What is capillarity? Obtain an expression for the surface tension of a liquid by capillary rise method.

Explanation

In a liquid whose angle of contact which solid is less than 90 degree, suffers capillary rise. On other hand, in a liquid whose angle of contact is greater than 90 degree suffers capillary fall. The rise or fall of a liquid in a narrow tube is called capillarity for capillary action.

Practical application of capillarity
(i) due to capillary action, oil rises in the cotton within an earthen lamp. Likewise, sap raises from the roots of a plant to its leaves and branches.
(ii) absorption of ink by a blotter paper.
(iii) capillary action is also essential for the tear fluid from the eye to drain constantly.
(iv) cotton dresses are preferred in summer because cotton dresses have find force which acts as capillaries for sweat.
Surface Tension by capillary rise method: the pressure difference across a curved liquid-air interface is is the basic factor behind the rise up of water in a narrow tube (influence of gravity is ignored). The capillary rise is more dominant in the case of of very fine tubes. But the phenomenon is the outcome of the force of surface tension. In order to arrive a relation between the capillary rise (h) and surface tension (T), consider a capillary to which is held vertically in a beaker containing water, the water rises in the capillary tube to a height h due to surface tension.
the surface tension force `F_T`, acts along the tangent at the point of contact downwards and its reaction for upwards. Surface tension t, it.s resolved into components (i) horizontal component `Tsin theta` and (ii) vertically component`T cos theta` acting awards, all along the whole circumstances of meniscus.
Where `theta` is the angle of contact, r is the radius of the tube. Let `rho` be the density of water and h be the height to which the liquid rises inside the tube. Then ,
`{:(("The volume of liquid"),("column in the tube, V")):}={:(("Volume of the liquid"),("column of radius r height h")):}+ {:(("Volume of liquid of radius r and height h"),("-volume of the hemisphere of radius r")):}`
`V= pi r^2 h + (pi r^2 xx r - 2/3 pi r^3)`
`implies V = pi r^2 h + 1/3 pi r^3`
The upwards force supports the weight of the liquid column above the free surface, therefore,
`2 pi r T cos theta = pi r^2 (h + 1/3 r) rho g implies T = (r (h + 1/3r)rho g)/(2 cos theta)`
If the capillary is a very fine tube of radius (i.e, radius is very small) then `r/3` can be neglected when it is compared to the height h. therefore,
`T = ( r rho gh)/(2 cos theta)`
Liquid rises through a height h
`h = (2 T cos theta)/(r rho g) implies h prop 1/r`. This implies that the capillary rise (h) is inversely, proportional to the radius (r ) of the tube i.e, the smaller the radus of the tube greater will be the capillarity.