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

Explanation

Consider a capillary tube 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 force upwards. Surface tension T, is resolved into two components.
(i) Horizontal component `T sin theta` and
(ii) Vertical component `T cos theta` acting upwards, all along the whole circumference of the meniscus.
Total upward force `= (T cos theta) (2pi r) = 2 pi rT cos theta` 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 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"),("r - Volume of the"),("hemisphere of radius r"))`

`V = pi r^(2) h + (pi r^(2) xx r - (2)/(3)pi r^(3)) rArr V = pi r^(2) h + (1)/(3) pi r^(3)`
The upward force supports the weight of the liquid column above the free surface, therefore,
`2pi rT cos theta = pi r^(2) (h + (1)/(3) r)rho g rArr T = (r(h+(1)/(3)r)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 g h)/(2 cos theta). h = (2T cos theta)/(r rho g) rArr h prop (1)/(r )`
h is inversely proportional to the radius of the tube. The smaller the radius of the tube greater will be the capillarity.