Capillary action

The rise of a column of liquid within a fine capillary tube is also due to surface tension. Capillary action causes liquid to soak upwards through a piece of blotting paper and it also partly explains the rise of water through the capillaries in the stems of plants. (In this last case osmotic pressure accounts for a large part of the rise.)

There are two alternative proofs for the formula for capillary rise and we will consider Figure 1(a) first.

Let the radius of the glass capillary tube be r, the coefficient of surface tension of the liquid he T, the density of the liquid be r, the angle of contact between the liquid and the walls of the tube be q and the height to which the liquid rises in the tube be h.
Consider the circumference of the liquid surface where it meets the glass.

Along this line the vertical component of the surface tension force will be 2pr cosqT.

This will draw the liquid up the tube until this force by the downward force due to the column of liquid of height h, that is just balanced at equilibrium:

Therefore
2pr cosq T = or2rgh which gives


Which for an angle of contact of 0^o becomes:


For the alternative proof consider Figure 1(b). We will assume that if the radius of the tube is small the shape of the liquid surface is very nearly hemispherical.

The pressure at A must be atmospheric, but since A is within a hemispherical surface the pressure at B must be less than A by an amount 2T/r. The pressure at C is also atmospheric but it is greater than the pressure at B by the hydrostatic pressure hrg. Therefore at equilibrium we have h = 2T/rrg, as above.

Both these methods show that the rise is greater in tubes with a narrow bore and for zero angles of contact. In fact when the coefficient of surface tension is measured by capillary rise in the laboratory the values obtained are nearly always too small because of the difficulty of getting perfectly clean apparatus. The angle of contact can rarely be made zero.

With a mercury-glass surface the angle of contact is >90^o and therefore cosq is negative. This means that the mercury level is not raised but depressed below the level of the surrounding liquid.