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 ρ, the angle of contact
between the liquid and the walls of the tube be θ 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 2πr cosθT.
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
2πr cosθ T = πr2ρgh
which gives
sure at C is also atmospheric but it is greater than the pressure at B by the
hydrostatic pressure hρg. 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 >90o and therefore cosθ is negative. This means
that the mercury level is not raised but depressed below the level of the surrounding liquid.