Rise of Liquid in a Capillary Tube

Rise of Liquid in a Capillary Tube:

glass capillary tube

Consider a glass capillary of radius R dipped in water as shown in Figure a. The pressure just above the liquid surface in the capillary is equal to atmospheric pressure po. As the meniscus is concave and nearly spherical, the pressure below the meniscus will be (po – 2T/r) where r is the radius of the meniscus. This is the condition of unstable equilibrium because the pressure in the liquid at this level just outside the capillary is po. Therefore, the liquid from the surroundings of the capillary will rush into the capillary till the pressure at that level in the capillary is equal to the pressure outside Figure b. As a result, the liquid level in the capillary rises to a height till the hydrostatic pressure of the liquid compensates for the decrease in pressure, i.e.

po = (po – 2T/r) + hρg
or h = 2T/rρg …………………….(a)

Where ‘ρ’ is the density of the liquid.

From the simple geometry, the radius of the meniscus r is related to the radius of the capillary through the relationship Figure (II).

R/r = cos θ
or r = R/cos θ

Substitute the value of r in equation (a), we get-

h = 2Tcos θ/Rρg

From this result we note-

(i) The capillarity depends on the nature of both the liquid and the solid. If θ > 90°, i.e., the meniscus is convex, h will be negative, i.e., the liquid will descend in the capillary (mercury in glass tube). However, if θ = 90°, i.e. the meniscus is plane, h = 0, i.e., no capillarity.

(ii) For a given liquid and solid at a given place as T, ρ and g are constants,

∴ h ∝ 1/R

i.e., the liquid will rise more in the narrow capillary and vice-versa.

(iii) The height h to which a liquid rises is independent of the shape of the capillary tube till the radius of the meniscus remains the same. It is because the difference in pressure (2T/r) is compensated by the hydrostatic pressure (hρg) so h depends only on r and the shape of the tube is of no significance.

(iv) In case of capillary of insufficient length, i.e., L < h, the liquid will rise in the capillary and on reaching the upper end, will increase the radius r’ of its meniscus Figure c such that hr = Lr’.

case of capillary of insufficient length

Thus, the liquid neither overflows from the upper end like a fountain nor does it trickles down along the vertical sides of the tube.

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