Capillary Rise and Fall

Capillarity is the name given to the rise (or, in some cases, the fall) of a liquid inside a sufficiently narrow tube, called a capillary tube, when the tube is dipped vertically into a wide, open reservoir of the same liquid. Water rises noticeably inside a fine glass capillary dipped into a dish of water, rising to a level well above the flat water surface in the dish outside; mercury, by contrast, is observed to be depressed -- pushed down below the level of the mercury outside -- inside a similar capillary dipped into a dish of mercury.

WBCHSE's own syllabus note for this topic specifies that capillary rise and fall is to be given only an analytical treatment here, without derivation -- so the formula below is stated directly, to be used and applied, rather than derived from first principles. For a capillary tube of internal radius $r$, dipped vertically in a liquid of surface tension $T$ and density $\rho$, which meets the glass of the tube at angle of contact $\theta$, the height $h$ to which the liquid rises (measured from the flat liquid level outside the tube up to the bottom of the meniscus inside it) is given by:

where $g$ is the acceleration due to gravity.

Reading the formula.

• $h \propto 1/r$: the height of rise is inversely proportional to the radius of the capillary tube -- a narrower tube produces a greater rise than a wider one, which is exactly why the effect is called "capillary" (from the Latin capillus, meaning a hair) -- it is most noticeable, and easiest to observe, in tubes as fine as a hair.
• $h \propto \cos\theta$: for a liquid such as water, with a small, acute angle of contact ($\theta$ close to $0°$, so $\cos\theta$ is close to $+1$), the formula gives a large positive $h$ -- a genuine rise. For a liquid such as mercury, with a large, obtuse angle of contact ($\theta$ well above $90°$, so $\cos\theta$ is negative), the formula gives a negative $h$ -- interpreted physically as a depression (a fall) of the liquid level inside the tube below the level outside, rather than a rise.
• $h \propto T$ and $h \propto 1/\rho$: a liquid with a larger surface tension shows a greater rise (or fall), while a denser liquid shows a smaller rise (or fall) for the same surface tension and tube radius, since a denser liquid's own weight opposes the effect of surface tension more strongly. …