Capillary Rise and Capillary Action: Formula and Examples

Dip a thin straw into a glass of water and the water inside the straw stands higher than the water outside. No pump is pushing it up. The climb is caused by surface tension pulling on the curved meniscus inside the narrow tube, and it is called capillary rise. The broader effect, liquid moving through narrow spaces on its own, is called capillary action.

It lifts water into roots, wax up wicks, ink into blotting paper. Narrower tubes lift higher, and the height follows a simple formula.

Key Concepts

• Capillary rise: the height h to which a liquid climbs in a narrow vertical tube above the surrounding liquid level.
• Capillary action: the general movement of liquids through narrow spaces, pores and fibres due to adhesion and surface tension, in any direction, not only upward.
• Meniscus: the curved surface of the liquid inside the tube. For a wetting liquid like water in glass it curves upward at the edges, and surface tension pulls upward along this curve.
• Why height depends on radius: the lifting force from surface tension acts around the tube's circumference (2 pi r), while the weight of the lifted column grows with its cross-sectional area (pi r^2). In a narrower tube there is less weight for each bit of circumference, so the liquid climbs higher.

Formulas

• Capillary rise: h = (2 T cos theta) / (r rho g). T is surface tension (N/m), theta is the angle of contact, r is the tube radius (m), rho is the liquid density (kg/m^3), and g = 9.8 m/s^2.
• Reading the formula: h grows with surface tension and shrinks with tube radius and liquid density. If theta is below 90 degrees, cos theta is positive and the liquid rises; above 90 degrees it is depressed.
• Assumptions and limits: the tube must be narrow, clean and vertical; the liquid must wet it evenly; and the formula assumes a spherical meniscus and neglects the small volume of liquid in the meniscus itself.

Worked Example

A glass capillary tube of radius 0.25 mm stands in water. T = 0.0728 N/m, rho = 1000 kg/m^3, and theta is about 0 degrees. Find the capillary rise.

1. r = 0.25 mm = 2.5 x 10^-4 m. cos 0 degrees = 1.
2. Numerator: 2 x 0.0728 x 1 = 0.1456.
3. Denominator: 2.5 x 10^-4 x 1000 x 9.8 = 2.45.
4. h = 0.1456 / 2.45 = 0.0594 m, about 5.9 cm.

Halving the radius doubles the rise, exactly as the formula predicts.

Common Misconceptions

• Capillary rise is not caused by air pressure pushing down outside. It is surface tension pulling up on the meniscus inside the tube. The effect works the same in a vacuum chamber.
• Plants do not rely on capillary rise alone. It helps water move through root hairs, but the main lift to treetops comes from transpiration pull as leaves lose water.
• A wider tube does not give a higher rise. The rise is inversely proportional to the radius, so finer tubes and pores lift liquid higher, not lower.

Real-World Applications

• Paper towels and blotting paper soak up spills through capillary action in the tiny gaps between fibres.
• Candle and oil-lamp wicks draw liquid fuel upward by capillary action to feed the flame.
• Soil scientists study capillary movement of water through soil pores to plan irrigation.

Capillary action is liquid climbing through narrow spaces under the pull of surface tension, with the rise given by h = 2 T cos theta / (r rho g). Finer tubes lift higher, moving liquids where pumps cannot reach.