Surface Tension by Capillary Rise A Complete Simple Guide

We know water's surface tension is about 0.0728 N/m, but how was that number ever measured? One of the simplest and most elegant methods needs nothing more than a thin glass tube, a ruler and a beaker of liquid. Dip the tube in, measure how high the liquid climbs, and the surface tension drops straight out of the capillary rise formula.

This method turns an invisible molecular force into a length anyone can read off a scale, and with care gives results accurate enough for real work.

Key Concepts

• Principle: in a narrow tube, surface tension pulls the curved meniscus upward until the weight of the lifted column balances it. Measuring the rise height h reveals the surface tension T.
• Working formula: T = (h r rho g) / (2 cos theta), where h is the rise (m), r the tube radius (m), rho the liquid density (kg/m^3), g = 9.8 m/s^2, and theta the angle of contact.
• Clean apparatus: the tube must be scrupulously clean, since grease changes the contact angle and ruins the result. Tubes are usually washed with chromic acid or detergent and rinsed in distilled water.
• Measuring the radius: the bore is so fine that r is measured with a travelling microscope or by weighing a mercury thread that fills a known length of the tube.

How It Works

1. Set up: mount a clean, uniform capillary tube vertically so its lower end dips into the test liquid in a wide beaker.
2. Wait for equilibrium: let the liquid climb and settle.
3. Measure h: read the height of the meniscus bottom above the outside liquid level.
4. Measure r: find the tube's internal radius with a travelling microscope.
5. Calculate: insert h, r, rho and theta into T = h r rho g / (2 cos theta). For water on clean glass, theta is taken as 0 degrees so cos theta = 1.

Worked Example

In a student experiment, water rises 2.97 cm in a capillary tube of radius 0.50 mm. Density of water rho = 1000 kg/m^3, theta = 0 degrees. Find the surface tension.

1. h = 0.0297 m, r = 5 x 10^-4 m, cos 0 degrees = 1.
2. Numerator: h x r x rho x g = 0.0297 x 5 x 10^-4 x 1000 x 9.8.
3. 0.0297 x 5 x 10^-4 = 1.485 x 10^-5; times 1000 = 0.01485; times 9.8 = 0.14553.
4. T = 0.14553 / 2 = 0.0728 N/m.

The experiment recovers the accepted 0.0728 N/m, showing how well the simple method works.

Common Misconceptions

• You cannot skip measuring the radius. The rise depends on 1/r, so even a small error in the bore size translates directly into the final value of T. The radius measurement is the most delicate step.
• Tap water gives a wrong answer. Dissolved salts and impurities change the surface tension, so distilled water and clean glassware are essential.

Real-World Applications

• School and undergraduate labs use it as the standard first experiment for measuring surface tension.
• Ink and coating manufacturers check surface tension this way to predict how liquids will spread and print.
• Pharmaceutical labs measure the surface tension of drug solutions, which affects drop size in sprays and eye drops.
• Detergent makers compare formulations by how much each lowers the capillary rise of water.

The capillary rise method turns surface tension into a length: measure how high a liquid climbs in a fine tube, measure the bore, and compute T = h r rho g / (2 cos theta). A molecular quantity from a ruler reading.