Capillary Rise Method: Measuring Surface Tension Simply

When a narrow glass tube is dipped in water, the water climbs up inside it, sometimes several centimeters high. This capillary rise happens because surface tension pulls the liquid surface upward along the tube walls, and measuring the rise height gives a clean, simple way to determine the liquid's surface tension.

The method matters because surface tension controls countless small-scale phenomena, from ink flowing in a pen to water climbing through plant roots. The capillary rise experiment needs no electronics: a tube, a ruler, and the formula are enough for a precision measurement.

Key Concepts

• Surface tension (T): force per unit length along a liquid surface, in newtons per meter; also interpretable as energy per unit area.
• Contact angle (theta): the angle between the liquid surface and the tube wall; nearly 0 degrees for water on clean glass, over 90 degrees for mercury.
• Capillary force: 2 pi r T cos(theta), the upward pull acting around the tube's inner circumference of radius r.
• Balancing weight: pi r squared h rho g, the weight of the risen liquid column of height h and density rho.
• Meniscus: the curved liquid surface; concave for water (rises), convex for mercury (falls).

Formulas

• Capillary rise height: h = 2 T cos(theta) / (r rho g).
• Surface tension from measurement: T = h r rho g / (2 cos theta).
• For water on clean glass, theta is about 0 degrees, so cos(theta) = 1 and T = h r rho g / 2.
• Units check: (m)(m)(kg/m cubed)(m/s squared) = kg/s squared = N/m. Correct for surface tension.
• Mercury has theta about 140 degrees, so cos(theta) is negative and the formula predicts a depression, not a rise.

Worked Example

Water at room temperature rises h = 5.9 cm in a clean glass tube of inner radius r = 0.25 mm. Find the surface tension.

1. Convert: h = 0.059 m, r = 2.5 times 10 to the power (minus 4) m, rho = 1000 kg/m cubed, g = 9.8 m/s squared, cos(theta) = 1.
2. Compute h times r = 0.059 times 2.5 times 10 to the power (minus 4) = 1.475 times 10 to the power (minus 5).
3. Multiply by rho: 1.475 times 10 to the power (minus 5) times 1000 = 0.01475.
4. Multiply by g and halve: 0.01475 times 9.8 / 2 = 0.14455 / 2 = 0.0723 N/m.
5. The handbook value for water at 20 degrees C is about 0.0728 N/m: the simple tube-and-ruler experiment lands within 1 percent.

Common Misconceptions

• "Wider tubes give higher rise." The opposite: h is proportional to 1/r, so halving the radius doubles the rise height.
• "All liquids rise." Mercury is depressed below the outside level because its contact angle exceeds 90 degrees, making cos(theta) negative.
• "A dirty tube works fine." Grease changes the contact angle and the effective radius; the tube must be chemically clean for the formula to hold.

Real-World Applications

• Plant biology: capillary action helps water climb through narrow xylem vessels in stems.
• Printing and pens: ink feed and paper absorption are tuned through surface tension and pore size.
• Detergents: surfactants lower water's surface tension so it penetrates fabrics instead of beading up.

The capillary rise method turns an everyday curiosity, water climbing a thin tube, into a quantitative measurement of surface tension. One height reading plus one formula gives a fundamental liquid property to percent accuracy.