Surface Tension — Formula, Capillarity & Applications

Surface tension makes a liquid surface resist expansion and pull toward the smallest area it can, and it is usually written $\gamma$ and measured in $\mathrm{N/m}$. Solving a surface-tension problem is really a matter of matching the situation to the correct formula, because three different geometries each have their own relation and using the wrong one is the most common error.

When To Use Which Formula

Three formulas appear again and again, each tied to a specific geometry:

The first gives force per unit length along a liquid surface. The second is the pressure difference across a spherical liquid droplet of radius $r$. The third is the capillary-rise height for a narrow cylindrical tube at equilibrium. A soap bubble is the exception, because it has two liquid surfaces, so its pressure difference carries an extra factor of two:

Use each formula only with its condition: the pressure formulas assume spherical shapes, and the capillary formula assumes a narrow cylindrical tube at equilibrium.

Step By Step

1. Start with the right model. Decide whether the problem is about force along a surface, pressure difference across a curved surface, or capillary rise in a narrow tube. This single choice selects the formula.

2. State the condition. Confirm the geometry the formula assumes, such as a cylindrical capillary tube at equilibrium, or a single-surface droplet versus a two-surface soap bubble.

3. Track the contact angle. In capillarity the sign and size of $\cos\theta$ matter. If the liquid wets the wall, $\theta < 90^\circ$ and $\cos\theta > 0$, so it rises (water in clean glass). If it does not wet the wall, $\theta > 90^\circ$ and $\cos\theta < 0$, so the level is depressed (mercury in glass).

4. Check units and meaning. Surface tension is commonly given in $\mathrm{N/m}$; a positive capillary height means a rise, a negative one a depression.

Full Worked Example: Capillary Rise In Water

Water rises in a clean glass capillary tube of radius $r = 0.50\ \mathrm{mm} = 5.0 \times 10^{-4}\ \mathrm{m}$. Take $\gamma = 0.072\ \mathrm{N/m}$, $\rho = 1000\ \mathrm{kg/m^3}$, and $g = 9.8\ \mathrm{m/s^2}$.

The model is capillary rise, so the right formula is the third one. Water wets glass well, so $\theta \approx 0^\circ$ and $\cos\theta \approx 1$. Then

So the water rises about $h \approx 2.9\ \mathrm{cm}$. The key trend is that a smaller tube gives a larger rise, because $h \propto 1/r$ when the other quantities are fixed.

Where Students Get Stuck, And How To Check Yourself

Picking the droplet formula for a soap bubble. Use $\Delta p = 2\gamma/r$ for a spherical droplet and $\Delta p = 4\gamma/r$ for a soap bubble; the extra factor of two appears because a bubble has an inner and an outer liquid surface while a droplet has one. Self-check: if your bubble pressure equals the droplet result, you dropped the factor of two.

Dropping the contact angle silently. Replacing $\cos\theta$ by $1$ assumes complete wetting. That is fine for many water-glass cases but not always true, so state the assumption.

Forgetting how radius scales the answer. A smaller tube radius means a larger magnitude of rise or depression, not a smaller one; the $1/r$ in the formula is the check. As a sanity test, doubling the tube radius should halve the capillary height.

Confusing surface tension with viscosity. Surface tension concerns the liquid surface; viscosity concerns resistance to flow within the bulk. They answer different questions.

For curved surfaces in general, remember the pressure jump grows as the radius shrinks, which is why these effects dominate at small scales.

What Surface Tension Means Physically

A molecule inside the liquid is surrounded by neighbors, but a molecule at the surface is not, and that imbalance raises the surface energy so the liquid reduces its area when it can. This is why small droplets become nearly spherical: for a given volume a sphere has the smallest surface area. The popular "stretched film" picture is a useful analogy, but the real cause is molecular interaction, not an elastic sheet. Surface tension matters in droplets, bubbles, wetting and coating, capillary action, detergents, inkjet printing, and microfluidic devices, where the competition between surface effects and gravity or pressure is sharpest at small length scales.