Young-Laplace Equation Calculator

This calculator answers two connected questions about a liquid standing inside a narrow tube: how much pressure the curved surface supports, and how high the liquid climbs because of it. Enter a tube radius, a contact angle and a liquid, and both numbers appear together.

It is the same physics behind a paper towel drawing up water, a wick feeding a lamp, a glass capillary in a laboratory, and the water held in the fine pores of soil. Anyone sizing a capillary, checking whether a liquid will wet a surface, or interpreting a measured rise or pressure difference can use it.

What you can solve for

• Forward: from the tube radius and the contact angle, get the meniscus radius, then the pressure difference, then the height of the column.
• Backwards: from a measured pressure difference or column height, get the meniscus radius, the tube radius, the contact angle, the density or the local gravity — whichever one you left out.
• With a custom liquid: from a measured pressure difference and a known meniscus radius, solve for the surface tension itself.

The two groups on the page follow the physics: the first group holds everything that sets the pressure across the curved surface, the second holds everything that turns that pressure into a height. Fill in the first, and the second fills itself in.

A first calculation: water in a 2 mm bore

Take a clean glass tube with a 2 mm bore, so the inner radius is 1 mm, and water at 20 °C with a contact angle of 20°. Pressed against a clean wall, water wets it, so the surface curves and the meniscus radius comes out slightly larger than the tube:

That curvature is what the surface tension pulls against:

and dividing by the weight of the liquid gives the rise:

Four short relationships describe everything on the page. The first two turn the geometry of the tube into a pressure, the third defines the pressure difference itself, and the fourth turns that pressure into a height.

1. The tube sets the curvature

A tube of inner radius $a$ with a contact angle $\theta$ holds a surface whose radius of curvature is $R$. A perfectly wetting liquid has $\theta = 0$, so $R = a$; the more the liquid beads up, the larger the radius of curvature becomes.

2. Surface tension turns curvature into pressure

This is the Young–Laplace relation for a spherical surface. It is the same balance that inflates a soap bubble: the sharper the curve, the more pressure the surface holds. Halving the meniscus radius doubles the pressure difference.

3. What the pressure difference means here

The difference is taken as the outside pressure minus the inside pressure, so a liquid that is drawn up the tube reports a positive value. Enter the outside pressure and the difference, and the pressure inside the liquid follows — it is the lower of the two.

4. Pressure becomes a height

A column of liquid weighs $\rho\,g\,h$ per unit area, and the rise stops when that weight exactly balances the pressure the curved surface can hold. Combine the four relationships and the familiar capillary-rise form appears, with the tube radius in the denominator:

The symbols

Gravity comes pre-filled with 9.80665 m/s², the standard acceleration of gravity that the international standard fixes exactly. It is not a rounded convenience figure, so leaving it alone means you are working with sea-level standard gravity; replace it with your local value when that matters.

Reading the equations backwards

Every one of the four can be rearranged, and the page does that for you: whichever boxes you leave empty are the ones solved for. The useful directions are worth knowing about explicitly.

• From a measured rise to the tube: a water column standing 30 mm high at a 20° contact angle implies $\Delta p = \rho\,g\,h$ of 294.2 Pa, a meniscus radius of 0.495854 mm and a tube radius of 0.466 mm — which is how you check an unknown capillary.
• From measurements to a surface tension: with the liquid set to Custom, a 240 Pa pressure difference across a 0.4 mm meniscus implies a surface tension of $\gamma = \tfrac{1}{2}\Delta p\,R$ = 0.048 N/m.
• From two pressures to a curvature: give both pressures and the pressure difference, and the meniscus radius that produced it follows.

What the liquid menu carries

These are the surface tensions and densities the seven tabulated entries load, in the base units the calculation uses. They are handbook figures for ordinary room-temperature conditions: good starting points, not measurements of your particular sample.

The calculation describes a static, idealised capillary. That is a good description of a clean glass tube in a laboratory, and a rougher one for everything else, so it is worth knowing which assumptions are doing the work.

For soil and plant work, the capillary pressure you compute here is the magnitude of the matric potential, with the opposite sign. Our water potential calculator combines that matric term with the osmotic, pressure, gravitational and overburden terms to give the total water potential of a soil or plant system, which is the number that decides whether water actually moves.

Treat the output as an estimate to reason with. Where a capillary is doing real work — a medical or laboratory device, a wick in a safety-critical system, a specification that has to be met — confirm it against a measurement or have it checked by the engineer or chemist responsible for the design.