Capillary Rise Calculator
Find how far a liquid climbs, or is pushed down, inside a narrow tube of a given radius.
What it does
Put a narrow tube into a beaker of water and the water climbs up the inside of the tube. Nobody had to push it. The same effect is why water climbs the walls of a narrow glass tube, why a paper towel soaks up a spill, and why trees can pull sap up a trunk that is a hundred metres tall.
The mechanism is the competition between two sets of forces. Molecules inside a liquid attract each other, and that cohesion tries to pull the surface flat. Molecules at the liquid surface are also attracted to the material of the tube, and that adhesion pulls the surface into a curve called a meniscus. OpenStax describes the effect directly: capillary action is the tendency of a fluid to be raised or suppressed in a narrow tube, and it comes from the combination of surface tension and the adhesive forces between the liquid and the container.
The height that results is remarkably simple, and it is derived in the standard text by balancing the upward pull of surface tension against the weight of the raised column. The answer is twice the surface tension times the cosine of the contact angle, over the product of the density, gravity and the radius of the tube. Everything about the shape of the tube drops out, because the same surface tension acts all the way round the rim.
This page runs that formula for eight liquids. Pick the liquid, give the radius of the tube and the contact angle, and it returns the height the column reaches, whether the liquid rises or is pushed down, and how much liquid is held above the level outside. The same answer in millimetres or inches.
The limits are worth understanding. The formula assumes the tube is narrow enough for the surface tension to dominate and wide enough for the meniscus to be well formed, it ignores the curvature of the meniscus beyond its contact angle, and it says nothing about how fast the liquid climbs, which is set by viscous drag rather than by the balance of forces.
How to use
- Choose metric or imperial for the tube radius and the answer.
- Pick the liquid. Each one carries its own surface tension and density.
- Enter the radius of the tube, not its diameter.
- Enter the contact angle. Use 0 degrees for a liquid that wets the tube completely, and 140 degrees for mercury in glass.
- Read the height, and check whether the liquid rises or is depressed.
- Read the volume held above the outside level, which matters if the tube is a pipette or a graduated cylinder.
Formula
- h
- Height of the liquid column above the level outside the tube, in metres.
- gamma
- Surface tension of the liquid in newtons per metre.
- theta
- Contact angle between the liquid surface and the tube wall, in degrees.
- rho
- Density of the liquid in kilograms per cubic metre.
- r
- Radius of the tube in metres.
A contact angle of 0 degrees means the liquid wets the wall completely, and the cosine is 1, which gives the largest possible rise. An angle of 90 degrees gives a rise of zero, and anything above 90 degrees pushes the liquid down, which is what happens with mercury in glass. The surface tensions used here are the values tabulated in the physics text, and the density of each liquid is taken from the same set of tables.
Worked example
Example: water in a 9 mm radius glass tube
Defaults: metric units, water at 20 degrees C, a tube radius of 9 mm and a contact angle of 0 degrees.
- Water at 20 degrees C has a surface tension of 0.0728 N per metre and a density of 1000 kg per cubic metre.
- The tube radius in metres is 9 / 1000 = 0.009 m.
- Numerator: 2 x 0.0728 x cos(0 degrees) = 0.1456.
- Denominator: 1000 x 9.8 x 0.009 = 88.2.
- Height: h = 0.1456 / 88.2 = 1.651e-3 m, so 1.651 mm.
- The volume held above the outside level is pi x 0.009 squared x 1.651e-3 = 420.076 mm3.
- The tool shows 1.651 mm, rises, 420.076 mm3 and a contact angle of 0.0 degrees.
Reference table
| Tube radius | Rise in mm | Rise in inches | Liquid held above the level |
|---|---|---|---|
| 1 mm | 14.857 mm | 0.5849 in | 46.7 mm3 |
| 2 mm | 7.429 mm | 0.2925 in | 93.4 mm3 |
| 5 mm | 2.971 mm | 0.1170 in | 233.4 mm3 |
| 9 mm | 1.651 mm | 0.0650 in | 420.1 mm3 |
| 10 mm | 1.486 mm | 0.0585 in | 466.8 mm3 |
Limits
- The formula assumes a narrow tube with a well formed meniscus, and it does not describe the shape of the meniscus itself.
- It gives the final height, not how long the liquid takes to get there, which is a viscous flow question.
- Very dirty or coated tubes change the contact angle, and the calculation has no way of knowing that.
- Air bubbles trapped in the tube block the rise, and nothing in the formula predicts that.
Common mistakes
- Using the diameter of the tube instead of the radius. That doubles the answer when it should halve it.
- Leaving the contact angle at 0 for a liquid that does not wet the tube, which is the case for mercury in glass.
- Forgetting to convert the radius into metres. The formula works in SI units, so a radius in millimetres gives a result a thousand times too small.
- Expecting the rise to scale with the surface tension alone. A denser liquid rises less, because it takes more weight to balance the pull.
- Ignoring that the contact angle changes with temperature and with contamination of the surface.
Try it yourself
Quick checks. Answers sit tucked under each question.
How high does water rise in a tube of 1 mm radius?
14.857 mm, which is 0.5849 in. The height is inversely proportional to the radius, so the 9 mm tube that gives 1.651 mm needs a tube nine times narrower to rise nine times as far.
What is the rise in a 2 mm radius tube?
7.429 mm. Halving the radius from 9 mm to 2 mm multiplies the rise by 4.5, and 1.651 mm x 4.5 gives 7.429 mm.
Why is mercury pushed down a glass tube instead of rising?
The contact angle of mercury with glass is about 140 degrees, so the cosine in the formula is negative. The cohesive forces inside the mercury beat the adhesive ones from the glass, and the column is depressed instead of raised.
Questions
What is capillary action?
The tendency of a liquid to move inside a narrow tube, or a porous material, against gravity. It comes from the pull of surface tension combined with the attraction between the liquid and the walls.
What decides how high the liquid rises?
The surface tension, the contact angle, the density of the liquid and the radius of the tube. A higher surface tension or a smaller tube means a taller column.
Why does a narrow tube give a taller column?
Because the weight of the raised column grows with the square of the radius while the upward pull of surface tension grows only with the radius, so the height ends up inversely proportional to it.
Can the liquid be pushed down instead of up?
Yes. If the contact angle is above 90 degrees the cosine turns negative and the column sits below the outside level. Mercury in glass is the classic case, with a contact angle near 140 degrees.
Does this explain how trees move water?
Only partly. The textbook example works out that a xylem tube would need a radius of about 0.0014 cm to raise sap 100 m, which is smaller than real xylem. So capillarity is not the whole story, and transpiration pull matters too.
Does the surface tension of the liquid change with temperature?
Yes, it falls as the liquid warms, which is why water climbs slightly less in a warm tube than a cold one. The list of liquids on this page gives the values near room temperature.
Sources
- OpenStax College Physics 2e, 11.8: surface tension and capillary action
- Physics LibreTexts: the surface tension table and the capillary rise formula
- Physics LibreTexts: College Physics 11.8, worked example of the capillary rise