Capillary Rise & Surface Tension Calculator
Work out how high a liquid climbs a narrow tube, or how far it is pushed down. Jurin’s law solves for the height, the tube radius, the surface tension or the contact angle, and the Laplace pressure mode gives the extra pressure inside a drop or a soap bubble.
How to Use
- Pick what to find: the rise height, the tube radius, the surface tension, the contact angle, or the Laplace pressure inside a drop or bubble.
- Choose a liquid to fill in its surface tension, density and contact angle (water, mercury, soapy water), or type your own.
- Enter the tube’s inside radius, half its bore, and the other known values, each with its own unit.
- Read the answer, the pressure under the meniscus, the force the surface pulls with, and the capillary length. A negative height means the liquid is pushed down.
- Press a preset to load an example, and open Show Work for every step.
Worked Example
Water in a glass capillary. A clean glass tube of 0.5 mm inside radius stands in water at 20 °C: γ = 0.0728 N/m, ρ = 998.2 kg/m³, and water wets clean glass, so θ = 0° and cos θ = 1. h = 2 × 0.0728 × 1 ÷ (998.2 × 9.80665 × 0.0005) = 29.75 mm. The pressure just under the curved surface is 2γ ÷ r = 291.2 Pa below the air above, exactly ρgh, which is what holds the column up.
Mercury in the same tube. γ = 0.485 N/m, ρ = 13,546 kg/m³ and θ = 140° (typical for mercury on glass), so cos θ = −0.766. h = 2 × 0.485 × (−0.766) ÷ (13,546 × 9.80665 × 0.0005) = −11.19 mm: the mercury stands 11.19 mm below the level outside, under a domed meniscus.
The common mistake: using the bore (diameter) as r. A tube sold as “0.5 mm bore” has r = 0.25 mm, and water climbs 59.50 mm in it. Putting 0.5 mm in for r gives 29.75 mm, half the real rise. The other slip is leaving out cos θ: for mercury that gives a 14.60 mm rise instead of an 11.19 mm drop.
Show Work
Formulas
Jurin, Young and Laplace
Liquids climbing thin glass tubes puzzled the early Royal Society. Francis Hauksbee demonstrated the effect around 1709, and in 1718 the physician James Jurin reported to the Society that the height is inversely proportional to the tube’s bore, whatever the thickness of the glass. That is the rule now called Jurin’s law.
The explanation came almost a century later. In 1805 Thomas Young described a liquid surface as a stretched membrane with a fixed contact angle where it meets a solid, and Pierre-Simon Laplace, independently in 1806, worked out the pressure jump across a curved surface. Together they give the Young–Laplace equation, from which both the capillary rise and the 2γ/r pressure inside a drop follow. Carl Friedrich Gauss put the theory on an energy footing in 1830.
Values used: water 72.8 mN/m at 20 °C and mercury about 0.485 N/m (CRC Handbook of Chemistry and Physics, typical); the contact angle of mercury on glass varies roughly from 130° to 150° with the glass and its cleanliness, and 140° is used here. Soapy-water surface tension depends on the soap; 25 mN/m is a typical value.
About This Tool
This calculator solves Jurin’s law for whichever of the rise height, tube radius, surface tension or contact angle you do not know, with a unit on every value, and reports the pressure under the meniscus, the force with which the surface holds the column, and the capillary length. Negative heights are depressions, as for mercury. A second mode gives the Laplace pressure inside a drop, a gas bubble in a liquid, or a soap bubble. Jurin’s law assumes a round tube much narrower than the capillary length and a clean, steady meniscus; it warns when the tube is too wide for that.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Bernoulli & Fluid Pressure Calculator, Stokes’ Law Settling Velocity Calculator, and Density Calculator.
Frequently Asked Questions
What is Jurin’s law?
The height a liquid climbs in a thin tube is h = 2γ cos θ ÷ (ρgr), where γ is the surface tension, θ the contact angle, ρ the density and r the tube radius. Water (72.8 mN/m, 998.2 kg/m³) in a clean glass tube of 0.5 mm radius climbs 2 × 0.0728 ÷ (998.2 × 9.80665 × 0.0005) = 29.75 mm.
Why does mercury go down instead of up?
Mercury does not wet glass: its contact angle is about 140° (typical), and cos 140° = −0.766 is negative, so h is negative. In a 0.5 mm-radius tube mercury (0.485 N/m, 13,546 kg/m³) sits 11.19 mm below the outside level, with a domed meniscus instead of a dished one.
Does a narrower tube really lift water higher?
Yes, in proportion to 1/r: the product h × r is fixed for a given liquid, 14.87 mm² for water. A 1 mm-radius tube lifts water 14.87 mm, a 0.05 mm one 297.5 mm. A 20 µm xylem vessel, typical of wood, would lift it 0.7437 m by capillarity alone, so tall trees need another mechanism: evaporation from the leaves pulls the water up.
Why is the pressure higher inside a small bubble?
A curved surface squeezes what is inside: Δp = 2γ/r for a drop, or for a gas bubble inside a liquid. A water droplet of 1 µm radius holds 2 × 0.0728 ÷ 0.000001 = 145.6 kPa above the outside pressure, 1.437 atm. A soap bubble has two surfaces, so Δp = 4γ/r: a 2 cm bubble of soapy water (about 25 mN/m, typical) holds only 5 Pa.
What is the capillary length?
It is √(γ ÷ ρg), the size below which surface tension beats gravity: 2.727 mm for water and 1.911 mm for mercury. Tubes much narrower than this follow Jurin’s law well; in wider ones the meniscus flattens out and the rise is smaller than the formula says.
How do I use the Capillary Rise & Surface Tension Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Lab glassware
Water in a 0.5 mm-radius capillary tube rises 29.75 mm, so readings in narrow tubes must allow for the meniscus.
Measuring surface tension
A liquid of density 789 kg/m³ that climbs 11.5 mm in a 0.5 mm-radius tube has γ = 789 × 9.80665 × 0.0115 × 0.0005 ÷ 2 = 22.25 mN/m.
Damp walls and soil
Pores of 0.1487 mm radius are enough to wick water 10 cm up a brick or through soil, which is why walls need a damp-proof course.
Barometers and manometers
In a mercury column of 0.5 mm radius the reading sits 11.19 mm low from capillary depression alone, which is why barometer tubes are wide.
Bubbles and sprays
A 1 µm water droplet holds 145.6 kPa of extra pressure inside; a 2 cm soap bubble holds just 5 Pa, a 0.51 mm water column.
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