The Young-Laplace calculator above works out the pressure difference sustained across a curved interface between two fluids, and the height to which a liquid climbs inside a narrow tube as a direct consequence of that pressure. Both come from the same physics: a surface under tension that is forced to curve pushes on the fluid inside it, and the tighter the curvature the harder it pushes. Enter a surface tension and a radius and the tool returns the pressure jump; add a tube radius, a contact angle and a liquid density and it returns the capillary rise as well.
Arb Digital publishes this as one of a set of free physical chemistry tools. The equation itself is one line, but it hides three traps that catch people routinely — the factor of two between a droplet and a soap bubble, the sign of the second radius on a saddle-shaped surface, and the difference between the tube radius and the radius of the meniscus when the contact angle is not zero. The tool handles all three explicitly rather than leaving them implicit.
What This Young-Laplace Calculator Does
A liquid surface behaves as though it were an elastic skin under tension. If the surface is flat, the tension pulls sideways and nothing happens to the pressure. If the surface curves, the tension acquires a component pointing toward the centre of curvature, and to keep the interface in mechanical equilibrium the pressure on the concave side must exceed the pressure on the convex side. The Young-Laplace equation quantifies exactly how much.
This page computes that jump for four cases: a spherical drop or a gas bubble held in a liquid, which has a single interface; a soap bubble in air, which has two interfaces and therefore double the pressure jump; a cylindrical interface such as a liquid bridge or a thread, where one principal radius is infinite; and a general surface where you supply both principal radii, including the negative second radius that describes a saddle. It then computes the capillary rise using the same equation applied to a meniscus.
One boundary worth stating: our live surface tension converter only changes surface tension between units — newtons per metre, dynes per centimetre and the imperial equivalents. It performs no physics. This page takes a surface tension as given and computes what it does to pressure and to liquid height.
How to Use It
- Enter the surface tension in mN/m for the specific liquid-fluid pair and temperature you are modelling. Surface tension falls noticeably as temperature rises, so a room-temperature value is wrong for a hot system.
- Pick the geometry. The droplet and bubble-in-liquid case has one surface; a free soap bubble has two; a cylinder has one infinite radius; the general option lets you set both.
- Enter the radius or radii in millimetres. For a saddle surface, enter the second radius as a negative number so the two curvatures partly cancel.
- Enter the capillary tube radius and contact angle. Zero degrees means perfect wetting; ninety means no rise at all; above ninety the liquid is depressed below the reservoir level.
- Enter the liquid density so the rise height can be balanced against gravity. Only the capillary results depend on it.
The Formula and How It's Calculated
The Young-Laplace equation is Δp = γ(1/R₁ + 1/R₂), where gamma is the surface tension and R₁ and R₂ are the two principal radii of curvature at the point on the surface. For a sphere both radii equal R, so the expression collapses to Δp = 2γ/R. For a cylinder one radius is infinite and its reciprocal vanishes, giving Δp = γ/R. A free soap film has a liquid-air interface on each side of the film, so the jumps add and you get Δp = 4γ/R.
Take water at 20 °C, with gamma of 72.8 mN/m, in a spherical droplet of radius 1 mm. Converting to base units, gamma is 0.0728 N/m and R is 0.001 m, so Δp = 2 × 0.0728 / 0.001 = 145.6 Pa. That is about 0.15% of atmospheric pressure — negligible for a millimetre drop, which is why raindrops are not noticeably compressed. Shrink the droplet to a micrometre and the same formula gives 145,600 Pa, or nearly one and a half atmospheres, which is why very small droplets evaporate so much faster than large ones.
Capillary rise follows by setting the Laplace pressure across the meniscus against the hydrostatic pressure of the raised column. The meniscus in a tube of radius a has a radius of curvature a/cosθ, so the balance gives h = 2γcosθ / (ρga), which is Jurin's law. Physics LibreTexts derives both results side by side in its section on surface tension effects in fluid mechanics.
A Worked Capillary Rise Example
Put water in a glass tube of 0.5 mm radius, a millimetre across. Glass is well wetted by clean water, so the contact angle is close to zero and cosθ is 1. Using gamma of 0.0728 N/m, density 998 kg/m³, gravitational acceleration 9.807 m/s² and a of 0.0005 m: h = 2 × 0.0728 / (998 × 9.807 × 0.0005) = 0.1456 / 4.894 = 0.0297 m, just under 30 mm. That matches the classic textbook demonstration, where water in a one-millimetre glass capillary climbs about three centimetres.
The inverse relationship with radius is the important part. Halve the tube to 0.25 mm radius and the rise doubles to about 60 mm. That is why fine-pored materials wick so strongly, why soil texture governs how far water moves upward from a water table, and why a paper towel with narrow fibre gaps outperforms a coarse cloth. It is also why the calculation stops being useful for very wide tubes: once the radius approaches the capillary length, the meniscus is no longer a spherical cap and Jurin's law overestimates the rise.
The Capillary Length and When the Equation Stops Working
The capillary length, √(γ/ρg), is the natural scale at which surface tension and gravity are comparable. For water it is about 2.7 mm, which the results grid reports. Below that scale, surface tension dominates: drops are near-spherical, menisci are near-spherical caps, and Jurin's law is accurate. Above it, gravity dominates: puddles flatten out, large drops sag, and a meniscus in a wide tube becomes a shallow curve near the walls with a flat centre.
That transition is where a naive application of these formulas goes wrong. Jurin's law assumes the meniscus is a spherical cap of radius a/cosθ, which is only true when a is much smaller than the capillary length. In a tube of comparable width the prediction runs high. NIST has published work testing precisely how far the classical treatment holds, reporting in Capillary Rise between Planar Surfaces that agreement with the Laplace-Young prediction is excellent provided the spacing between the walls is very narrow.
Contact Angle: the Input That Is Hardest to Pin Down
Surface tension and density are properties of the liquid and are tabulated to several significant figures. Contact angle is not: it is a property of the liquid, the solid and the vapour together, and it is notoriously sensitive to surface cleanliness, roughness and history. The same water on the same glass can show a contact angle near zero when the glass is freshly cleaned and thirty degrees or more once a monolayer of organic contamination has adsorbed.
Worse, the angle is not single-valued. An advancing meniscus shows a larger angle than a receding one, and the gap between them — contact angle hysteresis — can be tens of degrees on a rough or chemically heterogeneous surface. In practice this means a capillary rise prediction carries much more uncertainty from the angle than from anything else you type in. Because cosθ falls slowly near zero, small angles matter little; the sensitivity climbs steeply as the angle approaches ninety degrees, where the predicted rise passes through zero and then reverses.
Where the Equation Turns Up in Practice
Mercury porosimetry inverts Jurin's law: because mercury has a contact angle well above ninety degrees on most solids it must be pushed into pores rather than being drawn in, and the pressure needed to force it into a pore of a given radius is read straight off the Young-Laplace relation. Soil physics uses the same relation to convert a matric potential into an equivalent pore size. Inkjet printing depends on the pressure jump at the nozzle meniscus staying within a narrow window so the ink neither drips nor retracts.
In the lung, a thin liquid film lines alveoli of very small radius, where 2γ/R would be large enough to collapse them; surfactant lowers gamma dramatically and, crucially, lowers it further as the alveolus shrinks, which stabilises the system. And in plant physiology, the same capillary physics operating in xylem vessels a few tens of micrometres across is part — though only part — of the explanation for water transport to the top of tall trees. If you are working on the hydrostatic side of any of these problems, our hydrostatic pressure calculator handles the column-height term directly.
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Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Using 2γ/R for a soap bubble. A free film has two surfaces, so the correct expression is 4γ/R and the answer is twice as large.
- Entering a diameter where a radius is asked for, which halves the pressure and halves the predicted capillary rise.
- Making the second radius positive on a saddle surface. On a saddle the curvatures oppose, so one radius must be entered as negative.
- Using a room-temperature surface tension for a hot liquid. Gamma falls steadily with temperature and vanishes at the critical point.
- Applying Jurin's law to a wide tube. Once the radius approaches the capillary length the meniscus is no longer a spherical cap and the formula overpredicts.
Related Free Tools From Arb Digital
For unit work on gamma itself, use the surface tension converter. On the pressure side, the hydrostatic pressure calculator gives the depth term and the pressure converter moves between pascals, bar, psi and mmHg. Liquid properties come from the density calculator and the viscosity converter. For flow through the narrow channels where these effects matter, see the pipe flow calculator, and for the related membrane physics the osmotic pressure calculator. Browse the full free online tools hub for more.
Frequently Asked Questions
It states that the pressure difference across a curved fluid interface equals the surface tension multiplied by the sum of the reciprocals of the two principal radii of curvature. For a sphere this simplifies to twice the surface tension divided by the radius.
A soap bubble is a thin film with a liquid-air interface on both the inside and the outside. Each contributes its own pressure jump, so the total is four times the surface tension over the radius rather than twice.
Jurin's law gives the height as twice the surface tension times the cosine of the contact angle, divided by the liquid density, gravitational acceleration and the tube radius. Water in a clean glass tube of half a millimetre radius rises about 30 millimetres.
It indicates that the surface curves the opposite way along that principal direction, as on a saddle. Entering one radius as negative lets the two curvatures partly cancel, which can reduce the pressure jump to zero on a minimal surface.
It is the square root of surface tension divided by density times gravitational acceleration, and it marks the scale where surface tension and gravity are comparable. For water it is about 2.7 millimetres. Below that scale interfaces are dominated by surface tension.
When the contact angle exceeds ninety degrees the liquid does not wet the solid, the cosine term turns negative, and the meniscus curves the other way. Mercury in glass behaves this way and is depressed below the reservoir level rather than raised above it.
Yes, mainly through surface tension, which falls as temperature rises and reaches zero at the critical point. Density also falls slightly with temperature. Use property values quoted at your working temperature rather than at room temperature.
This calculator applies the published Young-Laplace and Jurin relationships and is provided for educational and engineering estimation purposes. Real interfaces are affected by surfactants, contamination, surface roughness and contact angle hysteresis, none of which appear in these formulas.
