Young-Laplace Equation Calculator
Curve a liquid surface and it pushes back: the pressure inside a droplet exceeds the pressure outside by an amount set by the surface tension and the radius — the smaller the droplet, the harder it squeezes. The Young–Laplace equation, ΔP = 2γ/R for a sphere, prices that squeeze exactly, and it quietly governs mist formation, inkjet printing, blood capillaries, and why small bubbles dissolve into big ones. This calculator evaluates it both ways — pressure jump from radius, and capillary rise from tube radius — with water’s surface tension as the default and any liquid’s γ welcome.
The Equation and Its Two Surfaces
Thomas Young (1805) and Pierre-Simon Laplace (1806) independently priced the curved surface: the pressure jump across an interface is ΔP = γ × (1/R₁ + 1/R₂), where R₁ and R₂ are the two principal radii of curvature. For a sphere both equal R, collapsing the sum to ΔP = 2γ/R — the form this calculator leads with, valid for a liquid droplet in gas and for a gas bubble inside a liquid (one interface each). A soap bubble is a thin liquid film with two interfaces, inner and outer, doubling the squeeze to 4γ/R — the calculator's second mode, and the resolution of the parlour paradox in its note: join two soap bubbles with a tube and the smaller one, at higher internal pressure, inflates the larger and collapses into it; intuition votes the other way and loses. Cylindrical surfaces (one finite radius, one infinite) halve the spherical figure — a cylindrical water jet of radius 25 µm carries ΔP = γ/R = 0.0728/0.000025 ≈ 2.91 kPa, exactly half what a droplet of the same radius holds; saddle surfaces, with curvatures of opposite sign, can cancel it entirely — the geometry menu of an equation that runs from raindrops to alveoli.
Worked Numbers Across the Radius Scale
The defaults put a 1 mm water droplet (R = 1,000 µm) under the knife: ΔP = 2 × 0.0728 / 0.001 = 145.6 Pa — a whisper, 0.14% of an atmosphere. Now walk down the scale the equation insists on. At R = 10 µm (a fog droplet), ΔP = 14,560 Pa ≈ 14.6 kPa, a seventh of an atmosphere of extra internal pressure. At R = 1 µm, 145.6 kPa — the droplet's interior is at two and a half atmospheres absolute. And the capillary mode at a 0.5 mm tube radius gives the household face of the same law: h = 2 × 0.0728 / (1,000 × 9.81 × 0.0005) = 29.7 mm of rise — the height water climbs a clean glass capillary unaided, the mechanism inside every soil pore, plant stem and paper towel. Surface tension is feeble at the scale of buckets and tyrannical at the scale of micrometres, and the Young–Laplace 1/R is the exact statement of where the tyranny begins.
Why Small Droplets Cannot Mind Their Own Business
The pressure gradient between sizes drives two famous instabilities. In a cloud or emulsion, the higher vapour pressure over small, highly curved droplets (the Kelvin consequence of Laplace pressure) lets big droplets grow at the small ones' expense — Ostwald ripening — so fogs, emulsions and alloys coarsen with age, small particles dissolving into large ones without any contact at all. In the lung, the same mathematics threatens collapse: the alveoli are microscopic interconnected bubbles whose Laplace pressures would empty the small into the large, were it not for pulmonary surfactant, which lowers γ preferentially as area shrinks, stabilising the size distribution — premature infants missing the surfactant face exactly the Young–Laplace instability, and its replacement therapy is among neonatology's triumphs. And in nucleation, the 1/R squeeze sets the critical radius a newborn bubble or droplet must exceed to survive: below it, Laplace pressure crushes the embryo back into solution, which is why champagne needs its scratches and crevices (ready-made radii) and why utterly smooth, clean water can be superheated past boiling in a microwave and erupt when disturbed — the equation, holding its breath. Boiling chips, etched glass and every rough kettle interior are, functionally, radius donors: they lend curvature the liquid cannot yet afford on its own, at exactly the moment the young bubble needs it most.
Surface Tension Values and Temperature
The calculator's default γ = 0.0728 N/m is water against air at 20 °C — near the top of the common-liquid league (mercury, at 0.465 N/m, is the outlier champion; most organic liquids sit at 0.02–0.03, ethanol 0.0223, which is why alcohols spread and wet so eagerly). The same 1 mm radius in ethanol (γ = 0.0223 N/m) squeezes by only 44.6 Pa — a third of water's figure — which is the droplet-scale receipt for alcohol's eagerness to spread, creep and evaporate. Temperature softens every value in the table: water falls to about 0.0589 N/m at 100 °C and vanishes entirely at the critical point, where the interface itself dissolves. Surfactants rewrite the table at a stroke — a soapy solution's γ near 0.03 N/m is why the soap-bubble mode of this calculator should be run with the solution's γ, not water's, and why a drop of detergent assassinates a mosquito larva's surface-tension life support. Treat γ as a property of the pair (liquid, surrounding fluid, temperature, contamination) and the equation's outputs will track the bench; treat it as a constant of nature and the first surprising result is already on its way.
Running the Equation Backwards: Measuring γ
The capillary mode doubles as the classic instrument for measuring the constant it consumes: raise a liquid in a tube of known radius, read the height, and γ = h·ρ·g·R/2 falls out — the 29.7 mm rise of water in a 0.5 mm tube is a surface-tension measurement anyone can repeat with a ruler. The extremes of the scale advertise the method's range: mercury, with γ ≈ 0.4655 N/m, would show a 1 mm droplet squeezed by 931 Pa — six times water's figure at the same radius, and one reason mercury refuses to wet, bead or behave like a polite liquid at all. And at the opposite pole, the near-zero effective tensions of microemulsions and critical-point mixtures are measured by watching Laplace pressures dwindle — the equation read backwards is a tensiometer, which is why Young and Laplace's two-hundred-year-old line of algebra still appears, unaltered, in the methods sections of papers about fluids that did not exist when it was written.
Related Tools
More from the Chemistry Calculators hub: the Boiling Point Calculator for vapour pressure's other household appearance, the Boiling Point at Altitude Calculator for pressure's effect on boiling, and the live Concentration Calculator for the solutions whose surfaces this page prices.
Frequently Asked Questions
Why 2γ/R for a droplet but 4γ/R for a soap bubble?
Count the interfaces. A droplet (or a gas bubble inside a liquid) is bounded by one surface, contributing γ(1/R + 1/R) = 2γ/R. A soap bubble is a liquid film with an inside surface and an outside surface, each squeezing — two times 2γ/R is 4γ/R. The calculator's geometry select exists because the factor is the commonest slip in the whole topic.
Does the small bubble really inflate the big one?
Yes — connect two soap bubbles by an open tube and the smaller, whose Laplace pressure is higher, drives air into the larger until it collapses to a film cap. It is the single most reliable counterintuitive demonstration in surface science, and the reason emulsion droplets and lung alveoli alike need stabilising mechanisms against their own physics.
What radius should I enter for a bubble in water?
The bubble's own radius — R in the formula is the radius of curvature of the surface, which for a spherical bubble or droplet is simply its radius (not its diameter; halving by accident is the second commonest slip). For a meniscus in a tube, the curvature radius is the tube radius divided by cos(contact angle) — the capillary mode here folds the clean-glass case (angle ≈ 0) into its formula.
How does this relate to capillary action in plants?
Directly: the rise height h = 2γcosθ/(ρgR) is Laplace pressure balancing a gravity column. In xylem vessels of radius ~20 µm the static rise is under a metre — real trees run on transpiration pull and cohesion, with capillarity a junior partner — but in soil pores, paper, sponges and microfluidic chips, where radii are micrometric and heights millimetric, the equation on this page is the whole engine.
Why does the pressure matter for very small droplets in medicine?
Aerosol droplets and nebuliser mists live at 1–10 µm, where Laplace pressures are a substantial fraction of an atmosphere: they set evaporation rates (through the Kelvin effect), deposition behaviour in airways, and the stability of inhaled formulations. The same range governs inkjet drops and spray-drying — micro-scale engineering is applied Young–Laplace with deadlines.
Is γ really a constant for my liquid?
Only at stated temperature and purity. Surface tension falls with temperature (about −0.15 mN/m per °C for water), collapses near the critical point, and is exquisitely sensitive to surface-active contamination — a fingerprint's worth of oil can measurably rewrite a water surface. Use the value for your conditions; the table in the temperature section gives water's landmarks.