How this instrument works
The Young-Laplace equation gives the pressure difference, delta P, across a curved interface between two fluids — such as the surface of a liquid droplet suspended in air, or a gas bubble inside a liquid — caused by surface tension. For a spherical interface, delta P = 2 x gamma / r, where gamma is the surface tension of the liquid and r is the radius of curvature. The pressure is always higher on the concave (inner) side of the curve, and the equation says that gap grows as the radius shrinks: a smaller droplet or bubble holds a larger internal pressure excess than a bigger one made of the same liquid.
The relationship falls out of a straightforward mechanical balance. Surface tension acts like a stretched elastic skin trying to minimize surface area, which for a sphere means pulling inward; that inward pull has to be balanced by an outward-pushing excess pressure inside, or the sphere would simply collapse. Working through the force balance across a hemisphere of the curved surface produces exactly delta P = 2gamma/r for a single spherical interface — the factor of 2 comes from the sphere's two principal radii of curvature both being equal to r and both contributing to the restoring force.
The equation is named for Thomas Young and Pierre-Simon Laplace, who worked out the mathematics of capillary action independently around 1805, and it explains a wide range of everyday and industrial phenomena: why smaller soap bubbles pop faster and merge into larger ones when two touch (the smaller bubble's higher internal pressure pushes air into the larger one), why capillary rise pulls liquid up a narrow tube higher than a wide one, and why the alveoli in human lungs need surfactant to keep their smallest, most tightly curved air sacs from collapsing under their own surface tension.
- Enter the liquid's surface tension into Surface tension (N/m) — water's value near room temperature, about 0.0728 N/m, loads by default.
- Enter the curve's radius into Radius of curvature (m) — the radius of the droplet, bubble, or curved surface in question.
- Read Pressure difference, delta P (Pa) — how much higher the pressure is on the concave (inner) side of the curved surface compared to the outer side.
- Shrink Radius of curvature (m) while holding Surface tension (N/m) fixed and watch Pressure difference, delta P (Pa) rise — the two are inversely proportional.
- Radius of curvature (m) must be greater than zero; a flat surface (infinite radius) has no pressure difference at all, and zero radius is not physically meaningful here.
Worked example — a 1 mm water droplet
Enter 0.0728 into Surface tension (N/m) — water's surface tension at about 20 degrees C — and 0.001 into Radius of curvature (m), a droplet 1 millimetre in radius. Pressure difference, delta P (Pa) reads 145.6000: the instrument computes (2 x 0.0728) / 0.001 = 145.6.
That means the pressure inside this droplet is about 145.6 pascals higher than the surrounding air — a small but measurable excess, roughly 0.14% of standard atmospheric pressure. Shrink Radius of curvature (m) tenfold, to 0.0001 m (a 100-micron droplet), and Pressure difference, delta P (Pa) rises tenfold in turn, to 1456.0000 Pa, since the equation is exactly inversely proportional to radius.
Questions
Why does a smaller droplet have a higher internal pressure than a larger one?
Because delta P is inversely proportional to radius — surface tension pulls inward on the curved surface with a force that depends on how sharply it's curved, and a smaller radius means a tighter curve. The same surface tension acting over a more sharply curved surface produces a larger restoring pressure, exactly the way a tightly inflated small balloon feels firmer than a loosely inflated large one made of the same material.
Why is the formula 2 gamma / r and not just gamma / r?
The factor of 2 comes from the geometry of a sphere specifically: a spherical surface has two equal principal radii of curvature (think of slicing the sphere through its center in any two perpendicular planes — both cuts show the same curve), and both contribute equally to the restoring pressure. A cylindrical interface, curved in only one direction, uses delta P = gamma/r instead, without the factor of 2, since it has only one radius of curvature contributing.
Does this formula apply to soap bubbles the same way?
Not quite as written — a soap bubble in air has two liquid-air interfaces, an inner surface and an outer surface, each contributing its own 2gamma/r pressure jump, so the total pressure difference across a soap bubble is 4gamma/r, double this calculator's single-interface result. A droplet of liquid suspended in air, or a gas bubble trapped inside a liquid, has only one interface, so 2gamma/r applies directly, which is the case this instrument is built for.
How does this relate to why alveoli in the lungs need surfactant?
The lungs' alveoli are tiny, curved, liquid-lined air sacs, and without help their surface tension alone would generate enough internal pressure difference, especially in the smallest alveoli, to make them collapse and empty into larger neighboring sacs — the same physics behind small soap bubbles merging into larger ones. Pulmonary surfactant, a substance the lungs produce, lowers effective surface tension disproportionately more in smaller, more tightly curved alveoli, keeping the whole system of differently sized air sacs mechanically stable.
What surface tension value should I use for a liquid other than water?
Look up that liquid's surface tension at your working temperature — surface tension is temperature-dependent and varies substantially between liquids: water's is comparatively high, around 0.0728 N/m near room temperature, while many organic solvents and oils sit several times lower, and liquid metals like mercury sit several times higher. Reference tables such as the CRC Handbook of Chemistry and Physics tabulate surface tension for common liquids across a range of temperatures.
Why does the instrument require the radius to be greater than zero?
Radius sits in the denominator of 2gamma/r, so a radius of exactly zero would make the pressure difference mathematically infinite, which isn't physically meaningful — no real droplet or bubble has zero size. A flat interface, by contrast, has an infinite radius of curvature and correctly gives a pressure difference of zero, since there's no curvature left to generate a restoring pressure.