Young-Laplace Equation Calculator

The Young-Laplace equation relates the pressure difference across a curved liquid interface to the surface tension and the radius of curvature. For a sphere with a single surface, such as a liquid droplet in air or a gas bubble in liquid, it is dP = 2*gamma/r, where gamma is the surface tension and r is the radius.

A soap bubble is a thin film of liquid with air on both sides, so it has two curved surfaces, an inner one and an outer one, each contributing a pressure jump of 2*gamma/r. Adding both surfaces gives a total pressure difference of dP = 4*gamma/r, twice that of a single-surface droplet of the same radius.

The pressure difference is inversely proportional to radius, dP is proportional to 1/r, so smaller droplets and bubbles have much higher internal pressure. Halving the radius doubles the pressure difference. This is why very small droplets in an aerosol or mist can have significant excess internal pressure.

Surface tension is the energy per unit area (equivalently, force per unit length) needed to increase the surface area of a liquid, caused by cohesive forces between molecules at the interface pulling inward. It is measured in N/m or equivalently J/m2, and typical values are around 0.07 N/m for water and 0.02 to 0.03 N/m for soap solutions.

Because Laplace pressure dP = 2 gamma/r (or 4 gamma/r for a soap film) is higher inside a smaller bubble, gas flows from the high-pressure small bubble into the lower-pressure large bubble when they are connected, causing the small one to shrink and the large one to grow. This is the same physics behind Ostwald ripening in foams and emulsions.

Surface tension gamma is typically in N/m, radius r in meters, and the resulting pressure difference dP comes out in pascals (Pa), since N/m divided by m equals N/m2 = Pa. This calculator accepts radius in millimeters and converts internally, then reports the pressure in both Pa and kPa.

The 2 gamma/r (or 4 gamma/r) form applies specifically to spheres. For a general curved surface with two principal radii of curvature R1 and R2, the full Young-Laplace equation is dP = gamma*(1/R1 + 1/R2). For a sphere R1 = R2 = r, which reduces to the simple 2 gamma/r form used here.

Capillary pressure is the pressure difference across a curved meniscus in a narrow tube or porous medium, and it is calculated using the same Young-Laplace relationship. In a cylindrical capillary tube it is often written dP = 2*gamma*cos(theta)/r, where theta is the contact angle between the liquid and the tube wall.

Surface tension decreases as temperature rises for most liquids, so the pressure difference predicted by the Young-Laplace equation decreases correspondingly at a fixed radius. For water, gamma falls from about 0.0756 N/m at 0 degrees C to about 0.0589 N/m near 100 degrees C, roughly a 22 percent drop.

The Laplace pressure needed to maintain a bubble scales as 1/r, so the pressure needed at the very start, when the bubble radius is tiny, is very high. As the bubble grows, the required pressure drops, which is why the first moment of blowing a bubble or a balloon requires noticeably more effort than continuing to inflate it.

With this sign convention dP is always positive because gamma and r are both positive, meaning the pressure inside a droplet or bubble is always higher than outside. A concave meniscus, such as water wetting a narrow glass capillary, effectively reverses the curvature direction and can produce a pressure inside the liquid that is lower than the surrounding pressure.