Young Laplace Equation Calculator - Capillary Pressure Solver

A young laplace equation calculator is a fluid-mechanics and surface-chemistry tool that solves the Young-Laplace equation Delta P = gamma (1/R1 + 1/R2) for the unknown among capillary pressure, surface tension, and the radius of a curved interface, so you can read off the pressure difference that surface tension creates across a droplet, bubble, or capillary meniscus.

• • Spherical droplet pressure: Compute the internal pressure of a raindrop, an inkjet droplet, or an emulsion droplet from its radius and the surface tension of the liquid.
• • Soap bubble excess pressure: Estimate the excess pressure inside a soap bubble or foam film, where the Young-Laplace equation gets a factor of two because the film has two surfaces.
• • Capillary rise height: Back out the equilibrium contact angle, surface tension, or tube radius for water or mercury rising in a glass capillary.
• • Pulmonary alveolus models: Use the equation as a first-pass model for the pressure needed to inflate a small alveolus in respiratory physiology.

The Young-Laplace equation links three measurable quantities: surface tension gamma, the principal radii of curvature R1 and R2, and the pressure jump Delta P across the interface. A more sharply curved surface produces a larger pressure difference, which is why inkjet nozzles, alveoli, and small raindrops evaporate or collapse faster than larger ones.

This calculator supports four interface geometries - a spherical single surface (a raindrop), a soap bubble film (two surfaces, so the curvature is doubled), a cylindrical interface (one principal radius, the other is infinite), and the fully general case where you supply both R1 and R2.

If you are tracking pressure-energy conversion along a streamline, the bernoulli equation calculator solves the matching fluid-mechanics energy balance in pascals.

The calculator evaluates the Young-Laplace equation in pressure form and rearranges it for the variable you selected. It uses the geometry you chose to build a curvature factor K, then solves Delta P = gamma K, gamma = Delta P / K, or R = f(gamma, Delta P) depending on the Solve For menu.

• Delta P: Pressure difference across the interface, in pascals (Pa). This is the inside minus outside pressure for a droplet or bubble.
• gamma: Surface tension of the liquid in newtons per metre (N/m). Pure water at 20 deg C is 0.0728 N/m; a soap film is closer to 0.025 N/m.
• R1, R2: Principal radii of curvature of the interface in metres. For a sphere they are equal to the sphere's radius; for a cylinder one of them is infinite.
• Interface mode: Switches between spherical single surface (K = 2/R), soap bubble film (K = 4/R), cylindrical (K = 1/R), and the general (K = 1/R1 + 1/R2) form.

The mean curvature output is reported in inverse metres, so you can verify the Laplace pressure by hand-multiplication. Capillary rise height is calculated only in the cylindrical mode and assumes a perfectly wetting contact angle of 0 deg for water, which is a useful upper bound for classroom comparison.

According to Wikipedia - Young-Laplace equation, The Young-Laplace equation states that the pressure difference across a curved interface equals the surface tension times the sum of the principal curvatures, Delta P = gamma (1/R1 + 1/R2).

According to Engineering Toolbox - Surface Tension, Dilute soapy water at 20 deg C has a surface tension of 0.0250 to 0.0450 N/m, much lower than pure water at 0.0728 N/m, while mercury sits near 0.485 N/m.

When the same capillary tube is part of a flowing network, the Reynolds number calculator tells you whether the flow stays laminar before capillary effects kick in.

Four ideas from surface science and differential geometry that make the Young-Laplace equation more than a one-line formula.

These four ideas reappear everywhere from industrial spray cooling to pulmonary surfactant research. The same physical intuition - sharper curves carry higher pressure - shows up in foam drainage and in the coalescence of small bubbles into bigger ones.

For gas-filled bubbles where the gas pressure itself changes with size, the ideal gas calculator gives the matching ideal-gas pressure-volume relationship to pair with this Laplace balance.

Use the young laplace equation calculator in five steps.

1. 1 Pick the interface mode: Open the Interface Mode menu and choose between spherical droplet, soap bubble film, cylindrical interface, or the general R1/R2 case.
2. 2 Choose what to solve for: Set Solve For to the variable you want returned: pressure difference (Delta P), surface tension (gamma), or radius (R).
3. 3 Enter the surface tension: Type gamma in newtons per metre. Use 0.0728 for water at 20 deg C, 0.025 for soap, or 0.485 for mercury.
4. 4 Enter the radius and known pressure difference: Type the radius R, and for the general mode type both R1 and R2. If you set Solve For to surface tension or radius, also type the known Delta P.
5. 5 Read the solved value and curvature: The primary output shows the solved variable in its natural unit, and the secondary outputs report the capillary pressure, mean curvature, and a capillary rise estimate.

For a 1 mm water droplet in air, set Interface Mode to spherical droplet, Solve For to pressure difference, gamma to 0.0728 N/m, and R to 0.001 m. The calculator returns Delta P = 145.6 Pa and a mean curvature of 2000 1/m.

Surface tension is temperature dependent, so when the problem involves a warm liquid the Arrhenius equation calculator helps quantify the underlying temperature change that drives a different gamma.

Practical reasons to use this young laplace equation calculator instead of rearranging the formula by hand.

• • One tool for all four interface shapes: Switch between spherical droplet, soap bubble, cylindrical, and general R1/R2 modes from a single menu.
• • Solve for any of the three core variables: Pick Delta P, gamma, or R from the Solve For menu and the calculator rearranges the equation for you, including the soap-bubble factor of 2.
• • Auditable curvature output: The mean curvature output is reported in 1/m, so you can verify the Laplace pressure by hand-multiplication.
• • Useful water and soap defaults: Common surface tension values for water, soap, and mercury are noted in the help text, so you can paste them in without leaving the page.
• • Connects to capillary rise and droplet dynamics: The same physical setup feeds textbook problems in capillary rise, emulsion stability, and respiratory physiology.

The calculator is intentionally narrow: it solves one equation correctly across four common interface geometries. For flows with surfactant transport or dynamic contact angle, you would still need a more detailed model on top of the pressure reported here.

What changes the pressure difference the young laplace equation calculator returns, and what it cannot capture.

• • The Young-Laplace equation assumes a single, well-defined surface tension and a static mechanical equilibrium, so it does not capture Marangoni flow or surfactant transport.
• • The general mode uses the sum 1/R1 + 1/R2 without a sign convention for concave vs convex curvature, so saddle-shaped interfaces need careful handling of the sign of Delta P.
• • The capillary rise estimate uses a perfectly wetting contact angle of 0 deg for water, so it is an upper bound. For mercury in glass the contact angle is closer to 140 deg.

The Young-Laplace equation is the starting point for capillary rise, droplet formation, and alveolus mechanics in physiology, but each of those applications adds extra physics on top of the static balance reported here.

According to LibreTexts Chemistry - Surface Tension and Capillarity, The Young-Laplace equation is the starting point for capillary rise, droplet formation, and alveolus mechanics in physiology, though each application adds extra physics on top of the static balance.

When the curved interface sits at an electrode surface, the Nernst equation calculator provides the electrochemical potential that the Laplace pressure must overcome for a bubble to nucleate.