Young-Laplace Equation Calculator
The Young-Laplace equation describes the pressure jump across a curved liquid-gas interface caused by surface tension. Enter the geometry mode, surface tension, and radius (or radii) of curvature to find the capillary pressure difference. Switch to General mode to specify two independent principal radii for non-spherical interfaces. The calculator also computes capillary rise height when you supply a tube radius and contact angle.
What the Young-Laplace equation tells you
The Young-Laplace equation relates the pressure difference across a curved liquid-gas (or liquid-liquid) interface to the surface tension and the geometry of that interface. In its most general form it reads: dP = gamma * (1/R1 + 1/R2), where gamma is the surface tension in N/m and R1, R2 are the two principal radii of curvature in metres. The curvature sum (1/R1 + 1/R2) is twice the mean curvature of the surface. A spherical interface has R1 = R2 = R, simplifying to dP = 2*gamma/R; a cylindrical interface has R2 approaching infinity, giving dP = gamma/R. The equation explains why small bubbles are harder to inflate than large ones, why water rises in a narrow tube, and why fog droplets remain suspended in air.
Capillary rise and the Jurin-Young-Laplace relation
When a narrow tube is placed in a liquid, the curved meniscus at the liquid-air interface produces a Laplace pressure that drives liquid up (or down) the tube. At equilibrium the Laplace pressure equals the hydrostatic pressure of the liquid column: 2*gamma*cos(theta)/a = rho*g*h. Solving for h gives the Jurin law: h = 2*gamma*cos(theta) / (rho*g*a). Here theta is the contact angle between the liquid and the tube wall, a is the tube radius, rho is the liquid density, and g is gravitational acceleration. A contact angle below 90 degrees means the liquid wets the surface and rises; above 90 degrees (mercury on glass, for example) the liquid is depressed. Capillary rise is exploited in thin-layer chromatography, inkjet nozzles, wicking materials, and the water transport system of trees.
Why radius matters so much - the inverse relationship
Because dP is inversely proportional to radius, the Laplace pressure grows rapidly as the interface shrinks. A water droplet 1 mm in radius has an excess pressure of about 146 Pa - barely noticeable. Shrink it to 1 micrometer and the pressure jumps to 146,000 Pa (about 1.4 atm). This steep dependence drives Ostwald ripening, where small droplets in an emulsion dissolve and feed larger ones over time. It also sets the minimum pressure needed to force gas through a porous membrane (the bubble-point pressure), which is used to qualify membrane filters in pharmaceutical and water-treatment applications.
Units, sign convention and practical ranges
The calculator works internally in SI (N/m for surface tension, metres for radii, pascals for pressure). Surface tension inputs are entered in mN/m because practical values for most liquids fall between 20 and 80 mN/m - a more comfortable numerical range. For a concave interface (theta greater than 90 degrees or a saddle surface with one negative principal curvature) the pressure difference can be negative, meaning the pressure is lower inside the curved surface than outside. The atm output helps connect the result to familiar pressures: 1 atm is 101,325 Pa. Micrometer-scale droplets reach pressures of several atmospheres; nanoscale bubbles in sonochemistry may reach hundreds of atmospheres.
Surface tension of common liquids at ~20 C
| Liquid | Surface tension (mN/m) | Density (kg/m^3) | Typical contact angle on glass |
|---|---|---|---|
| Water | 72.8 | 998 | 0-20 deg (wetting) |
| Seawater | 73.0 | 1025 | 0-20 deg (wetting) |
| Glycerine | 63.4 | 1261 | ~19 deg |
| Ethanol | 22.3 | 789 | ~0 deg (wetting) |
| Methanol | 22.7 | 792 | ~0 deg (wetting) |
| Benzene | 28.9 | 879 | ~0 deg (wetting) |
| Chloroform | 27.1 | 1489 | ~0 deg (wetting) |
| Mercury | 485 | 13534 | ~140 deg (non-wetting) |
| Olive oil | 32.0 | 900 | ~40 deg |
Values against air at approximately 20 degrees Celsius. Mercury is measured against vacuum.
Frequently asked questions
What is the Young-Laplace equation?
The Young-Laplace equation is dP = gamma * (1/R1 + 1/R2). It gives the pressure jump across a curved interface between two fluids (or a fluid and a gas) in terms of the interfacial tension gamma and the two principal radii of curvature R1 and R2. It was derived independently by Thomas Young (1805) and Pierre-Simon Laplace (1806) and underpins the entire field of capillary physics.
Why is the formula dP = 2*gamma/R for a sphere?
A sphere has two equal principal radii (R1 = R2 = R), so 1/R1 + 1/R2 = 2/R. Substituting into the general equation gives dP = gamma * (2/R) = 2*gamma/R. A cylinder has one finite radius and one infinite radius, giving 1/R1 + 1/R2 = 1/R, so dP = gamma/R - exactly half the spherical value for the same radius.
What is capillary rise and how does the formula change?
Capillary rise occurs when a tube is narrow enough that the Laplace pressure at the meniscus can support a column of liquid. Balancing capillary pressure against hydrostatic pressure gives h = 2*gamma*cos(theta) / (rho*g*a). Here h is the rise height, theta is the contact angle, rho is the liquid density, g is gravitational acceleration, and a is the tube radius. The factor cos(theta) accounts for the fact that a non-zero contact angle reduces the effective pressure the meniscus generates.
What contact angle should I use for water on glass?
Clean, freshly polished glass gives a contact angle close to 0 degrees for pure water (perfectly wetting). In practice, contamination and surface roughness raise this to 10-30 degrees. Mercury on glass has a contact angle of about 135-140 degrees, which is why mercury is depressed rather than elevated in glass capillaries.
Can the pressure difference be negative?
Yes. If the contact angle exceeds 90 degrees, cos(theta) is negative, so dP is negative in capillary mode, meaning the pressure inside is lower than outside - the liquid surface curves away from the solid, and the liquid is depressed in the tube. In General mode you can enter a negative R2 to represent a saddle-shaped surface where one principal curvature is concave, which can also make dP negative.
How does this relate to Ostwald ripening?
Ostwald ripening is driven directly by the Young-Laplace equation. Because smaller droplets have higher internal pressure, the dissolved concentration of the dispersed phase is higher near small droplets (via the Kelvin equation, which extends the Young-Laplace result to vapor pressure). This concentration gradient causes molecules to diffuse from small droplets to large ones, so small droplets shrink and large ones grow over time.
What units should I enter surface tension in?
This calculator accepts surface tension in mN/m (millinewtons per metre), which is the standard unit in most reference tables. It is numerically identical to dyn/cm (the older CGS unit), so values from older textbooks can be entered directly. The SI base unit N/m is also the same numerically scaled by 1000 - water at 20 C is 0.0728 N/m = 72.8 mN/m = 72.8 dyn/cm.