Laplace Pressure Calculator
Calculate the Laplace pressure (excess pressure) inside a droplet, soap bubble, or cylindrical liquid surface. Enter the surface tension and radius of curvature to get the pressure difference in Pa, kPa, psi, mmHg and more.
Find the Excess Pressure Inside a Droplet or Bubble
Laplace Pressure Formula
Where:
- \(\Delta P\) — pressure difference across the curved interface, in pascals (Pa); the pressure inside minus the pressure outside
- \(\gamma\) — surface tension of the liquid, in N/m
- \(r\) — radius of curvature of the interface, in m
These are forms of the Young–Laplace equation. A curved interface with surface tension behaves like a stretched membrane: the tighter the curvature (smaller \(r\)) and the stronger the surface tension, the larger the excess pressure needed on the concave side to keep the surface in mechanical equilibrium. The multiplier (2, 4 or 1) depends on the shape and on how many liquid–gas surfaces are present.
Laplace Pressure – Calculation Example
Example: a spherical water droplet in air with a radius of 1 mm. The surface tension of water at room temperature is about 72 mN/m (0.072 N/m), and the droplet has one curved surface, so the factor is 2.
\[ \Delta P = \frac{2\gamma}{r} = \frac{2 \times 0.072\ \text{N/m}}{0.001\ \text{m}} = 144\ \text{Pa} \]
Result: the pressure inside the droplet is about 144 Pa (0.144 kPa ≈ 1.44 mbar) higher than the surrounding air pressure.
How to use the calculator
- Choose the Interface type: a spherical droplet or gas bubble in liquid (one surface, factor 2), a soap bubble (two surfaces, factor 4), or a cylindrical surface such as a liquid jet or meniscus (factor 1).
- Enter the Surface tension (γ) of the liquid and select its unit — N/m, mN/m, dyn/cm or lbf/ft. Water is about 72 mN/m; the field info line lists other common values.
- Enter the Radius of curvature (r) and select its unit — mm, µm, cm, m or in.
- Read the Pressure difference across the interface, shown in Pa and converted to kPa, mbar, atm, psi and mmHg, together with a detailed breakdown of the calculation and its interpretation.
What is Laplace pressure?
Laplace pressure (also called capillary pressure or excess pressure) is the pressure jump that exists across any curved liquid–gas or liquid–liquid interface. It is a direct consequence of surface tension: molecules at a surface are pulled inward, so a curved surface behaves like a stretched elastic skin that squeezes the liquid on the concave side. The result is always a higher pressure on the inside of the curve — inside a droplet, inside a gas bubble, or inside a soap bubble.
The effect follows from the Young–Laplace equation, \(\Delta P = \gamma (1/R_1 + 1/R_2)\), where \(R_1\) and \(R_2\) are the two principal radii of curvature. For a sphere both radii equal \(r\), giving \(\Delta P = 2\gamma/r\). For a cylinder one radius is infinite, leaving \(\Delta P = \gamma/r\). A soap bubble is special: it has a thin liquid film with an inner and an outer surface, each contributing \(2\gamma/r\), so the total is \(4\gamma/r\).
The most important feature of the result is its inverse proportionality to radius. A 1 mm water droplet carries an excess pressure of only ~144 Pa, but shrink the same droplet to 1 µm and the excess pressure rises to ~144,000 Pa — about 1.4 atmospheres. This size dependence explains why tiny droplets evaporate faster than large ones (their higher internal vapour pressure), why small bubbles dissolve more readily in champagne, why pulmonary surfactant is needed to stabilize the small alveoli of the lungs, and why capillary forces dominate gravity at the millimetre scale and below. When analysing the motion of such small droplets through air, you may also find the Drag Force Calculator useful; the rate at which they lose heat while evaporating can be estimated with the Convection Heat Transfer Calculator.
Assumptions and limitations
- Static, ideal shape. The formulas assume a perfectly spherical or cylindrical interface at mechanical equilibrium. Oscillating, elongated or rapidly deforming droplets deviate from this.
- Uniform surface tension. Surfactants or temperature gradients make \(\gamma\) vary over the surface, which produces Marangoni flows not captured by the simple formula.
- Small Bond number. Gravity flattens large droplets; the spherical model is accurate only when the droplet is small compared with the capillary length (~2.7 mm for water). Large puddles and raindrops need shape corrections.
- Negligible film thickness. The soap-bubble formula assumes the film is much thinner than the bubble radius, so both surfaces share essentially the same radius.
- Constant values. Surface tension drops with temperature (water falls from 72 mN/m at 25 °C to about 59 mN/m at 100 °C), so use the value at your working temperature for accurate results.
Typical surface tension values
| Liquid (approx. 20–25 °C) | γ (mN/m) |
|---|---|
| Water | 72 |
| Soapy water | 25–35 |
| Ethanol | 22 |
| Mercury | 486 |
| Benzene | 29 |
| Glycerol | 63 |
To model the pressure inside a gas bubble surrounded by liquid, use the droplet option (one curved surface). For vapour bubbles forming during boiling or cavitation, the same \(2\gamma/r\) relation sets the pressure — and hence the temperature — at which the bubble can grow; heat delivery to the surrounding liquid can be checked with the Newton's Law of Cooling Calculator.
Frequently Asked Questions
How do I calculate Laplace pressure by hand?
First convert surface tension to N/m and radius to metres. Then apply the formula: for a droplet or gas bubble in liquid multiply \( 2\gamma \) and divide by \( r \); for a soap bubble use \( 4\gamma/r \); for a cylindrical surface use \( \gamma/r \). Example: a 0.5 mm droplet of water (\( \gamma = 0.072 \) N/m, \( r = 0.0005 \) m) gives \( \Delta P = 2 \times 0.072 / 0.0005 = 288 \) Pa. Always check that radius, not diameter, is used.
Why does a soap bubble have twice the pressure of a droplet of the same size?
A soap bubble is a thin film with two liquid–air interfaces: an inner surface and an outer surface. Each curved surface contributes \( 2\gamma/r \), and since the film is very thin both radii are essentially equal, so the contributions add to \( 4\gamma/r \). A droplet or a gas bubble in liquid has only one interface, hence \( 2\gamma/r \). This is why it is measurably harder to blow a small soap bubble than a large one.
Can Laplace pressure be negative?
Yes. The formulas give the magnitude of the pressure jump; pressure is always higher on the concave side of the interface. In a capillary tube the water meniscus is concave upward, so the pressure just under it is lower than atmospheric — this negative Laplace pressure is exactly what pulls water up the tube. Similarly, the liquid in a soap film between two plates, or inside a wetting wedge, can sit below ambient pressure.
What radius should I use for a surface that is not a perfect sphere or cylinder?
Use the mean curvature approach of the full Young–Laplace equation: \( \Delta P = \gamma (1/R_1 + 1/R_2) \), where \( R_1 \) and \( R_2 \) are the principal radii of curvature measured at the point of interest. For a saddle-shaped surface one radius is negative. For mildly non-spherical droplets, using an equivalent radius (same volume as a sphere) gives a good first estimate.
What are the most common mistakes when calculating Laplace pressure?
The top errors are: using the diameter instead of the radius; mixing units — remember that 1 dyn/cm = 1 mN/m = 0.001 N/m, so mercury at 486 dyn/cm is 0.486 N/m; applying the droplet formula (factor 2) to a soap bubble that needs factor 4; using a surface tension value measured at the wrong temperature; and forgetting that gravity deforms droplets larger than the capillary length (~2.7 mm for water), so the spherical formula loses accuracy.
How does temperature change the Laplace pressure?
Surface tension decreases almost linearly with temperature and reaches zero at the critical point. Water falls from about 72 mN/m at 25 °C to roughly 59 mN/m at 100 °C — an 18% drop that translates directly into an 18% lower Laplace pressure for the same droplet size. For accurate work, always look up \( \gamma \) at your working temperature rather than using the room-temperature default.
What are realistic Laplace pressure values, and where do they matter?
A 1 mm water droplet carries only ~144 Pa (≈ 1 mmHg above a millimetre of mercury), but at 10 µm the excess pressure reaches ~14 kPa and at 100 nm about 1.4 MPa. This size scaling matters in lung alveoli (surfactant lowers \( \gamma \) to keep them from collapsing), in emulsions and sprays (small droplets resist coalescence and dissolve faster), in cavitation and boiling, and in microfluidics, where capillary pressures routinely exceed gravity-driven pressures.