What this calculator does
A curved liquid surface sustains a pressure difference, with the inside of the curve at higher pressure. For a sphere the relationship is twice the surface tension divided by the radius, so a 1 mm water droplet holds about 146 Pa above ambient.
Smaller means higher, and steeply so. Shrink that droplet to 0.1 mm and the excess pressure rises tenfold to 1,456 Pa. This is why small bubbles are harder to create than large ones, why capillary action lifts water in narrow tubes, and why a lung needs surfactant to stop its smallest alveoli collapsing into the larger ones.
The formula
The pressure difference is the surface tension multiplied by the sum of the reciprocals of the two principal radii of curvature. For a sphere both radii are equal and the expression reduces to twice the tension over the radius. For a cylinder one radius is effectively infinite, so its reciprocal vanishes and the pressure is half the spherical value.
| Term | Meaning |
|---|---|
| Surface tension (γ) | Energy per unit area of interface. Water at 20 °C is 0.0728 N/m. |
| Principal radii | The two curvature radii of the surface. Equal for a sphere. |
| Laplace pressure | The pressure jump across the interface. |
| Capillary length | The scale at which surface tension and gravity balance, about 2.7 mm for water. |
The inputs explained
| Field | What to enter |
|---|---|
| Surface tension γ (N/m) | Surface tension in N/m. Water is 0.0728; most organic liquids are 0.02 to 0.03. |
| First principal radius R₁ (m) | First principal radius in metres. |
| Second principal radius R₂ (= R₁ for a sphere) (m) | Second principal radius. Set equal to the first for a sphere, or very large for a cylinder. |
When to use it
Understanding capillary rise
The curvature of a meniscus in a narrow tube generates the pressure that lifts the liquid column.
Bubble and foam behaviour
Small bubbles have higher internal pressure, which drives gas into larger ones and coarsens a foam over time.
Lung mechanics
Surfactant lowers surface tension in the smallest alveoli, preventing them emptying into larger neighbours.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does droplet size change the pressure?
A range of droplet radii.
| Radius (both principal radii equal) | ΔP (Pa) | ΔP (kPa) | Shape |
|---|---|---|---|
| 0.1 mm | 800.80 Pa | 0.8008 kPa | general curved interface |
| 1 mm | 145.60 Pa | 0.1456 kPa | sphere (droplet/bubble interface) |
| 10 mm | 80.08 Pa | 0.0801 kPa | general curved interface |
Questions
Why do small bubbles have higher pressure?
Because the pressure difference is inversely proportional to radius. A tighter curve means the surface tension pulls inward more effectively over a smaller area. Halving the radius doubles the excess pressure, which is why blowing the first small bubble is the hard part.
What is the difference between a droplet and a soap bubble?
A soap bubble has two surfaces, inner and outer, so its pressure difference is four times the tension over the radius rather than twice. A liquid droplet or a bubble inside a liquid has only one interface and uses the factor of two.
How does this relate to capillary action?
A liquid wetting a narrow tube forms a curved meniscus, and the Laplace pressure across it pulls the column upward until gravity balances it. The narrower the tube, the tighter the curve and the higher the rise, which is the same inverse relationship.
Why does the lung need surfactant?
Without it, the smallest alveoli would have the highest internal pressure and empty into larger ones, collapsing the lung. Surfactant reduces surface tension more strongly as an alveolus shrinks, which stabilises the small ones against exactly that.
For particle diffusion in a fluid, see the diffusion coefficient calculator. For osmotic pressure across a membrane, see the osmotic pressure calculator.