Hong Life CalculatorsUnderstanding the Young–Laplace equation calculator — concept and principle
A curved liquid surface acts like a stretched skin, so the pressure on its inner (concave) side is higher than outside. The Young–Laplace equation gives that pressure jump from the surface tension and the radii of curvature. This calculator handles drops and gas bubbles in liquid, soap bubbles, liquid jets and general surfaces, and reports the result in several pressure units.
How it is calculated
General surface: ΔP = γ × (1/R₁ + 1/R₂)
Sphere (drop or bubble in liquid): ΔP = 2γ ÷ R
Soap bubble (two surfaces): ΔP = 4γ ÷ R
Cylinder or jet: ΔP = γ ÷ R
- γ is surface tension, entered in mN/m and converted to N/m; radii are entered in µm and converted to m.
- ΔP is in pascals and also shown in kPa, mmHg (133.322387 Pa), cm of water (98.0665 Pa) and atm (101,325 Pa).
The principle
Molecules at a liquid surface are pulled inward by their neighbours, so the surface tries to shrink, like a stretched membrane. When the surface is curved, that tension has a component pointing toward the centre of curvature, and it must be balanced by a higher pressure inside. A balance of forces on a half sphere gives the classic result: pressure × πR² equals tension × 2πR, so ΔP = 2γ/R. Smaller radii mean larger pressure differences. A soap film has an inner and an outer surface, so its pressure jump is doubled. A cylinder curves in only one direction, so only one radius contributes.
Worked example
Default inputs: spherical drop, γ = 72 mN/m (water near room temperature), R = 100 µm.
- γ = 0.072 N/m; R = 100 × 10⁻⁶ = 0.0001 m.
- Curvature term = 2 ÷ 0.0001 = 20,000 1/m.
- ΔP = 0.072 × 20,000 = 1,440 Pa = 1.44 kPa.
- In other units: 10.8 mmHg, 14.68 cm of water, 0.01421 atm.
| Radius R or R₁ (µm) | Pressure difference ΔP (Pa) |
|---|---|
| 25 | 5,760 |
| 50 | 2,880 |
| 75 | 1,920 |
| 100 | 1,440 |
| 125 | 1,152 |
| 150 | 960 |
| 200 | 720 |
How to read the result
Pressure differences are tiny for large drops but become huge at the micrometre scale: halving the radius doubles ΔP. This is why small bubbles in a foam have higher pressure and gas diffuses from small bubbles into larger ones, why tiny water droplets evaporate more readily and why lungs need surfactant to keep small air sacs from collapsing. For saddle-shaped surfaces, enter radii with opposite signs.
Common mistakes
- Using 4γ/R for a gas bubble inside a liquid; that has only one surface.
- Forgetting unit conversion from mN/m and µm.
- Using pure-water surface tension when soap or contaminants are present; they lower γ considerably.
- Giving a flat direction a radius of zero instead of a very large value.
Frequently asked questions
Why is pressure higher inside a smaller bubble?
Tighter curvature turns more of the surface tension inward, which must be balanced by more pressure.
What is the surface tension of water?
About 72 mN/m near room temperature; it falls as temperature rises.
Why does a soap bubble use 4γ/R?
A soap film has two surfaces, inside and outside, each contributing 2γ/R.
Sources: Young–Laplace equation of capillarity ΔP = γ(1/R₁ + 1/R₂), as in standard physical chemistry and fluid mechanics texts