Young-Laplace Equation
ΔP = γ(1/R₁ + 1/R₂). Relates surface tension to capillary pressure. Droplet: 2γ/R; bubble: 4γ/R. Governs meniscus, capillary rise.
Why This Chemistry Calculation Matters
Why: Governs pressure in droplets, bubbles, capillary rise. Essential for lung mechanics, inkjet printing, emulsions.
How: General: ΔP = γ(1/R₁ + 1/R₂). Spherical droplet: 2γ/R. Bubble: 4γ/R (two interfaces).
- ●Water γ ≈ 72.8 mN/m at 20°C; soap ~25 mN/m.
- ●Smaller radius → higher internal pressure.
- ●Lung surfactant reduces alveolar surface tension.
Compact Examples
Inputs
⚠️For educational and informational purposes only. Verify with a qualified professional.
🔬 Chemistry Facts
ΔP = γ(1/R₁ + 1/R₂). General curved surface.
— IUPAC
Droplet: ΔP = 2γ/R. Bubble: ΔP = 4γ/R.
— Surface science
Mean curvature H = (1/R₁ + 1/R₂)/2.
— Geometry
Nanobubbles can exceed 2.9 MPa internal pressure.
— NIST
📋 Key Takeaways
- • General | ΔP = γ(1/R₁ + 1/R₂)
- • Droplet | ΔP = 2γ/R (single interface)
- • Bubble | ΔP = 4γ/R (two interfaces)
- • Smaller radius → higher pressure inside
Did You Know?
Soap bubbles have 4γ/R because of inner and outer liquid-air interfaces.
Source: Surface science
A 1 μm water droplet has ~145 kPa pressure inside—1.4× atmospheric.
Source: Capillarity
Lung surfactant reduces alveolar surface tension, preventing collapse.
Source: Physiology
Mean curvature H = (1/R₁ + 1/R₂)/2; Gaussian K = 1/(R₁·R₂).
Source: Differential geometry
Fog droplets (~10 μm) have ~14.5 kPa internal pressure.
Source: Meteorology
Nanobubbles (100 nm) can exceed 2.9 MPa internal pressure.
Source: Nanotechnology
How the Young-Laplace Equation Works
General Form
For a general curved surface with two principal radii of curvature:
ΔP = γ(1/R₁ + 1/R₂)
Spherical Droplet
For a spherical droplet (R₁ = R₂ = R), the equation simplifies to:
ΔP = 2γ/R
The factor of 2 accounts for the single liquid-gas interface.
Spherical Bubble
For a spherical bubble with two surfaces (inner and outer):
ΔP = 4γ/R
The factor of 4 accounts for two liquid-gas interfaces.
Expert Tips
Use SI Units
Convert γ to N/m, R to m for Pa.
Droplet vs Bubble
Bubble has 2× droplet pressure (two surfaces).
Temperature
Surface tension decreases with T.
NIST Data
Consult NIST for γ values.
FAQ
Why 2γ/R for droplet?
Spherical droplet: R₁=R₂=R, so 1/R₁+1/R₂=2/R. Single liquid-gas interface.
Why 4γ/R for bubble?
Two interfaces (inner and outer); each contributes 2γ/R.
Units for γ?
N/m or mN/m. 72.8 mN/m for water at 20°C.
When does it fail?
Very small radii (molecular scale), dynamic interfaces, surfactants.
Applications?
Capillary rise, lung mechanics, inkjet printing, emulsions.
Key Numbers
Formulas and Equations
| Case | Formula | Description |
|---|---|---|
| General Surface | ΔP = γ(1/R₁ + 1/R₂) | Two principal radii of curvature |
| Spherical Droplet | ΔP = 2γ/R | Single liquid-gas interface |
| Spherical Bubble | ΔP = 4γ/R | Two liquid-gas interfaces |
| Mean Curvature | H = (1/R₁ + 1/R₂)/2 | Average curvature |
| Gaussian Curvature | K = 1/(R₁·R₂) | Product of principal curvatures |
Variable Definitions:
- ΔP: Pressure difference (Pa)
- γ: Surface tension (N/m)
- R₁, R₂: Principal radii of curvature (m)
- R: Radius for spherical cases (m)
Common Surface Tension Values
| Liquid | Surface Tension (mN/m) | Temperature (°C) | Category |
|---|---|---|---|
| Water | 72.8 | 20 | Common Liquids |
| Water (37°C) | 69.6 | 37 | Biological |
| Ethanol | 22.3 | 20 | Organic Solvents |
| Methanol | 22.6 | 20 | Organic Solvents |
| Acetone | 23.7 | 20 | Organic Solvents |
| Benzene | 28.9 | 20 | Organic Solvents |
| Toluene | 28.4 | 20 | Organic Solvents |
| Hexane | 18.4 | 20 | Organic Solvents |
| Mercury | 485 | 20 | Metals |
| Glycerol | 63.4 | 20 | Common Liquids |
📚 Official Data Sources
⚠️ Disclaimer: This calculator uses IUPAC surface tension definitions and the Young-Laplace equation. For precise work, consult the IUPAC Gold Book (surface tension), NIST Surface Tension Data, and Adamson's Physical Chemistry of Surfaces. Actual results may vary with temperature, surface purity, and curvature conditions.