Young–Laplace Equation Calculator

Compute Laplace pressure jump across a fluid interface from surface tension and principal radii of curvature. Visualize droplets, bubbles, and cylindrical menisci.

Typical: water 0.072, ethanol 0.022, mercury 0.486
1 mm = 0.001 m, 1 µm = 1e-6 m
? Water droplet (R=1mm, γ=0.072)
? Soap bubble (R=2cm, γ=0.072)
? Lung alveolus (R=100µm, γ=0.025)
?️ Mercury droplet (R=0.5mm, γ=0.486)
? Cylindrical meniscus (R=0.5mm)
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The Young–Laplace Equation: Foundation of Capillarity

The Young–Laplace equation describes the pressure difference across a curved fluid interface due to surface tension. Proposed by Thomas Young (1805) and formalized by Pierre-Simon Laplace (1806), it is central to understanding droplets, bubbles, capillary rise, and wetting phenomena. It states: ΔP = γ (1/R₁ + 1/R₂), where γ is surface tension, and R₁, R₂ are principal radii of curvature.

ΔP = γ · (κ₁ + κ₂) = γ · ( 1⁄R₁ + 1⁄R₂ )

For a spherical droplet (R₁ = R₂ = R): ΔP = 2γ/R. A soap bubble has two liquid-air interfaces, thus ΔP = 4γ/R. Cylindrical interfaces (e.g., liquid in a narrow tube with cylindrical meniscus) obey ΔP = γ/R. This pressure jump explains why small droplets evaporate faster (higher internal pressure) and why bubbles coalesce.

? Real‑world Applications & Evidence

  • Pulmonary surfactant: In lung alveoli (radius ~100 µm), Laplace pressure would cause collapse without surfactant, which reduces γ from ~0.05 to 0.025 N/m, stabilising breathing.
  • Inkjet printing: Droplet formation depends on precise Laplace pressure to eject picolitre drops.
  • Microfluidics & Lab-on-chip: Capillary pressure drives fluid in channels – design relies on curvature control.
  • Emulsion stability: Laplace pressure gradients induce Ostwald ripening.
  • Plant hydraulics: Negative pressures (tension) in xylem follow similar curvature relations.
Case Study: Premature Infant Respiratory Distress Syndrome

Infant alveoli lack mature surfactant → high surface tension (≈0.05 N/m) and small radii (~50-100 µm) → Laplace pressure ΔP = 2γ/R up to 2000 Pa, causing alveolar collapse. Exogenous surfactant therapy lowers γ to ~0.025 N/m, reducing ΔP by 50% and enabling stable inflation. This direct application of the Young-Laplace equation saves lives. Our calculator models the pressure jump: for R=80 µm, γ=0.025 → ΔP=625 Pa vs γ=0.05 → ΔP=1250 Pa.

? Derivation & Theoretical Insights

The equation emerges from balancing mechanical equilibrium: surface tension forces integrated over curvature generate a net pressure difference. For a spherical cap, force balance gives ΔP·πR² = γ·2πR → ΔP = 2γ/R. Generalisation to any surface uses differential geometry: mean curvature H = (1/R₁+1/R₂)/2, so ΔP = 2γH. The equation is fundamental for contact angle measurements, capillary rise (Jurín's law), and the shape of pendant drops.

? Surface Tension Reference Table (20°C)

Liquid–Gas γ (N/m) Typical application
Water–air 0.0720 General capillary, droplets
Ethanol–air 0.0223 Low surface tension, wetting
Mercury–air 0.486 High cohesion, non‑wetting
Blood plasma–air 0.055 Biomedical relevance
Olive oil–air 0.032 Food science

⚙️ How to Use the Calculator Effectively

  1. Select interface geometry: sphere (droplet/bubble), soap bubble (thin film), cylinder, or general two radii.
  2. Enter surface tension γ in N/m (default water‑air).
  3. Provide radius (or radii) in meters – use scientific notation for micro/nano scales (1e-6).
  4. Click "Compute Laplace Pressure" – the pressure jump ΔP is displayed along with mean curvature.
  5. Observe the dynamic sketch: droplet size, bubble inflation visualisation, or cylindrical meniscus.
  6. Use preset examples to explore realistic physical scenarios.

Frequently Asked Questions

Surface tension pulls the interface inward, compressing the interior, leading to excess Laplace pressure. For a convex interface (droplet), interior pressure > exterior. For a bubble (concave from gas side), the pressure is higher inside the gas phase.

For saddle shapes (negative Gaussian curvature), principal radii can have opposite signs, reducing net ΔP. Our calculator assumes positive radii for ordinary droplets/bubbles; for advanced applications, enter positive values for convex curvature.

Down to ~10 nm, the Young-Laplace equation holds with modified Tolman length corrections. For typical microscale and above, it’s experimentally validated to high precision.

Yes, rearrange R = 2γ/ΔP for a sphere; you can manually compute using the results. We plan a dedicated solver in future updates.
Scientific references & credibility: Based on classical works: Young, T. (1805) "An Essay on the Cohesion of Fluids", Laplace, P.S. (1806) "Mécanique Céleste". Modern validation: de Gennes, Brochard-Wyart, Quéré "Capillarity and Wetting Phenomena" (2004). Data complies with NIST and IUPAC standards. 
? Last content review: April , 2026 | Based on standard Young-Laplace equation, quarterly review of physical constants and formulas.
? External references: NIST Surface Tension Database · IUPAC Gold Book – Laplace Pressure