All calculatorsYoung–Laplace Equation Calculator
Calculate the pressure difference (ΔP) across a curved interface using surface tension and curvature. Choose General (R₁ & R₂), or use the quick Spherical and Cylindrical cases. Results are unit-aware and include optional step-by-step.
Background
The Young–Laplace equation connects curvature to a pressure jump across an interface: smaller radii (more curved surface) produce a larger ΔP. This explains why tiny droplets/bubbles can have surprisingly large internal pressure.
How to use this calculator
- Choose a mode: General, Spherical, or Cylindrical.
- Enter surface tension γ and the required radius/radii.
- Pick your output unit (Pa, kPa, atm, etc.).
- Click Calculate to get ΔP and (optionally) the step-by-step.
How this calculator works
- Convert γ into N/m and radii into meters.
- Compute curvature: (1/R₁ + 1/R₂) (General) or (2/R) (Sphere) or (1/R) (Cylinder).
- Compute pressure jump: ΔP = γ × curvature.
- Convert ΔP from Pa into the output unit you selected.
Formula & Equation Used
General: ΔP = γ(1/R₁ + 1/R₂)
Sphere: ΔP = 2γ/R
Cylinder: ΔP = γ/R
Unit check: (N/m) × (1/m) = N/m² = Pa.
Example Problems & Step-by-Step Solutions
Example 1 — Spherical droplet
Water: γ = 72 mN/m, droplet radius R = 1 mm.
- Convert γ: 72 mN/m = 0.072 N/m.
- Convert R: 1 mm = 0.001 m.
- Compute ΔP = 2γ/R = 2(0.072)/0.001 = 144 Pa.
Example 2 — General curvature
γ = 50 mN/m, R₁ = 2 mm, R₂ = 6 mm.
- Convert γ: 0.050 N/m.
- Convert radii: 0.002 m and 0.006 m.
- Curvature sum: 1/0.002 + 1/0.006 = 500 + 166.7 = 666.7 1/m.
- ΔP = 0.050 × 666.7 ≈ 33.3 Pa.
Frequently Asked Questions
Q: What does ΔP mean here?
ΔP is the pressure difference across the interface caused by surface tension and curvature. More curvature (smaller radius) means a larger ΔP.
Q: Why does the spherical formula have 2γ/R?
A sphere has two equal principal curvatures: 1/R and 1/R, so (1/R₁ + 1/R₂) becomes 2/R.
Q: What radius should I use?
Use the interface’s radius of curvature at the point of interest. For ideal droplets/bubbles, that’s the droplet radius.
Q: Why is my ΔP huge for tiny radii?
Because curvature scales like 1/R. Micro-scale radii can produce large pressure differences—double-check units (mm vs µm).
Q: Does this include contact angle or gravity?
Not directly. This calculator uses the core Young–Laplace relationship. Real menisci can require geometry + boundary conditions to determine R₁ and R₂.
