Young-Laplace Equation Calculator

Calculate capillary pressure across curved interfaces and capillary rise heights in tubes and porous media. Enter surface tension, contact angle, and pore radius for instant results.

Surface tension of the liquid (water = 72.8 mN/m at 20°C)

Contact angle between liquid and solid surface (0° = perfect wetting, 180° = non-wetting)

Radius of the capillary tube or pore in millimeters

Density of the liquid (needed for height calculation)

Young-Laplace Equation: For a curved interface: ΔP = γ × (1/R₁ + 1/R₂) For a cylindrical capillary: ΔP = 2γ × cos(θ) / r Capillary Rise Height: h = 2γ × cos(θ) / (ρ × g × r) Where: • ΔP = Capillary pressure (Pa) • γ = Surface tension (N/m) • θ = Contact angle (degrees) • r = Pore/tube radius (m) • ρ = Liquid density (kg/m³) • g = Gravitational acceleration (9.81 m/s²) • h = Capillary rise height (m) Wettability Classification: • θ = 0°: Perfect wetting • θ < 90°: Hydrophilic (rising) • θ = 90°: Neutral • θ > 90°: Hydrophobic (depression) • θ = 180°: Perfect non-wetting Unit Conversions: 1 mN/m = 0.001 N/m 1 mm = 0.001 m 1 g/cm³ = 1000 kg/m³
Example 1: Water in Glass Tube Surface tension: 72.8 mN/m (water, 20°C) Contact angle: 20° (water-glass) Tube radius: 0.5 mm Density: 1.0 g/cm³ ΔP = 2 × 0.0728 × cos(20°) / 0.0005 = 273.8 Pa = 2.05 mmHg h = 273.8 / (1000 × 9.81) = 27.9 mm rise (Water rises ~28 mm in a 1 mm diameter tube) Example 2: Mercury in Glass (Non-Wetting) Surface tension: 486 mN/m (mercury) Contact angle: 140° (mercury-glass) Tube radius: 0.5 mm ΔP = 2 × 0.486 × cos(140°) / 0.0005 = -1488 Pa (negative = depression) h = -1488 / (13593 × 9.81) = -11.2 mm (mercury falls 11 mm) Example 3: Effect of Tube Radius Same water-glass (20° contact angle): r = 0.1 mm: h = 139.5 mm r = 0.2 mm: h = 69.8 mm r = 0.5 mm: h = 27.9 mm r = 1.0 mm: h = 13.9 mm r = 5.0 mm: h = 2.8 mm Capillary rise is significant only in tubes smaller than ~2 mm radius. Example 4: Common Surface Tensions • Water (20°C): 72.8 mN/m • Water (100°C): 58.9 mN/m • Ethanol: 22.1 mN/m • Mercury: 486 mN/m • Blood: 58.0 mN/m • Olive oil: 32.0 mN/m Common Contact Angles on Glass: • Water: 20-30° • Glycerol: 15-20° • Mercury: 140° • Hexane: ~0° (spreads) • PDMS: 100-120°

What is the Young-Laplace equation and what does it describe?

The Young-Laplace equation describes the pressure difference (ΔP) across a curved fluid interface, such as the surface of a droplet, bubble, or meniscus in a capillary tube. It states that ΔP = γ(1/R₁ + 1/R₂), where γ is surface tension and R₁, R₂ are principal radii of curvature. For a cylindrical capillary with contact angle θ, this simplifies to ΔP = 2γcosθ/r, governing capillary rise and pore pressure.

How does contact angle affect capillary rise?

The contact angle determines whether a liquid rises or falls in a capillary. For hydrophilic surfaces (θ < 90°), cosθ > 0, producing positive capillary rise - water rises in a glass tube. For hydrophobic surfaces (θ > 90°), cosθ < 0, producing capillary depression - mercury falls in a glass tube. At θ = 90°, cosθ = 0 and there is no capillary action. Superhydrophilic surfaces (θ ≈ 0°) give maximum capillary rise.

What is the practical significance of capillary pressure in porous media?

Capillary pressure in porous media determines how fluids move through materials like soil, rocks, and filters. It controls oil recovery from reservoirs (capillary trapping), water movement in soil (plant water availability), ink penetration in paper, and the performance of porous membranes. Higher surface tension and smaller pores produce stronger capillary forces. This is critical in hydrogeology, petroleum engineering, and materials science.

Why does water rise higher in a narrower tube?

The capillary rise height is inversely proportional to tube radius: h = 2γcosθ/(ρgr). A narrower tube has a more curved meniscus, creating a larger pressure difference that can support a taller column of liquid. For example, water (γ = 72.8 mN/m) in a 0.5 mm glass tube rises about 29.7 mm, but in a 0.1 mm tube it rises about 148.5 mm. This is why water wicks through thin paper towels better than thick ones.

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