Surface Tension Calculator — Capillary Rise (Jurin's Law)

Formula

How this is calculated

Surface tension arises because molecules at a liquid surface have fewer neighbours than those in the bulk, giving them higher potential energy. To minimise this energy, a liquid surface behaves like a stretched elastic membrane with a characteristic tension γ (in N/m or J/m²). Where this surface meets a solid wall, the balance between adhesive (solid–liquid) and cohesive (liquid–liquid) forces sets the contact angle θ: θ < 90° means the liquid wets the solid (water on glass); θ > 90° means it is repelled (mercury on glass).

In a narrow capillary tube, the curved meniscus creates a pressure difference across the interface — the Young–Laplace equation gives ΔP = 2γ/r for a spherical meniscus. This suction pulls the liquid upward until the weight of the lifted column exactly balances the pressure: h = 2γ cosθ / (ρgr). This is Jurin's law. Because h is inversely proportional to r, halving the tube radius doubles the rise — tree xylem vessels (r ≈ 10–100 μm) can lift water 10 m using only this mechanism.

The capillary length λ_c = √(γ/ρg) is the characteristic scale at which surface tension and gravity balance; above this scale (typically ~2–3 mm for water), gravity dominates droplet shape. All values here assume a perfectly circular, uniform tube and a static meniscus. Real capillaries are irregular, and wetting hysteresis means the advancing and receding contact angles differ.