§ 40. Surface Tension, Meniscus Curvature & Capillary Action (Grade 10)
Laplace pressure jump ΔP = 2σ cos θ / r under a concave or convex meniscus and equilibrium column height h. A narrow capillary tube makes visible the molecular cohesive forces capable of lifting a liquid column against gravity.
Theory Summary & Scientific Mechanism
Molecules at a liquid–gas interface experience an inward cohesive pull from bulk neighbors, endowing the surface with excess free energy Eₛ = σ S characterized by surface tension σ (N/m). When liquid–solid adhesion exceeds liquid–liquid cohesion (wetting, contact angle θ 90^°, such as mercury in glass), the meniscus is convex.
In a cylindrical capillary tube of inner radius r, the spherical meniscus of radius of curvature Rₘ = (r)/(cos θ) sustains a pressure jump across the interface given by the Young–Laplace equation: Δ p = (2σ)/(Rₘ) = (2σcos θ)/(r).
Equating this Laplace capillary suction to the hydrostatic pressure ρ g h of the liquid column yields Jurin’s Law: h = (2σcos θ)/(ρ g r) (positive rise for θ 90^°). Because h ∝ 1/r, narrower capillaries draw liquids to dramatically greater heights, driving sap transport in plants and wicking in porous media.
Everyday Analogy
How does a paper towel soak up spilled tea, or a tall tree draw water to its leaves? Inside hair-thin capillary tubes, water clings to the walls and surface tension pulls the liquid column upward against gravity!
Interactive Question & Derivation
What happens to the liquid level inside a narrow glass capillary if the liquid does not wet the walls (contact angle θ > 90^°, e.g., mercury on glass)?
- A: A convex meniscus forms, Laplace pressure points downward (cos θ < 0), and the liquid level inside the capillary drops below the reservoir level (h < 0)
- B: The liquid rises even higher than for complete wetting
- C: The capillary level remains equal to the wide vessel level
Study Hint: By Jurin’s law h = (2σcos θ)/(ρ g r): how does capillary rise height h change when tube radius r becomes narrower?
Proof & Explanation: By Jurin's formula h = (2σcos θ)/(ρ g r), the sign of h is governed by cos θ. For wetting liquids (θ 0), the concave meniscus pulls the column upward (h > 0), whereas for non-wetting liquids (θ > 90^°, cos θ < 0), the convex meniscus depresses the column downward (h < 0).
Key Equations & 3D Simulation Parameters
h = (2σ cos θ)/(ρ g r), Δ P = (2σ cos θ)/(r)
- Surface tension coefficient σ: 25 … 120 mN/m (default: 72.8 mN/m)
- Contact wetting angle θ: 0 … 140 ° (default: 20 °)
- Inner capillary radius r: 0.3 … 1.5 mm (default: 0.6 mm)
- Liquid density ρ: 0.7 … 2.5 g/cm³ (default: 1 g/cm³)