Capillary Rise Formula (Surface Tension)
Calculate capillary rise height using h = 2 gamma cos(theta) / (rho g r).
Explains meniscus, wicking, and liquid behavior in narrow tubes.
The Formula
The capillary rise formula, also known as Jurin's law, describes how high a liquid climbs inside a narrow tube due to surface tension. When you dip a thin glass tube into water, the water rises above the surrounding level because the attractive forces between water molecules and glass pull the liquid upward. The thinner the tube, the higher the liquid climbs.
Surface tension is a property of liquids that causes their surface to behave like a stretched elastic membrane. It arises because molecules at the surface experience a net inward pull from neighboring molecules, while molecules deeper in the liquid are pulled equally in all directions. This inward pull creates a minimum-surface-area tendency that manifests as surface tension, measured in force per unit length (N/m).
The contact angle θ (theta) describes how the liquid meets the tube wall. For water in a glass tube, the contact angle is nearly 0 degrees because water is strongly attracted to glass (hydrophilic). This creates a concave meniscus that curves upward at the edges. For mercury in glass, the contact angle is about 140 degrees because mercury does not wet glass. This creates a convex meniscus and the mercury actually depresses below the outside level (capillary depression).
The formula balances the upward force of surface tension against the downward weight of the liquid column. The surface tension acts along the circumference of the tube (2πr), pulling upward with force 2πrγcos(θ). The weight of the raised liquid column is ρgπr²h. Setting these equal and solving for h gives the capillary rise formula.
Capillary action is essential in nature and technology. It draws water from soil into plant roots, moves ink through pen nibs, drives liquid through wicks in candles and oil lamps, and causes moisture to seep into porous building materials. Understanding capillary forces is also crucial in microfluidics, where scientists manipulate tiny volumes of liquid in channels often narrower than a human hair.
Variables
| Symbol | Meaning |
|---|---|
| h | Height of capillary rise (meters, m) |
| γ | Surface tension of the liquid (N/m or J/m²) |
| θ | Contact angle between liquid and tube wall (degrees) |
| ρ | Density of the liquid (kg/m³) |
| g | Acceleration due to gravity (9.81 m/s²) |
| r | Inner radius of the capillary tube (meters, m) |
Common Surface Tension Values (at 20 degrees C)
| Liquid | Surface Tension (mN/m) | Contact Angle on Glass |
|---|---|---|
| Water | 72.8 | ~0° |
| Ethanol | 22.1 | ~0° |
| Mercury | 485 | ~140° |
| Acetone | 25.2 | ~0° |
Example 1
Water (γ = 0.0728 N/m, ρ = 1000 kg/m³, θ ≈ 0°) is placed in a glass tube with inner radius 0.5 mm. How high does the water rise?
h = 2γcos(θ) / (ρgr)
h = 2 × 0.0728 × cos(0°) / (1000 × 9.81 × 0.0005)
h = 0.1456 / 4.905
h ≈ 0.0297 m = 29.7 mm (about 3 cm)
Example 2
Mercury (γ = 0.485 N/m, ρ = 13,534 kg/m³, θ = 140°) is in a glass tube with radius 1 mm. What is the capillary depression?
h = 2 × 0.485 × cos(140°) / (13,534 × 9.81 × 0.001)
cos(140°) = -0.766
h = 2 × 0.485 × (-0.766) / (132.77)
h = -0.743 / 132.77
h ≈ -0.0056 m = -5.6 mm (negative means depression below surface level)
When to Use It
Use the capillary rise formula when you need to:
- Predict how high water rises in soil pores or narrow cracks
- Design microfluidic devices and lab-on-a-chip systems
- Understand moisture movement in building materials and textiles
- Calculate mercury depression in barometer tubes for correction
- Determine wick performance in heat pipes and candles
The formula assumes a perfectly cylindrical tube and uniform contact angle. For non-circular cross-sections, replace r with the hydraulic radius (2 × area / perimeter).