Angle of contact and capillary rise
A liquid rises in a narrow tube if it wets the wall (θ below 90°), and falls if not. The rise is h = 2T cos θ / (rρg).
Meniscus and angle of contact
Look at water in a glass tube. Its surface curves up at the walls. Mercury in glass curves down. This curved surface is the meniscus. To describe it we use the angle of contact (θ). It is the angle between the tangent to the liquid surface and the wall where they meet, measured inside the liquid. It depends on the liquid, the solid wall and the medium above the liquid. Capillarity (capillary action) is the rise or fall of a liquid in a capillary tube. This is a tube with a bore as fine as a hair.
What decides the shape of the meniscus
| Forces at the wall | Angle of contact | Meniscus | In a capillary | Example |
|---|---|---|---|---|
| Adhesion stronger than cohesion (your lesson: Fa > Fc sin θ) | Acute (less than 90°) | Concave (curves up) | Liquid rises | Water in glass |
| Cohesion stronger than adhesion (Fa < Fc sin θ) | Obtuse (more than 90°) | Convex (curves down) | Liquid falls | Mercury in glass; water in a paraffin wax tube |
| Balanced (Fa = Fc sin θ) | 90° | Flat | No rise or fall | Water in a silver tube (nearly) |
Derivation: height of capillary rise
Dip a glass capillary tube of radius r in a liquid of density ρ and surface tension T. The meniscus is concave, with radius of curvature R.
- 1Across a curved surface, the pressure on the concave side is more by 2T/R. Here the concave side faces up, so just below the meniscus the pressure is 2T/R less than atmospheric.
- 2Outside the tube, the flat surface is at atmospheric pressure. At the same level, the pressure inside the tube is lower, so liquid is pushed up the tube.
- 3The liquid rises until the column of height h makes up the shortfall: hρg = 2T/R.
- 4From the geometry of the meniscus, R = r/cos θ.
- 5Substitute: hρg = 2T cos θ / r.
- 6★So, h = 2T cos θ / (rρg). A finer tube gives a higher rise (h ∝ 1/r). If θ is more than 90°, cos θ is negative and the liquid falls, as mercury does.
★ key step
How to draw capillary rise and fall
- 1Draw a beaker of water. Dip a thin glass tube in it.
- 2Show the water inside the tube higher than outside, with a concave meniscus curving up at the walls. Mark the height h between the two levels.
- 3At the wall, draw the tangent to the meniscus and mark the acute angle θ inside the water.
- 4Beside it, draw a beaker of mercury with a glass tube. Show the mercury inside lower than outside, with a convex meniscus.
- 5Mark the obtuse angle θ inside the mercury, and label the two figures rise and fall.
Worked example: rise of water in a capillary
A glass capillary tube of radius 0.50 mm stands in water. How high does the water rise? (T = 7.2 × 10⁻² N m⁻¹, θ = 0°, ρ = 1000 kg m⁻³, g = 9.8 m s⁻²)
- r = 0.50 mm = 5.0 × 10⁻⁴ m; cos 0° = 1.
- h = 2T cos θ / (rρg).
- h = (2 × 7.2 × 10⁻² × 1) ÷ (5.0 × 10⁻⁴ × 1000 × 9.8) m.
- h = 0.144 ÷ 4.9 m = 0.0294 m.
About 2.9 cm. A tube of half this radius would give twice the rise, about 5.9 cm
Capillarity around you
- Blotting paper soaks up ink through the fine gaps between its fibres.
- A cotton wick in an oil diya or a kerosene lantern keeps drawing up oil to the flame.
- A towel, or the wet hem of a cloth, soaks water upward.
- Capillary action helps water climb up the fine channels in plant stems.
- Farmers plough after rain. This breaks the fine channels in the top soil, so water stays in the ground instead of rising and drying off.
- A tube shorter than h does not overflow. The liquid reaches the top and stops; its meniscus flattens (R grows) until hρg = 2T/R again.
- Filling a thermometer with mercury is hard, because mercury is pushed down in a fine glass tube.
A cotton wick in an oil diya or a kerosene lantern keeps drawing oil up to the flame by capillary action. Farmers plough fields after the rains to break the fine soil channels, so water stays in the soil instead of rising and drying off.
3 marks: “Derive an expression for the height to which a liquid rises in a capillary tube.” Pressure just below the concave meniscus is 2T/R less than above it. The liquid rises until hρg = 2T/R. With R = r/cos θ, h = 2T cos θ / (rρg).
2 marks: “Water rises in a glass capillary tube of radius 0.10 mm. Find the height of the rise. (T = 7.2 × 10⁻² N m⁻¹, θ = 0°, ρ = 1000 kg m⁻³, g = 9.8 m s⁻²)” h = 2T/(rρg) = 0.144 ÷ (1.0 × 10⁻⁴ × 1000 × 9.8) = 0.144 ÷ 0.98 ≈ 0.147 m, about 14.7 cm.
1 marks: “Why does mercury fall in a glass capillary tube?” Its angle of contact with glass is obtuse, so cos θ is negative and h comes out negative: the level falls.
Thin tube, tall rise: h ∝ 1/r. Wets the wall (θ less than 90°), it rises; won't wet (θ more than 90°), it falls.