Surface tension, surface energy, drops and bubbles
A liquid surface acts like a stretched skin. Surface tension T = F/L equals surface energy per unit area, and gives excess pressure 2T/r inside a drop.
Cohesion, adhesion and the stretched skin
Cohesion is the pull between molecules of the same substance. Adhesion is the pull between molecules of different substances; glue sticks by strong adhesion. A molecule deep inside a liquid is pulled equally from all sides. A molecule at the surface has liquid only below it, so it feels a net inward pull. Work must be done against this pull to bring a molecule up to the surface. So the surface layer stores extra energy, called surface energy. A system settles where its energy is least, so a liquid surface keeps its area as small as possible. The surface then behaves like a stretched membrane. This property is surface tension. It lets a steel needle, laid gently, float on water.
Derivation: surface tension equals surface energy per unit area
Take a rectangular wire frame with one side, of length L, free to slide. Dip it in soap solution to form a film on it.
- 1The film has two surfaces (front and back), and each pulls the sliding wire with force TL. Total pull F = 2TL.
- 2Pull the wire out slowly by a small distance Δx, against this force.
- 3Work done W = F × Δx = 2TL × Δx.
- 4The increase in area of both surfaces together is A = 2L × Δx.
- 5So W = T × A. This work is stored as surface energy of the new surface.
- 6★So, T = W/A: surface tension equals the surface energy per unit area, the work done to increase the area by one unit.
★ key step
Excess pressure inside curved surfaces
| Case | Number of surfaces | Excess pressure (inside minus outside) |
|---|---|---|
| Liquid drop in air, such as a raindrop | 1 (outer surface) | P = 2T/r |
| Air bubble inside a liquid | 1 (inner surface) | P = 2T/r |
| Soap bubble floating in air | 2 (outer and inner) | P = 4T/r |
Derivation: excess pressure inside a liquid drop
A curved surface pulls towards its centre of curvature, so the pressure on its **concave (inner) side is always more**. Let a drop of radius r have excess pressure P = P(inside) − P(outside).
- 1Let the excess pressure P push the surface out, so the radius grows from r to r + Δr.
- 2Increase in area: ΔA = 4π(r + Δr)² − 4πr² ≈ 8πrΔr (the tiny Δr² term is dropped).
- 3Extra surface energy = T × ΔA = T × 8πrΔr.
- 4Work done by the excess pressure = P × increase in volume = P × 4πr²Δr.
- 5These are equal: P × 4πr²Δr = T × 8πrΔr.
- 6★So, P = 2T/r. A soap bubble has two surfaces, so for it P = 4T/r. The smaller the drop or bubble, the larger the excess pressure.
★ key step
Worked example: excess pressure in a drop and a bubble
Find the excess pressure inside (a) a water drop of radius 2.0 mm, (b) a soap bubble of radius 2.0 mm. (T of water = 7.2 × 10⁻² N m⁻¹, T of soap solution = 2.5 × 10⁻² N m⁻¹)
- r = 2.0 mm = 2.0 × 10⁻³ m.
- (a) Water drop: P = 2T/r = (2 × 7.2 × 10⁻² N m⁻¹) ÷ (2.0 × 10⁻³ m) = 72 N m⁻².
- (b) Soap bubble: P = 4T/r = (4 × 2.5 × 10⁻² N m⁻¹) ÷ (2.0 × 10⁻³ m) = 50 N m⁻².
(a) 72 Pa; (b) 50 Pa. The soap bubble has two surfaces, but soap solution has a much lower surface tension
Surface tension in daily life, and how detergents work
- Drops are round: for a given volume, a sphere has the least surface area.
- Mosquitoes stand on still water: the surface dips under each leg. The upward part of surface tension along the edge holds up the insect's weight.
- Detergents lower the surface tension of water. Each detergent molecule has one end that is attracted to water and one end that is attracted to oil.
- The oil-loving ends grip greasy dirt and the water-loving ends face the water. The dirt is surrounded and lifts off when the cloth is squeezed or rinsed.
- A detergent solution has a lower surface tension than a soap solution, so detergents clean oily clothes better.
- Camphor dissolving near a toy wax duck lowers the surface tension on one side. The stronger pull on the other side moves the duck around.
- Try it: lay a loop of thread on a soap film in a wire ring. Prick the film inside the loop. The loop jumps into a circle, because the outer film pulls on it equally all round to make its own area least.
Detergent powder lowers the surface tension of water, so the water soaks into cloth fibres and lifts off oily dirt. Raindrops and drops from a tap are round because surface tension pulls them into the smallest surface.
5 marks: “Define surface tension and surface energy. Show that surface tension is numerically equal to the surface energy per unit area. Write its SI unit and dimensions.” T = F/L, force per unit length on a line in the surface. Surface energy: extra energy of the surface molecules. Frame with sliding wire L: the film has two surfaces, F = 2TL. Pull by Δx: W = 2TLΔx and new area A = 2LΔx, so W = TA and T = W/A. Unit N m⁻¹ (= J m⁻²), dimensions [MT⁻²].
2 marks: “Two soap bubbles have radii 1 cm and 2 cm. Find the ratio of the excess pressures inside them.” P = 4T/r, so P ∝ 1/r. P₁ : P₂ = r₂ : r₁ = 2 : 1. The smaller bubble has twice the excess pressure.
2 marks: “Why do detergents clean oily clothes better than plain water?” Detergents lower the surface tension of water, so it enters between the fibres. Their molecules hold oil at one end and water at the other, surrounding the dirt so it washes away.
One skin, 2T/r (a drop, or a bubble in water). Two skins, 4T/r (a soap bubble in air). Smaller r, bigger push.