Surface Energy & Pressure in Drops and Bubbles
Making new surface costs energy; curved surfaces squeeze what's inside.
Lesson at a glanceShow the key points
- Molecules in the surface have extra potential energy: surface energy. Surface tension = surface energy per unit area (14.14); 1 N/m = 1 J/m².
- A soap film has two surfaces, so pulling a wire of length l through x creates area 2lx and needs work 2Slx.
- Breaking a drop into many droplets increases the total area and so needs energy.
- Inside a spherical drop (or an air bubble in a liquid) the pressure exceeds the outside by 2S/R (14.15): the concave side is at higher pressure.
- A soap bubble has two surfaces: excess pressure 4S/R (14.16). Smaller bubbles have higher pressure.
Blowing a soap bubble takes effort, and it’s hardest right at the start, when the bubble is tiny. Surface tension explains both why making surface costs energy and why small bubbles push back harder.
Surface energy
A molecule in the surface feels a net inward pull (fewer neighbours above it). Bringing a molecule from the interior up to the surface means working against that pull, so surface molecules have extra potential energy. This extra energy of the surface layer is the surface energy. Molecules constantly swap between surface and interior, but the energy cost of having surface remains.
Take the U-frame and sliding wire (length l) again. A soap film, though it looks thin, is hundreds of thousands of molecules thick, so it has two surfaces, and the wire is pulled with . Pull the wire out slowly through x: the work done is , and new surface of area 2lx is created. With no change in kinetic energy, this work is stored as surface energy:
A mercury drop of radius 2.0 mm is split into 8 equal droplets. How much surface energy does this require? (S = 0.465 N/m)
Excess pressure inside a drop
Cut a small spherical drop of radius R into two hemispheres along a diameter and consider the forces on one of them (its surface layer):
- Surface tension from the other half, acting all round the rim of length 2πR, parallel to the axis: .
- The air outside pushes on every bit of the curved surface. By symmetry only components along the axis survive, and each element’s component equals P₀ times the element’s projection on the flat diametric plane. The projections fill the disc of area πR², so .
- The liquid inside, at pressure Pᵢ, pushes outward the same way: .
The same holds for an air bubble inside a liquid: there is one surface, with air on the concave side, so the air inside is at a pressure 2S/R above the surrounding liquid.
Find the excess pressure inside a water droplet of radius 0.10 mm (S = 0.075 N/m).
Excess pressure in a soap bubble
A soap bubble has air both inside and outside, and a film with two surfaces. Let P₀ be the outside air pressure, P′ the pressure in the film and Pᵢ the air pressure inside. The outer surface has the film on its concave side: . The inner surface has the inside air on its concave side: . Adding:
What height of water column would balance the excess pressure inside a soap bubble of radius 1.0 cm? (S = 0.030 N/m)
- 1Surface energy per unit area = surface tension (N/m = J/m²); creating new surface costs energy.
- 2One curved surface: , higher on the concave side (drops, air bubbles in liquid).
- 3Soap bubble (two surfaces): .
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