doPhysics · Fluids
Surface Tension
Why the surface of a liquid pulls like a stretched skin, and what that pull does. Surface tension and surface energy, the excess pressure inside drops and bubbles, bubbles that touch and merge, the angle of contact, and liquids that climb or sink in narrow tubes, between plates and in a wick.
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Why this chapter
Why this chapter exists
A steel needle rests on water. A raindrop is round. Blowing a soap bubble is hardest at the first puff. Oil climbs a cotton wick to the flame of a diya, and soap helps water wash clothes. Each of these comes from one fact: a molecule at the surface of a liquid is pulled inward by its neighbours. So a liquid surface pulls like a stretched skin, with a force per unit length that is also its energy per unit area. This chapter measures that pull and follows it into drops, bubbles and narrow tubes.
You will learn
By the end of this chapter you will
- Explain surface tension from the pull of molecules at a surface, and use \(T = F/L\) for films, needles and rings.
- Say how temperature and soap change surface tension, and why oil spreads on water.
- Show that surface tension is the surface energy per unit area, and find the energy to blow a bubble or to break up a drop.
- Derive the excess pressure \(2T/r\) in a drop and \(4T/r\) in a soap bubble, and find the pressure in an air bubble under water.
- Predict what happens to two bubbles joined by a tube. Find the radius of the wall between two bubbles, and the size of two bubbles merged in a vacuum.
- Define the angle of contact, and derive and use Jurin's law \(h = 2T\cos\theta/\rho g r\), for a rise and for the depression of mercury.
- Find the rise in a short tube, a tilted tube and a tube in a lift. Find it between plates, in a ring-shaped gap and between plates meeting at an angle.
Before you begin
Before you begin
From the chapter on Pressure in Fluids, you should know how pressure grows with depth, \(P = P_0 + \rho g h\). You should also know that points at one level of a liquid at rest share one pressure. From the chapters on Newton's laws, you should be able to balance the forces on a body at rest. You should know that work is force times distance, and that work done is stored as energy.
Lessons
1
Why a Surface Pulls
A molecule at the surface of a liquid is pulled inward, so the surface behaves like a stretched skin. Surface tension is the pull per unit length along any line in the surface. It holds up a needle and a pond skater, it lifts a ring, and heat and soap make it weaker.
Molecular origin of surface tensionSurface tension as force per lengthWhat changes surface tension
2
Surface Energy
Making new surface takes work, and the work per unit area is the surface tension itself. The energy to blow a soap bubble, to break a drop into many, and the energy set free when drops or bubbles merge all follow from \(W = T\,\Delta A\).
Surface energy and surface tensionEnergy to blow a bubbleSplitting and merging drops
3
Excess Pressure in Drops and Bubbles
A curved surface squeezes what it holds, so the pressure inside a drop is higher than outside by 2T/r, and inside a soap bubble, with two surfaces, by 4T/r. The smaller the drop, the larger the step. An air bubble under water adds this step to the pressure of the water around it.
Excess pressure in a dropExcess pressure in a soap bubbleAir bubble under water
4
Bubbles in Contact
Two soap bubbles joined by a tube do not share their air evenly: the small one, at the higher pressure, empties into the large one. Two bubbles pressed together share a wall that bulges into the larger, with radius \(r_1r_2/(r_2 - r_1)\). Bubbles that merge in a vacuum keep their total surface, \(R^2 = r_1^2 + r_2^2\).
Two bubbles joined by a tubeThe common wall of two bubblesBubbles merging
5
Contact Angle and Capillary Rise
Where a liquid meets a solid, cohesion and adhesion decide the angle of contact. Water wets glass and climbs a narrow tube; mercury does not wet glass and is pushed down. The height is \(h = 2T\cos\theta/\rho gr\), found from the pull of the rim and again from the pressure under the curved surface.
Angle of contactCapillary riseCapillary depression of mercury
6
Capillarity in Odd Shapes
A tube too short for the rise does not overflow; its meniscus flattens instead. A tilted tube fills to the same vertical height, and a tube in a lift feels \(g + a\). Between two plates a gap \(d\) apart, water rises half as high as in a tube of diameter \(d\), and between plates meeting at a small angle its edge is a hyperbola, as in a wick or a sheet of blotting paper.
A tube too shortTilted tubes and a tube in a liftRise between plates and in a ringPlates meeting at a small angle
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- adhesion
- The attraction between molecules of two different substances, such as between water and glass. When it is stronger than cohesion, a liquid wets the solid.
- angle of contact
- The angle between a solid surface and the tangent to a liquid surface where the two meet, measured through the liquid. Water on clean glass has nearly \(0^\circ\); mercury on glass about \(135^\circ\).
- capillary
- A tube with a very narrow bore, often less than a millimetre across. The word comes from the Latin for a hair.
- capillary depression
- The fall of a liquid that does not wet a narrow tube, such as mercury in glass, below the level outside. Jurin's law gives a negative \(h\).
- capillary rise
- The rise of a liquid that wets a narrow tube above the level of the liquid outside, to the height \(h = 2T\cos\theta/\rho g r\) given by Jurin's law.
- cohesion
- The attraction between molecules of the same substance, such as between the molecules of water. It acts only over a very short distance, about a nanometre.
- excess pressure
- The amount by which the pressure on the hollow side of a curved liquid surface exceeds the pressure on its other side. It is \(2T/r\) for a drop and \(4T/r\) for a soap bubble.
- Jurin's law
- The rise of a liquid in a capillary is inversely proportional to the radius of the tube, \(h = 2T\cos\theta/\rho g r\). So \(h r\) is the same for every tube of one material in one liquid.
- meniscus
- The curved surface of a liquid in a narrow tube. It is concave, hollow on top, for a liquid that wets the tube, and convex for one that does not.
- surface energy
- The extra energy stored in the surface of a liquid, because each surface molecule has been brought up against the inward pull of its neighbours. Per unit area, it equals the surface tension.
- surface tension
- The force per unit length across any line drawn in the surface of a liquid, \(T = F/L\). It acts along the surface, at right angles to the line. It is also the surface energy per unit area. Water has about \(0.07\,\text{N/m}\).