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Surface Tension

The fact that there is an extra energy associated with a surface in liquids means that creating more surface keeping other things like volume fixed costs more energy. To appreciate this, consider a horizontal liquid film ending in bar free to slide over parallel guides. This means (at equilibrium) that there is a force (F) due to the liquid acting on the bar equal and opposite to the applied force. This force is proportional to the surface tension of the liquid. We now relate this force to the extra energy of the liquid surface.

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Figure (i) - Stretching a film. (a) A film in equilibrium;
 (b) The film stretched an extra distance

Suppose at equilibrium, we move the bar by a small distance d as shown. The work done by the applied force is F.d= Fd. From conservation of energy, this is the additional energy the film now has. If the surface energy of the film is S per unit area, the extra area is 2 d l. A film has two sides and liquid in between, so there are two surfaces and the extra energy is
    S(2 d l) = Fd
        F/2 l = S
The quantity (F/2l) or S is the magnitude of surface tension. Clearly it is equal to the surface energy per unit area of the liquid interface, and is property of the interface between the liquid and something else (air, solid, another liquid).
We make the following observations
  1. Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of the interface between the liquid and any other substance; it is the extra energy that the molecules at the interface have as compared to molecules in the interior.
  2. Surface tension acts perpendicular to any line bounding this interface and acts inward, i.e., towards the liquid; thus tending to reduce the interface area.
  3. At any point on the interface besides the boundary, we can draw a line and imagine equal and opposite surface tension forces S per unit length of the line acting perpendicular to the line, in the plane of the interface. The line is in equilibrium. To be more specific, imagine a line of atoms or molecules at the surface. The atoms to the left pull this line of atoms towards them; those to the right pull it towards them! This line of atoms is in equilibrium under tension.
  4. Table gives the surface tension of various liquids. The value of the surface tension depends on the temperature. Like viscosity, the surface tension of a liquid usually falls with temperature.
Table: Surface tension of some liquids at the temperature indicated; heats of vapourization of the same liquids




Surface Tension(N/m)

Heat of Vapourization





















If the line really marks an end of the interface, as in Figs. 10.16 (a) and (b) there is only the force S per unit length acting inwards.
The force of tension is quite real, and can be directly measured by the vertical version of Fig.10.16 depicted in Fig. 10.17. A flat vertical glass plate, below which a vessel of some liquid is kept, forms one arm of a balance. The plate is balanced by weights on the other side, with its horizontal edge just above water. The vessel is raised slightly till the liquid just touches the glass plate and pulls it down a little because of surface tension. Weights are added till the plate just clears water.

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Figure (ii) - Measuring Surface Tension

Suppose the additional weight required is W. The surface tension of the liquid-air interface is given by
  Sla = (W/2l) = (mg/2l)
Where m is the extra mass and l is the length of the plate edge. The subscript l a emphasizes the fact that the liquid-air interface tension is involved.

Drops and Bubbles

One consequence of surface tension is that free liquid drops and bubbles are spherical if effects of gravity can be neglected. You must have seen this especially clearly in small drops just formed in a high-speed spray or jet, and in soap bubbles blown by most of us in childhood. Why are drops and bubbles spherical? What keeps soap bubbles stable?
As we have been saying repeatedly, a liquid-air interface costs energy, so that for a given volume the most stable surface is the one that has the least area. The sphere has this property. We cannot prove this here, but you can check that the sphere is better than at least the cube in this respect! So, if gravity and other forces (e.g., air resistance) were ineffective, liquid drops would be spherical.
Even a spherical drop has some surface area. This fact has a consequence that the pressure inside the drop is more than the pressure outside. Suppose a spherical drop is in equilibrium with radius r. Increase its radius by Δr. The extra surface energy is
[4π (r + Δr)2 - 4πr2]Sla = 8πrΔr Sla
If the drop is in equilibrium this energy cost is balanced by the energy gain due to expansion under the pressure difference (Pi - Po) between the inside of the bubble and the outside. The work done is
W = (Pi - Po) 4πr2 Δr
So that
 (Pi - Po) = (2 Sla / r)
In general, for a liquid gas interface, the convex side has a higher pressure than the concave side. For example, if one had an air bubble in a liquid, this air bubble would be at a higher pressure.

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Figure (iii) - Drop, Cavity and bubble of radius r

A bubble differs from a drop and a cavity; in this it has two interfaces. Applying the above argument we have for a bubble
(Pi - Po) = (4 Sla / r)
This is probably why you had to blow hard, but not too hard, to form a soap bubble. A little extra air pressure is needed inside.

Capillary Rise

One consequence of the pressure difference across a curved liquid-air interface is the well-known effect that water rises up in a narrow tube in spite of gravity. The word capilla means hair in Latin; if the tube were hair thin, the rise would be very large. To see this, consider a vertical capillary tube of circular cross section (radius a) inserted into an open vessel of water.

From what we have said above
(Pi - Po) = (2 S / r) 
            = 2S / (a sec â
            = (2S / a) cos â
Thus the pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure. Consider the two points A and B in Fig. They must be at the same pressure, namely
         Po + h ρ g = Pi PA
Where ρ is the density of water and h is called the capillary rise. We have
         h ρ g = (Pi - Po) = (2 S cos θ/a)
Now we can make it clear that the capillary rise is due to surface tension. It is larger, for a smaller a. Typically it is of the order of a few cm for fine capillaries. For example, if a = 0.05 cm, using the value of surface tension for water, we find that

h = 2S / (
ρ g a) = 2 x (0.073)(N / m) / {(103 kg / m3) (9.8 m / s2) (5 x 10 -4 m)}
    = 2.98 x 10-2 m = 2.98 cm
Notice that if the liquid meniscus is convex, as for Mercury (Hg), i.e. if cos θ is negative, it is clear that the liquid will be lower in the capillary!

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