#### Surface Tension, Physics tutorial

Concept of Surface Tension:

Surface tension is estimated as energy needed to increase surface area of the liquid by unit of area. Surface tension of the liquid results from the imbalance of intermolecular attractive forces, cohesive forces between molecules:

• The molecule in bulk liquid experiences cohesive forces with other molecules in every direction.
• The molecule at the surface of the liquid experiences only net inward cohesive forces.

Molecular Explanation of Surface Tension:

The molecule like A inside body of the liquid will suffer a force of attraction because of all other molecules within a small distance of A, this is known as sphere of molecular activity. Molecule A should be in equilibrium and so resultant force on A is hence zero. Though, another molecule like B very close to surface will suffer a net inward force; so too will the molecule like C which is really in surface.

Molecules can't move downwards as there are molecules below them but they do oppose being separated from each other, therefore giving skin effect. To call it the elastic skin is misleading, as surface tension doesn't differ with size of the surface as it would in case of the elastic sheet.

Coefficient of Surface Tension:

Surface tension is the property of the liquid. Magnitude of a surface tension across/perpendicular to a unit length is called as coefficient of surface tension. Surface tension differs with purity of liquid and temperature. Coefficient of surface tension γ is stated as force per unit length in surface acting perpendicular to one side of the line in the surface.

γ = [Force/Length]

Surface tension is independent of surface area. Unit of γ is Nm-1. Dimension of surface tension is MT-2.

τ differs with temperature of liquid. At 20oC, γ is value of 7.2 x 10-2Nm-1 for water, while at same temperature, its value is 4.5 x 10-1Nm-1 for mercury.

Surface tension γ, of liquid decrease with temperatures, molecules have higher kinetic energy. They are therefore, relatively further apart from neighboring molecules. This thus decreases inward force therefore work done against force is less. Molecules in surface now have less energy that at lower temperatures. I.e., surface tension is less at high temperatures than at lower temperature.

If γ = T/1

Thus T =γ1

Force of Cohesion and Force of Adhesion:

Molecular forces between like molecules are known as forces of cohesion. For instance two water molecules have cohesion forces between them. Molecular forces between unlike molecules are known as forces of adhesion. For instance, force between molecules of water and those of glass vessel will comprises adhesive forces. Water wets are clings to clean glass as adhesive forces between glass and water molecules are greater than cohesive forces of water. On the other hand, mercury doesn't wet clean glass but shrinks away from it as adhesive forces between mercury and glass molecules are less than cohesive forces of mercury.

Angle of Contact:

The angle of contact is another property utilized to demonstrate different types of meniscus (concave and convex) we get when the liquid is contained in the tube. Angle of contact (Θ) between the liquid and solid is stated as angle measured through liquid between tangent to the liquid surface where it touches solid surface and the solid surface itself. For instance angle of contact for pure water in contact with clean glass taken as is zero Θ = 0

Angle of contact for olive oil is 15o and angle of contact for mercury is 150o. Liquids which make obtuse angles (θ>90o) of contact with glass or any other solid surface don't wet surfaces (that is mercury and glass. Though, liquids with acute angles (θ<90o) of contact wet surface (that is water and glass). In manufacture of raincoats, garment fibers are treated chemically so that acute angle made by water with fiber is changed to the obtuse angle, therefore water then forms drops, that can be shaken off from raincoat.

Excess Pressure in Bubbles and Curved Liquid Surfaces:

Curvature of the liquid or bubble formed in liquid is associated to surface tension of liquid. Relate surface tension with excess pressure which exists in formation of bubbles in liquids or in air and curved liquid surfaces. Let the bubble formed inside the liquid.

Consider equilibrium of one half of B of bubble. Total force because of surface tension = 2Πrγ where 2Πr is circumference around which γ the surface tension acts.

Force because of Pressure P1 is given by P1Πr2.

The total force from left to right = γ2πr + P1Πr2

The force from right to left = P2Πr2 Since the bubble is at equilibrium

∴ P2Πr2 = γ2Πr + P1Πr2

On rearranging the terms,

(P2 - P1) Πr2 = 2Πrγ

Therefore (P2 - P1) = 2Πrγ/Πr2 we get (P2 - P1) = 2γ/r2

Therefore Excess Pressure P = (P2 - P1) = 2γ/r

Excess Pressure on Curved Liquid Surfaces:

Due to property of surface tension, the liquid surface always tends to have minimum surface area. Small drops and bubbles are found to have spherical shapes. Though, a big drop of liquid is not spherical in shape. Reason is that in case of the small drop of liquid, effect of gravity is negligible. As small drops and bubbles have spherical shapes, it implies that spherical surfaces should be having minimum surface area. Further, as the drop or a bubble doesn't collapse under effect of force of surface tension (that tends to minimize surface area), it indicates that pressure inside drop or bubble should be greater than that outside it.

The excess pressure is also expressed as P = 2γ/r. If the angle of contact is θ then the excess pressure is given as

P = 2γcosΘ/r

Excess Pressure in Soap Bubbles:

A soap bubble has two liquid surfaces in contact with air. One is inside the bubble while the other is outside the bubble. The force of one half B of the bubble due to the surface tension is thus:

γ X 2Πr x 2 = 4Πrγ

Force due to Pressure P1 = P1πr2

Force due to surface tension = 4πrγ

Therefore Total force from left to right = P1Πr2 + 4Πrγ

Total force due to Pressure P2 = P2Πr2

At equilibrium, therefore, we can write

P2Πr2 = 4Πrγ + P1Πr2

P2Πr2 - P1Πr2 = 4Πrγ

∴ (P2 - P1)Πr2 = 4Πrγ

On rearranging, we obtain the expression for excess pressure as

Therefore (P2-P1) = 4Πrγ/Πr2

Therefore Pressure P = (P2 - P1) = 4γ/r

From the above expressions on excess pressure, it will be observed that the excess pressure (P) is inversely proportional to the radius of the bubble or the curved surface. A smaller soap bubble has a greater pressure inside it than a larger bubble. This is demonstrated when such bubbles are formed on the ends of a tube. The smaller decreases in size while the larger grows slightly due to airflow.

When the air flow ceases, the air pressure inside has equalized. Thus the radius of the film to which the small bubble collapse is equal to the radius of large bubble on the other side. Communication between them is being prevented by a closed tap in the middle. If the tap is opened, the smaller bubble is observed to collapse gradually and the size of the larger bubble increases. The explanation for this observation is that the pressure P1 in the smaller bubble is greater than P2, the pressure in the large bubble. Consequently, air flows from P1 toP2.

Capillarity: The Rise and Fall of Liquids in Capillary Tubes:

When different kinds of tubes (of varying diameter) are dipped in water, water rises in a different way in such tubes. Height is greatest with tubes with smallest diameter. Such tubes are known as capillary tubes. In other words, the tube with the fine and uniform bore throughout length is known as a capillary tube

The behavior of liquids in capillary tubes can be related with surface tension of liquids. For instance, water rises in capillary tubes. With knowledge of angle of contact Θ, and excess pressure on one side of the curved liquid surface, we can work out that some liquids will rise in the capillary tube while some will be depressed, i.e., will fall.

Excess pressure formula

P2 - P1 = 2γ/r

For the bubble in water also applies to excess pressure across and curved liquid surface of radius r that is spherical in shape. We shall thus apply relationship to meniscus of water in the clean glass capillary tube when angle of contact is zero.

The meniscus curves upwards with radius r equivalent to that of capillary tube. Pressure P2 above meniscus is greater than pressure P1 just below by r2γ. But P2 = H, where H is atmospheric pressure and P1 = H, as level of water is same inside and outside tube. This, clearly, is impossible therefore meniscus is unstable. To P2 make greater than P1 by r2γ, water will rise to the height h, above outside. Here pressure above meniscus is still H but pressure is now H-hpg from hydrostatic pressure where p is density of liquid.

Hence, P2 - P1 = H-(H - hρg) = 2γ/r

Therefore P2 - P1 = hρg = 2γ/r

Therefore hρg = 2γ/r

Assume liquid has angle of contact Θ liquid then rises to the height h given by hρg = 2γ/r cosΘ

This is obtained from fact that vertical component of γ is γcosΘ

What happens when capillary tube is pushed down in water so that height h1 of tube above water outside is less than h. Note that liquid doesn't overflow. The meniscus at top of tube now forms the surface of radius R bigger than r of capillary tube and makes the acute angle of contacts Θ with glass.

Excess pressure = h1ρg = 2γ/R

Excess pressure = hρg = 2γ/r

Therefore h1/h = 2γ/r = cos Θ

From geometrical point of view, it can be shown that

Cos Θ = r/R

Therefore, the angle of contact can be determined.

Variation of Surface Tension with Temperature Jaeger's Method:

The rise in a capillary tube is not suitable for measuring the variation of surface tension of water with temperature. This is because the water in the beaker can be varied in temperature but the water in the capillary tube will not be at the same temperature. Thus, Jaeger used excess pressure in a bubble formed inside water to measure how the surface tension γ for water varies with temperature.

a bubble is formed slowly at the end of the tube A dipped into a beaker of water B by air from a vessel W. an oil manometer M is used to measure the pressure inside the bubble as it grows, by forming a bubble inside a liquid, and measuring the excess pressure, Jaeger found how the surface tension of a liquid varied with temperature.

Excess pressure = hρg - h1ρ1g = 2γ/r

Therefore γ = rg(hρ - h1ρ1)

Thus γ can be determined from the values of r, g, h, h1, ρ and ρ1

When a capillary tube A is connected to the vessel W containing the funnel C, so that air is driven slowly through A when water goes in W through C. Capillary tube A is placed in beaker containing liquid L. When air is passed through it at slow rate, bubble forms slowly at end of A. Bubble will be at three possible stages of growth. Radius grows from a to b. Pressure is larger at a as radius is smaller when it grows to c, radius of c is greater than b. Therefore it can't contain increasing pressure. Downward force of bubble because of pressure, in fact, would be greater than upward force because of surface tension. Therefore bubble becomes unstable and breaks away from A when it radius is same as A. Therefore as bubble grows, pressure in it grows to maximum and then breaks away. Maximum pressure is observed from the manometer < M having a light oil of density ρ. Series of observations are taken as numerous bubbles grow.

Maximum pressure inside bubble = H + hρg

Where, h = maximum difference in levels in manometer M

H = atmospheric pressure

Pressure outside bubble = H + h1ρ1g Where, h1 = depth of orifice of A below level of liquid L ρ1 = density of liquid L

Excess pressure = (H + hρg) - (H + h1ρ1g) = hρg - h1ρ1g = 2γ/r

Therefore 2γ/r = g(hρ + h1ρ1g)

Therefore = rg/2 = (hρ - h1ρ1g)

By adding warm liquid to vessel containing L. Variation of the surface tension with temperature can be found. In brief, it can be established that surface tension of the liquid reduces as temperature increases. There has been no acceptable formula to associate surface tension with temperature. On clarification about observation is that decrease in surface tension with temperature may be attributed to greater separation of molecules at higher temperatures. Force of attraction is therefore decreased; therefore, surface tension is decreased.

Surface Tension and Surface Energy:

Surface tension of the liquid as the force per unit length where force is present in the system, there may also is energy. For instance, energy exists in gravitational fields. Consider surface energy of the liquid and relationship to surface tension γ.

Consider the film liquid stretched across the horizontal frame ABCD. As γ is force per unit length force on rod BC of length l = γ x 2l. Length is 2l as there are two soap film surfaces in contact with air. Suppose rod is now moved through distance b from BC to CB′′ against surface tension forces, so that surface area of film increases. Temperature of film then generally decreases in which case surface tension modifies. If surface area increases under isothermal condition (isothermal means constant temperature) though, surface tension is constant we can then say that if γ is surface tension at that temperature:

Work done in enlarging surface area (W) = force x distance

= 2γlxb

= 2γl x b

= γ x 2lb

But 2lb is total increase in surface area.

Hence Work done in enlarging area/a = γ

W/2lb = γ

W/A = γ

Thus γ is stated as work done in increasing surface area by unit amount if there is no temperature change of surface as change is made. I.e. change is isothermal. This is also known as the free surface energy as mechanical work done can be released when surface contracts. Therefore "γ can be stated as work done per unit area in increasing surface area of the liquid under isothermal conditions"

If film temperature fell while the area was increased heat would flow in the film. Increase in total energy of area would then be greater than γ.

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