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12

PM2SURFACE TENSION

The swan can swim while sitting down,For pure conceit he takes the crown,He looks in the mirror over and over,and claims to have never heard of Pavlova.

OBJECTIVESAimsIn this chapter you will look at the behaviour of liquid surfaces and the explanation of that behaviourboth in terms of forces and in terms of energy. The principle of minimum potential energy can beinvoked to explain many surface phenomenaMinimum Learning GoalsWhen you have finished studying this chapter you should be able to do all of the following.1. Explain, interpret and use the terms

intermolecular forces, capillarity, angle of contact, wetting.2 (i) Describe an experimental determination of the surface tension of a liquid by the

measurement of the force on a glass slide in contact with the liquid.(ii) Perform simple numerical calculations associated with such a determination.

3 (i) Use a model of the microscopic structure of liquids to explain the phenomenon ofsurface tension in terms of potential energy.(ii) Extend this argument to explain why liquids tend to assume a shape which minimises thesurface area of the liquid.(iii) Do simple numerical calculations associated with energy per area.

4 (i) Explain how the surface tension of a liquid can be measured either in terms of force perlength or of energy per area.(ii) Demonstrate that these two descriptions are dimensionally equivalent.

5 (i) Explain how the phenomenon of capillarity results from forces between solid (e.g. glass)and liquid (e.g. water) molecules.

(ii) Recall, explain and use the relationship h = 2Trgr for capillary rise.

6 Give examples of how the wetting characteristics of surfaces can be altered.7 Explain, by identifying the relevant forces and using scaling arguments, why insects can walk

on water but larger animals cannot.8 Recall that the surface tension of water has a magnitude of 0.1 N.m-1.

PM2: Surface Tension 13

PRE-LECTURE

Recall from earlier lectures, particularly chapters FE2 and FE3 the following facts about the generalnature of forces.(i) The molecules of any substance - solid, liquid or gas - attract one another if they are far apart;

at short distances, the intermolecular force is repulsive. There is a crossover point where theforce is zero - neither attractive nor repulsive.

(ii) When a system is in equilibrium, then the sum of all the forces acting on the system is zero. Inparticular, the molecules of a substance tend to come together (pulled by the intermolecularattraction) until on the average their distances apart correspond to the cross over point betweenattraction and repulsion. This means the normal state of a substance is an average kind ofequilibrium.

(iii) Equilibrium can be discussed in terms of potential energy. The equilibrium configuration isone in which the potential energy is least.For a simple two body system you can see this by considering the diagrams on pages 17 and

59 of the Forces and Energy book.

LECTURE2-1 PHENOMENON OF SURFACE TENSIONThe surface of any liquid behaves as though it is covered by a stretched membrane.

Small insects can walk on water without getting wet.Demonstration

The membrane used is obviously quite strong: it will support dense objects, provided they are small and ofthe right shape:

a needle, a small square of aluminium sheet (weighted), a container made of fine wire gauze.The strength of the membrane varies for different liquids, e.g. it is much less for soapy water than pure

water.Demonstration

Ducks swim on water without getting very wet. However, they cannot swim on soapy water. [There arecases on record where ducks have drowned in farmyard ponds into which washing water was emptied, or instreams polluted with non degradable detergents.]

2-2 MEASUREMENT AND DEFINITION OF SURFACE TENSIONThe strength of the surface membrane can be imagined to arise from a set of forces acting on eachpoint of the surface, parallel to the surface, like the skin of a drum.


DemonstrationThe easiest way to measure these forces is with the following apparatus

BALANCE

ADJUSTABLE WEIGHT AND SAND GLASS

SLIDE

WATER

FIXEDCOUNTER WEIGHT

Fig 2.1 Experimental measurement of surface tension

Note that because the water surface curves up near the glass slide the surface tension forces between theglass and the water are vertical rather than horizontal.

SLIDE

WATER

MENISCUS

Fig 2.2 Shape of liquid meniscus

A first experiment yielded this result:A certain amount of sand (weight, W) was needed to keep slide just in contact with water; when the water

was removed this amount of sand plus a 0.55 g (extra weight 5.4 mN) was needed to have the slide in the sameposition

The difference, 5.4 mN, is a measure of the force due to the pull of the water on the slide.

A second experiment tested whether the force depended on the length of the slide (recall that on the surfaceof a drum, a bigger cut is harder to repair than a smaller one).

Length of slide used in first experiment: 38 mmLength of slide used in second experiment: 76 mmResult of second experiment: the force due to the pull of surface increases to 10 mN

Deduction: The force which a liquid surface exerts on any body with which it is in intimatecontact (as described above) is directly proportional to the length of the line of contact.

Force = T ¥ length.The constant of proportionality, T, is called the surface tension of the liquid.


DemonstrationIn the second experiment the width of the slide was 1 mm, so the total length of the line of contact

between the glass and the water was (76 + 1 + 76 + 1)mm. These values give a value for the surface tensionof water of 0.06 N.m-1.

[Most books of tables quote 0.07 N.m-1.]

Other liquids have different surface tensions (see post lecture material). Demonstration

A little detergent added to the water lowers it surface tension considerably.

As defined here the dimensions of surface tension are force per length. Its units in the S.I.system are N.m-1.2-3 MICROSCOPIC EXPLANATION AND SURFACE ENERGYTo understand why the phenomenon of surface tension arises, you must think of intermolecularattraction as recalled in the pre-lecture material.

Molecules of any substance want to pack together so that their average separation is low.In solids, this separation is fixed, whereas in gases, the random motion due to heat

predominates. In liquids, there is some random motion but, on the average, the molecular separationis low.

Consider a fixed number of liquid molecules. If they are packed so that they have a largesurface area, their average intermolecular separation is relatively high. If they have small surfacearea, the average intermolecular separation is relatively low. Their total potential energy is lower inthe latter case.

A logical conclusion from this is that energy has to be added in order to increase the surfacearea of a liquid. The bigger the change in surface area, the more energy has to be put in. Associatedwith the surface there is a potential energy that depends on the area of the surface. This means thatan alternative approach is to consider surface tension as an energy per surface area.

Since the equilibrium configuration of any system is that in which the potential energy is least,a liquid left to itself will assume a shape which minimises surface area, thereby minimising the totalsurface potential energy.

DemonstrationDrops of water are sphericalLoop of thread on water; detergent added inside loop; loop takes a circular shape.

Shaded area here is greater than shaded area here

CONTAINERLOOP OFTHREAD

PURE WATER

WATER ANDDETERGENT

Fig 2.3 Effect of placing a drop of detergent inside a loop of string that is floating on thesurface of water

(Surface tension of detergent and water is much lower than that of water.)


The dimensions of energy are force ¥ length, so energyarea has the same dimensions as

forcelength .

Sometimes it is easiest to explain surface phenomena in terms of energy considerations,sometimes in terms of force considerations

Demonstration Three matches on water:

MATCHES

CONTAINER

PURE WATER

DETERGENT ADDED

becomes

Fig 2.4 Effect of placing a drop of detergent inside a triangle of matches that are floatingon the surface of water

This is basically the same as the loop of thread demonstration, but it is easier to explain whyeach match moved in terms of forces as thus for the match at the top of the diagram:

larger force(water: highersurface tension)

smaller force(detergent: lower surface tension)

Fig 2.5 The net force acting on the match pushes it away from the detergent

2-4 CAPILLARITYA consequence of the phenomenon of surface tension is that many liquids will "creep up" tubes, anobservation made readily with glass tubes of very narrow bore.

h

WATER (DYED)Fig 2.6 Capillary rise

The height of the water in the capillary above the level of the liquid in the surrounding liquid,as indicated by h in the diagram, is called the capillary rise.


Demonstration Glass tube of narrow bore in water.It can be demonstrated that:(i) the capillary rise is larger for liquids of higher surface tension than of lower surface tension (e.g. larger

for pure water than for water and detergent) ;(ii) the height increases as the radius of the bore of the tube gets smaller.In fact, the height varies inversely as r.

Demonstration Glass wedge in water:

TWO GLASS SHEETS

RUBBER BAND

PLAN ELEVATION

WIDE END

NARROW END

HYPERBOLIC !! !SHAPE

WATER (DYED)

Fig 2.7 The rise of water in a wedge between two flat glass sheets(iii) We would like to have shown that height decreased with increasing density, but we could not

find two common liquids with roughly the same surface tension and vastly different densities.

The relation between capillary rise, surface tension and density (see post lecture) is

h = 2Trgr

The tube used in the demonstration had a bore of radius 0.50 mm and the measured rise was28!mm. For a tube of this radius, the calculated rise is

h = 2!¥ !0.06!N.m-1

1!¥ !103 !kg.m-3!¥ !9.8!m.s-2!¥ !0.50!¥ !10-3!m

= 2 cm.Specific Applications:(i) Rise of water through soils.

Demonstration Although water rising in a column of soil is not rising through a tube of uniform bore it is moving

through spaces roughly the same size as the soil grains. So the same kind of capillarity formula will apply.A consequence is that water rises highest in column with finest grains.

[Note water rises fastest in column with largest grains. We return to this in the post lecture of chapterPM4.]

(ii) Chromatography.Demonstration

This is a method of chemical analysis which can be done by eye. See post lecture material for a morecareful description.


2-5 WETTINGA question we have skimmed over is: why is there an attractive force between water and glasscausing the rise of water in a glass capillary tube? This is a question about intermolecular forceswhich only chemists can answer properly. But certainly different liquids are attracted to differentsolids in different degrees. For example, the level of mercury will fall in a glass capillary tube.

Demonstration Drops on solid surfaces.

MERCURYWATER

GLASS

MERCURYWATER

LEADFig 2.8 Water and mercury drops on glass and lead surfaces

Laboratory workers measure the intersurface forces in terms of the angle of contact definedas follows.

ANGLE OFCONTACT f

tangent line

Fig 2.9 Definition of f, the angle of contact between a liquid and a solid surfaceThe concept of angle of contact is treated further in the post lecture.This phenomenon is called wetting. Water is said to wet glass completely (the angle of contact

is virtually zero).The wetting characteristics of surfaces can be changed by putting a layer of a different material

on the surface.Demonstration

Oil on glass will repel water.

GLASS

OIL

WATER

Fig 2.10 The presence of oil results in the water forming a drop rather than spreading overthe glass surface

DemonstrationWaterproofing of material (this usually involves coating fibres with oil or polymers).

DemonstrationPreening of birds.Water birds spread oil on their feathers to make them water resistant.

DemonstrationWater resistant sands.

Some West Australian sands are virtually impervious to water as a result of fibrous materialbetween the grains making them water resistant. This leads to bad run off conditions in vast areas ofthe state.


DetergentsThe properties of detergents arise from their complicated molecular structure. This can be illustratedschematically thus:

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

C

H

H

H C O-O

This end is repelled by water molecules [hydrophobic] and is attracted to oils, fats [lipiphilic]

This end is attracted to water molecules [hydrophilic]

Fig 2.11 A detergent molecule(i) When detergent is put into water this happens:

Fig 2.12 Detergent molecules in water (schematic)Note that along the surface there are water molecules and hydrophobic ends. The surface

tension is lower than that of pure water. It is easier to pull this surface apart than it is to pull asurface of pure water apart

(ii) In washing up water the following sequence occurs as the water is stirred up.

watergreaseDETERGENT ADDED STIRRED

Fig 2.13 Stirring of soapy water during "washing up"


The particles of organic matter are rendered soluble by being coated with detergent molecules:lipophilic ends stick to the particles and hydrophilic ends point outwards.

Emulsification.Many organic substances which are insoluble in water (DDT is a good example) can be mixed intoan emulsion with water by the addition of a little detergent.

DemonstrationOil and water.

POST-LECTURE2-6 UNITS AND DIMENSIONSA couple of statements were made (or implied) in 2-3 above, which may not be all that obvious.Q2.1 The loop of thread changed its shape to a circle because a circle is the geometrical shape which has

maximum area for a fixed circumference.

This is not easy to prove in general but consider the following concrete example: assume that the length ofthread in the loop was 0.l!m and work out which, of the following possible shapes the loop could have, hasthe largest area.

2.5 cm

2.5 cm

1 cm

4 cm

3.3 cm

3.2 cm

3.3 cm 3.3 cm

Fig 2.14 Diagram for Q2.1Q2.2 Energy/area is the same as force/length. The following example illustrates this fact.

Imagine you are increasing the area of a rectangular soap film; as indicated the original dimensions of the film are aand Ú. The surface tension of the soapy water is T.

F

a

Ú

Fig 2.15 Diagram for Q2.2


Suppose that to stretch the film at a constant speed a uniform force F equal (and opposite) to the force associated withsurface tension is applied.

Since the film has two surfaces, the relation between F and T is

F = 2ÚT .

Calculate the total work done in increasing the distance a by an amount b, and show it is proportional to the changein area of the soap film.

2-7 MORE ON CAPILLARITYThe law quoted in 2-4 can be derived theoretically as follows. Ask yourself first, why should waterrise up inside the tube? It is an effect of the surface tension at the top of the water column,particularly where it meets the glass wall.

Glassmolecules

Water moleculesFig 2.16 Interaction of water and glass molecules

Each water surface molecule exerts forces on those near it Since there is equilibrium the lastwater molecule must also have a force exerted on it by the glass molecule near it. Therefore, allaround the top of the water, the glass is exerting a force on the water. Because is so happens thatwater wets glass so well, this force is a vertical force.

So that it why the water rises in the tube: because the glass is pulling it up. The length of theline of contact between the water and the glass is 2p times the radius of tube, so the magnitude of theupward force is:

= T ¥ (2p radius of tube)= 2 p rT.

The next question is: why does not the water keep rising indefinitely?The answer is that the higher the column the more the weight of the water in the column pulls it

back. Thus there is a downward force equal to r (pr2h) g.The two forces are in equilibrium so

2prT = rπr2hgand, therefore, for this situation, where the water wets the glass completely, the final height of thewater column can be written

h = 2Trgr

Q2.3 In the experiment with soil, we found that for the coarse grained soils (radius of soil grains ~ 0.3!mm) aftera long time the water finally stopped rising at a height of ~ 150 mm.

Although soil is by no means a series of uniform bore capillary tubes, it cannot be too bad an approximation toapply the above relation. Apply the relation and find how much error is in fact introduced.


2-8 ANGLE OF CONTACTThe angle of contact is defined to be the angle between the surface of the liquid and the solidsurface at the point of contact.

ANGLE OFCONTACT f

tangent line

Fig 2.17 Angle of contact for a liquid that does not "wet" the solid surfaceYou will observe that for a water-glass contact, as in the next diagram, the angle of contact is

much smaller;angle of contact

f

small

tangent line

Fig 2.18 Angle of contact for water-glass contactfor mercury-glass, as in the next diagram, it is almost 180°.

angle of contact

f

large

tangent line

Fig 2.19 Angle of contact for mercury-glass contactWhen the angle of contact is less than 90°, the liquid is said to wet the solid surface, while it is

said not to wet the surface if the angle of contact is greater than 90°.When the angle of contact is not 0° or 180°, the angle explicitly enters those equations which

directly or indirectly involve the force exerted by a solid on a liquid due to surface tension.


Forces associatedwith surface tension

Angle

f

Fig 2.20 Close up of part of Fig 2.16Redrawing an earlier diagram in a more general way, we note that the force the liquid exerts on

the wall (and vice versa) is not vertical. There is a horizontal component, T sin f (which for f equalto 0 ̊or 180˚ is zero), which results in a usually imperceptible distortion of the wall. There is avertical component, T cos f (which for f equal to 0 ̊ or 180˚ is T), which causes the liquid in acapillary tube to rise.

So the equation for capillary rise that we wrote is not complete. The general form is

h = 2Tcosf

rgr

For clean glass-water contacts f ª 0 and cos f ª 1. So the equation was suitable for water ina clean glass tube.Q2.4 For mercury-glass we saw f ª 180° and we know that cos 180° = -1. The formula for capillary height will

therefore have a minus sign in it. Does this mean that if you put a glass tube in mercury the level of thesurface would be lower inside the tube?


2-9 SCALING QUESTIONSQ2.5 Why can insects walk on water, but larger animals (no matter how much water repellent material they put on

themselves) cannot?

Similarly, why will a needle float on water, but a much larger piece of metal of exactly the same shape will not?

Try to answer this question as follows:

(i) Consider a nice simple geometric shape for the needle, say a rectangular bar. Take the length to be 40 mmand the width 0.50 mm.

(ii) Calculate its weight (the density of iron is 7.8 ¥ 103 kg.m-3).

(iii) Now assume it is on top of the water with an angle, f, as shown.Needle

f

Fig 2.21 Needle "floating" on water

Calculate the total upward force (remember the force associated with surface tension acts right around the contact linebetween the needle and the water).

(iv) Can the weight of the needle be supported?

(v) How does the angle of contact depend on the weight?

(vi) Now assume the "needle" is 4 m in length and 5 cm thick.

Will its weight be supported by surface tension?

(vii) See if you can use the kind of scaling argument which was employed in chapter FE8 to answer the originalquestion succinctly.


<< 2-10 CHROMATOGRAPHY

Chromatography is a technique for separating out the chemical constituents of mixtures. It is particularlyuseful in biological contexts. There are two commonly used forms.

Paper Chromatography : Here a few drops of the chemical mixture are put onto a piece of filter paper andallowed to dry. Next the paper is touched to a reservoir of some solvent which will dissolve the chemical substanceyou hope to detect. The solvent is sucked up into the filter paper (by capillary action), and as it flows past the driedmixture, it dissolves out the chemical constituents and carries them along. However, different chemical substancesadhere more or less strongly to the paper (i.e. the surface tension between the surface of the solution and the fibres ofthe paper differs) and so different chemical substances are carried along at different rates. So if you remove the paperfrom the solvent after a while the various chemical constituents of the original mixture will be at different positionson the filter paper.

Colour Chromatography (This is the experiment we filmed.) Here the solvent is put on top of themixture, and allowed to flow through a plug composed of grains of cellulose. Again, the adhesion between thechemical constituents of the sample (spinach leaf) and the cellulose grains is different and they all sink at differentrates. In our experiment (which we filmed in the Department of Agricultural Chemistry with the help of Dr BobCaldwell) the final order of chemical constituents is

TOP: Flavonoid (Yellow)Chlorophyll B (Green)Xanthophyll (Yellow)Chlorophyll S (Green)Pheophytin (Purple)

BOTTOM Carotenoids (Yellow)Only the two chlorophyll bands show up well on the TV screen. >>

2-11 VALUES OF SURFACE TENSIONHere are the values of surface tension of some common liquids. They are listed heremerely for the purpose of showing you what range the values of surface tension canhave.

Liquid Surface Tension/N.m-1

water (20°C) 0.073water (100°C) 0.059alcohol 0.022glycerine 0.063turpentine 0.027mercury 0.513

2-12 REFERENCES "Surface tension in the lungs"Scientific American, p 120, Dec 1962.

"Synthetic detergents"Kushner & Hoffman, Scientific American, p 26, Oct 1951.

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