G2 Molecular phenomena when fluids meet. Gases and solids
Compaction of gases at solids
As a companion to solution of gases in fluids, one has their absorption on the surface of solids. Gases are compacted at them (mostly those, which solve most readily in water) and form there a adhesive layer. The more extended is the surface, the more gas is absorbed. Hence porous substances absorb most, because also the insides of the pores belong to the surface and make it enormously large compared with the volume. Especially strongly absorbs freshly and strongly heated charcoal (box tree, coconut); 1 cm² compacts at normal temperature:
|ammonia||hydrochloric acid||carbonic acid||hydrogen|
|90 cm³||85 cm³||35 cm³||1.7 cm³|
Very fine platinum powder absorbs very strongly. It compacts especially hydrogen so strongly that it becomes heated and ignites the hydrogen (cigarette lighter). The ignition primers for household gas work in a similar manner.
Porosity of the surface is not necessarily required for absorption: Gas sticking to glass, even in a vacuum and strongly heated, is not readily removed. - Certain substances absorb internally from the air mainly water vapour (hygroscopic materials), for example, phosphoric anhydride, potash, chlor-calcium,concentrated sulphuric acid. - Platinum, when heated in hydrogen, occludes (Thomas Graham 1805-1869) large quantities and even holds on to them in a vacuum. Even more occludes palladium; according to Graham, when forged, it absorbs at normal temperature 376 cm³ of gas in 1cm³. Forged iron, cast iron and steel always contain gas inside them, which they can only expel in a vacuum at about 800º C.
Tension in the bounding surface of a fluid (Surface tension)
Diffusion and osmosis depend largely on the fact that substances in contact tend to penetrate each other totally (molecularly). On the diametrically opposite tendency depend the surface tension of fluids and the capillary phenomena, which arise from it. However, the substances, which participate in it, while touching each other at a common interface, do not mix, like water and oil or water and mercury or water and air; in fact, they separate again, if you shake them and leave them alone. The sharp interfaces are the conditio sine qua non (Latin: Condition, without which not) for the tension in bounding surfaces. The molecules, which lie further away from the surface, are surrounded by neighbours if the same substance, but not those at the interface (Fig. 274). Hence the interface is in a special state, which manifests itself by characteristic actions. Some of these are known to everyone; for example, a needle made out of metal, when slightly greased (to avoid it being moistened) and placed carefully on water, stays there, or many insects (while their legs are covered by a layer of fat) can run on water, without breaking the surface, or a water skin can become a large sphere - a soap bubble. We will start from the last phenomenon.
A soap bubble is a very thin skin of water (toughened by soap). Let it hang from one end of a thin tube, close the other end to keep it in shape; if you now open the tube, it contracts and drives with perceptible pressure the air out of the tube (Fig. 270). Hence the water skin - it is bounded on both sides by air - is in tension, that is, has potential energy. Not the spherical shape of the soap bladder is the cause of the tension, it is also present in a plane water skin.
You can see this (Fig. 271): AA1, BB1 is a vertical wire frame, DC a readily moved wire link. If you create between AB and DC a soap water skin, you can suspend from about 70 milligram before the skin will tear. If you increase the load carefully, you can determine the weight, which will just balance the tension in the skin. The tension in the skin is everywhere and in all directions the same; you can discover that also: For example, dip a plane wire frame (Fig. 272) into soap water; as it is withdrawn, it surrounds a plane skin of fluid. Place a silk thread loop, moistened by soap water, on top of it, it will form an arbitrary curve. However, if you puncture the skin inside the loop, it will form a circle - the fluid skin exerts around the loop (I) tension and this is the same at each point (II) where the thread touches the fluid ( G.L. van der Mensbrugghe 1873),
There exists also tension in a concave interface - it is present in all interfaces. Imagine you have drawn a straight line in the plane surface, in order to confirm: There acts a constant tension from both sides at right angle. We will employ this constant value, expressed in force units (dyn) - recall that 1 dyn = 1 mg* - and related to 1 cm length as the unit of the surface tension. You can determine its magnitude in different ways, for example, by finding the value of the weight in milligram, as in Fig. 271, at which the skin tears. At room temperature, you will find
|water||72.8||larger than for other substances except mercury|
Note: These values only apply, when the fluid is bounded by air (as is assumed in the following). We find then
|mercury - water||418|
|mercury - alcohol||399|
|mercury - olive oil||335|
|water - olive oil||21|
|alcohol - olive oil||2.3|
The surface tension is largest at the freezing point, drops quickly as the temperature rises and disappears whenever the fluid state changes into the gaseous state (cf. critical temperature)
The reduction of the surface tension as the temperature increases explains why a spot of fat in a cloth disappears, when you place a hot flatiron on top of it and underneath the spot a porous substance (blotting paper): The surface tension of the fat on the hot side decreases, whence the fat is drawn to the colder side and gradually enters the blotting paper.
The idea that energy is stored in the surface is fully justified, because, in order to create a water skin of a given area, you must perform work in overcoming the surface tension, which grows with the area (Fig. 273). Let AB and AC - covered with soap water - be in touch initially; in order to pull them apart (along AB and CD) to form ABCD and a soap skin in between, let the required force be F; then the work performed is F·AC, which is the energy contained in the surface. Hence you ascribe to the molecular surface layer of the fluid a special form of energy and say: It possesses surface energy and measure it by its magnitude per unit of area. The dimensional formula is then energy/area = lmt-2·l / l2 = [m·t-2]. If S is the surface tension energy per unit area, that the energy content of the surface is S·AB·AC, whence S·AB·AC = F·AC. The force is obviously as large as the force, at which the skin tends to contract at right angle to PQ, that is, PQ times the surface tension T, that is, T·PQ. Hence S·AB·AC = T·PQ·AC or, since PQ = AB, S = T, that is, the numerical value of the surface tension per unit length is equal to the surface energy per unit area. The tension in a fluid skin therefore does not depend at all on its extent, because it remains the same per unit area, but it is proportional to the surface. Thus, a fluid skin behaves completely differently from an extended rubber membrane, the tension of which depends on how fat it has been extended. Moreover, the tension in a fluid skin is at each point and in each direction equal.
Relationship between surface tension and molecular volume of a fluid (Eötvös' Law)
Surface tension, in nature a molecular force, is remarkable related to a fluid's molecular volume, the volume occupied by a single mol. Let a1, a2 denote the surface tensions at temperatures t1, t2 and v1,v2 the corresponding molecular volumes, then (Eötvös)
(a1v1 - a2v2)2/3/( t1 - t2) = k , a constant.
Hence the molecular surface energy changes independently of the nature of the fluid proportional to the temperature. In other words, if v is the volume, v2/3 its surface, then a v2/3 is its molecular surface energy. Eötvös has formulated this law in a still different manner. Let T0 be the absolute temperature, at which a v2/3 = = 0, that is, at which the surface tension vanishes, then the law becomes: a v2/3 = k (T0 - T). The temperature T0 almost coincides with the critical temperature.
Eötvös' Law is very important for Physical Chemistry. If you take the relationship between the molecular volume v and the molecular weight m into account, v = m /s, where s is the density of the fluid, this law yields a method for the determination of the molecular weight at different temperatures from capillary observations.
In order to protect the surface of the fluids against changes by contamination of any kind, Eötvös worked only with glass vessels, which had been closed by melting, and with a strange optical method of observation; he was thus able to observe and measure at temperatures beyond the boiling point up to the critical temperature. He thus showed that the molecular surface energy of fluids only depends on the temperature.
Boundary layer of a fluid. Internal pressure depending on the surface's curvature
There exists surface tension in every bounding surface, independently of its form; however, it depends on its form - more exactly, on its curvature -, whether it has a component, perpendicular to the surface or not, and how large is the pressure, which it exerts on the fluid, it bounds.
In what respect differ physically upwards concave and upwards convex fluid surfaces compared with the plane? Assume that there is only the mutual attraction between neighbouring fluid particles and that every particle is attracted equally strongly by all neighbours. The distance, up to which the attraction from a given particle acts, is called the radius of the molecular sphere of action. (It is about equally large for all substances, approximately 1/20,000 mm.) Let there be drawn in the fluid a spherical surface with this radius drawn about the particle M (Fig. 274). M has neighbours in all directions; two oppositely located particles, say antipodes, attract M in opposite directions equally strongly, so that effectively no force acts. It is different for particles in the interface! The sphere of action of the particle A, the distance of which from the interface is less than the sphere of action, reaches beyond, that is, it is not filled completely with its fluid particles. The particles in the section fgh lack the antipodes, whence the attraction of these particles on A generates a pressure, directed into the fluid. This is true for every particle, the distance of which from the interface is smaller than its radius of action, whence the fluid layer immediately below the interface, which consists of all of them, the boundary layer, must exert a pressure on the fluid.
This pressure and the contribution which the particles in the surface make, differs with the shape of the surface (Fig. 275). If the surface is flat, d1l1, the sphere of action of the particle lacks a certain number of antipodes, namely those, which would have their locations in the outside spherical segment; if it is outwards convex, dl, a larger number of antipodes is lacking; in contrast, if it is outwards concave, d2l2, a smaller number of antipodes is missing, that is, there exists a smaller inward pressure that in the case of d1l1. Briefly speaking, the pressure of a boundary layer on the fluid it bounds is smaller than where it is plane. In all cases , the boundary layer generates an internal pressure, whatever is its curvature. However, the (tangential) surface tension does not always contribute. Wherever the surface is plane, there does not exist a perpendicular component, where it is outwards convex (concave), the component is directed inwards (outwards).
What is the magnitude of the inwards normal pressure on a curved surface? We will give the answer without proof (Fig. 276). Let O be a point on the surface and ON the local normal and R and R' the radii of the circles through O with largest and smallest curvature, then the pressure per unit area acting along ON is N = F(1/R + 1/R'), where F denotes the surface tension. (all capillary actions can be referred to this expression!) The excess pressure inside a soap bubble (Fig. 270) is according to this formula N = 4F/R, since the skin has two faces and for a sphere R = R'.)
The formula N = F(1/R + 1/R') demonstrates the form of the surface of a fluid at rest, left to itself, so that only the molecular forces act on it, that is, it is not exposed to gravity: At each point of its surface, the pressure must be perpendicular to it and equally large. Hence 1/R + 1/R' = const. This condition is met by all surfaces of constant mean curvature, for example, the sphere, plane and circular cylinder. The influence of gravity can be removed from the fluid to be examined by placing it into a second fluid with the same specific weight. Joseph Antoin Ferdinand Plateau 1801-1883 1843 placed olive oil into a corresponding mixture of water and alcohol and obtained spheres with diameters up to 10 cm. Using such spheres, he demonstrated rotation phenomena similar to those, which according to the hypothesis of Kant and Laplace were to explain the formation of planetary systems, for example, the flattening at the poles and the formation of Saturn's rings. If you place into the mixture of water and alcohol with oil moistened wire frames and ensure, that the interface (oil) passes through given points of them, you can also obtain different forms of equilibrium, which correspond to the equation 1/R + 1/R' = const.
Very small quantities of fluid form spherical drops, as you know from everyday life. However, the fluid must not moisten the base and the mass must be so small that the molecular forces overcome the gravity; the surface tension then acts unimpeded and makes the surface as small as is possible given the volume if the bounding mass, that is, it forms a spherical surface. The strictest proof for the perfect spherical form of drops free from external forces is supplied by the rain bow: The slightest deviation from the perfect spherical form of the drops would change its appearance totally, larger deviations would make it altogether impossible.
If the surface of a mercury drop is polarized electrically, its tension changes and hence the shape of the drop. This process is the basis of a very sensitive electrometer (the capillary electrometer of Lippmann 1873).
Meeting of three interfaces
The difference in magnitude in the surface tension, displayed by the figures above, manifests it self in a characteristic manner, when the interfaces of so different surface tensions meet. If two fluids touch each other, which simultaneously border on air, for example, a drop of oil on a water surface, touched by air (Fig. 277), we have three substances which touch each other and three interfaces: Water-air, water-oil, oil-air. They meet on the water along the circle, surrounding the oil drop; hence there act at each point of this circle the forces: The surface tensions Tab,Tbc,Tca, denoting oil, water, air by a, b, c. Fig. 277 is an instant vertical view and displays the points O and O', where it intersects the circle.
For example, for the three forces T, acting at O, to be in equilibrium, each of them must balance the other two, just as the three forces in Fig. 24. Equilibrium becomes impossible, if even only one of them is larger than the other two together. In this case, it is impossible, because Twater-air = 0.073 is larger than Twater-oil = 0.073 and Toil-air = 0.073 together. Hence the interface water-air pulls apart the oil drop and spreads it over the water surface. During this process, the angle at the rim of the drop becomes ever sharper and approaches the angle 0º. This process can continue, if the water surface is large enough, until the thickness of the oil layer reaches the radius of molecular action, when it disintegrates completely and cannot be viewed anymore as a fluid. The motion, due to the excess of one of the three forces, can be characterized as follows: One of the fluids intercedes between the other two and separates them. If the separating fluid is air (it is oil here), one of the other two forms drops, and these drops rest on the other fluid (for example, water on fat) without moistening it, the separating skin of air stops the drops from touching the fluid.
Th excess weight of the one border tension over the sum of the other two also explains why there does not exist equlibrium, when c is a solid, a perfectly clean glass plate, and a drop of fluid: For example, if a is pure water, it is spread out over the entire surface and thereby presses the air away from the glass. If a is pure mercury, it withdraws completely from the glass, it forms spherical drops on top of it, the air spreads all over the glass and the mercury does not moisten the glass, a skin of air between the drop and the glass separates both (as in the case of rain drops on a material, which has been impregnated against contact with water).
If the forces T are in equilibrium, you can construct from the vectors T a triangle (Fig. 277). It can only be constructed , that is, the vectors close a triangle, if the sum of two sides is larger than the third (two of the vectors T together are larger than the third). Otherwise, the triangle and equilibrium are impossible, like when oil and water meet. (The external angles of the triangle yield the angles, at which the three bounding faces meet You find that Tbc/sin A = Tca/sin B = Tab/sin C. Hence the angles between the interfaces depend only on the surface tensions. Interfaces of the same three fluids, which are in equilibrium, form therefore always the same angles.
Note the special case, when a fluid and a solid meet, as in Fig. 278 the fluid borders partly against a solid, partly - as one says: With its free surface - against glass, for example, atmospheric air. If a, b, c are water, air and glass, then Tglass-air and Tglass-water act along the wall in opposite directions, the tension Twater-air acts along OP. The equilibrium of that part of the fluid tangential to the solid wall depends on the surface tensions Tab, Tbc ,Tca, for experiments show that OP takes on eventually a direction such that the component OQ of the surface tension balances the difference Tbc - Tca (the component perpendicularly to the wall becomes ineffective by the wall). Hence Tabcosa = Tbc - Tca. The angle a (POQ), defined by (Tbc - Tca)/Tab , is called the edge angle. It is sharp, if Tbc > Tca, blunt if Tca > Tbc. For example, the angle ABC is blunt, if the fluid a is mercury (Fig. 279); it is then 128º 52' (Georg Hermann Quincke 1834-1924). In the case of water, the angle at a perfectly clean water face vanishes completely, because then Tbc>Tab+Tca, that is, the water spreads over the entire wall and presses the air away, whence the edge angle gradually vanishes. If the glass is not perfectly clean, the angle can grow to 90º or more.
The curvature of the the bounding surfaces explains the capillary action. In Fig. 280, let AB and CD , respectively, be water and mercury surfaces at rest; they are horizontal, because they are at rest. Let E and F be two narrow glass tubes, 1 - 2 mm wide and open at both ends. If one end is dipped into the water, the water inside rises above the initial level and is bounded above by a upwards concave miniscus (Greek mhnh = Moon), dipped into the mercury, the mercury drops in the tube below the initial level and is on top bounded by an upwards concave meniscus.
These two phenomena have the explanation: As the water touches the glass wall, the particles of the boundary layer are pulled upwards and the initially plane interface is transformed into a concave surface. (The curvature becomes more visible, when a narrower tube is used.) That is the first action; the rising of the water is the result of this transformation. Analogously, as the mercury touches the wall, it forms its initially flat surface into a drop and thereby generates the upwards convex surface. Also here the transformation is the primary action, the sinking of the mercury in the tube the result of the transformation. The pressure difference (between the plane surface outside and the curved one inside the tube Fig. 280) drives the water up and the mercury down the tube. The difference in the heights inside and outside the tube is the larger, the narrower is the tube (the narrower the tube, the stronger the curvature of the enclosed surface, whence the pressure difference between inside and outside the tube is larger). The narrow tubes are called capillary tubes (Latin: capilla = hair) and the phenomena capillary actions. For example, the rising of a fluid in a porous body, for example, in lump sugar, in a sponge, in blotting paper, in the wick of a candle is due to capillary action.
If a fluid moistens the (clean!) surface of a body completely, it is covered totally by a skin of fluid, for example, if one dips it into the fluid and extracts it. The solid only serves the fluid skin as support and otherwise has not role. If a fluid stands in a tube, it has moistened (as the water in Fig. 280), it does not touch the glass; this is done by a skin of fluid, made out of it. The skin pulls the free surface of the fluid column in the tube upwards (the edge angle becomes zero, as the moistening is perfect), the surface of the column becomes therefore upwards concave and the fluid rises in the tube, corresponding to the inwards acting excess pressure; it rises so long until the weight of the fluid column above the outside level balances the upwards pull;; of the concave fluid skin. Hence: Hs=a·2/r, where H is the height of the column above the external level - you can neglect the differences inside the meniscus - s is the density of the fluid, a the capillary constant, r the radius of the circular tube and of the meniscus, which forms a hemisphere of radius r (as the edge angle is zero!). The formula a = rHs/2 leads to very exact measurements of a.
We realize that for the same fluid, the increased heights H are inversely proportional to the radii of the tubes (James Jurin 1718). In a tube with a diameter of 1 mm, at 8.5ºC, the capillary rise of distilled water is 30.05 mm, in a tube with a diameter of several m (= 0.001 mm) it is several metres. (This explains, for example, that in the sap tubes of plants - their diameters lie between 20m and 1m - water rises high, also that walls on a wet foundation become damp to the top unless the upper layers have been appropriately isolated from the lower ones.) The rise outside a moistened cylinder takes place just as inside a tube (for example, this effects appreciably the accuracy of the scale hydrometer) and also between two parallel, sufficiently close plates. The elevation is here half as high as in a capillary tube with the diameter of the distance between the plates. According to the theory, the elevation between free standing, vertical, moistened planes is h = (2a/s)1/2.
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