the opening through which it passes. The actual volume of liquid that escapes is only about 62 per cent. of the volume calculated from the expression a√2gH, while the cross section of the vena contracta is about .62 times the cross section of the opening. The quantity of liquid which escapes can be considerably increased if a small cylindrical tube or ajutage, of the same diameter as the opening, is fitted to the aperture. In this case the outflow may be increased to about 82 per cent. of the calculated amount. If the ajutage is of considerable length, the outflow is again reduced, this being due to viscosity, i.e. friction between the different parts of the liquid. CHAPTER XVIII MOLECULAR PHENOMENA IN LIQUIDS 156. Cohesion.—If a rod or tube of glass is dipped into water and is then withdrawn, a drop of the liquid will be left hanging to the end of the rod. If more water is carefully added, the size of the drop will increase until its weight is sufficient to tear it away from the glass. In the same way, if a clean metal ring is dipped into a solution of soap and then withdrawn, a film of the liquid will remain stretched across the ring. In both these cases the effects are said to be due to the cohesion of the liquid. The term adhesion is, however, sometimes used to indicate the attraction manifested between a liquid and a solid, and the term cohesion restricted to the attraction between the different particles of a mass of liquid. This cohesive force is in most cases masked by the action of gravity, and hence to observe its effects we require to reduce the effects of gravity to a minimum. Thus, if a large drop of oil is placed on the surface of water it immediately spreads. If, however, a mixture of alcohol and water is prepared of exactly the same density as the oil, and a drop, or even a considerable volume, of oil is introduced in the water, it immediately gathers itself into a sphere which remains suspended in the alcohol and water. By floating the oil in a liquid of the same density as itself we remove it from the influence of gravity, and then the cohesion between the liquid particles causes the drop to assume the spherical form. 157. Surface Tension. In the case of the globule of oil floating in a liquid of the same density, the shape assumed is the same as the oil would take had it been enclosed in an elastic membrane or skin. The presence of such an elastic skin would also serve to explain the formation of the drops on the end of the glass rod or the soap film. We can explain these facts on the molecular hypothesis in which it is assumed that in a liquid the molecules exert on one another an attractive force; this force, however, being only appreciable when the molecules are within a short distance of one another, which is called the range of molecular attraction. If we describe a sphere with any particle as centre having a radius equal to the range of molecular attraction, then we may neglect the effects of all the molecules which lie outside this sphere on the molecules at the centre. E In the case of a molecule A (Fig. 130) well within a liquid, the whole sphere will lie within the liquid, and hence the molecule A will be attracted by the neighbouring molecules equally in all directions. If, however, the molecule (B) is so near the surface of the liquid, E F, that the sphere would in all directions. The the molecule B, will have a resultant which is zero. The attractions of the molecules within the shaded part will, however, have a resultant directed towards the inside of the liquid mass, and perpendicular to the surface. In the case of a molecule actually on the surface, as at C, this resultant is a maximum. The effect of these unbalanced molecular forces acting on the molecules near the surface is to exert a pressure on the interior of a liquid mass, similar to that which would be caused by an elastic skin, and it is frequently convenient to speak as if such an elastic skin really existed, and to say that this pressure within a liquid mass is due to the surface tension of the liquid. The magnitude of the pressure due to the surface tension depends on the form of the liquid surface. Let us take the case of three molecules, E E E A Fig. 131. A, B, and C (Fig. 131), at equal distances, less than the radius of molecular attraction, from the surface EF, which in the first case is plane, in the second concave, and the third convex, and, as before, let us indicate by shading the part of the sphere of molecular attraction which is efficacious in producing an inwardly directed force on the molecule. If the surface is concave as at B, then, although the molecule B is at the same distance below the surface as is A, where the surface is plane, the shaded part is less, so that the molecular force acting on B towards the inside of the liquid is less than that on A. In the case where the liquid surface is convex (C), the shaded part is larger than in A, and hence the force is larger. Looking at it from the point of view of an elastic membrane, it is evident that at B the elasticity of the membrane would diminish the pressure within the liquid, while at C it would increase the pressure. The existence of this pressure due to molecular, as distinct from gravitational attractions, cannot be experimentally demonstrated, but there are many striking phenomena depending on the fact that the surface of a liquid is in a state of tension. Thus if a metal ring is dipped in a solution of soap, and a small loop of cotton, which has been previously moistened with the solution, is placed on the film left on the ring, this loop can be made to take up any form such as A (Fig. 132), and will retain this form. If, however, the film within the loop is broken, the loop immediately takes up the circular form shown at B; and if it is now deformed in any way, on being released it immediately springs back to the circular form. This behaviour is due to the fact that, in the first case, the surface tension of Fig. 132. B the liquid film acts equally on both sides of the cotton, but when the film inside the loop is broken, the surface tension only acts on one side, and hence draws the loop out into a circle. Another method of showing the surface tension is by means of a bent wire ABC (Fig. 133) and a straight wire DE, which simply rests against this. If D B -E a soap film is formed in the enclosed space DEE, it will be found that the surface tension acting on DE is able to support not only the weight of the wire DE, but also a small weight w. This arrangement might also be used to obtain a rough measure of the amount of the surface tension. If W is the mass of D' the cross wire DE and its attached weight, then the surface tension of the film supports weight W, and therefore exerts a force of Wg units of force. The surface tension of the film acts all along the portion of the wire DE, intercepted between the legs of the bent wire, and acts at right angles to the wire. Since the film has two surfaces, if the force exerted on unit length of DE due to the surface tension of one side of the film be 7, then the whole upward force on A W Fig. 133. DE due to surface tension is 277, where 7 is the length of DE in contact with the film. Hence if there is equilibrium The quantity Tis called the surface tension of the liquid, and is the force exerted across unit length taken along the surface of the liquid. In the c.g.s. system the surface tension is measured in dynes per centimetre. The dimensions of surface tension, are [Force]÷[Length] or [MT-2]. B F In the arrangement shown in Fig. 133, the two limbs AB, BC are not parallel, for if they were the arrangement would not be in stable equilibrium, but in neutral. For in this case (Fig. 134) the length / of the film in contact with the movable rod EF is constant, and Chence the force 27 exerted by the film is independent of the position of the rod EF. Since the downward force 'g is also independent of the position of EF, if these two forces are exactly equal the rod EF will remain wherever it is put. If, however, we have not succeeded in exactly adjusting W to the right value, then if W is too small EF will be drawn up till it is in contact with BC, or if Wis too great EF will fall till the ends A and D are reached, when the film will break. When the side wires are inclined as in Fig. 133, the length of the film in contact with DE, and hence the force exerted by surface tension varies with the position of the cross bar. If, when the bar is at DE, Wg is greater than 27, the bar will fall to some such position as D'E'; so that the new value of 1, say l, exactly fulfils the condition 27 Wg. If, on the other hand, I is too small, the bar will rise and diminish till this relation is fulfilled. A Wg FIG. 134. In the case of the arrangement shown in Fig. 134, if we start with EF in contact with BC, and then pull it down into the position shown, we shall in doing this have to do work, since we are moving EF against a force of 27.EF. The work done is 2 T. EFX BE, since BE is the distance through which EF has been moved against the force. The energy corresponding to this work is stored up in the film, and may be recovered by allowing the film to contract. Hence if E is the energy of the film due to the surface tension, or the superficial energy, we have E=2T. EFX BC. |