CAPILLARY ACTION 59 some cases diffusion takes place to a limited extent, after which the resulting mixtures do not mix with each other. The same substance may be able to exist in two different states at the same temperature and pressure, as when water and its saturated vapour are contained in the same vessel. The conditions under which the thermal and mechanical equilibrium of two fluids, two mixtures, or the same sub stance in two physical states in contact with each other, is possible belong to thermodynamics. All that we have to observe at present is that, in the cases in which the fluids do not mix of themselves, the potential energy of the system must be greater when the fluids are mixed than when they are separate It is found by experiment that it is only very close to the bounding surface of a liquid that the forces arising from the mutual action of its parts have any resultant effect on one of its particles. The experiments of Quincke and others seem to show that the extreme range of the forces which produce capillary action lies between a thousandth and a twenty thousandth part of a millimetre We shall use the symbol e to denote this extreme range, beyond which the action of these forces may be regarded as insensible. If x denotes the potential energy of unit of mass of the substance, we may treat x as sensibly con stant except within a distance e of the bounding surface of tbe fluid. In the interior of the fluid it has the uniform value x^ hi like manner the density, p, is sensibly equal to the constant quantity o , which is its value in the interior of the liquid, except within a distance of the bounding surface. Hence if V is the volume of a mass M of liquid bounded by a surface whose area is S, the integral M ^ (1), where the integration is to be extended throughout the volume V, may be divided into two parts by considering (separately the thin shell or skin extending from the outer surface to a depth c, within which the density and other properties of the liquid vary with the depth, and the interior portion of the liquid within which its properties are constant. Since e is a line of insensible magnitude compared with the dimensions of the mass of liquid and the principal radii of curvature of its surface, the volume of the shell whose surface is S and thickness e will be Se, and that of the interior space will be Y - St. If we suppose a normal v less than t to be drawn from the sur face S into the liquid, we may divide the shell into elementary shells whose thickness is dv, in each of which the density and other pro perties of the liquid will be constant. The volume of one of these shells will be Sdv. Its mass will be Spd*. The mass of the whole shell will therefore be sf f pdv, and that of the interior part of the liquid (V - Sf)p . "We thus find for the whole mass of the liquid To find the potential energy we have to integrate (3). Substituting xf f r P in the process we have just gone through, we find Multiplying equation (2) by xo an( i subtracting it from (4), E-M Xo = Xo S/ o ( X - Xo W". . . . (5). In this expression M and X o are both constant, so that the varia tions of the right hand side of the equation is the same as that of the energy E, and expresses that part of the energy which depends on the area of the bounding surface of the liquid. We may call this the surface energy. The symbol x expresses the energy of unit of mass of the liquid at a depth v within the bounding surface. When the liquid is in contact with a rare medium, such as its own vapour or any other gas, x ls greater than xo> an< i the surface energy is positive. By the principle of the conservation of energy, any displacement of the liquid by which its energy is diminished will tend to take place of itself. Hence if the energy is the greater, the greater the area of the exposed surface, the liquid will tend to move in such a v. ay as to diminish the area of the exposed surface, or in other words, the exposed surface will tend to diminish if it can do so consistently with the other conditions. This tendency of the surface to contract itself is called the surface-tension of liquids. M. Dupre has described an arrangement by which the surface- tension of a liquid film may be illustrated. A piece of sheet metal is cut out in the form AA (fig. 1). A very fine slip of metal is laid on it in the position BB, and the whole is dipped into a solution of soap, or M. Plateau s glycerine mix ture. When it is taken out the rectangle AACC is filled up by a liquid film. This film, however, tends to contract on itself, and the loose strip of metal BB will, if it is let go, be drawn up towards AA, provided it is sufficiently light and smooth. Let T be the surface energy per unit of aiea; then the energy of a surface of area S will be ST. If, in the rectangle AACC, AA -a, and CC-ft, its area is S = ab, and its energy Tab. Hence if F is the force by which the slip BB is pulled towards AA 1 , =T Tn6 = Ta (6), or the force arising from the surface-tension acting on a length a of the strip is Ta, so that T represents the surface-tension acting transversely on every unit of length of the periphery of the liquid surface. Hence if we write ve may define T either as the surface-energy per unit of area, or as the surface-tension per unit of contour, for the numerical values of these two quantities are equal. If the liquid is bounded by a dense substance, whether liquid or solid, the value of x may be different from its value when the liquid has a free surface. If the liquid is in contact with another liquid, let as distinguish quantities belonging to the two liquids by suffixes. We shall then have E 1 -M lX oi = S <1 (x 1 -Xoi)pAi (8), Adding these expressions, and dividing the second member by S, we obtain for the tension of the surface of contact of the two liquids TI,, = /* (Xi - XoiM", +/**(Xi - XwWfri 0>- J e J o If this quantity is positive, the surface of contact will tend to contract, and the liquids will remain distinct. If, however, it were negative, the displacement of the liquids which tends to enlarge the surface of contact would be aided by the molecular forces, so that the liquids, if not kept separate by gravity, would at length become thoroughly mixed. No instance, however, of a phenomenon of this kind has been discovered, for those liquids which mix of themselves do so by the process of diffusion, which is a molecular motion, and not by the spontaneous puckering and replication of the bounding surface as would be the case if T were negative. It is probable, however, that there are many cases in which the integral belonging to the less dense fluid is negative. If the denser body be solid we can often demonstrate this; for the liquid tends to spread itself over the surface of the solid, so as to increase the area of the surface of contact, even although in so doing it is obliged to increase the free surface in opposition to the surface-tension. Thus water spreads itself out on a clean surface of glass. This shows that/* (x - XoW" must be negative for water in contact with glass. J o Ox THE TENSION OF LIQUID FILMS. The method already given for the investigation of the surface-tension of a liquid, all whose dimensions are sensible, fails in the case of a liquid film such as a soap- bubble. In such a film it is possible that no part of the liquid may be so far from the surface as to have the potential and density corresponding to what we have called
the interior of a liquid mass, and measurements of the