Page:Encyclopædia Britannica, Ninth Edition, v. 5.djvu/72

Rh GO CAPILLARY ACTION tension of trie film when drawn out to different degrees of thinness may possibly lead to an estimate of the range of the molecular forces, or at least of the depth within a liquid mass, at which its properties become sensibly uni form. We shall therefore indicate a method of iuvestigat- iu (T the tension of such films. O Let S be the area of the film, M its mass, and E its energy ; a the mass, and e the energy of unit of area ; then M = Sr ....... (11), E^Se ....... (12). Let us now suppose that by some change in the form of the boundary of the film its area is changed from S to S -I- dS. If its tension is T the work required to effect this increase of surface will be IdS, and the energy of the film will be increased by this amount. Hence But since II is constant, (13). (14). Eliminating dS from equations (13) and (14), and dividing by S, we find &amp;lt; In this expression a denotes the mass of unit of area of the film, and e the energy of unit of area. If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth 2, then &amp;lt;j=f C edz ....... (16), J o and Both p and x are Junctions of z, the value of which remains the eame when z - c is substituted for z. If the thickness of the film is greater than 2f, there will be a stratum of thickness c - 2t in the middle of the film, within which the values of p and x will be/&amp;gt; and Xo- In the two strata on either side of this the law, according to which p and x depend on the depth, will be the same as in a liquid mass of large dimensions. Hence in this case o = (c-2)p + 2/*pck ...... (18), e = (c-2 f )x po + 2*xpdv ..... (19), dff de
 * u

Hence the tension of a thick film is equal to the sum of the ten sions of its two surfaces as already calculated (equation 7). On the hypothesis of uniform density we shall find that this is true for films whose thickness exceeds. The symbol x is defined as the energy of unit of mass of the sub stance. A knowledge of the absolute value of this energy is not required, since in every expression in which it occurs it is under the form x - Xo&amp;gt; that is to say, the difference between the energy in two different states. The only cases, however, in which we have experi mental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. It is impossible to make direct measurements of the properties of particles of the substance within the insensible distance of the bounding surface. When a liquid is in thermal and dynamical equilibrium with its vapour, then if p and x are the values of p and x for the vapour, and p and x u those for the liquid, where J is the dynamical equivalent of heat, L is the latent heat of unit of mass of the vapour, and p is the pressure. At points in the liquid very near its surface it is probable that x is greater than Xo, and at points in the gas very near the surface of the liquid it is probable that x is less than x, but this has not as yet been as certained experimentally. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics. We have next to determine the value of x in terms of the action between one particle and another. Let us suppose that the force between two particles m and in at the distance /is .... (22), being reckoned positive when the force is attractive. The actual force between the particles arises in part from their mutual gravita tion, which is inversely as the square of the distance. This foice is expressed by mm -^. It is easy to show that a force subject to this law would not account for capillary action. We shall, therefore, iu what follows, consider only that part of the force which depends on &amp;lt;(/), where &amp;lt;p(f) is a function of / which is insensible for all sensible values of/, but which becomes sensible and even enormously great when /is exceedingly small. If we next introduce a new function of /and write (23), then in m H(f) will represent 1. The work done by the attractive force on the particle in, while it is brought from an infinite distance from m to the distance/ from m ; or 2. The attraction of a particle m on a narrow straight rod resolved in the direction of the length of the rod, one extremity of the rod being at a distance /from m, and the other at an infinite distance, the mass of unit of length ot the rod being m. The function n(/) is also insensible for sensible values of/, but for insensible values of /it may become sensible and even very great. If we next write f (24), then 2Trmal/(z) will represent 1. The work done by the attractive force while a particle in is brought from an infinite distance to a distance 2 from an infinitely thin stratum of the substance whose mass per unit of area is a ; 2. The attraction of a particle m placed at a distance 2 from the plane surface of an infinite solid whose density is cr. Let us examine the case in which the particle m is placed at a distance 2 from a curved stratum of the sub stance, whose principal radii of cur vature are Rj and U 8. Let P (fig. 2) be the particle and PB a normal to the surface. Let the plane of the paper be a normal section of the surface of the stratum at the point B, making an angle u with the sec tion whose radius of curvature is R,. Then if is the centre of curvature in the plane of the paper, and BO = u, Fig. 2. ^c^^sinA, (25) _ Let POQ =, P0 = r , PQ=/ , BP-z, /- -u* + r*- 2ur cos. 6 (26). The element of the stratum at Q may be expressed by &amp;lt;ru* sin. 6 d6du , or expressing dB in terms of df by (26), u Multiplying this by m and by n/, we obtain for the work done by the attraction of this element when m is brought from an infinite distance to Pj, Integrating with respect to/ from fz iofa, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work Cma ~ r ( ty (c) - f,(a) ]d&amp;lt;a, and since ty(a) is an insensible quantity we may omit it. We may also write since z is very small compared with u, and expressing u in terms or so by (25), we find r lir , I /COS. 2 o&amp;gt; sin. 8 a&amp;gt; ) / maty(z) +z  ,-7 + 5 } du&amp;gt; o n I V K, l&amp;gt;a / } This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a