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I. The Attraction of Unlike Molecules.-I. The Diffusion of
Gases. By WILLIAM SUTHERLAND *.

IN my previous papers on the laws of molecular force the

attractions of like molecules have been under consideration, and the results seemed to indicate that the laws of the attractions of unlike molecules would not be difficult to ascertain. With the hope of determining the general law of attraction of any two molecules, I adopted two methods as being at present available for giving values of the attractions of unlike molecules, namely, that of the Diffusion of Gases and that of the Surface-Tension of Mixed Liquids. Both methods have led to the same result, viz., that if the attraction between two molecules M, of mass m, at distance r apart be denoted by 3A1m12/r, and that between two molecules M2 by 3A2m22/4, then the attraction between an M, and an M, is

or the attraction of two unlike molecules is equal to the square root of the product of the attractions of the corresponding like molecules at the same distance apart. As the expression 3A,m,2 for two like molecules can be split into two parts 3Am1, the general law of the attraction of any two molecules, like or unlike, can be stated thus:-Any two molecules attract one another with a force inversely proportional to the fourth power of the distance between them

* Communicated by the Author. Phil. Mag. S, 5. Vol. 38. No. 230. July 1894.

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and directly proportional to the product of the parameters √3Am characteristic of each. Although the parameter is written in the form √3Am apparently involving the mass m it will be shown that 3Am is independent of the mass m but is a function of the size of the molecule; it would therefore be better denoted by a single symbol a, so that the attraction between any two molecules M1 and M2 is a1a/r4, or between two M1 is a12/4, the parameter a being a function of the size of the molecule but not directly of its mass. Thus, with Gm1m/r2 to denote the gravitational attraction of two molecules M1 and M2, the general expression for the force between them is

Gm1m2/r2+a1a2/r4.

The dependence of the coefficient of diffusion of two gases on the attraction between their molecules was indicated in general terms in a recent paper on the Viscosity of Gases and Molecular Force (Phil. Mag., Dec. 1893). In that paper it was shown that in those parts of the kinetic theory of gases which depend on the number of encounters of a molecule per second (or, in other words, on its mean free path), the effect of molecular force cannot be neglected as of only secondary importance; it is fundamental. Thus the complete expression for the coefficient of diffusion of two gases will involve the attractions between their molecules in a manner now to be established; but as the kinetic theory of the diffusion of gases, even when simplified by treating the molecules as forceless, is in a little confusion (there being at least three forms of expression for the diffusion-coefficient in the field), it may be desirable to recapitulate briefly the theories of the diffusion of forceless molecules from the three points of view.

The first in time is that of Stefan, accepted by Maxwell; the second is O. E. Meyer's, given in his book on the 'Kinetic Theory of Gases ;' and the third is that of Tait (Trans. Roy. Soc. Edin. xxxiii.), who has treated the diffusion of gases rather elaborately.

Stefan's theory is this :-If two gases are diffusing into one another, then at any point one has a general velocity a in one direction, and the other a velocity as in the other, the density of the first diminishes in the direction of a1, of the second in that of a2. Consider, then, an element of the first of section unity and length 8x in the direction of a the partial pressure due to its molecules at one end is p1, and at the other pi+dx dp1/dx, so that there is a driving pressure Sx dp fde which is resisted by a resistance like friction offered by the other gas in the length da, which may be denoted by

:

R Sa, and then any acceleration of the element of the first gas is due to the force dx (dp1/dx-R), but this acceleration can be neglected in comparison with either dx dp1/dx or R dx,

and we have

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R has now to be evaluated; it is the resistance offered by the molecules of the two sets in unit volume at x to one another's motion. Let v be the number of encounters per second between n1 molecules of the first set in unit volume and n, of the second, and let μ be the average value of the momentum communicated when a molecule of the first set with velocity a collides with one of the second with velocity a in the opposite direction, then

R=vμ.

If α1 and a, are the radii of the molecules of the two sets treated as spheres, and 3,2/2 and 322/2 are their mean square velocities, then (see for instance Tait, Trans. Roy. Soc. Edin. xxxiii.)

2

v=2n ̧n2(α1+a2)2π3 (k,2+k22)3,

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Now in diffusion the pressure remains constant, so that as many molecules of one sort pass in one direction as of the other in the other, or n11=na1⁄2, and accordingly

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but n1m1=p1, the density of the first gas, and

P1=n1112/2=p112/2, so that (1) becomes

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The coefficient of dp/dx is by definition the coefficient of diffusion D; remembering that K12/K22=m2/m1, we get

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Meyer's method of proceeding is quite different. He says that as the density of one of the diffusing gases diminishes in one direction, then, if a plane is drawn anywhere at right angles to this direction, the density increases on one side and diminishes on the other, so that more molecules cross the

plane from the side of increasing density than from that of decreasing density, and diffusion results. Thus if n, is the number of molecules per unit volume at the plane, that at a small distance x from it will be n1+xdn1/dx. The number of molecules leaving an element de after encounter in it to cross the plane before the next encounter must be proportional to n+xdn/dx, to da, to the mean number of collisions per second v11, where A is the mean free path of the molecules of the first set near a, and finally to e- the probability of a path greater than , so that the number of molecules of the first set which cross the plane from one side in unit time is proportional to

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(although was stipulated to be small to justify the expression n1+xdn1/da, no harm can come of integrating to co because the value of the integral becomes negligible for all values of x greater than a few times λ). The number of molecules crossing from the other side is proportional to

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so that the excess accumulating in unit time on one side is proportional to

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that is to 2vdn/da. The number of molecules of the other set crossing in the opposite direction is proportional to 2v2λdn2/dx. As these two expressions are not equal, there is a gain of molecules on one side of the plane and a loss on the other proportional to

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and to preserve the uniformity of pressure Meyer supposes a bodily motion of the mixed gases to take place so as to carry this number of molecules in the opposite direction, of which the fraction n1/(n1 + n2) belongs to the first set and n2/(n1+n2) to the second thus the diffusion-stream of the first gas is proportional to

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that of the other being equal and opposite. On account of the uniformity of pressure, dn,/dx=dng/da and the diffusion

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The defect of Meyer's theory is that it takes no account of the actually existing diffusion velocities in the two sets. Supposing the process of diffusion to be arrested suddenly by some cause which then suddenly ceases to act, Meyer's method shows how the diffusion-streams would begin to flow again, but it cannot follow the process after that, because it takes no cognizance of the bodily motions existing in the two sets.

Tait's theory takes account of both Stefan's and Meyer's causes; he supposes the molecules of each medium besides their velocities of agitation to have velocities of translation en masse a1 and a2, and then calculates the quantities of each flowing in unit time across unit section, these quantities depending on aj on a1 and a, and on expressions similar to Meyer's. The velocities a1 and a, are determined exactly as in Stefan's method, so that Tait's method labours under this difficulty, that he supposes each molecule of each set to have a certain velocity combined with the velocity of agitation, and yet this velocity is different from that of the set as a whole.

On theoretical grounds, therefore, Stefan's theory appears. not to have been improved by the later attempts, and, further, it seems to me not to have been sufficiently recognized that Stefan has given satisfactory experimental proof of the soundness of his method of treating the diffusion problem; for exactly on the lines of his theory of the diffusion of two gases into one another he constructed a theory of the evaporation of a liquid into a gas, which led to a striking formula for the velocity of evaporation of a liquid, a formula verified by his own and Winkelmann's experiments. As Stefan's elegant theory of evaporation will only take a few lines to reproduce here, and as it gives a valuable method of determining diffusion-coefficients, it may as well be reproduced in the present connexion.

Suppose a tube half filled with a liquid whose properties are to be denoted by suffix 1 evaporating into an atmosphere with suffix 2, but with fresh liquid added from below so as always to keep the free surface of the liquid at a fixed mark on the tube; then, when a stationary state is established, there is a steady diffusion-stream of the vapour through the upper

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