amount of positive electrification within a sphere of radius 6 :

4π

3

thus ve/b is equal to P, where p is the density of the positive electrification in the sphere: thus, if the density of the electrification be kept constant, the radius of the ring will be independent of the size of the sphere. Now let us take a large sphere and place within it a ring of such a size that the ring would be in stable equilibrium if its centre were at the centre of the sphere. To fix our ideas, let us take the case of three corpuscles at the corners of an equilateral triangle, and place this triangle so that its centre O' is no longer at the centre of the sphere: we can easily see that the corpuscles will remain at the corners of an equilateral triangle of the same size, and that the triangle will move like a rigid body acted upon by a force proportional to the distance of its centre from O the centre of the sphere. To prove this we notice that the repulsion between the corpuscles is the same as when the centre of the triangle is at O. The attraction of the sphere on a corpuscle P is proportional to OP, and so may be resolved into two forces, one proportional to O'P along PO' (O' is the centre of the triangle) and the other proportional to 00' acting along O'O. Now the corpuscles are by hypothesis in equilibrium under their mutual repulsions, and the attraction to the centre proportional to O'P: thus the relative position of the corpuscles will remain unaltered, and the system of three corpuscles will move as a rigid body under a central force acting on its centre of gravity proportional to the distance of that point from the centre of the sphere.

be ex

The three corpuscles will, at a point whose distance from their centre is large compared with a side of the triangle, produce the same effect as if the charges on the three corpuscles were condensed at the centre of the triangle; they will thus at such points act like a unit, and the results we have previously obtained for single corpuscles may tended to the case when the single corpuscles are replaced by rings of corpuscles which would by themselves be in equilibrium. It should be noted that the atom in which these systems are placed must be large enough to allow these rings of corpuscles-sub-atoms we may call them, to be separated by distances considerably greater than the distance between the corpuscles in one of the rings.

If we regard the atoms of the heavier elements as produced by the coalescence of lighter atoms, it is reasonable to suppose that the corpuscles in the heavier atoms may be arranged in secondary groups or sub-atoms, each of these groups acting

as a unit. When the corpuscles are done up in bundles in this way, it is possible to have stability when these bundles are arranged in a ring with a smaller number of corpuscles inside than when the corpuscles in the bundles are arranged at equal intervals round the circumference of the ring. Thus, take the case of a ring of 30 corpuscles; if these were arranged at equal intervals, 101 corpuscles would be required inside the ring to make it stable. If, however, the 30 corpuscles were grouped in ten sets of three each, only 3x3=9 corpuscles in the interior would be required to make the arrangement stable.

Constitution of the Atom of a Radioactive Element.

Our study of the stability of systems of corpuscles has made us acquainted with systems which are stable when the corpuscles are rotating with an angular velocity greater than a certain value, but which become unstable when the velocity falls below this value. Thus, to take an instance, we saw (p. 249) that four corpuscles can be stable in one plane at the corners of a square, if they are rotating with an angular velocity greater than 325ve/mb3, but become unstable if the velocity falls below this velocity, the corpuscles in this case tending to place themselves at the corners of a tetrahedron. Consider now the properties of an atom containing a system of corpuscles of this kind, suppose the corpuscles were originally moving with velocities far exceeding the critical velocity; in consequence of the radiation from the moving corpuscles, their velocities will slowly-very slowly-diminish; when, after a long interval, the velocity reaches the critical velocity, there will be what is equivalent to an explosion of the corpuscles, the corpuscles will move far away from their original positions, their potential energy will decrease, while their kinetic energy will increase. The kinetic energy gained in this way might be sufficient to carry the system out of the atom, and we should have, as in the case of radium, a part of the atom shot off. In consequence of the very slow dissipation of energy by radiation the life of the atom would be very long. We have taken the case of the four corpuscles as the type of a system which, like a top, requires for its stability a certain amount of rotation. Any system possessing this property would, in consequence of the gradual dissipation of energy by radiation, give to the atom containing it radioactive properties similar to those conferred by the four corpuscles.

XXV. The Solubility and Diffusion in Solution of Dissociated Gases. By O. W. RICHARDSON, M.A., B.Sc., Clerk Maxwell Student and Fellow of Trinity College, Cambridge*

THE

HE solubility of gases which act chemically on the solvent, or which dissociate in solution, is treated in Van t'Hoff's Lectures on Theoretical Chemistry, pt. ii. p. 28 et seq. It is there pointed out, on theoretical grounds, that gases which dissolve without chemical action, or which associate with the solvent in such a way that each aggregate contains only one molecule of the dissolved gas, obey Henry's Law of proportionality between the pressure and the mass of gas dissolved. As all gases which have been examined appear to obey Henry's Law, with the exception of ammonia, sulphur dioxide, and hydrochloric acid in water, this subject has not hitherto attracted much attention. Phenomena of this kind appear, however, to characterize the absorption of hydrogen by palladium and platinum, and, probably, of other gases by other metals (for instance, carbon monoxide by iron). The recent experiments of Winkelmann on the variation of the rate of diffusion of hydrogen through hot palladium and platinum, with the driving pressure, led him to the conclusion that the hydrogen was partly dissociated into atoms, and that only the atoms were capable of passing through the hot metal. A series of experiments on the rate of diffusion of hydrogen through platinum at different pressures and temperatures has just been carried out by the author in conjunction with Messrs J. Nicol and T. Parnell, and will shortly be published. This investigation, so far as the pressure relations are concerned, has yielded results. similar to those of Winkelmann, and most of the phenomena appear to be capable of explanation on the view that the hydrogen dissolves in the platinum and then dissociates (partially, at any rate) into atoms. It was in seeking an explanation of these results that the author was led to examine into the theory of the solubility of a dissociating gas and to the results which are given in the present

communication.

§ 1. Solubility Relations.

We shall confine our attention to the case of a gas in which each molecule dissociates into n similar molecules. The same methods could, of course, be applied to a more complicated case if it should occur. The reaction which we are considering is symbolized by a chemical equation of the Communicated by the Author.

+ Drude's Annalen, vol. vi. p. 104.

Ibid. vol. viii. p. 388.

type X=nX and takes place, to a greater or less extent, both in the solution and in the surrounding gas. In the investigation given by Van t'Hoff (loc. cit.) certain relations are deduced by making the dissociated portion of the gas obey Henry's Law. It is evident, however, that for a steady state not only must there be equilibrium between the free and dissolved parts of the dissociated gas, but the undissociated portion must also be governed by a similar relation. It is not sufficient for equilibrium merely to postulate equality between the total amount of gas entering and leaving the solvent in a given time. It is necessary that the amount entering and leaving should be the same for each constituent. The only alternative is to suppose the gas to enter the solution in one form, to dissociate or recombine there and leave in the other form. Such processes involve a continuous transfer of heat at a rate depending on the value of the heat of dissociation. It is thus necessary that there should be a separate relation between the concentrations of the free and dissolved portions of each constituent; this reasoning is true whatever be the nature of the relation, quite apart from its assuming the special form of Henry's Law.

We have to take into account then four different equilibrium conditions. We have two equations which determine the relation between the undissociated and dissociated constituents of the dissolved, and undissolved, gas respectively,. and two more equations which make the internal concentration proportional to the external concentration of each constituent. If these relations do not hold it is easy to see that perpetual motion is obtained.

Let the suffix o denote the gas outside, and inside the solution. Let C be the concentration of the undissociated, and c of the dissociated portion. The equations which determine the equilibrium between the dissociated and undissociated portions of the gas inside and outside the solution respectively are then :

where k and k, are the dissociation constants of the freeand dissolved gas respectively. In general ko will not be equal to ki, as for instance in the case of an acid gas like HC where electrolytic dissociation occurs in solution. Applying Henry's Law to each of the two constituents we get two further equations, viz.:

C=AC1_and_c=acı,

where A and a are the inverses of the solubilities of the undissociated gas and of the products of dissociation respectively.

By eliminating the concentration from these equations we obtain an interesting relation between the constants, viz. :

In other words, the solubility of the products of dissociation is determined absolutely by the solubility of the undissociated substance, together with the two dissociation constants. In the simplest case, where the two dissociation constants are equal, the solubility of the dissociation products is the nth root of that of the original substance.

These results may be confirmed and extended by treating the subject thermodynamically. We can obtain a reversible cycle, at constant temperature, as follows:-Suppose we have a cylinder whose walls are perfect conductors of heat and supplied with a piston at each end. Across the middle of the cylinder is a slice of the solution we are considering. The two sides of the slice are bounded by semipermeable membranes, one end allowing only undissociated, and the other only dissociated molecules to pass. Initially the external gas is in equilibrium with that inside the solution at both ends. Since the diaphragms are only permeable to one of the two gases present, this does not necessarily imply equilibrium between the internal and external gas at any one end as regards both constituents, but only as regards one constituent. According to the result we obtained before, this would involve equality in the total pressures as well; since, as we have already seen, there is one total pressure for which the constituent gases are in equilibrium with the internal ones. We shall show that this equality follows thermodynamically; although any further proof cannot be regarded as strictly necessary, since the result is merely a particular case of Gibbs's general theorem regarding the equilibrium of mixed systems.

Let us suppose the partial pressures of the undissociated gas (X) and of the dissociated gas (X) on the side permeable to X are P2 and p2 respectively, the corresponding quantities on the other side being P, and p1. Then a volume V, of gas is forced through the X, diaphragm, into the solution, a corresponding quantity being withdrawn through the X side so as to maintain the total internal pressure constant. Owing to the supposed difference of pressure on the two sides, the volume V, withdrawn will not be the same as V2, but is given by the modified law of Boyle and Charles for a dissociating gas, viz. :