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its constituents in any state of relative motion, no atom will fly away from it, provided the total kinetic energy of the given initial motion does not exceed some definite limit. A gas is a vast assemblage of molecules thus defined, each moving freely through space, except when in collision with another cluster, and each retaining all its own constituents unaltered, or only altered by interchange of similar atoms ■between two clusters in collision.
§ 20. For simplicity we may suppose that each atom, A, has a definite radius of activity, a, and that atoms of different kinds, A, A', have different radii of activity, as, a!; such that A exercises no force on any other atom, A7, A", when the distance between their centres is greater than a. + a! or a!1. We need not perplex our minds with the inconceivable idea of “ virtue,” whether for force or for inertia, residing in a mathematical point* the centre of the atom; and without mental strain we can distinctly believe that the substance (the “ substratum ” of qualities) resides, not in a point, nor vaguely through all space, but definitely in the spherical volume of space bounded by the spherical surface whose radius is the radius of activity of the atom, and whose centre is the centre of the atom. In our intermolecular forces thus defined, we have no violation of the old scholastic law, “Matter cannot act where it is not,” but we explicitly violate the other scholastic law, “ Two portions of matter cannot simultaneously occupy the same space.” We leave to gravitation, and possibly to electricity (probably not to magnetism), the at present very unpopular idea of action at a distance.
§ 21. We need not now (as in § 16, when we wished to keep as near as we could to the old idea of colliding elastic globes) suppose the mutual force to become infinite repulsion before the centres of two atoms, approaching one another, meet. Following Boscovich, we may assume the force to vary according to any law of alternate attraction and repulsion, but without supposing any infinitely great force, whether of repulsion or attraction, at any particular distance; but we must assume the force to be zero when the centres are coincident. We may even admit the idea of the centres being absolutely coincident, in at all events some cases of a chemical combination of two or more atoms; although we might consider it more probable that in most cases the chemical combination is a cluster, in which the volumes of the constituent atoms overlap without any two centres absolutely coinciding.
§ 22. The word “collision” used without definition in § 19 may now, in virtue of §§ 20, 21, be unambiguously defined
* See Math, and Phys. Papers, vol. iii. nrt. xcvii. “ Molecular Consti-iution of Matter/’ § 14.
thus : Two atoms are said to be in collision during all the time their volumes overlap after coming into contact. They necessarily in virtue of inertia separate again, unless some third body intervenes with action which causes them to remain overlapping; that is to say, causes combination to result from collision. Two clusters of atoms are said to be in collision when, after being separate, some atom or atoms of one cluster come to overlap some atom or atoms of the other. In virtue of inertia the collision must be followed either by the two clusters separating, as described in the last sentence of § 19, or by some atom or atoms of one or both systems being sent flying away. This last supposition is a matter-of-fact statement belonging to the magnificent theory of dissociation, discovered and worked out by Sainte-Clair Deville without any guidance from the kinetic theory of gases. In gases approximately fulfilling the gaseous laws (Boyle's and Charles'), two clusters must in general fly asunder after collision. Two clusters could not possibly remain permanently in combination without at least one atom-being sent flying away after collision between two clusters with no third body intervening *.
§ 23. Now for the application of the Boltzmann-Maxwell doctrine to the kinetic theory of gases: consider first a homogeneous single gas, that is, a vast assemblage of similar clusters of atoms moving and colliding as described in the last sentence of § 19; the assemblage being so sparse that the time during which each cluster is in collision is very short in comparison with the time during which it is unacted on by other clusters, and its centre of inertia, therefore,, moves uniformly in a straight line. If thore are i atoms in each cluster, it has 3i freedoms to move, that is to say, freedoms in three rectangular directions for each atom. The Boltzmann-Maxwell doctrine asserts that the mean kinetic energies of these Si motions are all equal, whatever be the mutual forces between the atoms. From this, when the durations of the collisions are not included in the time-averages, it is easy to prove algebraically (with exceptions noted below) that the time-average of the kinetic energy of the component translational velocity of the inertial centre t, in any direction, is equal to any one of the 3i mean kinetic energies asserted to be equal to one another in the preceding statement. There are exceptions to the algebraic proof
* See Kelvin’s Math, and Phys. Papers, vol. iii. Art. xcvn. § 33. In this reference, for “scarcely” substitute “not.”
t This expression I use for brevity to signify the kinetic energy of the whole mass ideally collected at the centre of inertia.