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 there 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 3 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 timeaverages, 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 †, 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. xcvi. § 33. In this reference, for "scarcely" substitute "not."

This expression I use for brevity to signify the kinetic energy of the whole mass ideally collected at the centre of inertia.

1

corresponding to the particular exception referred to in the last footnote to § 18 above; but, nevertheless, the general Boltzmann-Maxwell doctrine includes the proposition, even in those cases in which it is not deducible algebraically from the equality of the 3i energies. Thus, without exception, the average kinetic energy of any component of the motion of the inertial centre is, according to the Boltzmann-Maxwell doctrine, equal to of the whole average kinetic energy of 3i the system. This makes the total average energy, potential and kinetic, of the whole motion of the system, translational and relative, to be 3i(1+P) times the mean kinetic energy of one component of the motion of the inertial centre, where P denotes the ratio of the mean potential energy of the relative displacements of the parts to the mean kinetic energy of the whole system. Now, according to Clausius' splendid and easily proved theorem regarding the partition of energy in the kinetic theory of gases, the ratio of the difference between the two thermal capacities to the constant-volume thermal capacity is equal to the ratio of twice a single component of the translational energy to the total energy. Hence, if according to our usual notation we denote the ratio of the thermal capacity, pressure constant, to the thermal capacity, volume constant, by k, we have,

§ 24. Example 1.-For first and simplest example, consider a monatomic gas. We have i=1, and according to our supposition (the supposition generally, perhaps universally, made) regarding atoms, we have P=0. Hence, k-1=}.

This is merely a fundamental theorem in the kinetic theory of for the case of no rotational or vibrational energy of gases the molecule; in which there is no scope either for Clausius' theorem or for the Boltzmann-Maxwell doctrine. It is beautifully illustrated by mercury vapour, a monatomic gas according to chemists, for which many years ago Kundt, in an admirably designed experiment, found k-1 to be very approximately; and by the newly discovered gases argon, helium, and krypton, for which also k-1 has been found to have approximately the same value, by Rayleigh and Ramsay. But each of these four gases has a large number of spectrum lines, and therefore a large number of vibrational freedoms, and therefore, if the Boltzmann-Maxwell doctrine were true, k-1 would have some exceedingly small value, such as that

shown in the ideal example of § 26 below. On the other hand, Clausius' theorem presents no difficulty; it merely asserts that k-1 is necessarily less than in each of these four cases, as in every case in which there is any rotational or vibrational energy whatever; and proves, from the values found experimentally for k-1 in the four gases, that in each case the total of rotational and vibrational energy is exceedingly small in comparison with the translational energy. It justifies admirably the chemical doctrine that mercury vapour is practically a monatomic gas, and it proves that argon, helium, and krypton, are also practically monatomic, though none of these gases has hitherto shown any chemical affinity or action of any kind from which chemists could draw any such conclusion.

But Clausius' theorem, taken in connection with Stokes' and Kirchhoff's dynamics of spectrum analysis, throws a new light on what we are now calling a "practically monatomic gas." It shows that, unless we admit that the atom can be set into rotation or vibration by mutual collisions (a most unacceptable hypothesis), it must have satellites connected with it (or ether condensed into it or around it) and kept, by the collisions, in motion relatively to it with total energy exceedingly small in comparison with the translational energy of the whole system of atom and satellites. The satellites must in all probability be of exceedingly small mass in comparison with that of the chief atom. Can they be the "ions by which J. J. Thomson explains the electric conductivity induced in air and other gases by ultra-violet light, Röntgen rays, and Becquerel rays?

Finally, it is interesting to remark that all the values of k-1 found by Rayleigh and Ramsay are somewhat less than ; argon 64, 61; helium 652; krypton 666. If the deviation from 667 were accidental they would probably have been some in defect and some in excess.

Example 2.-As a next simplest example let i=2, and as a very simplest case let the two atoms be in stable equili brium when concentric, and be infinitely nearly concentric when the clusters move about, constituting a homogeneous gas. This supposition makes P, because the average potential energy is equal to the average kinetic energy in simple harmonic vibrations; and in our present case half the whole kinetic energy, according to the Boltzmann-Maxwell doctrine, is vibrational, the other half being translational. We find k-1=4=2222.

Example 3.-Let i=2; let there be stable equilibrium, with the centres C, C' of the two atoms at a finite distance a

asunder, and let the atoms be always very nearly at this distance asunder when the clusters are not in collision. The relative motions of the two atoms will be according to three freedoms, one vibrational, consisting of very small shortenings and lengthenings of the distance C C', and two rotational, consisting of rotations round one or other of two lines perpendicular to each other and perpendicular to CC' through the inertial centre. With these conditions and limitations, and with the supposition that half the average kinetic energy of the rotation is comparable with the average kinetic energy of the vibrations, or exactly equal to it as according to the Boltzmann-Maxwell doctrine, it is easily proved that in rotation the excess of CC' above the equilibrium distance a, due to centrifugal force, must be exceedingly small in comparison with the maximum value of CC-a due to the vibration. Hence the average potential energy of the rotation is negligible in comparison with the potential energy of the vibration. Hence, of the three freedoms for relative motion there is only one contributory to P, and therefore we have P. Thus we find k-1=4=2857.

The best way of experimentally determining the ratio of the two thermal capacities for any gas is by comparison between the observed and the Newtonian velocities of sound. It has thus been ascertained that, at ordinary temperatures and pressures, k-1 differs but little from 406 for common air, which is a mixture of the two gases nitrogen and oxygen, each diatomic according to modern chemical theory; and the greatest value that the Boltzmann-Maxwell doctrine can give for a diatomic gas is the 2857 of Ex. 3. This notable discrepance from observation suffices to absolutely disprove the Boltzmann-Maxwell doctrine. What is really established in respect to partition of energy is what Clausius' theorem tells us (§ 23 above). We find, as a result of observation and true theory, that the average kinetic energy of translation of the molecules of common air is 609 of the total energy, potential and kinetic, of the relative motion of the constituents of the molecules.

§ 25. The method of treatment of Ex. 3 above, carried out for a cluster of any number of atoms greater than two not in one line, j+2 atoms, let us say, shows us that there are three translational freedoms; three rotational freedoms, relatively to axes through the inertial centre; and 3j vibrational freedoms. Hence we have P and we find k-1=

=

j

j+2'

1

3(1+j) The values of k 1 thus calculated for a triatomic and tetratomic gas, and calculated as above in Ex. 3 for a diatomic

gas, are shown in the following table, and compared with the results of observation for several such gases:

It is interesting to see how the dynamics of Clausius' theorem is verified by the results of observation shown in the table. The values of k-1 for all the gases are less than 3, as they must be when there is any appreciable energy of rotation or vibration in the molecule. They are different for different diatomic gases; ranging from 41 for oxygen to 32 for chlorine, which is quite as might be expected, when we consider that the laws of force between the two atoms may differ largely for the different kinds of atoms. The values of k-1 are, on the whole, smaller for the tetratomic and triatomic than for the diatomic gases, as might be expected from consideration of Clausius' principle. It is probable that the differences of k-1 for the different diatomic gases are real, although there is considerable uncertainty with regard to the observational results for all or some of the gases other than air. It is certain that the discrepancies from the values, calculated according to the Boltzmann-Maxwell doctrine, are real and great; and that in each case, diatomic, triatomic, and tetratomic, the doctrine gives a value for k-1 much smaller than the truth.

§ 26. But, in reality, the Boitzmann-Maxwell doctrine errs enormously more than is shown in the preceding table. Spectrum analysis showing vast numbers of lines for each gas makes it certain that the numbers of freedoms of the constituents of each molecule is enormously greater than those which we have been counting, and therefore that unless we attribute vibratile quality to each individual atom, the Phil. Mag. S. 6. Vol. 2. No. 7. July 1901.

C