|1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40|
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 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 triato-mic 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 Boltzmann-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
molecule of every one of the ordinary gases must have a vastly greater number of atoms in its constitution than those hitherto reckoned in regular chemical doctrine. Suppose, for example, there are forty-one atoms in the molecule of any particular gas; if the doctrine were true, we should have y=S9. Hence there are 117 vibrational freedoms, so that there might be 117 visible lines in the spectrum of the gas;
and we have k — 1 = “ ’0083. There is, in fact, no xJi\)
possibility of reconciling the Boltzmann-Maxwell doctrine with the truth regarding the specific heats of gases.
§ 27. It is, however, not quite possible to rest contented with the mathematical verdict not proven, and the experimental verdict not true, in respect to the Boltzmann-Maxwell doctrine. I have always felt that it should be mathematically tested by the consideration of some particular case. Even if the theorem were true, stated as it was somewhat vaguely, and in such general terms that great difficulty has been felt as to what it is really meant to express, it would be very desirable to see even one other simple caseT besides that original one of Waterston’s, clearly stated and tested by pure mathematics. Ten years ago*, I suggested a number .of test-cases, some of which have been courteously considered by Boltzmann ; but no demonstration either of the truth or untruth of the doctrine as applied to any one of them has hitherto been given- A year later, I suggested what seemed to me a decisive test-case disproving the doctrine; but my statement was quickly and justly criticised b}r Boltzmann and Poincare; and more recently Lord Rayleigh f has shown very clearly that my simple test-case was quite indecisive. This last article of Rayleigh’s has led me to resume the consideration of several classes of dynamical problems, which had occupied me more or less at various times during the last twenty years, each presenting exceedingly interesting features in connection with the double question: Is this a case which admits of the application of the Boltzmann-Maxwell doctrine; and if so, is the doctrine true for it?
§ 28. Premising that the mean kinetic energies with which the Boltzmann-Maxwell doctrine is concerned are time-integrals of energies divided by totals of the times, we may conveniently divide the whole cla^s of problems, with
* ■' On some Test Cases for the Maxwell-Boltzmann Doctrine regarding' Distribution of Energy/ Proc. Roy. Soc., June 11, 1891.
t Phil. Mag., vol. xxxiii. 1892, p. 350. “Remarks on Maxwell’s Investigation respecting Boltzmann’s Theorem.”