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4. On the Foundations of the Kinetic Theory of Gases

Published online by Cambridge University Press:  15 September 2014

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Extract

The following note was written at the instance of Sir W. Thomson. It is directed mainly to one extremely important point connected with the Kinetic Gas Theory; and is designed to show to the ordinary reader the nature of the investigation, and of the real evidence for the result, without the imposing array of symbols which is usually marshalled in papers on that Theory.

Type
Proceedings 1885-86
Copyright
Copyright © Royal Society of Edinburgh 1886

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References

note * page 388 The mean value, of the square of the distance of any point on a sphere from an internal or external point A, is the sum of the squares of the radius of the sphere and of the distance of A from the centre. Divide the spherical surface into pairs of elements by double cones, of very small angle, whose vertices are at the centre. For each pair of these the theorem is obviously true. Hence if the speeds of two points be p and q, their mean square relative speed is p2 + q2. From this the above statement follows at once; provided that all directions are equally likely for each amount of speed.

Here we meet with a quasi-metaphysical difficulty, which must be men-tioned in passing. For, it may be said, since there is perfect reversibility, the mere instantaneous reversal of a state which is approaching finality will give a state whose tendency is to depart from finality, i.e., to get back to the exact reverse of its original condition. True, and most important, but not fatal to the conclusion; unless an infinite time has elapsed since the start. For, when the reversal has brought the system back to the same configuration as at start-ing, but with velocities reversed, it is a new departure: —which will lead towards, but never to, its own state of finality.

note * page 391 There is no inconsistency between the two expressions above, viz., “great number of simultaneous impacts,” and “small percentage of each system which is involved in any simultaneous collisions.” For we must remember that the whole number of particles is very great; and even a “small per-centage ”of a very great number may itself be “a great number.”

note * page 393 [Inserted Jan. 8, 1886.] I hope to show at the next meeting of the Society that, though neither of these assumptions is correct, Maxwell's Theorem is rigorously true. Neither in Maxwell's paper nor in this has any account been taken of the fact that collisions are more frequent as the relative speed is greater. This consideration affects only numerically the results of §§ 7, 9, 10 above, and does not interfere with the argument based on them.

note * page 402 Dissertatio de gasorum theorid, 1866.Google Scholar Quoted in his work Die Kinetische Theorie der Gase, 1877, p. 294.Google Scholar