A system of physical chemistry - Index of

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A system of physical chemistry - Index of

APPENDIX II 177

lapse of time. The difficulty of accepting this suggestion is the difficulty

of believing that such a compromise should be found to be so completely

independent of the nature of the walls as experiment demonstrates.

The formula, however, does agree with fact very well for long

wave-lengths, just as Wien's does for short.

Faced with this discrepancy, there are two alternatives which offer

themselves as obvious methods of escape from it. We may deny the

validity of the calculation of the degrees of freedom, or we may urge

objections to the law of equipartition of the energy. As a matter of

fact, Planck's quantum hypothesis arises from his adoption of the second

alternative, and a denial of the validity of equipartition in the case of

vibratory motion. The principle of equipartition was first deduced in

connection with the kinetic theory of gases. Now even in this original

and limited sphere the principle does not stand on absolutely undebatable

foundations. A great deal depends on the interpretation of the meaning

to be attributed to the word " average ". It is usually assumed that the

average condition, say, of a molecule, is a "time" average; the sum of

successive values of a quantity connected with the molecule, over a long

period of change in the system, divided by the number of such values.

But it is doubtful if the principle of equipartition based on such an

interpretation of the word "average " is really proved at all by dynamical

principles. The is point fully treated by Jeans in his Dynamical Theory

of Gases. It would appear that the "average" really referred to is an

average extended over all conceivable conditions or " complexions " of

a system (excepting a negligible number) and not merely over such condi-

tions or "complexions" through which the system passes on any

particular "path ". The extension to ''time" average cannot be made

unless by. the introduction of Maxwell's assumption of " continuity of

path," viz. that the system will in process of time pass through all conceivable

" complexions ". This assumption is of doubtful vaHdity, and

is known to be unsound in certain of the problems treated in general

dynamics, e.g. the periodic orbits of astronomy (a type of vibratory

motion).

To proceed, Planck, denying the applicability of equipartition to

radiation theory, works out from special considerations (to be dealt

with presently) that in the vibratory motion obtaining in wave-trains,

each degree of freedom corresponding to a frequency v should possess

on the average not kT ergs of kinetic and potential energy, but an

amount—

1^

hv

^""gs

(where h is a new universal constant, determined by experiment to be

approximately 6-55 x iq-^^). If we make use of the Rayleigh-Jeans

calculation of the degrees of freedom, we find that the energy-density

of the radiation in the range v, v -f hv is—

VOL. III. 12

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