in the normal spectrum, first derived by W. Wien from molecular-kinetic considerations and later by me from the theory of electromagnetic radiation, is not valid
generally.
In any case, the theory requires a correction, and I shall attempt in the following to
accomplish this based on the theory of electromagnetic radiation, which I developed.
For this purpose, it will be necessary first to find in the set of conditions leading to
Wien’s energy distribution law that term which can be changed; thereafter it will be a
matter of removing this term from the set and making an appropriate substitution
for it.
In my last article
14 I showed that the physical foundations of the electromagnetic
radiation theory, including the hypothesis of “natural radiation,” withstand the most
severe criticism; and since to my knowledge there are no errors in the calculations,
the principle persists that the law of energy distribution in the normal spectrum is
completely determined when one succeeds in calculating the entropy S of an
irradiated, monochromatic, vibrating resonator as a function of its vibrational energy
U. Since one then obtains, from the relationship dS/dU ¼ 1/θ, the dependence of the
energy U on the temperature θ, and since the energy is related to the density of
radiation at the corresponding frequency by a simple relation,
15 one also obtains the
dependence of this density of radiation on the temperature. The normal energy
distribution is then the one in which the radiation densities of all different frequencies have the same temperature.
Consequently, the entire problem is reduced to determining S as a function of U,
and it is to this task that the most essential part of the following analysis is devoted.
In my first treatment of this subject, I had expressed S, by definition, as a simple
function of U without further foundation, and I was satisfied to show that this form of
entropy meets all the requirements imposed on it by thermodynamics. At that time I
believed that this was the only possible expression and that consequently Wein’s
law, which follows from it, necessarily had general validity. In a later, closer
analysis,
16 however, it appeared to me that there must be other expressions which
yield the same result, and that in any case one needs another condition in order to be
able to calculate S uniquely. I believed I had found such a condition in the principle,
which at the time seemed to me perfectly plausible, that in an infinitely small
irreversible change in a system, near thermal equilibrium, of N identical resonators
in the same stationary radiation field, the increase in the total entropy S N ¼ NS with
which it is associated depends only on its total energy U N ¼ NU and the changes in
this quantity, but not on the energy U of individual resonators. This theorem leads
again to Wien’s energy distribution law. Nevertheless, since the latter is not confirmed by experience, one is forced to conclude that even this principle cannot be
generally valid and thus must be eliminated from the theory.
17
14 M. Planck, Ann. d. Phys. 1 (1900), p. 719.
15 Compare with equation (8).
16 M. Planck, loc. cit., pp. 730 ff.
17 Moreover one should compare the critiques previously made of this theorem by W. Wien (Report
of the Paris Congress 2, 1900, p. 40) and by O. Lummer (loc. cit., 1900, p. 92).
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4 Unified Mechanics Theory
generally.
In any case, the theory requires a correction, and I shall attempt in the following to
accomplish this based on the theory of electromagnetic radiation, which I developed.
For this purpose, it will be necessary first to find in the set of conditions leading to
Wien’s energy distribution law that term which can be changed; thereafter it will be a
matter of removing this term from the set and making an appropriate substitution
for it.
In my last article
14 I showed that the physical foundations of the electromagnetic
radiation theory, including the hypothesis of “natural radiation,” withstand the most
severe criticism; and since to my knowledge there are no errors in the calculations,
the principle persists that the law of energy distribution in the normal spectrum is
completely determined when one succeeds in calculating the entropy S of an
irradiated, monochromatic, vibrating resonator as a function of its vibrational energy
U. Since one then obtains, from the relationship dS/dU ¼ 1/θ, the dependence of the
energy U on the temperature θ, and since the energy is related to the density of
radiation at the corresponding frequency by a simple relation,
15 one also obtains the
dependence of this density of radiation on the temperature. The normal energy
distribution is then the one in which the radiation densities of all different frequencies have the same temperature.
Consequently, the entire problem is reduced to determining S as a function of U,
and it is to this task that the most essential part of the following analysis is devoted.
In my first treatment of this subject, I had expressed S, by definition, as a simple
function of U without further foundation, and I was satisfied to show that this form of
entropy meets all the requirements imposed on it by thermodynamics. At that time I
believed that this was the only possible expression and that consequently Wein’s
law, which follows from it, necessarily had general validity. In a later, closer
analysis,
16 however, it appeared to me that there must be other expressions which
yield the same result, and that in any case one needs another condition in order to be
able to calculate S uniquely. I believed I had found such a condition in the principle,
which at the time seemed to me perfectly plausible, that in an infinitely small
irreversible change in a system, near thermal equilibrium, of N identical resonators
in the same stationary radiation field, the increase in the total entropy S N ¼ NS with
which it is associated depends only on its total energy U N ¼ NU and the changes in
this quantity, but not on the energy U of individual resonators. This theorem leads
again to Wien’s energy distribution law. Nevertheless, since the latter is not confirmed by experience, one is forced to conclude that even this principle cannot be
generally valid and thus must be eliminated from the theory.
17
14 M. Planck, Ann. d. Phys. 1 (1900), p. 719.
15 Compare with equation (8).
16 M. Planck, loc. cit., pp. 730 ff.
17 Moreover one should compare the critiques previously made of this theorem by W. Wien (Report
of the Paris Congress 2, 1900, p. 40) and by O. Lummer (loc. cit., 1900, p. 92).
182
4 Unified Mechanics Theory