The theory of the Energy Distribution Law of the Normal Spectrum

The chosen energy element c for a given group of resonators must be proportional to the frequency v follows immediately from

• The original states “degrees centigrade” which is clearly a slip [D. t. H.].

t F. Kurlbaum {Ann. Phys. 65 [ = 301], 759 (1898)) gives SlQo~S0 = 0-0731 Watt cm- 2 , while O. Lummer and E. Pringsheim (Verh. Deutsch. Physik Ges. 2, 176 (1900)) give A,„r> = 2940 /* degree.

the extremely important so called Wien displacement law.42

The relation between u and U is one of the basic equations of the electromagnetic theory of radiation.

The whole deduction is based upon the single theorem that the entropy [radiance] of a system of resonators with given energy is proportional to the logarithm of the total number of possible complexions for the given energy.

This theorem can be split into 2 other theorems:

1. The entropy [radiance] of the system in a given state is proportional to the logarithm of the probability of that state

This is just a definition of the probability of the state, insofar as we have for energy radiation no other a priori way to define the probability than the determination of its entropy.

This is one of the distinctions43 from the corresponding situation in the kinetic theory of gases.

2. The probability of any state is proportional to the number of corresponding complexions, or, in other words, any given complexion is equally probable as any other given complexion.

This is the core of the whole theory.

Its proof can only be given empirically.

It is a more detailed definition of my hypothesis of natural radiation: the energy of the radiation is completely “randomly” distributed over the various partial vibrations present in the radiation.*

When Mr. W. Wien in his Paris report (Rapports II, p. 38, 1900) about the theoretical radiation laws did not find my theory on the irreversible radiation phenomena satisfactory since it did not give the proof that the hypothesis of natural radiation is the only one which leads to irreversibility, he demanded too much of this hypothesis.

If one could prove the hypothesis, it would no longer be a hypothesis, and one did not have to formulate it at all. However, one could then not derive anything essentially new from it.

The kinetic theory of gases is unsatisfactory since nobody has proven that the atomistic hypothesis is the only one which explains irreversibility.

To conclude I may point to an important consequence of this theory which at the same time makes possible a further test of its admissibility. Mr.

Boltzmann* has shown that the entropy of a monatomic gas in equilibrium is equal to coRlnP0, where:

• P0 is the number of possible complexions (the “permutability”) corresponding to the most probable velocity distribution,
• R is the well known gas constant (8-31 x 107 for 0 = 16), w the ratio of the mass of a real molecule to the mass of a mole, which is the same for all substances.

If there are any radiating resonators present in the gas, the entropy of the total system must according to the theory developed here be proportional to the logarithm of the number of all possible complexions, including both velocities and radiation.

According to the electromagnetic theory of radiation, the velocities of the atoms are completely independent of the distribution of the radiation energy.

The total numbers of complexions is simply equal to the product of the numbers relating to the velocities and the number relating to the radiation. For the total entropy we have thus:

f In (P0R0) = f In P0 + f In R0,

where/is a factor of proportionality. The first part of the sum is the kinetic, the second part the radiation entropy. Comparing this with the earlier expressions we find

f=<aR=k, k

or

co = - = l - 6 2 x l 0 - 2 4 ,

that is, a real molecule is 1 -62 x 1024 of a mole, or, a hydrogen atom weighs44 T64x 1024 g, since H = 1-01, or, in a mole of any substance there are l/cu = 6T75 x 1023 real molecules.45

Mr. O. E. Meyerf gives for this number 640 xlO2 1 which agrees closely.45

Loschmidt’s number L, that is, the number of gas molecules in 1 cm3 at 0°C and 1 atm is46

, 1 013 200 R . 273 . to

Mr. Drudet finds L = 2-l x 1019 = 2-76 xlO1 9 .

The Boltzmann-Drude constant a, that is, the average kinetic energy of an atom at the absolute temperature 1 is a = fa)jK = fAi = 2-02x \0~i6.

Mr. Drude* finds a = 2-65x lO"1 6 .

The elementary quantum of electricity e, that is, the electrical charge of a positive monovalent ion or of an electron is, if e is the known charge of a monovalent mole,47

e = e a j = 4-69xl0-! O e.s.u.

Mr. F. Richarzf finds l-29x l f r 1 0 and Mr. J. J. Thomson:! recently 6-5 x l O " 1 0 .

If the theory is at all correct, all these relations should be not approximately, but absolutely valid.

The accuracy of the calculated numbers is thus essentially the same as that of the relatively worst known, the radiation constant k, and is thus much better than all determinations of those quantities up to now. To test it by more direct methods should be both an important and a necessary task for further research.