On the Theory of the Energy Distribution Law of the Normal Spectrum.

by (Max) (Planck), 1900

(\Huge{Gentlemen:}) when some weeks ago I had the honour to draw your attention to a new formula which seemed to me to be suited to express the law of the distribution of radiation energy over the whole range of the normal spectrum [Planck, 1900], I mentioned already then that in my opinion the usefulness of this equation was not based only on the apparently close agreement of the few numbers, which I could then communicate, with the available experimental data, 1 but mainly on the simple structure of the formula and especially on the fact that it gave a very simple logarithmic expression for the dependence of the entropy of an irradiated monochromatic vibrating resonator on its vibrational energy. This formula seemed to promise in any case the possibility of a general interpretation much rather than other equations which have been proposed, apart from Wien's formula which, however, was not confirmed by experiment.

Entropy means disorder, and I thought that one should find this disorder in the irregularity with which even in a completely stationary radiation field the vibrations of the resonator change their amplitude and phase, as long as considers time intervals long compared to the period of one vibration, but short compared to the duration of a measurement. The constant energy of the stationary vibrating resonator can thus only be considered to be a time average, or, put differently, to be an instantaneous average of the energies of a large number of identical resonators which are in the same stationary radiation field, but far enough from one another not to influence each other. Since the entropy of a resonator is thus determined by the way in which the energy is distributed at one time over many resonators, I suspected that one should evaluate this quantity in the electromagnetic radiation theory by introducing probability considerations, the importance of which for the second law of thermodynamics was first of all discovered by Mr. Boltzmann [Boltzmann, 1877]. This suspicion has been confirmed; I have been able to derive deductively an expression for the entropy of a monochromatically vibrating resonator and thus for the energy distribution in a stationary radiation state, that is, in the normal spectrum. To do this it was only necessary to extend somewhat the interpretation of the hypothesis of "natural radiation" which is introduced in electromagnetic theory. Apart from this I have obtained other relations which seem to me to be of considerable importance for other branches of physics and also of chemistry.

I do not wish to give today this deduction --which is based on the laws of electromagnetic radiation, thermodynamics and probability calculus-- systematically in all details, but rather to explain as clearly as possible the real core of the theory. This can be done most easily by describing to you a new, completely elementary treatment through which one can evaluate --without knowing anything about a spectral formula or about any theory-- the distribution of a given amount of energy over the different colours of the normal spectrum using one constant of nature only and after that the value of the temperature of this energy radiation using a second constant of nature. You will find many points in the treatment to be presented arbitrary and complicated, but as I said a moment ago I do not want to pay attention to a proof of the necessity and the simple, practical details, but to the clarity and uniqueness of the given prescriptions for the solution of the problem.

Let us consider a large number of monochromatically vibrating resonator --$N$ of frequency $\nu$ $($per second$)$, $N^{\prime}$ of frequency $\nu^{\prime}$, $N^{\prime \prime}$ of frequency $\nu^{\prime \prime}$, ..., with all $N$ large number -- which are at large distances apart and are enclosed in a diathermic medium with light velocity c and bounded by reflecting walls. Let the system contain a certain amount of energy, the total energy $E_t$ $($erg$)$ which is present partly in the medium as travelling radiation and partly in the resonators as vibrational energy. The question is how in a stationary state this energy is distributed over the vibrations of the resonator and over the various of the radiation present in the medium, and what will be the temperature of the total system.

To answer this question we first of all consider the vibrations of the resonators and assign to them arbitrary definite energies, for instance, an energy $E$ to the $N$ resonators $\nu$, $E^{\prime}$ to the $N^{\prime}$ resonators $\nu^{\prime}, \ldots$ . The sum

\begin{equation*} E + E^{\prime} + E^{\prime \prime} + \ldots = E_0 \end{equation*}

must, of course, be less than $E_t$. The remainder $E_t - E_0$ pertains then to the radiation present in the medium. We must now give the distribution of the energy over the separate resonators of each group, first of all the distribution of the energy $E$ over the $N$ resonators of frequency $\nu$. If $E$ considered to be continuously divisible quantity, this distribution is possible in infinitely many ways. We consider, however -- this is the most essential point of the whole calculation -- $E$ to be composed of a very definite number of equal parts and use thereto the constant of nature $h = 6.55 \times 10^{-27}$ erg $\cdot$ sec. This constant multiplied by the common frequency $\nu$ of the resonators gives us the energy element $\varepsilon$ in erg, and dividing $E$ by $\varepsilon$ we get the number $P$ of energy elements which must be divided over the $N$ resonators. If the ratio is not an integer, we take for $P$ an integer in the neighbourhood.

It is clear that the distribution of $P$ energy elements over $N$ resonators can only take place in a finite, well--defined number of ways. Each of these ways of distribution we call a "complexion", using an expression introduced by Mr. Boltzmann for a similar quantity. If we denote the resonators by the numbers 1, 2, 3, $\ldots$, $N$, and write these in a row, and if we under each resonator put the number of its energy elements, we get for each complexion a symbol of the following form

\begin{equation*} \frac{1 \phantom{xX} 2\phantom{X} 3\phantom{Xx} 4\phantom{X} 5\phantom{X} 6\phantom{X} 7\phantom{X} 8\phantom{X} 9\phantom{X} 10}{7 \phantom{X} 38 \phantom{X} 11\phantom{X} 0\phantom{X} 9\phantom{X} 2\phantom{X} 20\phantom{X} 4\phantom{X} 4\phantom{X} 5} \end{equation*}

We have taken here $N = 10$, $P = 100$. The number of all possible complexions is clearly equal to the number of all possible sets of number which one can obtain for lower sequence for given $N$ and $P$. To exclude all misunderstandings, we remark that two complexions must be considered to be different if the corresponding sequences contain the same numbers, but in different order. From the theory of permutations we get for the number of all possible complexions

\begin{equation*} \frac{N(N + 1) \cdot (N + 2) \ldots (N + P - 1)}{1 \cdot 2 \cdot 3 \ldots P} = \frac{(N + P - 1) !}{(N - 1) ! P !} \end{equation*}

or to a sufficient approximations,

\begin{equation*} = \frac{(N + P)^{N + P}}{N^N P^P}. \end{equation*}

We perform the same calculation for the resonators of the other groups, by determining for each group of resonators the number of possible complexions for the energy given to the group. The multiplication of all numbers obtained in this way gives us then the total number $R$ of all possible complexions for the arbitrary assigned energy distribution over all resonators.

In the same way any other arbitrarily chosen energy distribution $E, E', E'' \ldots$ will correspond to a definite number $R$ of all possible complexions which is evaluated in the above manner. Among all energy distributions which are possible for a constant $E_0 = E + E^{\prime} + E^{\prime \prime} + \ldots $ there is one well--defined one for which the number of possible complexions $R_0$ is larger than for any other distribution. We look for this distribution, if necessary by trial, since this will just be the distribution taken up by the resonators in the stationary radiation field, if they together possess the energy $E_0$. This quantities $E, E^{\prime}, E^{\prime \prime}, \ldots $ can then be expressed in terms of $E_0$. Dividing $E$ by $N, E^{\prime}$ by $N^{\prime}, \ldots$ we obtain the stationary value of the energy $U_{\nu}, U^{\prime}_{\nu^{\prime}}, U^{\prime \prime}_{\nu^{\prime \prime}} \ldots$ of a single resonator of each group, and thus also the spatial density of the corresponding radiation energy in a diathermic medium in the spectral range $\nu$ to $\nu + d\nu$,

\begin{equation*} u_{\nu} d\nu = \frac{8 \pi \nu^2}{c^3} \cdot U_{\nu} d \nu, \end{equation*}

so that the energy of the medium is also determined.

Of all quantities which occur only $E_0$ seems now still to be arbitrary. One sees easily, however, how one can finally evaluate $E_0$ from the total energy $E_t$, since if the chosen value of $E_0$ leads, for instance, to too large a value of $E_t$, we must decrease it, and the other way round.

After the stationary energy distribution is thus determined using a constant $h$, we can find the corresponding temperature $\vartheta$ in degrees absolute 2 using a second constant of nature $k = 1.346 \times 10^{-6}$ erg $\cdot$ degree$^{-1}$ through the equation

\begin{equation*} \frac{1}{\vartheta} = k~ \frac{d~ ln ~R_0}{dE_0}. \end{equation*}

The product $k ~\mbox{ln}\cdot R_0$ is the entropy of the system of resonators; it is the sum of the entropy of all separate resonators.

It would, to be sure, be very complicated to perform explicitly the above-- mentioned calculations, although it would not be without some interest to test the truth of the attainable degree of approximation in a simple case. A more general calculation which is performed very simply, using the above prescriptions shows much more directly that the normal energy distribution determined in this way for a medium containing radiation is given by expression

\begin{equation*} u_{\nu} d \nu = \frac{8 \pi \nu^3}{c^3} \frac{d \nu}{e^{h \nu / k \vartheta} - 1} \end{equation*}

which corresponds exactly to the spectral formula which I give earlier

\begin{equation*} E_{\lambda} d \lambda = \frac{c_1 \lambda^{-5}} {e^{c_2/\lambda \vartheta} -1} d \lambda. \end{equation*}

The formal differences are due to the differences in the definitions of $u_{\nu}$ and $E_{\lambda}$. The first equation is somewhat more general inasfar as it is valid for arbitrary diathermic medium with light velocity $c$. The numerical values of $h$ and $k$ which I mentioned were calculated from that equation using the measurements by $F$. Kurlbaum and by O. Lummer and E. Pringsheim. 3

I shall now make a few short remarks about the question of the necessity of the above given deduction. The fact that the chosen energy element $\varepsilon$ for a given group of resonators must be proportional to the frequency $\nu$ follows immediately from the extremely important Wien displacement law. The relation between $u$ and $U$ is one of the basic equations of the electromagnetic theory of radiation. Apart from that, the whole deduction is based upon the theorem that the entropy 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 two other theorems: $($1$)$ The entropy of the system in a given state is proportional to the logarithm of the probability of that state, and $($2$)$ The probability of any state is proportional to the number of corresponding complexions, or, in other words, any definite complexion is equally probable as any other complexion. The first theorem is, as for as radiative phenomena are concerned, 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 that the definition of its entropy. We have here a distinction from the corresponding situation in the kinetic theory of gases. The second theorem is the core of the whole of the theory presented here: in the last resort its proof can only be given empirically. It can also be understood as a more detailed definition of the hypothesis of natural radiation which I have introduced. This hypothesis I have expressed before [Planck, 1900b] only in the form that the energy of the radiation is completely "randomly" distributed over the various partial vibrations present in the radiation.4 I plan to communicate elsewhere in detail the considerations, which have only been sketched here, with all calculations and with a survey of the development of the theory up to the present.

To conclude I may point to an important consequence of this theory which at the same time makes possible a further test of its reliability. Mr. Boltzmann [Boltzmann, 1877b] has shown that the entropy of a monatomic gas in equilibrium is equal to $\omega R \ln P_0$, where $P_0$ is the number of possible complexions $($the "permutability"$)$ corresponding to the most probable velocity distribution, $R$ being the well known gas constant $($$8.31 \times 10^7$ for O = 16$)$, $\omega$ 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. Since according to the electromagnetic theory of the radiation the velocities of the atoms are completely independent of the distribution of the radiation energy, the total number of complexions is simply equal to the product of the number relating to the velocities and the number relating to the radiation. For the total entropy we have thus

\begin{equation*} f ~\mbox{ln}~(P_0 R_0) = f ~\mbox{ln}~ P_0 + f~ \mbox{ln}~ R_0, \end{equation*}

where $f$ is a factor of proportionality. Comparing this with the earlier expressions we find

\begin{equation*} f = \omega R = k, \end{equation*}

or

\begin{equation*} \omega = \frac{k}{R} = 1.62 \times 10^{-24}, \end{equation*}

that is, a real molecule is $1.62 \times 10^{-24}$ of a mole or, a hydrogen atom weighs $1.64 \times 10^{-24}$ g, since H = 1.01, or, in a mole of any substance there are $1/ \omega = 6.175 \times 10^{23}$ real molecules. Mr. O.E Mayer [Mayer, 1899] gives for this number $640 \times 10^{21}$ which agrees closely.

Loschmidt's number $L$, that is, the number of gas molecules in 1 cm$^3$ at $0^{\circ}$ C and 1 atm is

\begin{equation*} L = \frac{1~ 013~ 200}{R \cdot 273 \cdot \omega} = 2.76 \times 10^{19}. \end{equation*}

Mr. Drude [Drude, 1900] finds $L = 2.1 \times 10^{19}.$

The Boltzmann-Drude constant $\alpha$, that is, the average kinetic energy of an atom at the absolute temperature 1 is

\begin{equation*} \alpha = \frac{3}{2}~ \omega R = \frac{3}{2}~ k = 2.02 \times 10^{-16}. \end{equation*}

Mr. Drude [Drude, 1900] finds $\alpha = 2.65 \times 10^{-16}$.

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

\begin{equation*} e = \varepsilon \omega = 4.69 \times 10^{-10} \mbox{c.s.u.} \end{equation*}

Mr. F. Richarz [Richarz, 1894] finds $1.29 \times 10^{-10}$ and Mr. Thomson [Thomson, 1898] recently $6.5 \times 10^{-10}$.

If the theory is at all correct, all these relations should be not approximately, but absolutely, valid. The accuracy of the calculated number is thus essentially the same as that of the relatively worst known, the radiation constant $k$, and is thus much better than all determinations up to now. To test it by more direct methods should be both an important and a necessary task for further research.


FOOTNOTES

  1. In the meantime Mr. H. Rubens and Mr. F. Kurlbaum [Rubens and Kurlbaum, 1900] have given a direct confirmation for very long wave lengths. back

  2. The original states "degrees centigrade" which is clearly a slip [D. tar Haar]. back

  3. F. Kurlbaum [Kurlbaum, 1898] gives $S_{100} - S_0 = 0.0731$ Watt cm$^{-2}$, while O. Lummer and E. Pringsheim [Lummer and Pringsheim, 1900] give $\lambda_m \vartheta = 2940 \mu \cdot \mbox{degree}$. back

  4. When Mr. Wien in his Paris report 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 surely demanded, in my opinion, 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. However, one could then not derive anything new from it. From the same point of view one should also declare the kinetic theory of gases to be unsatisfactory since nobody has yet proved that the atomistic hypothesis is the only which explains irreversibility. A similar objection could with more or less justice be raised against all inductively obtained theories. back

References


  1. M. Planck, Verh. D. Physik. Ges. Berlin 2, 202 $($1900$)$ $($reprinted as Paper 2 on p. 82 in "The Old Quantum Theory", D. tar Haar$, 1967)$.
  2. H. Rubens and F. Kurlbaum, S.B. Preuss. Akad. Wiss. p. 929 $($1900$)$.
  3. L. Boltzmann, S.B. Akad. Wiss. Wien 76, 373 $($1877$)$.
  4. F. Kurlbaum, Ann. Physik 65, 759 $($1898$)$.
  5. O. Lummer and E. Pringsheim, Verh. D. Physik. Ges. Berlin 2, 176 $($1900$)$.
  6. M. Planck, Ann. Physik 1, 73 $($1900$)$.
  7. L. Boltzmann, S.B. Akad. Wiss. Wien 76, 428 $($1877$)$.
  8. O.E. Mayer, Die Kinetische Theorie der Gase, 2nd ed., p. 337 $($1899$)$.
  9. P. Drude, Ann. Physik 1, 578 $($1900$)$.
  10. F. Richarz, Ann. Physik 52, 397 $($1894$)$.
  11. J.J. Thomson, Phil. Mag. 46, 528 $($1898$)$.

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