Results · §2 · what spectral constants determine

Pages 136, 137

Planck's constants and the number of molecules

For large wavelengths and radiation densities Planck's formula becomes $\varrho_\nu = \frac{\alpha}{\beta}\nu^2 T$ , the relation of § 1, and equating the coefficients gives $N = 6{,}17 \cdot 10^{23}$ , exactly Planck's value; so Planck's determination of the elementary quanta is, to a certain degree, independent of his theory of black radiation.

As printed

Für große Werte von $T/\nu$ , d. h. für große Wellenlängen und Strahlungsdichten geht diese Formel in der Grenze in folgende über: $\varrho_\nu = \frac{\alpha}{\beta} \nu^2 T.$ Spectral energy densitySpectral constant ASpectral constant BFrequency of the lightTemperature of the radiationExplain this equation Man erkennt, daß diese Formel mit der in § 1 aus der Maxwellschen und der Elektronentheorie entwickelten übereinstimmt. Durch Gleichsetzung der Koeffizienten beider Formeln erhält man: $\frac{R}{N} \frac{8 \pi}{L^3} = \frac{\alpha}{\beta}$ Gas constantAvogadro's numberSpeed of lightSpectral constant ASpectral constant BExplain this equation oder $N = \frac{\beta}{\alpha} \frac{8 \pi R}{L^3} = 6{,}17 \cdot 10^{23},$ Avogadro's numberSpectral constant BSpectral constant AGas constantSpeed of lightExplain this equation d. h. ein Atom Wasserstoff wiegt $1/N$ Gramm $= 1{,}62 \cdot 10^{-24}$ g. Dies ist genau der von Hrn. Planck gefundene Wert, welcher mit den auf anderen Wegen gefundenen Werten für diese Größe befriedigend übereinstimmt.

Text on pages 136–137 of the German source, § 2, paragraph 2

In modern notation

\frac{A\,\nu^{3}}{\exp\left(\frac{B\,\nu}{T}\right) - 1} \approx \frac{A}{B}\,\nu^{2}\,T

\frac{A}{B} = \frac{8\,\pi\,R}{N_A\,c^{3}}

N_A = \frac{B}{A}\,\frac{8\,\pi\,R}{c^{3}}

Approximation: The two formulas agree only in the limit of large $T/\nu$ ; at high frequencies they part.
A comparison, not a premise: The value is compared with Planck's, which it matches exactly, and with values found in other ways, which it matches satisfactorily; it is a consistency check, not a new measurement.
An inference beyond the argument: What the paper concludes is limited: Planck's determination of the elementary quanta is independent of his theory of black radiation "bis zu einem gewissen Grade", to a certain degree.

Decoder

$\alpha$ , $\beta$: The two constants of Planck's formula as the paper prints them, $\alpha = 6{,}10 \cdot 10^{-56}$ and $\beta = 4{,}866 \cdot 10^{-11}$ .
large $T/\nu$: Large wavelengths and radiation densities, where $e^{\beta\nu/T} - 1$ is close to $\beta\nu/T$ and Planck's formula takes the form of § 1.
$N$: The number of real molecules in a gram-equivalent. Its reciprocal is the mass of a hydrogen atom in grams, which the paper gives as $1{,}62 \cdot 10^{-24}$ g.

Try it

LQ-03: At long wavelengths, where do Planck's formula and the relation of § 1 agree, and where do they part?

Where this stops: Modern exact SI constants give a consistency check, not a new measurement of N. A fit to the Planck spectrum does not by itself establish independently moving light elements. Read the argument

Common wrong turns

Planck had already proposed that light consists of quanta.

Argument: A comparison of two formulas in a limit
Model: Exact in the limit of large T/ν
Evidence: A consistency check against Planck's constants
History: Planck's formula and constants are earlier work the paper cites; the comparison is this paper's

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§2 · what spectral constants determine

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