Annus MirabilisEinstein in 1905
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light quanta

The equation explained

∫0∞αν3e−βνTdν∫0∞NRβναν3e−βνTdν=3RNT.\frac{\int_0^\infty \alpha \nu^3 e^{-\frac{\beta \nu}{T}} d \nu}{\int_0^\infty \frac{N}{R \beta \nu} \alpha \nu^3 e^{-\frac{\beta \nu}{T}} d \nu} = 3 \frac{R}{N} T.
In wordsThe integral over all frequencies of the constant α times the frequency cubed, times e to minus the constant β times the frequency over the absolute temperature, which is the energy of black radiation per unit volume, divided by the same integral with each frequency's share divided by the size of one quantum there, which counts the quanta, equals three times the gas constant R over the number N, times the temperature.OverviewWith Wien's formula, the mean size of an energy quantum in black radiation is 3(R/N)T: the total energy divided by the number of quanta. The paper sets it beside the mean kinetic energy of a molecule's motion at the same temperature, (3/2)(R/N)T.Full explanationThe paper compares the mean size of the energy quanta of black radiation with the mean kinetic energy of the centre-of-mass motion of a molecule at the same temperature, which it gives as 32RNT\frac{3}{2} \frac{R}{N} T. The mean size of a quantum is the energy per unit volume divided by the number of quanta per unit volume. The numerator adds up Wien's energy density over all frequencies. In the denominator each frequency's energy is divided by the size of one quantum at that frequency, RβνN\frac{R \beta \nu}{N}, which counts the quanta there. The paper names Wien's formula as the basis of the calculation, and the integrals use it over the whole spectrum, although section 4 said it holds only for large ν/T\nu/T. Both integrals are standard, and the ratio is 3RNT3 \frac{R}{N} T, twice the molecule's figure. The paper sets the two side by side and draws no further conclusion from the factor here.Every step∫0∞αν3e−βνTdν\int_0^\infty \alpha \nu^3 e^{-\frac{\beta \nu}{T}} d \nuThe energy of black radiation per unit volume: Wien's energy density, section 4's first display, added up over all frequencies.NRβναν3e−βνT\frac{N}{R \beta \nu} \alpha \nu^3 e^{-\frac{\beta \nu}{T}}The energy density at the frequency ν\nu divided by the size of one quantum there, RβνN\frac{R \beta \nu}{N}: the number of quanta per unit volume and unit frequency. Integrated over all frequencies it is the denominator.∫0∞ν3e−βνTdν=6(Tβ)4\int_0^\infty \nu^3 e^{-\frac{\beta \nu}{T}} d \nu = 6 \left( \frac{T}{\beta} \right)^4Take βνT\frac{\beta \nu}{T} as the variable of integration. The integral of its cube times the exponential of its negative, from zero to infinity, is 6, and the change of variable brings out four factors of Tβ\frac{T}{\beta}.∫0∞ν2e−βνTdν=2(Tβ)3\int_0^\infty \nu^2 e^{-\frac{\beta \nu}{T}} d \nu = 2 \left( \frac{T}{\beta} \right)^3The same with the square in place of the cube: the integral is 2, with three factors of Tβ\frac{T}{\beta}. The ν\nu in the quantum's size has cancelled one power of the frequency.6α(Tβ)4NRβ⋅2α(Tβ)3=3RNT\frac{6 \alpha \left( \frac{T}{\beta} \right)^4}{\frac{N}{R \beta} \cdot 2 \alpha \left( \frac{T}{\beta} \right)^3} = 3 \frac{R}{N} TDivide. α\alpha cancels, 6 over 2 is 3, and one factor of Tβ\frac{T}{\beta} is left, whose β\beta cancels the β\beta from the quantum's size: the printed value.Historian's marginThe provenance receipt records that the denominator of the small printed fraction in (3/2)(R/N)T, on p. 143, does not resolve on the pinned scan. It also records that the edition's 3/2 rests on physics, the mean kinetic energy of a molecule's centre-of-mass motion, and not on a reading of the plate: watch-three-halves-fraction-p143 (docs/provenance/ap-17-132.md). The value 3(R/N)T of the display itself is printed in full size, and its digits and letters are legible on the plate.Also

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