Annus MirabilisEinstein in 1905
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The equation explained

S−S0=Eβνlg⁡(vv0).S - S_0 = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right).
In wordsThe entropy of monochromatic radiation in the volume v, less its entropy in the volume v₀, equals its energy over the constant β times the frequency, times the natural logarithm of v over v₀.OverviewSection 6 starts again from section 4's result: how the entropy of monochromatic radiation of low density changes with its volume. It is the same logarithm of the volume ratio that section 5 found for a gas.Full explanationThis is section 4's last display, repeated so that section 6 can interpret it by Boltzmann's principle. It holds for monochromatic radiation of energy EE in a narrow band at the frequency ν\nu, dilute enough for Wien's law to apply, when its volume changes from v0v_0 to vv with its energy unchanged. Section 5 found for nn independent points, a gas or the molecules of a dilute solution, the same logarithm of the volume ratio with a different coefficient. The next display rewrites the radiation's coefficient so that the two can be compared letter for letter.Every stepS=−Eβν{lg⁡Evαν3dν−1}S = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - 1 \right\}Section 4's entropy of radiation of energy EE in the band dνd \nu filling the volume vv, from Wien's law.S−S0=−Eβν{lg⁡Evαν3dν−lg⁡Ev0αν3dν}S - S_0 = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - \lg \frac{E}{v_0 \alpha \nu^3 d \nu} \right\}Subtract the same expression for the volume v0v_0. The −1-1 in the two brackets cancels.S−S0=Eβνlg⁡(vv0)S - S_0 = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right)Everything but the two volumes cancels in the quotient of the logarithms: the printed line.Also

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