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 the 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 the ratio of the two volumes.OverviewKeep the energy and the band and change only the volume: the entropy grows by E/(βν) times the logarithm of the ratio of the volumes. The paper goes on to say that monochromatic radiation of low density varies its entropy with the volume by the same law as an ideal gas or a dilute solution.Full explanationSection 4 now looks only at how the entropy depends on the volume. Write S0S_0 for the entropy of the same radiation, with the same energy and band, when it fills the volume v0v_0, and subtract. Everything that does not involve the volume cancels, and what is left is the logarithm of the ratio of the volumes: doubling the volume adds Eβνlg⁡2\frac{E}{\beta \nu} \lg 2, and doubling it again adds the same amount again. The paper draws the conclusion in its next sentence: the entropy of monochromatic radiation of sufficiently small density varies with the volume by the same law as the entropy of an ideal gas or of a dilute solution. Section 5 works that law out for a gas, and section 6 interprets the radiation's version of it by Boltzmann's principle.Every stepS=−Eβν{lg⁡Evαν3dν−1}S = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v \alpha \nu^3 d \nu} - 1 \right\}The entropy in the volume vv (the preceding display).S0=−Eβν{lg⁡Ev0αν3dν−1}S_0 = -\frac{E}{\beta \nu} \left\{ \lg \frac{E}{v_0 \alpha \nu^3 d \nu} - 1 \right\}The same radiation, with the same energy and band, in the volume v0v_0.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 −1-1 in the two brackets cancels.lg⁡Evαν3dν−lg⁡Ev0αν3dν=lg⁡v0v\lg \frac{E}{v \alpha \nu^3 d \nu} - \lg \frac{E}{v_0 \alpha \nu^3 d \nu} = \lg \frac{v_0}{v}A difference of logarithms is the logarithm of the quotient, and everything except the two volumes cancels in the quotient.S−S0=−Eβνlg⁡v0v=Eβνlg⁡(vv0)S - S_0 = -\frac{E}{\beta \nu} \lg \frac{v_0}{v} = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right)Turning a fraction upside down changes the sign of its logarithm: the printed line.Also

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