light quanta

The equation explained

S - S_0 = \frac{R}{N} \lg \left[ \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}} \right]

In wordsThe same entropy difference, written as the gas constant R over the number N, times the natural logarithm of the ratio of the volumes v and v₀ raised to the power N over R times the radiation's energy over the constant β times the frequency.OverviewThe same entropy difference, rewritten: R/N times the logarithm of the volume ratio raised to a power. Nothing changes but the form, and the form now matches Boltzmann's principle.Full explanationThe coefficient

\frac{E}{\beta \nu}

is multiplied and divided by

\frac{R}{N}

, and the part that is left over,

\frac{N}{R} \frac{E}{\beta \nu}

, goes into the logarithm as an exponent, because a number times a logarithm is the logarithm of the power. The value of the entropy difference is the same as in the preceding line. What changes is its shape:

\frac{R}{N}

times the logarithm of something, which is the shape of Boltzmann's principle in the next display.Every step

\frac{E}{\beta \nu} = \frac{R}{N} \cdot \frac{N}{R} \frac{E}{\beta \nu}

Multiply and divide by

\frac{R}{N}

, which changes nothing.

S - S_0 = \frac{R}{N} \cdot \frac{N}{R} \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right)

Put that into the preceding display.

\frac{N}{R} \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right) = \lg \left[ \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}} \right]

A number times a logarithm is the logarithm of the power with that number as exponent, the rule of section 5 read backwards.

S - S_0 = \frac{R}{N} \lg \left[ \left( \frac{v}{v_0} \right)^{\frac{N}{R} \frac{E}{\beta \nu}} \right]

The printed line.AlsoOverviewFull explanationEvery step

Read it in the paper