Page:Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.pdf/11

 We ask: how large is the probability of the last-mentioned state relative to the original state? Or, what is the probability that at some point in time all n independently moving points in a volume v0 have by chance ended up in the volume v?

For this probability, which is a "statistical probability" one obtains the value:


 * $$W = \left( \frac{v}{v_0} \right) ^n \ ;$$

one derives from this, applying Boltzmann's principle:


 * $$S - S_0 = R \left(\frac{n}{N}\right)\lg\left(\frac{v}{v_0}\right).$$

It's noteworthy that for this derivation, from which the Boyle-Gay-Lussac law and the identical law of osmotic pressure can be easily derived thermodynamically, there is no need to make any assumption regarding the way the molecules move.

Interpretation of the Volume Dependence of the Entropy of Monochromatic Radiation using Boltzmann's Principle
In paragraph 4 we found for the dependence of Entropy of the monochromatic radiation on volume the expression:


 * $$S - S_0 = \frac{E}{\beta\nu}\lg\left(\frac{v}{v_0}\right).$$

This formula can be recast as follows:


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

$$