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 ^[1], 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:

$S-S_{0}={\frac {R}{N}}\lg \left[{\left({\tfrac {v}{v_{0}}}\right)}^{{\tfrac {N}{R}}{\tfrac {E}{\beta \nu }}}\right]$

1. If E is the energy of the system, then one obtains:

   $-d(E-TS)=pdv=TdS=TR{\frac {n}{N}}{\frac {dv}{v}}$;

   therefore

   $pv=R{\frac {n}{N}}T.$