V O L . 3 , D O C . 1 0 a B O L T Z M A N N S P R I N C I P L E 7 where the constant “const.” can depend on the temperature but not on the volume. From this we immediately obtain the force that the N molecules are able to exert on a wall that forces them to remain within volume V. For, if the energy of the system is independent of V, and if G signifies the work received upon infinitesimally enlarging volume V along a reversible path, then holds, hence . We thus have the equation of ideal gases and osmotic pressure. At the same time it is revealed that the universal constant kN of this equation is equal to constant R of the gas equation. In my opinion, the main importance of the Boltzmann equation does not lie in that with its help one is able to calculate the entropy for a known molecular sce- nario. Rather, the most important application is, conversely, that by means of Boltz- mann’s equation one can get the statistical probabilit[ies] of the individual states from the empirically established entropy function S. Thus it is possible to assess how much the systems’ behaviors differ from the behavior required by thermody- namics. Example.[7] A particle that is slightly heavier than the liquid in which it is sus- pended and which it displaces. Such a particle should, according to thermodynamics, sink to the bottom of the vessel and stay there. According to Boltzmann’s equation, however, a probability W is ascribed to each height z above the bottom the particle incessantly changes its height in an irregular way. We want to determine S and from it W. If is the parti- cle’s mass and is that of the liquid it displaces, then the work has to be expended to raise the particle to height z above the bottom. In order for the energy of the system to stay constant, the amount of heat has to be removed from the system, whereupon the entropy diminishes by . There- fore, . From the Boltzmann equation it follows, if one substitutes the value for k: . S kNlg V V0 ----- - kNlgV const., + = = p V d G +TdS + kNT------, dV V - = = = pV kNT = [p. 8] μ μ0 A μ μ0)gz ( = G A = G T --- - A T --- = S const 1 T -- - μ μ0)gz ( –= R N --- - W conste N RT -(μ μ0)gz –------ =
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