On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures.

Fabien Paillusson

doi:10.3390/e21060599

Abstract

Most undergraduate students who have followed a thermodynamics course would have been asked to evaluate the volume occupied by one mole of air under standard conditions of pressure and temperature. However, what is this task exactly referring to? If air is to be regarded as a mixture, under what circumstances can this mixture be considered as comprising only one component called “air” in classical statistical mechanics? Furthermore, following the paradigmatic Gibbs’ mixing thought experiment, if one mixes air from a container with air from another container, all other things being equal, should there be a change in entropy? The present paper addresses these questions by developing a prior-based statistical mechanics framework to characterise binary mixtures’ composition realisations and their effect on thermodynamic free energies and entropies. It is found that (a) there exist circumstances for which an ideal binary mixture is thermodynamically equivalent to a single component ideal gas and (b) even when mixing two substances identical in their underlying composition, entropy increase does occur for finite size systems. The nature of the contributions to this increase is then discussed.

Highlights

Whether one thinks of ubiquitous substances such as air or drinking water or more specific fluids such as petroleum or milk, they all count as multicomponent systems, i.e., they all comprise more than one identifiable type of constituent
References therein) on Gibbs’ original work [2,3] on mixtures, the latter are thought to play a key role in the—quantum [4,5,6,7,8,9,10,11,12,13] or classical [14,15,16,17,18,19,20,21,22,23,24,25]—foundations of classical statistical mechanics; through thefamous Gibbs paradox
After having briefly presented what we called the textbook treatment of binary mixtures, we looked at a heuristic generalisation which is grounded in the idea that a size-independent definition of a substance necessitates the existence of a prior underlying probability distribution from which are drawn the particles identity

Summary

Introduction

Whether one thinks of ubiquitous substances such as air or drinking water or more specific fluids such as petroleum or milk, they all count as multicomponent systems, i.e., they all comprise more than one identifiable type of constituent. Mixtures are usually treated but as an exception to the identical particle paradigm The reasons for this lack of visibility of multicomponent systems, despite their omnipresence in natural and artificial settings, are often rather elusive so that we could be left with—wrongly—attributing motives to their authors ranging from “mixtures do not matter as much as we think” to “mixtures are too complicated to treat in all their details anyways”. Gibbs originally developed the notions of grand (and petit) canonical ensembles to elucidate the statistical thermodynamics of systems with varying particle numbers, including mixing problems [3]. The entropy of mixtures and Gibbs’ paradoxes were revisited within a more contemporary framework involving probabilities and particle exchanges protocols equivalent to the grand canonical ensemble for non-interacting systems [28]. For finite size systems, the grand canonical ensemble may not always give the same result as the canonical ensemble and, in practice, many mixing scenarios do not involve any external reservoir with which to exchange

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Discussion

Conclusion