Fluctuations and mass action law breakdown in statistical thermodynamics of small systems

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Abstract
The statistical equilibrium of chemical reaction of the type P Q⇄P+Q, taking place within small but macroscopic closed vesicle, is considered using statistical physics approach. It is shown, that mass action law, being the result of mean‐field‐type approximation, breaks down for sufficiently small vesicle volume and/or equilibrium constant of reaction, when mean number of free ‘‘P’’ particles, 〈p〉, within the vesicle becomes of order one or less. At the same time, the Nernst equation is shown to be applicable for systems of arbitrary volume and it gives the relation ΔF∼−ln〈p〉 for free energy ‘‘payment’’ for one ‘‘P’’ particle liberation from the vesicle. Due to fluctuations the true 〈p〉 and (−ΔF) values are essentially lower than corresponding mass action law predictions. The same effect of fluctuations leads to essential random inhomogeneity in the ensemble of vesicles, prepared under equivalent macroscopic conditions, and to non‐Gaussian distributions of these vesicles over total number of particles within them or over ΔF value. Estimations show the effects of fluctuations to be essential for vesicles with sizes of order 102–103 Å, which is just the typical order of magnitude for many biological vesicles. Corresponding possible explanations of some experimental results in bioenergetics are discussed briefly.

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