Stochasticity of the transfer of reactant molecules between nano-reactors affecting the reversible association A + B ⇆ C

Breakdown of Langmuir Adsorption Isotherm in Small Closed Systems

For more than a century, monolayer adsorptions in which adsorbate molecules and adsorbing sites behave ideally have been successfully described by Langmuir's adsorption isotherm. For example, the amount of adsorbed material, as a function of concentration of the material which is not adsorbed, obeys Langmuir's equation. In this paper, we argue that this relation is valid only for macroscopic systems. However, when particle numbers of adsorbate molecules and/or adsorbing sites are small, Langmuir's model fails to describe the chemical equilibrium of the system. This is because the kinetics of forming, or the probability of observing, occupied sites arises from two-body interactions, and as such, ought to include cross-correlations between particle numbers of the adsorbate and adsorbing sites. The effect of these correlations, as reflected by deviations in predicting composition when correlations are ignored, increases with decreasing particle numbers and becomes substantial when only few adsorbate molecules, or adsorbing sites, are present in the system. In addition, any change that augments the fraction of occupied sites at equilibrium (e.g., smaller volume, lower temperature, or stronger adsorption energy) further increases the discrepancy between observed properties of small systems and those predicted by Langmuir's theory. In contrast, for large systems, these cross-correlations become negligible, and therefore when expressing properties involving two-body processes, it is possible to consider independently the concentration of each component. By applying statistical mechanics concepts, we derive a general expression of the equilibrium constant for adsorption. It is also demonstrated that in ensembles in which total numbers of particles are fixed, the magnitudes of fluctuations in particle numbers alone can predict the average chemical composition of the system. Moreover, an alternative adsorption equation, predicting the average fraction of occupied sites from the value of the equilibrium constant, is proposed. All derived relations were tested against results obtained by Monte Carlo simulations.

Stochastic or Not? Method To Predict and Quantify the Stochastic Effects on the Association Reaction Equilibria in Nanoscopic Systems

The stochastic nature of chemical reaction and impact of the stochasticity on their evolution is soundly documented. Both theoretical predictions and emerging experimental evidence indicate the influence of stochastic effects on the equilibrium state of association reaction. In this work simple mathematical formulas are introduced to estimate these effects. First, the dependence of the ratio of observed reactants (apparent association constant, equivalent of macroscopic association constant in stochastic analysis) on the volume and the number of molecules of reagents is discussed and the limiting factors of this effect are shown. Next, the apparent association constant is approximated for nanoscale systems by closed-form formulas derived for this purpose. Finally, an estimation for the macroscopic constant value from the apparent one is provided and validated on the published experimental data. This work was inspired by chemical reactions occurring in biological compartments, but the results can be used for all systems belonging to the stochastic regime of chemical reactions.