Binding Reactions at Finite Systems

09 December 2021, Version 1
This content is a preprint and has not undergone peer review at the time of posting.

Abstract

A perpetual yearn exists among computational scientists to scale-down the size of physical systems, a desire shared as well with experimentalists able to track single molecules. A question then arises whether averages observed at small systems are the same as those observed at large, or macroscopic, systems. Utilizing statistical-mechanics formulations in ensembles in which the total numbers of particles are fixed, we demonstrate that system's properties of binding reactions are not homogeneous functions. That means averages of intensive properties, such as the concentration of the bound-state, at finite-systems are different than those at large-systems. The discrepancy increases with decreasing numbers of particles, temperature, and volume. As perplexing as it may sound, despite these variations in average quantities, extracting the equilibrium constant from systems of different sizes does yield the same value. The reason is that correlations in reactants' concentrations are ought be accounted for in the expression of the equilibrium constant, being negligible at large-scale but significant at small-scale. Similar arguments pertain to the calculations of the reaction rate-constants, more specifically, the bimolecular rate of the forward reaction is related to the average of the product (and not to the product of the averages) of the reactants' concentrations. Furthermore, we derive relations aiming to predict the composition of the system only from the value of the equilibrium constant. All predictions are validated by Monte-Carlo and molecular dynamics simulations. An important significance of these findings is that the expression of the equilibrium constant at finite systems is not dictated solely by the chemical equation but requires knowledge of the elementary processes involved.

Keywords

Equilibrium Constant
Standard Gibbs Free Energy
Association Reactions
Bimolecular Rate Constants
Statistical Mechanics

Supplementary materials

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Description
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Title
Supporting Information - Binding Reactions at Finite Systems
Description
1) Statistical mechanics derivation of the equilibrium constant and its consequences at finite systems. 2) Derivations of the relation between relative fluctuations and average number of bound particles. 3) Relations predicting composition from the equilibrium constant. 4) Computational details. 5) Additional figures and tables. 6) Comparisons with analytical/numerical results.
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