# ReactionEquilibrium:Background:Reaction Equilibrium Model

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This module permits simulation of a system in which atomic species can combine to form long-lived dimers, comprising two atoms. Under proper conditions, a dimer can dissociate back into its atoms, which can then recombine or "react" with other free atoms in the system to form new dimers. The aim of this module is to show that this stochastic process can be modeled using an equilibrium constant that can be computed from the parameters of the atomic-interaction model.

A complete quantum mechanical description of the reaction process is too difficult to include, so we resort to a classical mechanical model that has the appropriate features. In particular, we introduce a strong, short ranged, isotropic attractive interaction that functions to keep two atoms bound together. We further prohibit formation of oligomers (bound clusters of more than two atoms) by permitting an atom to have this interaction with only one other atom at a time.

So the model is this Two atoms that are not bound to any other atoms will mutually interact with a short ranged square well potential. The total diameter of interaction is designated ${\displaystyle \sigma }$. Within this range is a hard repulsive core of diameter ${\displaystyle \kappa \sigma }$, ${\displaystyle \kappa <1}$.Atoms separated by distances between ${\displaystyle \sigma }$ and ${\displaystyle \kappa \sigma }$ have an attractive (negative) energy of magnitude ${\displaystyle \varepsilon }$. If an atom is bound to another atom, meaning it is at a separation between ${\displaystyle \sigma }$ and ${\displaystyle \kappa \sigma }$, it interacts with all other atoms as purely repulsive hard sphere of diameter ${\displaystyle \sigma }$.

The following figure illustrates the definition of the size parameters of the potentials

The following three images show a pair of atoms about to interact, inside the range of attraction, and at the hard-core collision

In the following, the third atom is repelled from the others when separated from one of them by a distance ${\displaystyle \sigma }$

In the module, the atoms are colored as red or black spheres of diameter ${\displaystyle \sigma }$; the inner hard core is not illustrated. They look like this