  In recent years the advance of chemical synthesis has made it possible to
obtain \textquotedblleft naked\textquotedblright clusters of different
transition metals. It is well known that cluster experiments allow studying the
fundamental reactive behavior of catalytic materials in an environment that
avoids the complications present in extended solid-phase research. In
physicochemical terms, the question that arises is the chemical reduction of
metallic clusters could be affected by the presence of H$_{_{2}}$ molecules,
that is, by the probability of encounter that these small metal atomic
agglomerates can have with these reducing species. Therefore, we consider the
stochastic movement of $N$ molecules of hydrogen in a cubic box containing $M$
metallic atomic clusters in a confined region of the box. We use a Wiener
process to simulate the stochastic process, with $\sigma$ given by the
Maxwell-Boltzmann relationships, which enabled us to obtain an analytical
expression for the probability density function. This expression is an exact
expression, obtained under an original proposal outlined in this work,
\textit{i.e.} obtained from considerations of \textit{mathematical rebounds}.
On this basis, we obtained the probability of encounter for three different
volumes, $0.1^{^{3}}$, $ 0.2^{^{3}} $ and $ 0.4^{^{3}} $ m$^{^{3}}$, at three
different temperatures in each case, $293$, $373$ and $473$ K, for $
10^{^{1}}\leq N \leq10^{^{10}}$, comparing the results with those obtained
considering the distribution of the position as a Truncated Normal
Distribution. Finally, we observe that the probability is significantly
affected by the number $N$ of molecules and by the size of the box, not by the
temperature.

Materials Science;Chemical Physics