Stochastic approaches: modeling the probability of encounters between H-molecules and metallic atomic clusters in a cubic box

Abstract

In recent years the advance of chemical synthesis has made it possible to obtain “naked”clusters ofdifferent transition metals. It is well known that cluster experiments allow studying the fundamentalreactive behavior of catalytic materials in an environment that avoids the complications present inextended solid-phase research. In physicochemical terms, the question that arises is the chemicalreduction of metallic clusters could be affected by the presence of H2 molecules, that is, by theprobability of encounter that these small metal atomic agglomerates can have with these reducingspecies. Therefore, we consider the stochastic movement of N molecules of hydrogen in a cubicbox containing M metallic atomic clusters in a confined region of the box. We use a Wiener processto simulate the stochastic process, with σ given by the Maxwell-Boltzmann relationships, whichenabled us to obtain an analytical expression for the probability density function. This expression isan exact expression, obtained under an original proposal outlined in this work, i.e. obtained fromconsiderations of mathematical rebounds. On this basis, we obtained the probability of encounterfor three different volumes, 0.1^3, 0.2^3and 0.4^3 m3, at three different temperatures in each case, 293, 373 and 473 K, for 101≤N≤1010, 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 bythe temperature.