  Reaction-diffusion systems where transition rates exhibit quenched disorder
are common in physical and chemical systems. We study pair reactions on a
periodic two-dimensional lattice, including continuous deposition and
spontaneous desorption of particles. Hopping and desorption are taken to be
thermally activated processes. The activation energies are drawn from a binary
distribution of well depths, corresponding to `shallow' and `deep' sites. This
is the simplest non-trivial distribution, which we use to examine and explain
fundamental features of the system. We simulate the system using kinetic Monte
Carlo methods and provide a thorough understanding of our findings. We show
that the combination of shallow and deep sites broadens the temperature window
in which the reaction is efficient, compared to either homogeneous system. We
also examine the role of spatial correlations, including systems where one type
of site is arranged in a cluster or a sublattice. Finally, we show that a
simple rate equation model reproduces simulation results with very good
accuracy.
