  We study the steady state of a stochastic particle system on a
two-dimensional lattice, with particle influx, diffusion and desorption, and
the formation of a dimer when particles meet. Surface processes are thermally
activated, with (quenched) binding energies drawn from a \emph{continuous}
distribution. We show that sites in this model provide either coverage or
mobility, depending on their energy. We use this to analytically map the system
to an effective \emph{binary} model in a temperature-dependent way. The
behavior of the effective model is well-understood and accurately describes key
quantities of the system: Compared with discrete distributions, the temperature
window of efficient reaction is broadened, and the efficiency decays more
slowly at its ends. The mapping also explains in what parameter regimes the
system exhibits realization dependence.
