We study a monomer-dimer reaction model in two dimensions in which the adsorption process of a dimer on the surface sites is separated into two steps (i.e., B2+*-->B2 *, and B2 * +*-->2B*, where * is an empty site), as first introduced by Evans and co-workers. It is clear that the dissociation of a dimer is dependent on the complicated configuration of the adsorbate. We show that the continuous transition from the reactive state to the O-passivated state and the discontinuous transition to the CO-passivated state both shift toward higher values of the fraction p of the monomer in gas phase but the reaction window decreases compared to the Ziff-Gulari-Barshad model. For the model studied here, the critical exponents of the continuous phase transition still exhibit a directed percolation character as expected.