First principles based density functional methods and theoretical models have been used to investigate the reactivity of different RhCu alloys toward molecular hydrogen dissociation. For pure Rh(111), it is predicted that H-2 can either dissociate spontaneously or be physisorbed as molecular hydrogen. These predictions agree with recent experimental findings. The simple addition of a Cu atom to a Rh-rich environment is enough to eliminate the presence of the energy minimum associated with molecularly adsorbed hydrogen and to convert H-2 dissociation in an activated process. From the present results it is concluded that increasing the Cu content may lead to a change in the active site and permits us to speculate about the enhancement on the irreversible hydrogen uptake on going from Rh to RhCu alloys with a small Cu content.