The abstraction of bromine atoms from the Si(lll) surface has been studied by scanning tunneling microscopy (STM), Vacant dangling bond (DB) sites created by H-atom abstraction of surface Br are identified as are the subsequent reaction of these DB sites with atomic hydrogen. The kinetics of Br abstraction and the H-atom-DB reaction are shown to be dependent on the local surface structure. Bromine abstraction rates on the 1x1 regions of the surface are about twice that of the 7x7 regions. Surprisingly, we also find that the DB reaction rate is significantly different on the two regions of the surface : reaction on the 7x7 regions being about six times more efficient than on the 1x1 regions. In substantial agreement with earlier studies we find an abstraction-to-DB reaction ratio for H atoms of about 0.2 for the 1x1 regions of the surface. This ratio is much smaller (about 0.01) for the 7x7 regions of the surface, suggesting different types of H-atom-surface interactions in these regions. Abstraction of chemisorbed hydrogen is about 50 times more efficient than abstraction of chemisorbed bromine. Although the low reaction barrier (similar to 1 kcal/mol) we find is consistent with a direct Eley-Rideal mechanism, this mechanism cannot readily account for the structural sensitivity observed. The possibility of a hot-atom precursor mechanism is discussed; At large exposures definite evidence for an etching reaction is presented, which precludes analysis of the data over a wide exposure range.