A quantum mechanical model for the dissociative sticking, and scattering of a diatomic molecule on a potential surface containing both elastic and inelastic couplings is solved exactly. The inelastic potential can be due to the excitation of either electron-hole pairs, or phonons. The model is parametrized for vibrationally cold H-2 on copper assuming that electron-hole pairs are the source of the inelastic potential. The main effect is a dynamical polarization of the adiabatic electron system, which increases the sticking coefficient over that calculated with the elastic potential alone, for most couplings. It is a no-loss contribution. Within this model the real energy loss contributes little to the sticking coefficient because a negligible fraction of reaction exothermicity is dissipated before the two atoms are separated on the surface. If the electron-hole pair coupling is very strong and highly localized within the molecule bond, the effect of inelasticity is to act as a reflection barrier reducing the sticking coefficient compared to the elastic calculation. The mass dependence of these effects is also discussed.