Quasiclassical molecular dynamics studies are made of H or D atoms incident from the gas phase onto D or H- covered Cu(111) surfaces. Two detailed model potential energy surfaces are used, both based on the results of extensive total energy calculations using the density functional method. The incident H (D) atoms can react directly to form HD via the Eley-Rideal mechanism, or trap onto the surface. These trapped hot atoms can react with the adsorbates to form HD or can eventually dissipate enough energy through collisions with the adsorbates to become immobile. We also observe the formation of D-2 (H-2). Probabilities for these various processes, as well as the rotational, vibrational, and translational energy distributions of the products are computed and compared with experiment. Hot-atom pathways to product formation are shown to make significant contributions. One of the potentials gives excellent agreement with experiment, while the other is less successful.