A series of model catalysts were prepared by depositing different size Pd_n clusters on alumina films grown to variable thickness on a Ta(110) support. Samples were characterized by a combination of X-ray photoelectron spectroscopy, low energy He⁺ scattering, and temperature-programmed reaction and desorption (TPR/TPD). For the activity studies, the samples were first exposed to ¹⁸O₂ at T_ox, and then to ¹³CO at 180 K, where CO sticks to Pd, but not to the alumina support. CO oxidation activity increased with increasing thickness of the alumina support up to 4.5 nm, but was constant for greater thicknesses. Activity increased, with T_ox up to 400 K, but then declined for T_ox = 500 K. Activity was also found to be non-monotonically dependent on deposited cluster size, with Pd_n (n ≤ 6) being generally more reactive than the larger clusters studied. Activity was only weakly correlated with exposed Pd binding sites, which decreased with increasing cluster size, however, there does appear to be a correlation between activity and electronic structure, as probed via the Pd 3d binding energy. Unlike previous systems we have studied, the activity of small Pd_n on these alumina films was quite stable, with essentially no changes observed in up to eight successive TPR experiments.