Chemical differences between K and Na in alkali cobaltates

K_xCoO₂ shares many similarities with Na_xCoO₂, as well as some important differences (no hydration-induced superconductivity has been reported). At T_c2=20 K, K_0.5CoO₂ becomes an insulator with a tiny optical gap, as happens in Na_0.5CoO₂ at 52 K. This similarity, with a known common structure, enables direct comparisons to be made. Using the K-zigzag structure recently reported and the local density approximation, we compare and contrast these cobaltates at x=0.5. Although the electronic structures are quite similar as expected, substantial differences are observed near the Fermi level. These differences are found to be attributable mostly to the chemical rather than the structural difference. Although Na is normally considered to be fully ionic, K has a somewhat more highly ionic character than does Na in these cobaltates.