The density functional study of the activation and adsorption of several H-2 molecules on Pd-n clusters is extended to n = 5. Detailed mechanisms-of H-H activation by the trigonal bipyramidal (tbp) Pd-5 cluster are described. They encompass various reaction pathways initiated by binding the first and second H-2 to the axial/equatorial atoms of the Pd-5 core, leading to the formation of Pd-5(H)(2) and Pd-5(H)(4) activation products, respectively. The Pd-5 + H-2 --> Pd-5(H)(2) reaction involves spin crossing, and the most stable product, which is a singlet, prefers both hydrogens at cap sites of the nonadjacent faces of Pd-5. By contrast to the cases for n = 3 and 4 studied previously, the activation of the second H-2 through the reaction Pd-5(H)(2) + H-2 --> Pd-5(H)(4) is found to be exothermic with an associated barrier (DeltaH (298.15 K)) lying below the Pd-5(H)(2) + H-2 asymptote. The subsequent steps of the hydrogenation of Pd-5 take place via molecular adsorption onto the Pd-5(H)(4) dissociative complex, ultimately giving the hydrogen-rich cluster Pd-5(H)(4)(H-2)(5), consistent with experiment. A comprehensive set of the non-hydrogen-saturated species that contain the Pd-5(H)(4) core structure have also been predicted. The thermodynamic stabilities of the Pd-5(H)(4)(H-2)(m) (m = 1-5) hydrogenated clusters are discussed in terms of DeltaH and DeltaG values at T = 298.15 and 70 K.