The structural evolution of small copper clusters of up to 15 atoms and the dissociative chemisorption of H2 on the minimum energy clusters are studied systematically using density functional theory. The preferred copper sites for chemisorption are identified and the transition state structures and activation barriers for clusters four to nine atoms are determined and found to be inconsistent with the empirical Bronsted-Evans-Polanyi relationship. The physicochemical properties of the clusters are computed and compared with the bulk and surface values. The results indicate that a phase transition must occur in the going from cluster to bulk.