Activated dissociative adsorption of a hydrogen molecule on metal surfaces is studied in the framework of the precursor mediated dissociation (PMD) mechanism according to which the dissociation proceeds through a molecular-adsorbed state (precursor) which then dissociates to H atoms. The PMD activation energies and rate constants are calculated by considering quantum transitions from various vibration levels of initial state into a continuous manifold of energies of the final dissociated state. The geometric parameters required are taken from the literature for a nickel surface while on other metals the same geometry is assumed to hold but the energies differ. This approach is applied to gain an insight into specific details of the reaction mechanism. The Brønsted-Evans-Polanyi relationship coefficients are obtained and used for prediction of the catalytic reactivity of a wide range of (111) metals and alloys. Hydrogen kinetic isotope effect (KIE) of the reaction on the palladium surface and its temperature dependence are investigated, and numerical estimations are compared with experimental data. The KIE values of the reaction on two other (111) metal surfaces are also studied and correlated with the fcc H adsorption energies. It is shown that the reaction on Ag and Cu surfaces is highly activated, and the adsorption frequency at T = 300 K on Ag is significantly below 1 s-1. The H2 dissociation rates on palladium/silver alloy surfaces, popular hydrogen membranes, are estimated from published the fcc H adsorption energies on PdnAgm surfaces. The dependence of the H adsorption energy on coverage (as calculated by supercell density functional theory method) is used to show that the hydrogen dissociation rate on some (111) metal surfaces declines with coverage.