The desorption of molecular hydrogen during low-energy electron irradiation of self-assembled monolayers containing n-alkanethiols has been previously reported, yet to date, there is no consensus as to the mechanism for the formation of this ubiquitous product. In this study, mixed monolayers containing known ratios of perhydrogenated and perdeuterated alkanethiols were chemisorbed to Au(111)/mica substrates and used as targets for low-energy electron irradiation; by measuring the electron-stimulated production of H-2, D-2, and HD as a function of the film composition, we unambiguously show that the desorbing molecular hydrogen is formed via a two-step bimolecular reaction process. The initial electron-molecule scattering event produces a reactive atomic fragment, which then abstracts a hydrogen atom from a nearby molecular site to produce the measured bimolecular yields; the contribution of one-step unimolecular dissociation channels to the overall molecular hydrogen yields is below the similar to 5% detection limit. The dependence of the electron-induced modifications to the film on the incident electron energy suggests that the primary event is dissociative electron attachment, and that the primary reactive fragment is most likely H-. Quantitative analysis of the product yields shows that while similar to 80% of the molecular hydrogen is formed by this bimolecular mechanism within the film, the remaining 20% is formed from reactive atomic fragments that are ejected from the film and subsequently react with residual H2O adsorbed on the chamber walls.