Aims: The gas-phase abundance of methanol in dark quiescent cores in the interstellar medium cannot be explained by gas-phase chemistry. In fact, the only possible synthesis of this species appears to be production on the surfaces of dust grains followed by desorption into the gas. Yet, evaporation is inefficient for heavy molecules such as methanol at the typical temperature of 10 K. It is necessary then to consider non-thermal mechanisms for desorption. But, if such mechanisms are considered for the production of methanol, they must be considered for all surface species. Methods: Our gas-grain network of reactions has been altered by the inclusion of a non-thermal desorption mechanism in which the exothermicity of surface addition reactions is utilized to break the bond between the product species and the surface. Our estimated rate for this process derives from a simple version of classical unimolecular rate theory with a variable parameter only loosely contrained by theoretical work. Results: Our results show that the chemistry of dark clouds is altered slightly at times up to 10^6 yr, mainly by the enhancement in the gas-phase abundances of hydrogen-rich species such as methanol that are formed on grain surfaces. At later times, however, there is a rather strong change. Instead of the continuing accretion of most gas-phase species onto dust particles, a steady-state is reached for both gas-phase and grain-surface species, with significant abundances for the former. Nevertheless, most of the carbon is contained in an undetermined assortment of heavy surface hydrocarbons. Conclusions: The desorption mechanism discussed here will be better constrained by observational data on pre-stellar cores, where a significant accretion of species such as CO has already occurred.< Réduire