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When monoenergetic atoms are continuously introduced into a thermal gas, they can undergo deactivating, activating, and reactive collisions. The net result of such collisions is to establish a steady-state distribution of laboratory energies which, although not as sharp as the initial distribution, preserves some of its features, such as being centred at about the initial energy. The reactive collisions which occur under these conditions are characterized by the associated relative energy distribution function and the energy-dependent reaction cross section. As a result, as the initial laboratory energy of the atoms is experimentally varied, the relative energy distribution function can be made to sample appropriately the reaction cross section curve. Therefore, from measurements of the competition between reaction and thermalization processes as a function of initial atom laboratory energies, and from a knowledge of the non-reactive differential scattering cross section, it is possible to obtain information about the dependence on relative energy of the rotationally averaged reaction cross section. The appropriate Boltzmann steady-state equation needed to obtain this information is derived in this paper and solved for an assumed set of reactive and non-reactive cross sections. Distribution functions of relative energies are thereby obtained and used to indicate the usefulness of the suggested measurements.