The relaxation of an internal state distribution in the presence of an excess of an inert gas is considered. The explicit time dependence of the nonequilibrium contributions to the transition rate coefficients is approximated using the Kapral-Hudson-Ross method. The resulting solution contains cross-correlation terms which do not appear when a single reaction is considered. It is shown that the first term of a perturbation expansion of an exact formal solution gives the Kapral-Hudson-Ross solution for short times, and the Chapman-Enskog solution at long times if there is a wide separation in time scales. The Kapral-Hudson-Ross, Chapman-Enskog, and exact solutions are compared for a two-state, hard-sphere model system.