We present in this article an efficient numerical solver for the time integration of atmospheric multiphase chemical kinetics. This solver is based on a second-order Rosenbrock scheme, that has been proposed by Verwer et al \cite{verwer7}. for gas-phase chemical kinetics. We show that the stiff time dependence of cloudy events (through liquid water content) has to be solved by the numerical scheme and a non-autonomous version has to be used. We benchmark our non-autonomous {\sc ros2} scheme with the classical {\sc lsode} solver for two kinetic schemes. For detailed schemes such as {\sc radm2}, the speed-up is of magnitude 5 for the same accuracy.