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. (SIAM J. Sci. Comput. 20 (4) (1999) 1456) 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 the non-autonomous ROS2 scheme with the classical LSODE solver for two kinetic schemes. For detailed schemes such as RADM2, the speed-up is of magnitude 5 for the same accuracy.