Abstract: A process for cooling a hot reaction gas, as formed, in the preparation of phthalic anhydride by oxidation of o-xylene with hot air in a reactor (213), to a predetermined inlet temperature for introduction into a separator (221), in which the separation into crude phthalic anhydride and air is then effected, is described. A simple and economical design, which can be used for comparable processes, too, of a system for cooling the hot reaction gas, which simultaneously gives rise to low maintenance costs, is obtained by locating upstream of the separator (221) a heat exchanger (227) through which a cooling medium flows, the hot reaction gas being passed through the heat exchanger (227) for interaction with the cooling medium. Especially such a system comprises in addition to a main reactor (213) a downstream reactor (241), which my have a means for intermediate cooling (243), and the gaseous cooling medium leaving the heat exchanger 227 will be fed to the main reactor (213) as reactant.