From the general mole balance on some species , where for a CSTR steady state and perfect mixing are assumed,
Assuming a constant volumetric flow rate , which is the case for a liquid reactor or a gas phase reaction with no net generation of moles,
where the space-time is defined to be the ratio of the reactor volume to volumetric flow rate. It is the time required for a slug of fluid to pass through the reactor. For a decomposition reaction, the rate of reaction is proportional to some power of the concentration of . In addition, for a single reaction a  conversion may be defined in terms of the limiting reactant, for the simple decomposition that is species 
As can be seen, as the Damköhler number increases, the other term must decrease. The ensuing polynomial can be solved and the conversion for the rule of thumb Damköhler numbers found. Alternatively, one can graph the expressions and see where they intersect with the line given by the inverse Damköhler number to see the solution for conversion. In the plot below, the y-axis is the inverse Damköhler number and the x-axis the conversion. The rule-of-thumb Damköhler numbers have been placed as dashed horizontal lines.