We report a robust quadratic relation between the inverse substrate concentration and the second moment, < t(2)>, of the catalytic turnover time distribution for enzyme reactions. The results hold irrespective of the mechanism and dynamics of the enzyme reaction and suggest a novel single molecule experimental analysis that provides information about reaction processes of the enzyme-substrate complex and ergodicity of the enzyme reaction system, which is beyond the reach of the conventional analysis for the mean reaction time, < t >. It turns out that < t >(2)> - 2 < t >(2) is linear in inverse substrate concentration for an ergodic homogeneous enzyme system given that the enzyme substrate encounter is a simple rate process, and its value at the high substrate concentration limit provides direct information about if any non-Poisson reaction process of the enzyme-substrate complex. For a nonergodic heterogeneous reaction system, the corresponding quantity becomes a quadratic function of the inverse substrate concentration. This leads us to suggest an ergodicity measure for single enzyme reaction systems. We obtain a simple analytic expression of the randomness parameter for the single catalytic turnover time, which could provide a quantitative explanation about the previously reported randomness data of the beta-galactosidase enzyme. In obtaining the results, we introduce novel chemical kinetics applicable to a non-Poisson reaction network with arbitrary connectivity, as a generalization of the conventional chemical kinetics.