Uncertainty and Invariants of Inverse Problem Solutions for a First-Order Reaction

Abstract

The problem of uncertainty of inverse problem solution for a first-order reaction with rate constant distribution of reactive states is considered. Although every distribution determines the reaction kinetics unambiguously, the inverse problem solving recovers a set of distributions in accordance with the same kinetic data. A choice of one of them is shown to be impossible in principle because all distributions give the same description of the reaction kinetics, and uncertainty of the solution remains even if the accuracy and number of measurements are increased many times. A method for the calculation of a set of distributions is proposed. The method can be applied to systems with both discrete and continuous sets of states. The distributions belonging to the set of inverse problem solutions are shown to have invariants, which can be determined from the initial moments of normalized distributions or directly from data on the reaction kinetics. The physical meaning of the invariants is considered as statistical rate constants for states with high, medium, and low reactivities that characterize the width of the true distribution and the degree of its asymmetry. The application of the invariants is considered for particular examples of the fluorescence decay of Rhodamine 6G in porous glasses and of 5,10,15,20-tetrakis(pentafluorophenyl)porphyrin in the Langmuir–Blodgett layers. Despite uncertainty of the inverse problem solution, the application of the invariants gives rise to unambiguous characteristics of the reactivity of compounds in reactions with distributed parameters.