Use of the Taylor theorem to predict kinetic curves in an arbitrary mechanism

Abstract

This work serves as a proof of concept by reporting the development of a novel method for the solution of kinetic rate equations based on Taylor polynomials. The key observation is that mass action type rate equations are autonomous ordinary differential equations, which give the first derivative of a concentration as the polynomial of the concentrations of all substances appearing in the system. Two time scale transformations (based on first order and second order formal kinetics) are introduced to provide bounded polynomials in a way that the polynomial nature of the ordinary differential equation obtained with the transformed variable is conserved. Three chemical systems are used to test the performance of the method. The first system is a series of two consecutive reactions in which the first process is second order and the second process is first order with respect to its sole reactant. Kinetic curves in this scheme could be described by this method with a suitably chosen time transformation and a 100-degree polynomial with an accuracy better than 0.1%. The second system is a series of three reactions containing an autocatalytic step, it produces three extrema on the kinetic curve of one of the species (oligooscillation). The third system is the parallel-consecutive bimolecular reaction, for which high simulation accuracy was achieved with minimal effort. The simulations show that the time transformation used in the calculations has a central role, and the second order transformations typically works better than the other one. Also, the examples confirm that extrema on the kinetic traces are the problems.