Analytic solution to the rate law for a fundamental autocatalytic reaction mechanism operating in the “efficient” regime

• Published in: International journal of chemical kinetics 2022-10, Vol.54 (10), p.586-593
• Main Authors: Baird, James K., Barlow, Douglas A.
• Format: Article
• Language: English
• Subjects: analytical solution; autocatalysis; Chemical evolution; Closed form solutions; Exact solutions; Kermack‐McKendrick; Mathematical analysis; rate law integral; Reaction mechanisms; “efficient” regime
• ISSN: 0538-8066; 1097-4601
• DOI: 10.1002/kin.21598

Integrated rate laws in analytic form are scarce in the kinetics literature in cases where a reaction mechanism involves an autocatalytic step. As an example, we have taken up the challenge of integrating the rate law for a mechanism where an initial autocatalytic step competes with a subsequent step in which the catalyst decays unimolecularly to form the product. In addition to its chemical application, this mechanism forms the basis for theories of metabolism and chemical evolution. Even though it is one of the most elementary mechanisms possible, an exact analytic solution to the governing rate law has never been obtained in closed form. Of greatest chemical interest is the case where the autocatalytic step is much faster than the unimolecular step. This situation has the effect of enhancing the yield of the product and is known as the efficient regime. We derive a closed form analytical solution to the rate equation which is accurate in this limit. We use our analytic solution to obtain a formula for the induction time during which the rate of formation of product is accelerating.