Kinetics of Reversible Consecutive Reactions

Abstract

Rate equations are exactly solved for the reversible consecutive reaction of the first-order and the time- dependence of concentrations is analytically determined for species in the reaction. With the assumption of pseudo first-order reaction, the calculation applies and determines the concentration of product accurately and explicitly as a function of time in the unimolecular decomposition of Lindemann and in the enzyme catalysis of Michaelis-Menten whose rate laws have been approximated in terms of reactant concentrations by the steady-state approximation. Mechanisms and rate laws of most chemical reactions can be described based on reversible, consecutive or parallel processes whose reaction rates are determined in detail as functions of time provided that they consist of the first-order or pseudo first-order steps. Concentrations in terms of time for species involved in these simple processes are easily obtained mathematically and the exact results are found in usual chemical kinetics texts. However many realistic reac- tions in fact contain the variety of combinations of the three rudimentary processes aforementioned and this complicates solving rate equations analytically and makes it difficult determining time-dependence of rates accurately. Approxi- mate methods thus have been developed and introduced to handle these reactions. A typical example is the steady-state approximation (SSA) which is often used to describe reactions in which reversible processes are combined with consecutive steps. It has been developed from the first-order reversible consecutive reaction which is represented by . If the rate of consumption of B is much greater than its rate of formation in the reaction scheme, the concentration of B may be presumed to change little with time and can be estimated in terms of reactant concentration from the assumption. The overall rate of reac- tion is then evaluated using the approximate concentration without detailed integration of rate equations. This simple but efficient technique is widely applied in reactions consist- ing with many steps to determine rates approximately. Ex- amples are the unimolecular decomposition known for Lin- demann mechanism and the enzyme catalysis of Michaelis- Menten. 1 The rate of reaction , if it is obtained fully as a function of time, can be extended under the pseudo first-order reaction condition to precisely determine the time-dependent behavior of reactions obeying mechanism of Lindemann or of Michaelis-Menten. Analytical solution to the rate equations of these reactions, however, has not been reported and the detailed time-dependence of concentrations is not known yet within the author's knowledge. In this work, rate equations of the first-order reversible consecutive reaction are exactly solved and two sample calculations are performed with different set of rate con- stants to show how concentrations depend on time in detail. The derivations are applied to reactions of the named mech- anisms to calculate concentrations of products analytically and are used to check the approximate rate laws by SSA for validity and limitations. Then a brief summary concludes the article.