Systematic Application of the Principle of Detailed Balancing to Complex Homogeneous Chemical Reaction Mechanisms

Abstract

It is not uncommon for proposed complex reaction mechanisms to violate the principle of detailed balancing. Here, we draw attention to three ways in which such violations can occur: reversible reaction loops where the rate constants do not attain closure, illegal loops, and reversible steps having rate equations in the forward and reverse directions that are inconsistent with the equilibrium expressions. We present two simple methods to test whether a proposed mechanism is consistent with the first two aspects of the principle of detailed balancing. Both methods are restricted to closed homogeneous isothermal reactions having mechanisms that consist of stoichiometrically balanced reaction steps. The first method is restricted to mechanisms in which all reaction steps are reversible; values of Δ f G°are assigned to all reaction species, equilibrium constants are then computed for all steps, and all rate constants for elementary steps are constrained by the relationship K eq = k f /k r . The second method is applicable to mechanisms that can consist of a series of reversible and/or irreversible reaction steps. One first examines the subset of reversible steps to determine whether any of these steps are stoichiometrically equivalent to a combination of any of the other steps. If so, the forward and reverse rate expressions must yield equilibrium constants that are in agreement with the stoichiometric relationships. Next, the complete set of steps is examined to look for "illegal reaction loops". Both of these procedures are performed by constructing matrices that represent the stoichiometries of the various reaction steps and then performing row reductions to identify basis sets of loops. A method based on linear programming is described that determines whether a mechanism contains any illegal loops. These methods are applied in the analysis of several published reaction mechanisms.

Large-Scale Models of Radiation Chemistry and the Principle of Detailed Balancing

“…It is anticipated that DETBAL will be useful in testing large-scale kinetic models in other fields, such as advanced oxidation processes, , environmental chemistry, and metabolism. It should also simplify the task of checking smaller-scale models for compliance with the principle of detailed balancing such as those described earlier …”

“…DETBAL prepares for this analysis by isolating the submatrix of loops from the second row reduction, transposing it, and appending a row of −1. The linear programming analysis is completed by exporting this final matrix and analyzing it with Mathematica as described earlier …”

“…One of us recently showed that it is not uncommon for medium-scale kinetic models to violate the principle of detailed balancing. 2 Three types of violations were identified: (1) reversible reaction steps where the forms of the forward and reverse rate equations disagreed with the equilibrium constant expression, (2) sets of reversible reaction steps that formed a "loop" but had rate constants that disagreed with the equilibrium requirements, and (3) sets of irreversible and reversible reaction steps that formed illegal loops. It was found that mechanisms with increasing numbers of steps were more likely to include these violations.…”

The principle of detailed balancing, the iron-catalyzed disproportionation of hydrogen peroxide, and the Fenton reaction

“…We recently published a method based on linear algebra to find loops, both legal and illegal. 40 Subsequently we devised a computer software method, DETBAL, that greatly simplifies and automates the process. 41 It is not uncommon for mechanisms to have loops that are linked by shared steps; in such cases the complete set of loops is generated by linear combinations of a basis set, and there is no unique basis set.…”

“…The principle of detailed balancing has been the topic of several of the author's recent publications, [39][40][41][42] but its history extends back almost a century. It is a fundamental principle of chemical kinetics that has both qualitative and quantitative implications.…”