Analytical solutions for the rate equations of irreversible two-step consecutive processes with second order later steps

The same experimental data can often be equally well described by multiple mathematically equivalent kinetic schemes. In the present work, we investigate several model‐fitting algorithms and their ability to distinguish between mechanisms and derive the correct kinetic parameters for several different reaction classes involving consecutive reactions. We have conducted numerical experiments using synthetic experimental data for six classes of consecutive reactions involving different combinations of first‐ and second‐order processes. The synthetic data mimic time‐dependent absorption data as would be obtained from spectroscopic investigations of chemical kinetic processes. The connections between mathematically equivalent solutions are investigated, and analytical expressions describing these connections are derived. Ten optimization algorithms based on nonlinear least squares methods are compared in terms of their computational cost and frequency of convergence to global solutions. Performance is discussed, and a preferred method is recommended. A response surface visualization technique of projecting five‐dimensional data onto the three‐dimensional search space of the minimal function values is developed.

“…A detailed derivation of a solution for the component of B can be found in . Omitting intermediate calculus, we get

\begin{matrix} true \\ right {[normalB]}_{t} & left = \sqrt{\frac{k_{1} {[A]}_{0} e^{- k_{1} t}}{2 k_{2}}} \\ left \frac{2 K_{1} ((), \sqrt{8 k_{2} {[A]}_{0} e^{- k_{1} t} / k_{1}}) - ω I_{1} ((), \sqrt{8 k_{2} {[A]}_{0} e^{- k_{1} t} / k_{1}})}{2 K_{0} ((), \sqrt{8 k_{2} {[A]}_{0} e^{- k_{1} t} / k_{1}}) + ω I_{0} ((), \sqrt{8 k_{2} {[A]}_{0} e^{- k_{1} t} / k_{1}})} \end{matrix}

\begin{matrix} true \\ right ω = \frac{\sqrt{8 {[normalA]}_{0} / ((), k_{2} / k_{1})} K_{1} ((), \sqrt{8 (k_{2} / k_{1}) {[A]}_{0}})}{\sqrt{2 {[normalA]}_{0} / ((), k_{2} / k_{1})} I_{1} ((), \sqrt{8 (k_{2} / k_{1}) {[A]}_{0}})} \end{matrix}

\begin{matrix} true \\ right K_{n} (x) & = & left x^{n} \prod j \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

“…Lente describes in detail the process of obtaining analytical solutions for system (6). The final solution is

\begin{matrix} true \\ right {[normalA]}_{t} = \frac{{[A]}_{0}}{1 + 2 k_{1} {[normalA]}_{0} t} \end{matrix}

\begin{matrix} true \\ right {[normalB]}_{t} = \frac{k_{1} {[normalA]}_{0}}{2 k_{2}} \frac{((), 1 + α) {(\sqrt{1 + 2 k_{1} {[normalA]}_{0} t})}^{- 1 + α} + ω (1 - α) {(\sqrt{1 + 2 k_{1} {[normalA]}_{0} t})}^{- 1 - α}}{{(\sqrt{1 + 2 k_{1} {[normalA]}_{0} t})}^{1 + α} + ω {(\sqrt{1 + 2 k_{1} {[normalA]}_{0} t})}^{1 - α}} \end{matrix}

\begin{matrix} true \\ right ω = \frac{1 + α}{α - 1}, α = \sqrt{2 k_{2} / k_{1} + 1} \end{matrix}

…”

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Equivalency of Kinetic Schemes: Causes and an Analysis of Some Model Fitting Algorithms

Петров

Dergachev

2017

“…It seems that the rate equations given in Eq. (5) do not have in easily accessible closed-form solution, although analytical solutions for a number of two-step schemes are available in general [2,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In cases like this, it is occasionally useful to seek a solution that gives one of the concentrations as a function of the other rather than time [2].…”

Section: Coupled Consecutive Reactionsmentioning

confidence: 99%

Reaction Schemes That Are Easily Confused with a Reversible First-Order Reaction

Balogh

Lente

Kalmár³

et al. 2015

Self Cite

A detailed kinetic analysis of two schemes, one involving coupled consecutive processes and another featuring parallel reactions and decay of ???, is presented here using Taylor series expansion. It is shown that both of these schemes are easily confused with the reversible second order reaction in a routine kinetic study. The kinetic traces predicted by both schemes are sufficiently close to pseudo-first order curves so that it is practically impossible to identify the deviations based on data with the usual experimental errors, which was also demonstrated by fitting simulated theoretical curves to exponential functions. The dependence of the pseudo-first order rate constants on the concentration of the excess reagent features the same trend as in the case of a reversible reaction: a straight line with an intercept is observed. This analysis emphasizes that the reversible nature of reactions should be demonstrated by direct equilibrium studies, kinetic observations alone might be misleading.

Double Exponential Evaluation under Non-Pseudo-First-Order Conditions: A Mixed Second-Order Process Followed by a First-Order Reaction

Kiss

Ősz

2017

In this paper, a double exponential approximating approach is described for a quite common kinetic model (mixed second‐order formation of an intermediate followed by its first‐order decay) under non‐pseudo–first‐order conditions (i.e., when the initial ratio of the two reactants is between 1 and 10). For the evaluation, first the exact kinetic curves predicted by the two‐step model were calculated and then fitted to a double exponential function. The goodness of the fits and the estimated parameters of the double exponential function for both I and P concentrations were determined as a function of the rate constants and initial concentrations in the two‐step model. It was found that the fit of the double exponential function is acceptable or very good under these conditions despite the fact that none of the reagents is in a large excess. Since UV–vis absorption spectroscopy is probably the most common technique to follow kinetic traces, we also made efforts to deal with the typical properties of monitoring the process through UV–vis. It was found that the experimental curves can be fitted quite well with a double exponential function if the reagents have minor absorption compared to the intermediate and/or product. The connection between the observed rate constants and the rate constants of the above‐mentioned mechanism is also studied.