A new method of processing the time-concentration data of reaction kinetics

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Abstract

Experimental data of reaction kinetics are usually in the form of concentration versus time. For kinetics investigation it is more convenient to have the data in the form of reaction rate versus concentration. Converting time-concentration data into concentration-reaction rate data is an ill-posed problem in the sense that if inappropriate methods are used the noise in the original data will be amplified leading to unreliable results. This paper describes a conversion procedure, independent of reaction rate model or mechanism, that manages to keep noise amplification under control. The performance of this procedure is demonstrated by applying it to several sets of published kinetic data. Since these data are accompanied by their rate equations, the computed rates are used to obtain the unknown parameters in these equations. Comparison of these parameters with published figures and the ease with which they are obtained highlights the advantages of the new procedure.

Introduction

In a typical reaction kinetics investigation, the progress of the reaction is monitored by measuring the concentration of one or more of the products or reactants as a function of time. Guided by prior experience or the likely reaction mechanism, a reaction rate model is then adopted that gives the rate as a function of concentration. The next step is to determine all the rate constants in the model from the experimental kinetic data. A variety of procedures has been developed to perform this task. For relatively simple reactions it is possible to obtain these constants by graphical construction of one kind or another (Connors, 1990). For other reactions it may be possible to deduce the rate constants by examining the initial slope of the time-concentration data. A more general technique is to treat the rate equations as ordinary differential equations. These equations are integrated to give the concentration of the reactants and products as a function of time with the rate constants appearing as unknown parameters. These parameters are then adjusted to minimize the deviation of the computed time-concentration profile from its experimentally observed counterpart.

Integration of simple rate equations can be performed analytically leading to simple expressions for the time-concentration profiles. However, for many rate equations analytical solution cannot be found and they have to be integrated numerically. Determination of the rate constants to minimize the deviation from experimental data becomes correspondingly more complicated. A variety of minimization schemes has been developed to find these parameters (Press, Teukolsky, Vetterling, & Flannery, 1992). But there is often another problem. Because of the complexity of the integrated time-concentration profile, there may be more than one set of rate constants that will lead to a local minimum of the deviation. It is also possible that the minimum may not be sufficiently sharp to allow the set of parameters to be determined to the degree of accuracy required. Faced with these complications one can return to the laboratory to increase the number of data points or to explore new experimental methods. Another approach is to modify or simplify the kinetic model if this can be physically justified. It is also not uncommon for the investigator to explore alternative methods of searching for the optimum set of parameters (Belohlav, Zamostny, Kluson, & Volf, 1997; Luus, 2000).

The present investigation suggests a different approach. Instead of integrating the rate equation and then matching the resulting time-concentration profile with experimental observations, the experimental data are converted into a concentration-reaction rate profile. This is then compared against the proposed reaction rate model directly. Since the expressions for the reaction rate models are usually simpler than that for the integrated time-concentration profiles, this allows the parameters in the model to be determined with greater ease and possibly also with a higher degree of certainty.

Thus the key step in this new procedure is the evaluation of the reaction rate from experimental time-concentration data. Direct numerical differentiation of experimental data usually does not lead to reliable reaction rates. This is because differentiation amplifies the unavoidable noise in the experimental data. Special procedures have been developed to perform this task (Engl, Hanke, & Neubauer, 2000). Recently, Yeow and Taylor (2002) showed that Tikhonov regularization is a reliable procedure for obtaining the derivative of velocity profiles from various experimentally measured velocity data. This procedure has been successful in keeping noise amplification under control and meets the important requirement that it does not require the assumption of a model to describe the original experimental data. This procedure can, with appropriate modifications, be applied to convert time-concentration data into concentration-reaction rate data. The aim of this paper is to demonstrate the performance of this procedure by applying it to a number of well-documented kinetic data taken from the literature and to show how the subsequent determination of the rate constants, based on rate models proposed by the originators of the data, can be simplified as a consequence of the results of Tikhonov regularization.

The derivation of the working equations of Tikhonov regularization may appear complicated but the computational steps associated with the procedure are relatively simple. Implementation of these steps will be described in detail. Using the resulting concentration-rate data to locate the optimum set of parameters for the assumed kinetic model can be performed using any of the proven numerical minimization techniques. Details of these techniques, apart from a brief reference to the software used, will be omitted.

Section snippets

The governing equation

The relationship between the reaction rate r(t)=dc(t)/dt and the time-concentration profile c(t) can be rewritten asc(t)=t′=0tr(t′)dt′+c0,where c0 is the initial concentration. This equation can be regarded as a Volterra integral equation for the unknown reaction rate r(t) and initial concentration c0 if this quantity is not measured directly or if the experimental measurement is considered to be unreliable. This is an integral equation of the first kind. The mathematical nature of this

Discretized equation

In discretized form Eq. (2) becomesciC=c0+tir0+tj′=0tj′=tiαij(ti−tj′)fjΔt′,i=1,2,…,ND,j=1,2,…,NK,where f1,f2,f3,…,fNK are the discretized f(t). The independent variable 0⩽t′⩽tmax is divided into NK uniformly spaced discretization points with step size Δt′=tmax/(NK−1). In the present investigation NK is typically of the order of 101 to 401. tmax=tND is the largest ti in the data set. αij is the coefficient arising from the numerical scheme used to approximate the integral in Eq. (2). For

Tikhonov regularization

Because of the noise in the experimental data, minimizing δTδ will not in general result in a smooth f(t). To ensure smoothness, additional conditions have to be imposed. In the present investigation, the additional condition is the minimization of the sum of squares of the second derivative d2f/dt2 at the internal discretization points. In terms of the column vector f, this condition takes on the form of minimizingj=2NK−1d2fdt′2j2=(βf)T(βf)=fTβTβf,where β is the tri-diagonal matrix of

Thermal decomposition of ammonium dinitramide (ADN)

Oxley, Smith, Zheng, Rogers, and Coburn (1997) reported the kinetics of thermal decomposition of ammonium dinitramide (ADN). Their data, for T=160°C, showing XRem(t), the fraction of ADN left, as a function of time are reproduced in Fig. 1(a).Eq. (14) is used to obtain f(t)=d2XRem/dt2 from these data. The rate of change r(t)=dXRem/dt and XRem(t) are then obtained, without relying on a reaction rate model, by numerical integration using the computed r0 and c0 as initial conditions. The rate r(t)

Discussion

The procedure based on Tikhonov regularization has successfully converted time-concentration data into concentration-reaction rate data. It is applicable to different types of reactions and to kinetic data presented in a variety of formats. The same equation, Eq. (14), is used to perform all the data conversion. The numerical operations in setting up the matrices in this equation and all the subsequent manipulations are relatively simple and can be implemented using many of the popular computer

Conclusions

Tikhonov regularization provides a reliable way of converting the time-concentration data of reaction kinetics into concentration-reaction rate data. The procedure is independent of reaction rate model and is applicable to a wide variety of reactions. As noise amplification is kept under control, the resulting concentration-reaction rate curve allows the rate constants in any rate model used to describe the reaction to be determined with relative ease and reliability.

Acknowledgements

The example on hydrogenation of toluene was suggested by an anonymous reviewer of an earlier version of this paper. This is gratefully acknowledged.

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