A Methodology: The Determination of Kinetic Parameters for Single Independent Chemical Reaction Sequences
	Kenneth M. Maloney* and Ana Clare Frye Chemistry and Physics Division, Baton Rouge Community College, 201 Community College Drive, Baton Rouge, LA 70806
	Given the basic equation, WF = ¡•tr^p •exp(-SL·tr) [WF is the proportion reacted, ¡ is the scale reaction factor, tr is the reduced time, p and SL are the curvature and reaction terminus factors respectively], an essential consistency test requirement, for a given chemical reaction to be a true independent sequence, is that the curvature reaction factor, p, and the terminus reaction factor, SL, must be equivalent(i.e., p = SL). The single replacement Mg(s) + 2HCl(aq) ® MgCl2(aq) + H2(g) reaction is an irreversible reaction with a well defined and easily determined endpoint; therefore, it was used to test the validity of this important requirement. The experimental results confirmed the boundary condition requirement that p = SL for this single sequence reaction.
Introduction Ideally, chemical reactions are performed in well-defined, fixed reaction volumes where the initial concentrations of the reactants, at the initiation of the reaction sequences, are equivalent to the initial concentration measurements. Practically, however, many reactions necessarily occur in instantaneously generated reaction volumes where the volumetric capacities differ substantially from the original containment volumes of the reactants prior to the initiation of reaction. Realization of a methodology, in which these circumstances are simply treated as added dimensions to the analysis and determination of kinetic parameters, would be highly desirable. It has been found that such a methodology is inherently a derivative of the consistency tests and basic equation previously outlined.^1 In this paper, this methodology is detailed and the utility of the approach demonstrated by an evaluation and analysis of the experimental data obtained from a study of the single replacement Mg(s) + 2HCl(aq) ® MgCl2(aq) + H2(g) irreversible reaction system. Basic Considerations For the reaction A + B + C….etc ® Products, we have the simple n-order rate law

Rate = - d[A]/dt = k×[A]^n

(1)

for the case of equal initial concentrations of the individual reactants(i.e., [A] = [B] = [C] = …etc.) where k is the specific rate constant. Expressed in terms of the proportion reacted(WF)¹ and the initial concentration([A₀]) , one has

WF = 1 - [A]/[A0]

(2)

Integration of eq(1) and substitution into eq(2) gives
WF = 1 – {1 - [(1 - n)×k×t /[A]0 ^(1 - n) ] } ^1/(1-n)	(3)
where the units of k are conc^-(n-1) sec^-1. Eq(3) becomes

WF = 1 – (1 + z×tr)^e

(4)

where z = (n – 1)× k/[A₀] ^{(1 – n)}, k = k×t’, t_r = t/t’, and e = 1/(1-n). t_r is the corresponding reduced independent variable where the quantity t’ represents the value used to reduce the independent variable(t) sequence. 0n the occasion that one simply wishes to determine the order of the reaction, eqs 3 and 4 reduces to

WF = 1 - {1 - [(1 - n)×t/tC] }^1/(1-n)

(5)

where t_C = k^-1 ∙[A]₀ ^{(1 - n)}. For the singularity at n = 1(i.e., first-order),

WF = 1 – exp(-k×tr)

(6)

Figure 1 illustrates the proportion reacted vs time for the conventional definition of reaction order-n(i.e., eq 1).

Figure 1. Graphical representation of the proportion reacted vs reduced time for values of the conventional reaction order factor over a range of n = 0 – 3.

Many important chemical reactions, however, proceed through a number of complex pathways. For the vast majority of cases, the reactants A, B, C,….. etc are not present in equal concentrations. Therefore, the rate law is, routinely, much more complicated. Fortunately, many of these complex systems are amenable to meaningful simplifications. For example, Lavabre, et. al.² examined seven general kinetic schemes encompassing reversible, mixed first- and second-order as well as autocatalytic reactions and found them all to be governed by a single rate law. A single dimensionless variable Y(t), which represents the characteristics of the course of the reaction, was introduced. Y(t) is given by the expression

Y(t) = (1 – m)/[exp(a×t) – m]

(7)

where the parameter m characterizes the geometrical shape and the value of the a parameter depends on the particular reaction scheme under consideration². In effect, Y(t) is the normalized amplitude of the kinetics. Thus,

WF(t) = 1 – Y(t)

(8)

Separately¹, it was shown that a particular reaction sequence maybe uniquely represented by

WF(t) = ¡r • tr^p •exp(-SL×tr)

(9)

The independent variable, t_r, can be time, concentration or any other independent variable with the same functional dependence relative to WF. ¡_r is a scale factor, p is a geometrical shape factor, and S_L is a reaction terminus factor. Together, the parameters ¡_r, p, and S_L define a unique set of reaction field factors(RxnF-factors) for the particular reaction sequence of interest. Given that any reaction sequence can be expressed in terms of the proportion reacted versus an appropriate independent variable, the order/curvature factors n, m, and p factors(hereinafter call geometry factors) can be systematically determined¹ by the nonlinear regression of the measured experimental data with eqs 5,6, 8, and 9. Illustrated in Fig 2 is the functional relationship between n, m, and p.