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Complex Kinetic Diagram

A complex kinetic diagram is introduced in Figure 3 to understand the interactions that result in the oscillations shown in Figure 1. The complex kinetic diagram shows a snapshot at 360 sec of the rates of reactions leading to the production of chemical species. The right-hand side of the diagram is a bar chart indicating the percent change in the concentration of the nine chemical species per unit time. On the left-hand side of this diagram, the fastest rate of all seven reactions is shown to be [Graphics:../Images/index_gr_124.gif], and that value represents unity. All other rates are scaled to that value. The nine black circles shown each have a radius of 1. The red circles indicate net production of a species while the blue circles indicate a net loss of a species. The rate of loss or production is proportional to the radius of the circle shown (i.e., as a fraction of unity). We can say, for example, that d[[Graphics:../Images/index_gr_125.gif]]/dt is being lost at [Graphics:../Images/index_gr_126.gif] whereas Ce(III) is being produced at approximately 75% of this rate. The dashed lines show connections between products and reactants. The product side of the line is indicated by the heavy black mark. Thus, in this scheme, [Graphics:../Images/index_gr_127.gif] is potentially produced from [Graphics:../Images/index_gr_128.gif], [Graphics:../Images/index_gr_129.gif], and [Graphics:../Images/index_gr_130.gif]. In turn, [Graphics:../Images/index_gr_131.gif] produces Ce(IV) and [Graphics:../Images/index_gr_132.gif]. How much [Graphics:../Images/index_gr_133.gif] is actually being produced? The length of the red line covering the heavy black line indicates the rate at which a species is actually being produced. It is apparent that very little [Graphics:../Images/index_gr_134.gif] is being produced. In contrast, Ce(III) is rapidly being produced from organic and Ce(IV). We can also see that HOBr is being produced rapidly from [Graphics:../Images/index_gr_135.gif], which is likewise being produced rapidly from Ce(IV). Very little is producing Ce(IV), and its net loss rate (circle in blue) is seen to be substantial.

[Graphics:../Images/index_gr_136.gif]

Figure 3. Complex kinetic diagram showing a snapshot of the Belousov-Zhabotinskii reaction at 360 sec. See further explanation in the text.

In Figure 4, the concentrations of Ce(IV), [Graphics:../Images/index_gr_137.gif], and [Graphics:../Images/index_gr_138.gif] are overlaid through several chemical cycles. The drop in [[Graphics:../Images/index_gr_139.gif]] is seen to be followed by a rapid increase in [[Graphics:../Images/index_gr_140.gif]]. Similarly, the rise in [Ce(IV)] is coincident with the rise in the [[Graphics:../Images/index_gr_141.gif]]. A specific timepoint (e.g., 360 sec) can be selected and analyzed in greater detail, as was done in Figure 3. Timepoints of 309, 325, 332, and 342 sec are also selected and shown as complex kinetic diagrams in Figure 5. Figure 4 shows that these timepoints cover the initiation, peak, and quiescence of one chemical cycle.


[Graphics:../Images/index_gr_142.gif]


Figure 4. Concentrations of Ce(IV), [Graphics:../Images/index_gr_143.gif], and [Graphics:../Images/index_gr_144.gif] from 200 to 500 sec. The gray bars are shown at timepoints of 309, 325, 332, 342, and 360 sec and correspond to the complex kinetics diagrams shown in Figures 3 and 5.

The Mathematica notebook is highly automated. Subsequent to the numerical solutions obtained in Figures 1 and 2, the user can execute the following command to display a complex kinetics diagram at a given time point. The user enters the parameters shown in blue, that is, drawrates[1,2,3,4,5] where the first parameter is the timepoint and the other parameters are options not employed in this article but discussed in the notebook itself.

[Graphics:../Images/index_gr_145.gif]

[Graphics:../Images/index_gr_146.gif]

Figure 5. Complex kinetic diagrams of the Belousov-Zhabotinskii reaction at 309, 325, 332, and 342 sec. The timepoint at 360 sec is shown in Figure 3.


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