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Kinetic Equations

A Mathematica notebook accompanying this article, Kinetics.nb, contains the instructions to process the kinetic input and generate the graphs. To translate the chemical equations into a system of coupled differential equations, the user interface proceeds by identifying the chemical species, by providing initial conditions and the integration time, and by specifying each kinetic step. Every place a user should change information to specify the kinetic system is shown in blue. The detailed procedure is as follows.

Names of Species in Kinetic Analysis

[Graphics:../Images/index_gr_34.gif]

Initial Concentrations (M) of Chemical Species

[Graphics:../Images/index_gr_35.gif]

Length of Time (sec) Over Which to Obtain Kinetic Solution

[Graphics:../Images/index_gr_36.gif]

Elementary Rate Equation for Each Species

The entry of the kinetic equations is now shown by employing reaction 2 as an example, as follows.

2:  [Graphics:../Images/index_gr_37.gif]+ [Graphics:../Images/index_gr_38.gif]+ [Graphics:../Images/index_gr_39.gif] -> 2 HOBr         [Graphics:../Images/index_gr_40.gif][Graphics:../Images/index_gr_41.gif][Graphics:../Images/index_gr_42.gif]

The following entries are made for this elementary step:


[Graphics:../Images/index_gr_43.gif]

The first line is interpreted as follows: "In reaction 2, species #4 (i.e., [Graphics:../Images/index_gr_44.gif]) is lost according to a rate law including species #4 ([Graphics:../Images/index_gr_45.gif]), #1 ([Graphics:../Images/index_gr_46.gif]), and #3 ([Graphics:../Images/index_gr_47.gif]) with a rate constant of [Graphics:../Images/index_gr_48.gif]." The corresponding differential equation is as follows:

[Graphics:../Images/index_gr_49.gif]

This entry is one term in the complete expression for [Graphics:../Images/index_gr_50.gif]. Reactions 3 to 6 also contribute to [Graphics:../Images/index_gr_51.gif].

The six entries in each sublist correspond to the following format:

1.reaction number
2.species number for time-derivative
3."loss" or "production" of species
4.{first species, second species, ... and so on} of kinetic rate equation
5.rate multiplier [account for stoichiometry]
6.rate constant

Reactions that do not fit this mold (e.g., surface reactions) can be manually adjusted in the kinetic equations later.

The complete set of entries corresponding to reactions 1 to 7 of the Belousov-Zhabotinskii chemical mechanism are shown below.

[Graphics:../Images/index_gr_52.gif]

Special Instructions

There is occasionally the need for special instructions to be appended to the elementary rate equations. For example, in the Belousov-Zhabotinskii reaction, there is the parameter f. What value should it take?

[Graphics:../Images/index_gr_53.gif]

Input Verification

After providing this input to Mathematica, the user verifies the interpretation. The species numbers, labels, and initial concentrations are displayed, as follows:

[Graphics:../Images/index_gr_54.gif]
1 [Graphics:../Images/index_gr_55.gif] [Graphics:../Images/index_gr_56.gif]
2 HOBr [Graphics:../Images/index_gr_57.gif]
3 [Graphics:../Images/index_gr_58.gif] 2
4 [Graphics:../Images/index_gr_59.gif] [Graphics:../Images/index_gr_60.gif]
5 [Graphics:../Images/index_gr_61.gif] [Graphics:../Images/index_gr_62.gif]
6 [Graphics:../Images/index_gr_63.gif] [Graphics:../Images/index_gr_64.gif]
7 Ce(III) [Graphics:../Images/index_gr_65.gif]
8 Ce(IV) [Graphics:../Images/index_gr_66.gif]
9 organic [Graphics:../Images/index_gr_67.gif]

The individual kinetic terms are also displayed, as shown below for sorting by species. A complete kinetic term for the time rate of change of a species is the sum of the individual terms.

[Graphics:../Images/index_gr_68.gif]
1 [Graphics:../Images/index_gr_69.gif] 0.5 f x 1. [organic][Ce(IV)]
1 [Graphics:../Images/index_gr_70.gif] [Graphics:../Images/index_gr_71.gif]
1 [Graphics:../Images/index_gr_72.gif] [Graphics:../Images/index_gr_73.gif]
1 [Graphics:../Images/index_gr_74.gif] [Graphics:../Images/index_gr_75.gif]
2 d[HOBr]/dt = [Graphics:../Images/index_gr_76.gif]
2 d[HOBr]/dt = [Graphics:../Images/index_gr_77.gif]
2 d[HOBr]/dt = [Graphics:../Images/index_gr_78.gif]
2 d[HOBr]/dt = [Graphics:../Images/index_gr_79.gif]
3 [Graphics:../Images/index_gr_80.gif] [Graphics:../Images/index_gr_81.gif]
3 [Graphics:../Images/index_gr_82.gif] [Graphics:../Images/index_gr_83.gif]
3 [Graphics:../Images/index_gr_84.gif] [Graphics:../Images/index_gr_85.gif]
3 [Graphics:../Images/index_gr_86.gif] [Graphics:../Images/index_gr_87.gif]
3 [Graphics:../Images/index_gr_88.gif] [Graphics:../Images/index_gr_89.gif]
3 [Graphics:../Images/index_gr_90.gif] [Graphics:../Images/index_gr_91.gif]
4 [Graphics:../Images/index_gr_92.gif] [Graphics:../Images/index_gr_93.gif]
4 [Graphics:../Images/index_gr_94.gif] [Graphics:../Images/index_gr_95.gif]
4 [Graphics:../Images/index_gr_96.gif] [Graphics:../Images/index_gr_97.gif]
4 [Graphics:../Images/index_gr_98.gif] [Graphics:../Images/index_gr_99.gif]
4 [Graphics:../Images/index_gr_100.gif] [Graphics:../Images/index_gr_101.gif]
5 [Graphics:../Images/index_gr_102.gif] [Graphics:../Images/index_gr_103.gif]
5 [Graphics:../Images/index_gr_104.gif] [Graphics:../Images/index_gr_105.gif]
5 [Graphics:../Images/index_gr_106.gif] [Graphics:../Images/index_gr_107.gif]
6 [Graphics:../Images/index_gr_108.gif] [Graphics:../Images/index_gr_109.gif]
6 [Graphics:../Images/index_gr_110.gif] [Graphics:../Images/index_gr_111.gif]
7 d[Ce(III)]/dt = 1. [organic][Ce(IV)]
7 d[Ce(III)]/dt = [Graphics:../Images/index_gr_112.gif]
8 d[Ce(IV)]/dt = -1. [organic][Ce(IV)]
8 d[Ce(IV)]/dt = [Graphics:../Images/index_gr_113.gif]
9 d[organic]/dt = -1. [organic][Ce(IV)]


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