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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


Initial Concentrations (M) of Chemical Species


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


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:


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:


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.


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?


Input Verification

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

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.

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)]

Converted by Mathematica     

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