Multi-Step Reactions: The Methods for Analytical Solving the Direct Problem
If a reaction proceeds by a large number of elementary steps and involves many different substances, developing its mathematical model “by hand” turns into a quite exhausting procedure fraught with different possible errors, especially provided complicate
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Multi-Step Reactions: The Methods for Analytical Solving the Direct Problem
2.1
Developing a Mathematical Model of a Reaction
If a reaction proceeds by a large number of elementary steps and involves many different substances, developing its mathematical model “by hand” turns into a quite exhausting procedure fraught with different possible errors, especially provided complicated reaction stoichiometry. This stage can be considerably simplified by using matrix algebra suits. Let us consider a reversible reaction consisting of two elementary steps: k1
ðIÞ
A þ B ! AB;
ðIIÞ
AB ! A þ B:
k2
A rate of each of the steps is written as ~ r ¼ k1 CA ðtÞCB ðtÞ; r ¼ k2 CAB ðtÞ: Obviously, a reaction mathematical model is an equation system dCA ðtÞ ¼ ~ r þ r ¼ k1 CA ðtÞCB ðtÞ þ k2 CAB ðtÞ; dt dCB ðtÞ ¼ ~ r þ r ¼ k1 CA ðtÞCB ðtÞ þ k2 CAB ðtÞ; dt dCAB ðtÞ ¼~ r r ¼ k1 CA ðtÞCB ðtÞ k2 CAB ðtÞ: dt
V.I. Korobov and V.F. Ochkov, Chemical Kinetics with Mathcad and Maple, DOI 10.1007/978-3-7091-0531-3_2, # Springer-Verlag/Wien 2011
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2 Multi-Step Reactions: The Methods for Analytical Solving the Direct Problem
Let us define a stoichiometric matrix a for the given kinetic scheme as
1 a¼ 1
1 1 : 1 1
The number of rows in a stoichiometric matrix corresponds to the number of elementary steps and the number of columns is equal to the total number of substances taking part in a reaction. Each matrix element is a stoichiometric factor of a definite substance at a given step. At the same time, a minus sign is ascribed to stoichiometric factors of reactants and ones of products have a positive sign. If a substance is not involved in a given stage, a corresponding stoichiometric factor is equal to zero. Thus, the middle column of the stoichiometric matrix reflects participation of substance B in the overall process: it is consumed at the step (I) and accumulated at the step (II). Let us place expressions for the rates of each of the steps in a rate vector r r¼
~ r k C C ¼ 1 A B : r k2 CAB
Let us now find a product of the transposed matrix a and the vector r 2
1 aT r ¼ 4 1 1
3 2 3 1 k1 CA ðtÞCB ðtÞ þ k2 CAB ðtÞ C ðtÞC ðtÞ k B ¼ 4 k1 CA ðtÞCB ðtÞ k2 CAB ðtÞ 5: 1 5 1 A k2 CAB ðtÞ 1 k1 CA ðtÞCB ðtÞ k2 CAB ðtÞ
It is easy to see that this yields a vector formed by right parts of a differential equation system for a kinetic scheme under consideration. Thereby, a kinetic equation for a multi-step reaction in a matrix form is written as: d CðtÞ ¼ aT rðtÞ: dt
(2.1)
Naturally, the above multiplication of a stoichiometric matrix and a rate vector can be performed by suits of mathematical packages. Figure 2.1 shows the operation sequence of such automatic developing the mathematical model of the considered reaction in the Mathcad environment. Another way of developing a mathematical model can also be indicated. If an overall chain of conversions contains reversible steps, they can be considered as formally elementary. As a result, dimensions of a stoichiometric matrix and a rate vector can b
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