János TÓTH¹, Krisztián KOVÁCS², Béla VIZVÁRI³, Miklós RIEDEL²

¹ Department of Mathematical Analysis, Budapest University of Technology and Economics
H-1111 Budapest, Egry J. u. 1., HUNGARY
² Department of Physical Chemistry, Loránd Eötvös University
H-1117 Budapest, Pázmány P. sétány 1/ A., HUNGARY
³Department of Operations Research, Loránd Eötvös University
H-1117 Budapest, Pázmány P. sétány l/C., HUNGARY
 

The aim of the present work

is to decompose a complicated chemical reaction into elementary steps with full knowledge of the species figuring in the process. The elaborated computational method also permits to arrange reaction steps into right order therefore a complete reaction mechanism can be established. As all the possible decompositions can be generated, there is chance of finding mechanisms inaccessible to the chemical intuition.
The many times investigated but still elusive permanganate/oxalic acid reaction serves as an example to demonstrate the performance of the method.

The autocatalytic reaction

                                                                atomic matrix of the species

	C	H	Mn	O	e
H2C2O4	2	2	0	4	0
HC2O4-	2	1	0	4	1
H+	0	1	0	0	-1
C2O42-	2	0	0	4	2
CO2-	1	0	0	2	1
CO2	1	0	0	2	0
H2O	0	2	0	1	0
MnO4-	0	0	1	4	1
MnO2	0	0	1	2	0
Mn3+	0	0	1	0	-3
MnC2O4	2	0	1	4	0
Mn2+	0	0	1	0	-2
[MnC2O4, MnO4-, H+]	2	1	2	8	0
[MnC2O42+, MnO3-]+	2	0	2	7	-1
[MnC2O42+, MnO3-, H+]2+	2	1	2	7	-2
[H+, MnO2, H2C2O4]+	2	3	1	6	-1
[MnO2, H2C2O4]	2	2	1	6	0
[Mn(C2O4)]+	2	0	1	4	-1
[Mn(C2O4)2]-	4	0	1	8	1

Computation of reaction steps
Requirements
1. The steps fulfil the law of atomic balance and also the law of charge balance.
2. Elementary reaction step is either uni- or bimolecular
3. Reversible steps are considered as pairs of reaction and antireaction (i.e. they are two steps)
We are going to construct all the possible uni- and bimolecular steps fulfilling the first requirement.

Levels of species and reaction steps
There is another necessary condition to be fulfilled. At the very beginning of the reaction only reaction steps among the initial species can take place. Then, reaction steps using the products of these reactions can also enter the scene, and so on. This hierarchy can be formalised as follows.

• Let us assign the zero index to the initial species.
• Let us assign zero index to all elementary steps which can take place using only initial species.
• In the next step let us assign index 1 to all species which can be obtained as products of reaction steps of zero index without an index up to now,
• i.e. a species having zero index but also produced by some reactions of zero index, is not getting index 1,
• and let the same index 1 be given to all reaction steps without an index up to now, but being able to take place using species of index not greater than one.
• Let us assign the index +¥ to all remaining species and reaction steps.

It is obvious (1) that one has to adjoin MnC₂O₄ to the set of initial species. The new and final computation result (2) shows that the sequence of the reaction steps is acceptable.

Chemically interpreted computational results 1
By the algorithm we obtained 249 decompositions of the overall reaction. The following is chosen to demonstrate the course of the levelling process.

Levelling 1
                                                                                                                   equation
H₂C₂O₄ + MnO₂ ® [MnO₂, H₂C₂O₄]                                                             (3)
H₂C₂O₄ + [MnO₂, H₂C₂O₄] ® CO₂^- + CO₂ + 2H₂O + [Mn(C₂O₄)]⁺             (6)
H⁺ + HC₂O₄^{- ®}H₂C₂O₄                                                                                                               (4)
C₂O₄^2- + H⁺ ® HC₂O₄^-                                                                                   (7)
H⁺ + MnC₂O_{4 ®}HC₂O₄^- + Mn²⁺                                                                                             (2)
C₂O₄^2- + [Mn(C₂O₄)]⁺® [Mn(C₂O₄)₂]^-                                                           (8)
MnC₂O₄ + MnO₄^- ® CO₂^- + CO₂ + 2 MnO₂                                                  (1)
[Mn(C₂O₄)₂]^- ® CO₂^- + CO₂ + MnC₂O₄                                                         (9)
2 CO₂^- ® C₂O₄^2-                                                                                              (5)

Repeated computation also with MnC₂O₄ as initial species
Levelling 2
                                                                                                                        index             equation
MnC₂O₄ + MnO₄^- ® CO₂^- + CO₂ + 2 MnO₂                                                  0                     (1)
H⁺ + MnC₂O_{4 ®}HC₂O₄^- + Mn²⁺                                                                                             0                     (2)
H₂C₂O₄ + MnO₂ ® [MnO₂, H₂C₂O₄]                                                              1                     (3)
H⁺ + HC₂O₄^{- ®}H₂C₂O₄                                                                                                                1                     (4)
2 CO₂^- ® C₂O₄^2-                                                                                              1                     (5)
H₂C₂O₄ + [MnO₂, H₂C₂O₄] ® CO₂^- + CO₂ + 2H₂O + [Mn(C₂O₄)]⁺              2                     (6)
C₂O₄^2- + H⁺ ® HC₂O₄^-                                                                                    2                     (7)
C₂O₄^2- + [Mn(C₂O₄)]⁺® [Mn(C₂O₄)₂]^-                                                            3                     (8)
[Mn(C₂O₄)₂]^- ® CO₂^- + CO₂ + MnC₂O₄                                                          4                     (9)

Discussion

• The levelling cannot be carried out if the set of initial species consists only of reactants (H⁺, KMnO₄, H₂C₂O₄) of the overall reaction.
• Without MnC₂O₄ several species figuring in the mechanism has infinite index.
• This is consistent with the experimental fact that the process is autocatalytic, and does not take place unless a small quantity of the product is present at the beginning.

Perspectives
By this computation method

• the species, consequently also the reaction steps can be arranged in right order
• i.e. a possible reaction mechanism can be constructed
• which can be later on verified by experimental methods

• corroborate the occurrence of previously proposed reaction steps
• rule out some reaction steps
• find reaction steps inaccessible to the chemical intuition
• point out the direction of the experiments since further measurements are needed to decide which decomposition corresponds to the real mechanism of the reaction
• The method can be applied in case of other complex reactions, too.