Decomposition of the permanganate/oxalic acid overall reaction to elementary steps based on integer programming theory Krisztia ´n Kova ´cs, a Be ´la Vizva ´ri, b Miklo ´s Riedel c and Ja ´nos To ´th* d a Department of Physical Chemistry, Eo ¨ tvo ¨ s University, H-1117 Budapest, Pa ´ zma ´ny P. se ´ta ´ ny 1/A, Hungary. E-mail: Krisztian.Kovacs@iphysiol.unil.ch b Department of Operations Research, Eo ¨ tvo ¨ s University, H-1117 Budapest, Pa ´ zma ´ ny P. se ´ta ´ ny 1/C, Hungary. E-mail: vizvari@math.elte.hu c Department of Physical Chemistry, Eo ¨ tvo ¨ s University, H-1117 Budapest, Pa ´ zma ´ny P. se ´ta ´ ny 1/A, Hungary. E-mail: riedel@ludens.elte.hu d Department of Mathematical Analysis, Budapest University of Technology and Economics, H-1111 Budapest, Egry J. u. 1., Hungary. E-mail: jtoth@math.bme.hu Received 24th November 2003, Accepted 26th January 2004 F|rst published as an Advance Article on the web 23rd February 2004 A systematic method to obtain chemically acceptable decompositions of the overall autocatalytic reaction permanganate/oxalic acid into elementary steps is presented. The decomposition process was based on integer programming theory and chemical evidence, and it can be applied to elucidate other mechanisms as well. As a result, we obtained mathematical justification of the well-known autocatalytic nature and of the less-known crucial role of radical CO 2  . By inspecting the obtained decompositions, one can find reaction steps inaccessible to chemical intuition, and these point out the direction of further experimental investigations. 1 Introduction The principles for the automatic generation of chemical reac- tion mechanisms seem to be well elaborated. 1 However, while simulation programs are generally available, mechanism gen- eration programs are not available either on a commercial basis or in the public domain. In the present paper we provide a systematic and chemically acceptable decomposition of the overall autocatalytic reaction 2MnO 4  þ 6H þ þ 5H 2 C 2 O 4 ¼ 2Mn 2þ þ 8H 2 O þ 10CO 2 ð1Þ into elementary steps. It is a well-known reaction which has been studied since 1866, 2 and was recently investigated by several groups. 3–8 The basis of our decomposition method has been outlined in ref. 9; an alternative method can be found in ref. 10; a review of earlier methods is given in ref. 1. A reaction is considered elementary in this paper if no more than two species (mole- cules, ions, atoms, radicals) meet to react. The authors of ref. 6 have defined a mechanism consisting of eight reversible and six irreversible steps, of which three are not elementary in the sense defined above. The reaction mechan- ism includes the following 18 species: Here the square bracket [] refers to encounter pairs or tri- plets, 11 if they contains more than one species. Based upon chemical evidence (see Subsection 2.1), neither the entire set of species, nor the set of elementary steps are acceptable without alterations. We redefined species, reaction steps, and requirements. 2 Materials and methods: Computational and chemical 2.1 Species The above set of species is accepted with a few minor modifica- tions. We omit SO 4 2 , HSO 4  , and H 2 SO 4 , because sulfuric acid is not an essential component of the medium. On the other hand, there is a need to bridge the gap between Mn(VII) and Mn(IV). For this purpose we assume the presence of the hypothetical transient species ½MnC 2 O 4 ; MnO 4  ; H þ ½MnC 2 O 4 2þ ; MnO 3   þ ½MnC 2 O 4 2þ ; MnO 3  ; H þ  2þ ½H þ ; MnO 2 ; H 2 C 2 O 4  þ (of which three are binuclear manganese complexes). Thus, our set of 19 species is as follows: These species are made of five different atoms, if electrons are also considered as an atomic constituent: C, H, Mn, O, e. The transpose of the atomic matrix D of the species is as follows. H 2 C 2 O 4 HC 2 O 4  H þ C 2 O 4 2 Mn 2þ MnC 2 O 4 MnO 4  MnO 2 Mn 3þ CO 2 H 2 O [MnO 2 ,H 2 C 2 O 4 ] CO 2  [Mn(C 2 O 4 )] þ [Mn(C 2 O 4 ) 2 ]  H 2 SO 4 , HSO 4  , SO 4 2 , H 2 C 2 O 4 HC 2 O 4  H þ C 2 O 4 2 Mn 2þ MnC 2 O 4 MnO 4  MnO 2 Mn 3þ CO 2 H 2 O [MnO 2 ,H 2 C 2 O 4 ] CO 2  [Mn(C 2 O 4 )] þ [Mn(C 2 O 4 ) 2 ]  [MnC 2 O 4 , MnO 4  ,H þ ] [MnC 2 O 4 2þ , MnO 3  ] þ [MnC 2 O 4 2þ , MnO 3  ,H þ ] 2þ [H þ , MnO 2 , H 2 C 2 O 4 ] þ PCCP www.rsc.org/pccp RESEARCH PAPER 1236 Phys. Chem. Chem. Phys. , 2004, 6, 1236–1242 This journal is Q The Owner Societies 2004 DOI: 10.1039/b315211a