ON THE POSSIBILITY OF USING GRAPHS IN ORDER TO DERIVE RATE EQUATIONS FOR STRAIGHT CHAIN REACTIONS (PART I)

Transcription

1 ON TE POSSIBILITY OF USING GRAPS IN ORDER TO DERIVE RATE EQUATIONS FOR STRAIGT CAIN REACTIONS (PART I) E Segal* The author presents some results of an attempt to apply the graph theory for the ineti desription of straight hain reations Using some lassial examples already solved by help of the quasi-steady-state approximation, a ritial disussion on the apabilities of the graphtreatment is brought up The ineti analysis of straight hain reations by help of plane diagrams originates in the wor of Christiansen [] On the other hand, beginning with the wor of King and Altman [] the graph theory was suessfully applied in order to desribe inetially losed atalyti sequenes [ 6] Taing into aount that the straight hain reations are also based on losed sequenes, the idea of desribing them by help of graph theory appears as natural This is the reason why Temin suggested a method to derive rate equations for atalyti and straight hain with linear mehanism reations by help of graphs [7] In suh a theoretial frame the graphial shemes of Christiansen were transposed in graphs [8] Detailed ineti treatments of onrete straight hain reations are rather diffiult This is probably due to the nonlinear harater of the mehanisms assoiated to straight hain reations At least one elementary step of suh reations involves more than an ative enter As nown, rate equations for atalyti sequenes an be obtained relatively easy mainly for linear mehanisms The problem is whether the diffiulties related to the nonlinearity of the straight hain mehanisms ould be removed and onsequently whether graphs an be used in order to allow derivation of the orresponding rate equations [9] In the following, the apabilities of graph theory to desribe inetially straight hain reations are going to be disussed We are going to onsider gas phase straight hain reations whih were already analyzed by help of the quasi-steady-state approximation [QSSA] The Reation between ydrogen and omine It is a well nown reation whose inetis was investigated by Bodenstein and Lind [] and whose mehanism was suggested in the wors of Christiansen [], erzfeld [] and Polanyi [] They showed that the experimental rate equation for the reation: (g) + (g) = (g) * Department of Physial Chemistry, Faulty of Chemistry, University of Buharest, Blvd Elisabeta -, Buharest, ROMANIA The author supposes that the readers of this artile possess the neessary nowledge onerning the use of graphs in inetis of atalyti reations

2 E SEGAL an be aounted for if the following sequene of elementary steps is onsidered: (I) Atually sequene (I) orresponds to a mehanism with two basi routes, I and II [8] I II (II) + + with stoihiometri numbers of the elementary steps written in the orresponding olumns ' II I -' - - Fig : Graph assoiated to sequene II (aording to referene [8]) The graph assoiated to mehanism (II) has the following form [8] (figure ): The dotted seondary edges aount for the fat that two ative enters are generated in step, and two ative enters are onsumed in step For simpliity sae the frequenies of the elementary steps whih usually label the edges, i, have been substituted only by figure i Unlie graphs orresponding to atalyti sequenes the ative enter attending the first step, aording to Temin [7], was substituted by a so alled zero ompound whose

3 ON TE POSSIBILITY OF USING GRAPS (PART I) onentration is supposed equal to unity and whih is noted by The two mentioned routes orrespond to the two yles of the shown graph Cyle I ontains the propagation steps and while yle II ontains the initiation step, and termination step, As, aording to Emanuel and Knorre, the rate along the propagation route an be onsidered as the rate of the hain reation [], in order to get this rate one an treat yle I by help of King, Altman, Volenshtein and Goldshtein proedures Aording to the mentioned authors the reation rate an be alulated by help of the formula: idi i r () D i where D i is the basi determinant of the vertex i of the representative graph, i is the frequeny of the linear elementary step i whih equals its rate divided by the onentration of the ative enter attending it and is the onentration of the ative enter Formula () is valid for linear atalyti sequenes as well as for the linear propagation steps of straight hain reations Thus for yle I, taing into aount the rules of alulation of determinants D i [8], one obtains suessively: or taing into aount relations (), D D i D D r r Taing into aount the meanings of i we get: r It is nown that in the ourrene onditions of the reation [5], the rate determining ative enters are the bromine atoms ie, > Under suh onditions equation (5) taes the simple form: r In order to obtain equation (6) we managed to bypass the use of QSSA [6] Equation (6) ontains the unnown onentration of bromine atoms This an be eliminated taing into aount that, aording to the theory of straight hain reations, the rate of hain () () () (5) (6)

4 E SEGAL initiation (step ) should be equal to the rate of hain termination (step ) [7,8] This equality gives: (7) Introduing this last result in equation (6) one obtains: r (8) or r (9) As E = 76 al mole - and E = al mole - [9] it follows that and the rate equation (9) redues to: r () or d d t () Equation () is idential with that obtained using only QSSA as well as with the equation obtained experimentally by Bodenstein and Lind []

5 ON TE POSSIBILITY OF USING GRAPS (PART I) It is true that in order to get equation (9), besides the graph theory we had to use relation (7), but this equation is more general than the steady state one In a way this equation ould tae another partiular form for different onditions The Reation between Chlorine and Carbon Monoxide [7,8] For the reation: CO (g) + Cl (g) = COCl The following two route mehanism, similar to the previous one, an be onsidered: Cl Cl I II Cl + CO COCl - (III) COCl + Cl COCl + Cl Cl Cl The assoiated graph has the form shown in figure where taing into aount the previously mentioned reasons, only the propagation steps orresponding to route I are onsidered Cl - COCl Fig : Graph assoiated to sequene III By applying the same proedure as in the previous ase one obtains the following rate equation: or taing into aount that r Cl COCl ()

6 E SEGAL CO ; ; Cl and onsidering the hlorine atoms as rate determining ative enters ie, we get: and ; Cl COCl CO Cl r CO Cl Cl Considering again the equality of the rates of initiation and termination, it is easy to obtain: Cl Cl Cl () () By introduing this last result in equation (), the following form of the rate equation is obtained: If one taes into aount that r Cl Cl CO Cl (5) (step requires high ativation energy) as well as that ative enter COCl is relatively unstable, equation (5) turns into: CO Cl r (6) equation whih oinides with the experimental verified one [8] We managed to bypass the nonlinearity of step using the equality of the initiation and termination rates, in this ase too Photohemial Chlorination of Formi Aid [8] Aording to literature data, the reation: COO (g) + Cl (g) = Cl (g) + CO (g) is atually a straight hain reation whih ours through the following two route sequene:

7 ON TE POSSIBILITY OF USING GRAPS (PART I) 5 I II h Cl Cl Cl + COO Cl + COO (IV) COO + Cl Cl + CO + Cl Cl + stone Cl with the assoiated graph: Cl COO or Fig : Graph assoiated to sequene IV Using the same proedure as in previous ases one obtains suessively: D Cl =, D COO = r COO Cl COO Cl COOCl r (7) Cl COO Considering the hlorine atoms as rate determining ative enters or and Cl COO taing into aount the equality between the rates of initiation and termination one obtains finally: r Cl COO (8) rate equation whih oinides with the one obtained using only QSSA [8] Some other examples of treatment of straight hain reations by means of graphs will be presented in a next paper

8 6 E SEGAL Conlusions A mixed proedure, using graph theory for the propagation steps as well as the ondition of the equality between the initiation and termination steps, was applied in order to obtain rate equations of straight hain reations The obtained rate equations are more general than those based on the use of QSSA only and offer the possibility to predit hanges in the ineti behaviour at the hange of experimental onditions REFERENCES Christiansen JA, (95) Adv Catalysis, 5, King EI and Altman C, (956) J Phys Chem, 6, 75 Volenshtein MI and Goldshtein EN, (966) Dolady Aad Nau SSSR, 7, 96 Iatsimirsii KB, (97) Z Chem,, 5 Iablonsii GS and Byov VI, (977) Kinetia I Kataliz, 8, 96; (977) Dolady Aad Nau SSSR,, 6 6 Petrov LA, (99), Chemial Graph Theory: Reativity and Kinetis, vol II, Gordon and eah Siene Publishers, London, -5 7 Temin MI, (96) Dolady AadNau SSSR, 65, 65 8 Kiperman SL, (979), Osnovy himithesoi inetie v geterogennom atalize, Mosva, Chapter V 9 Temin ON, Zeigarni A and Bonhev D, (996), Chemial Reation Networs, CRC Press, Boa Raton Bodenstein M and Lind C, (96) Z Phys Chem, 57, 68 Christiansen JA, (99) Danse Vidensab Mat-Pys Medd No erzfeld KF, (99) Z Eletrohem, 5, Polanyi M, (9) Z Eletrohem, 6, 5 Emanuel N and Knorre D, (975), Cinétique Chimique, Mir, Mosva, 5 Boudart M, (968) Kinetis of Chemial Proesses, Prentie all, Englewood Cliffs New-Jersey, 6 Temin MI, (97), Mehanism and Kinetis of Complex Catalyti Reations, Naua, Mosow, 57 7 Isa V and urdu N, (99), Chimie Fizi (Cineti Chimi i Cataliz), Ed tiina, Chiinu, Cap I6 8 Panthenov GM and Lebedev VP, (96), Khimithesaia Kinetia I Kataliz, Izdatelstvo Mosovogo Universiteta, Mosva, 6 9 Laidler KJ, (965), Chemial Kinetis, MGraw-ill, London, Ch 8