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Boreskov-Horiuti Problem and General Law of Chemical Kinetics Gregory S. Yablonsky Saint Louis University and Washington University in St. Louis, USA What is a LAW? Dependence Correlation MODEL Equation LAW !!! Some definitions “A physical law is a scientific generalization based on empirical observations” (Encyclopedia) This definition is too fuzzy Physico-chemical law is a mathematical construction(functional dependence)with the following properties: 1) It describes experimental data in some domain 2) This domain is wide enough 3) It is supported by some basic considerations. 4) It contains not so many unknown parameters 5) It is quite elegant Our goal: We will discuss the general law of complex chemical reaction.A special attention will be paid to the analysis of the Horiuti - Boreskov’s problem: Famous Horiuti’s question was “How can a kinetic equation be found for a back reaction?” (Juro Horiuti,1939) Boreskov tried to find answer the same question analyzing “SO2 oxidation - SO3 decomposition” processes. General Kinetic Expression for the Complex Reaction R =R+ - R- , where (R+ / R- ) = Keq (f+(C) / f- (C)), A similar equation was tested for some oxidation reactions, especially for SO2 oxidation and for ammonia synthesis (Boreskov, 1950-1960ies). (The backward reaction rate was estimated using isotope exchange data, especially data of the homomolecular isotope exchange in the ammonia synthesis ) It was found that this equation did not fit experimental data. So, what to do? Other ideas, other assumptions SO2 oxidation: 1) 2 K + O2 2 KO; 1 2) KO + SO2 K + SO3 2 Overall reaction 2 SO2 + O2 2 SO3 Numbers on the right side (1,2) are Horiuti’s numbers () Assumptions: 1) Non-linearity of some step (the first step is nonlinear). 2) One step is limiting, other steps are under fast equilibrium conditions. New equation Boreskov’s molecularity was introduced in his hystorical paper by 1945) M R (C) Keq f (C) ; M (1/ lim . ); R (C) f (C) M f (c) R R (C)1 K f (c) eq Molecularity SO2 oxidation 1) 2 K + O2 2 KO; 1 2) KO + SO2 K + SO3 2 Overall reaction 2 SO2 + O2 2 SO3 In SO2 oxidation, molecularity is equal to 1/2. Therefore, the limiting step is the second one. Rigorous Status of Boreskov’s Equation The rigorous status of Boreskov’s equation was found based on algebraic ideas (concept of kinetic polynomial) See Marin, Yablonsky, “Kinetics of Chemical Reactions: Decoding Complexity”, 2011, J. Wiley Chemical Kinetics. Textbook Knowledge Detailed mechanism is a set of elementary reactions which law is assumed , e. g. the mass-action-law An example:Hydrogen Oxidation 2H2 +O2 = 2H2O 1)H2 + O2 = 2 OH ; 2) OH + H2= H2O + H ; 3) H + O2 = OH + O; 4) O + H2 = OH + H ; 5)O + H20 = 2OH; 6) 2H + M = H2 + M ; 7) 2O + M = O2 + M; 8)H + OH + M = H2O + M; 9) 2 OH + M = H2O2 + M; 10) OH + O + M = HO2 + M; 11) H + O2 + M = HO2 + M; 12) HO2 + H2 = H2O2 + H;13)HO2 +H2 = H2O +OH; 14) HO2 + H2O = H2O2 + OH; 15) 2HO2 = H2O2 + O2; 16) H + HO2 = 2 0H; 17) H + HO2 = H2O + O; 18) H + HO2 = H2 + O2; 19) O + HO2 = OH +H; 20) H + H2O2 = H2O + OH; 21) O + H2O2 = OH +HO2; 22) H2 + O2 = H2O + O; 23) H2 + O2 + M = H2O2 + M; 24) OH +M = O + H + M; 25) HO2+OH=H2O+O2; 26) H2 + O +M = H2O +M; 27) O + H2O + M = H2O2 + M; 28)O + H2O2 = H20 + O2; 29) H2 + H2O2 = 2H2O; 30) H + HO2 + M = H2O2 +M Simple reactions Simple reactions are rather exceptions than rule e.g., first-order reactions of monomolecular decomposition C2H5Br C2H4 + HBr or CH3-N2- CH3 C2H6 + N2; the second-order reaction 2 NOI 2 NO + I2 A Question: Imagine that hydrogen oxidation reaction is performed under steady-state or pseudo-steady-state conditions. It means that concentrations of intermediates are governed by gas concentrations. Do we know, how to present the reaction rate as a function of main reactant and product concentrations (hydrogen, oxygen, water)? The answer is: NO ! We do not know the law of chemical kinetics for the complex reaction, we even do not know its form. Nice Quotation Daniels: “Despite Eyring and Arrhenius, chemical kinetics is all-in-all confusion. But through all the confusion of complications some promising perspective can be seen. Numerous consecutive, competing and reverse reactions by themselves are simple mono- or bimolecular reactions that obey simple laws. Hence we are fighting not so much with primary steps as with the problem of their mutual coordination to interpret the observed facts and to make practical predictions”. Hidden History of Chemical Kinetics,I Gul’dberg and Waage , Norway, 1862-1867 Mass-Action-Law( M.A.L.) Equilibrium formulation “ In chemistry like in mechanics the most natural methods will be to determine forces in their equilibrium states”. Kpq Kp'q', where p, q,p'q' are the " action masses" Initially, Guldberg and Waage used an expression Kp q K(p') (q') Hidden History of Chemical Kinetics, II Gul’dberg and Waage, 1879 Dynamic Formulation of the Mass-Action-Law (M.A.L.) R = K pqr Hidden History of Chemical Kinetics, III Van’t Hoff, Netherlands, the first winner of the Nobel award (1901) on chemistry 1884, “Essays on chemical kinetics” Idea of normal transformations “The process of chemical transformations is characterized solely by the number of molecules whose interaction provides this transformation” (AB; 2A B; A+B C; 2A+B C) Strong discussion with Gul’dberg and Waage: “As a theoretical foundation I have accepted not the concept of mass action ( I had to leave this concept in the course of my experiment)”. Van’t Hoff tried to eliminate mechanics from chemistry. Hidden History of Chemical Kinetics, IV Van’t Hoff believed that he found the chemical (not mechanical) LAW OF CHEMICAL KINETICS However, his normal transformation dependences did not fit many real experimental data, e.g. hydrogen oxidation data Hidden History of Chemical Kinetics, V Idea of the complex mechanism: “Reaction is not a single act drama” (Schoenbein) Intermediates (X) and Pseudo-Steady-State-Hypothesis According to the P.S.S.H., Rate of intermediate generation = Rate of intermediate consumption Ri.gen (X, C) = Ri.cons(X, C) Then, X = F(C) and Reaction Rate R(X, C)=R (C, F(C))=R(C) The term “chemical mechanism” has an obvious “mechanical origin” In 1879, a vivid interpretation of complex systems as mechanical systems was given by Maxwell. “In an ordinary chime every bell has a rope that is drawn through a hole in the floor into the bell-ringer room. But let us imagine that every rope instead of putting into motion one bell participates in the motion of many parts of the mechanism and that the motion of every bell is determined not only by the motions of its own rope but the motions of several ropes; then let us assume that all this mechanism is hidden and absolutely unknown for the people standing near the ropes and capable of seeing only the holes ceiling above them”. Hidden History of Chemical Kinetics, VI In the XX century, the history of chemical kinetics is a series of attempts to hide the failure of Van’t Hoff’s paradigm P.S.S.H, or Bodenstein’s Principle A paradox of PSSH. Reflecting complexity, we are introducing new unobserved substances (intermediates). At the same time, we are eliminating intermediates searching for simplicity. “The first who applied this theory was S. Chapman and half the year later Bodenstein referred to it in the paper devoted to the hydrogen reaction with clorine. His efforts to confirm his view point were so energetic that this theory is quite naturally associated with his name” (Christiansen) P.S.S.H. has been applied in many areas of chemical kinetics: Reactions in gaseous phase Heterogeneous catalytic reactions Enzyme reactions Etc. Chain Reaction Fragment of the mechanism: 1) H + Cl2 HCl + Cl 2) Cl + H2 HCl +H Overall reaction: H2 + Cl2 2HCl R=(k 1k2CH2CCl2- k-1k-2C2HCl) / , where = k 1CH2 +k2CCl2 + k-1CHCl +k-2 CHCl Thermodynamic validity The equation R=(k 1k2CH2CCl2 - k-1k-2C2HCl) / , where = k 1CH2 +k2CCl2 + k-1CHCl +k-2 CHCl is valid from the thermodynamic point of view. Under equilibrium conditions, R=0,and (C2HCl / CH2 C Cl2) = (k1k2 /k-1k-2)= K eq(T) Forward and backward rates of complex reaction In the case H2 +Cl2 = 2 HCl reaction, Rate can be represented as a difference of two terms : R =R+ - R- , where (R+ / R- ) = Keq (f+(C) / f- (C)), and f+(C) and f- (C) correspond to the overall reaction, forward and backward respectively. Heterogeneous Catalytic Reaction Langmuir- Hinshelwood equation K ci ni R i 1 K jiCimi j i Langmuir ( Langmuir-Hinshelwood ) equation is related to the irreversible case: the“back term” is absent. For simple molecules oxidation (H2, CO) over noble catalysts (Pt, Pd), Langmuir made an assumption on the fast adsorption - desorption equilibrium. Hougen-Watson Equations. Typically, there is no assumptions about the detailed mechanism R K CC6H12 K CC6H10 CH 2 (K1C C6H10 K 2C H 2 K3C C6H12 )m R R Enzyme cycle (King, Altman,1956) First application of the graph theory 1) S + E X1; 2) X1 X2; 3)X2 X3 ;… n-1) Xn-2 Xn-1 ; n) Xn-1 P+ E E X1 X2 Xn Xn-1 … Overall reaction is : S P X = f (Cs, Cp ) According to the P.S.S.H, R=[ (k1Cs)(k2)…(kn) - (k-1) (k-2)… (k-n)Cp] / ( )= R= (K+CS - K-Cp) /( ), where = ( Ki(Ci)P) R = R+ - R- Catalytic mechanism. Two-step mechanism: Temkin-Boudart • • • • 1) Z + H2O ZO + H2; 1 2) ZO + CO CO2 + Z 1 The overall reaction is CO + H2O= CO2 +H2, R=[(k1Cco)(k2CH2O)- (k1CH2)(k2CCO2)] / , where = k1CH2O +k2CCO+ (k-1CH2)(k-2CCO2)], R = R+ - R- ; (R+/ R- ) = (K+CcoCH2O)/(KCH2CCO2) Conversion of methane 1) CH4 + Z ZCH2 +H2 ; 2) ZCH2 +H2O ZCHOH +H2 ; 3) ZCHOH ZCO +H2 ; 4) ZCO Z + CO Overall reaction : CH4 + H2O CO + 3 H2 R = (K+CCH4CH2O - K-CCOCH23) / ; R= R+ - R(R+ / R- ) = (K+CCH4CH2O) / ( K-CCOCH23) Linear mechanisms What is general about these different reactions: H2+Cl2 = 2 HCl, enzyme reaction, two-step catalytic reaction, catalytic conversion of methane? 1) All these reactions have a cycle (circuit) in the detailed mechanism. 2) In any reaction, only one molecule of the intermediate participates. Such mechanisms are called linear ones. The corresponding PSSH-equations for intermediated can be solved easily. One-route catalytic reaction with the linear mechanism. General expression (Yablonsky, Bykov, 1976) R = Cy / , where Cy is a “cyclic characteristics”, Cy = K+ f+(C) - K- f- (C) , Cy corresponds to the overall reaction; presents complexity of complex reaction; pi K j c ji j i Main Properties of the General Expression (I) 1. Cyclic Characteristic, Cy (numerator) does not depend on the details of the mechanism 2. All the information about the mechanism complexity is reflected by the denominator, , which is the complexity term. 3. Another form of the rate equation is: ( ) R = Cy, or ( ) R - Cy = 0 Main Properties of the General Expression(II) 4. R =R+ - R- , where (R+ / R- ) = Keq (f+(C) / f- (C)), 5. These equations can be derived very easily (with ‘a rate of pen’) using the methods of the graph theory. Life is Tough (that’s its stuff) A similar equation was tested for some oxidation reactions, especially for SO2 oxidation and for ammonia synthesis (Boreskov, 1950-1960ies). (The backward reaction rate was estimated using isotope exchange data, especially data of the homomolecular isotope exchange in the ammonia synthesis ) It was found that this equation did not fit experimental data. So, what to do? Other ideas, other assumptions SO2 oxidation: 1) 2 K + O2 2 KO; 1 2) KO + SO2 K + SO3 2 Overall reaction 2 SO2 + O2 2 SO3 Numbers on the right side (1,2) are Horiuti’s numbers, Assumptions: 1) Non-linearity of some step (the first step is nonlinear). 2) One step is limiting, other steps are under fast equilibrium conditions. New equation (Boreskov’s molecularity) M R (C) Keq f (C) ; M (1/ lim . ); R (C) f (C) M f (c) R R (C)1 K f (c) eq Molecularity SO2 oxidation 1) 2 K + O2 2 KO; 1 2) KO + SO2 K + SO3 2 Overall reaction 2 SO2 + O2 2 SO3 In SO2 oxidation, molecularity is equal to 1/2. Therefore, the limiting step is the second one. Life is even more tough than we expected In the kinetics of heterogeneous catalysis, in the late 1960ies and 1970ies, various critical phenomena under isothermic conditions were observed. A typical example of these phenomena is the multiplicity of steady states which was observed in many oxidation reactions, particularly CO oxidation, SO2 oxidation etc. This means that the different values of reaction rate can correspond to one or the same composition of reaction mixture Critical Phenomena in Heterogeneous Catalytic Kinetics: Multiplicity of Steady-States in Catalytic Oxidation (CO oxidation over Platinum) Reaction Rate C “Extinction” B A “Ignition” D CO Concentration Typical catalytic system with critical phenomena Adsorbed mechanism (Langmuir’s mechanism) 1) 2 K + O2 2 KO 2) K + B KB 3) KO + KB 2 K + BO Overall reaction 2B+O2 KB Particular cases of this mechanisms are mechanisms of CO oxidation over noble metals of VIII group and SO2 oxidation over Pt. Kinetic model of the adsorbed mechanism 1) 2 K + O2 2 KO 2) K + SO2 KSO2 3) KO + KSO2 2 K + SO3 Steady state (or pseudo-steady-state) kinetic model is KO : 2k1CO2(CK )2 - 2 k-1 (CKO )2 - k3 (CKO ) (CKSO2 )+ k-3 CSO3 (CK )2 = 0 ; KSO2: k2CSO2CK - k-2 (CKSO2) - k3 (CKO ) (CKSO2 )+ +k-3CSO3 (CK )2 = 0 ; CK + CKO +CKSO2 =1 The same question: What is the general form of kinetic description taking into account the thermodynamic validity and, ideally,with no assumptions about step limiting, fast equilibrium etc.? The obvious difficulty In general, i.e. non-linear, case, an explicit presentation for steady (or pseudo-steady-state) reaction rate can be obtained only for special non-linear kinetic models. Generally, it is impossible. The obvious requirement! Any representation should be valid from the thermodynamic point of view, i. e. under equlibrium conditions (R=0), we should have Keq = (f+(C) / f- (C)) Thermodynamics does not care about our mathematical difficulties Our strategy Our strategy was to perform an analysis of the concrete non-linear kinetic models, especially kinetic model corresponding to adsorbed mechanism of CO oxidation, and to develop the general theory based on the ideas of algebraic geometry. Mathematical basis Our basis is algebraic geometry, which provides the ideas of variable elimination 1. 2. 3. 4. 5. Aizenberg L.A., and Juzhakov,A.P. “Integral representations and residues in multi-dimensional complex analysis”, Nauka, Novosibirsk, 1979 Tsikh, A.K., Multidimensional residues and their applications, Trans. Math. Monographs, AMS, Providence, R.I., 1992 Gelfand,I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, Resultants, and Multidimensional Determinants, Birkhauser, Boston, 1994 Emiris, I.Z., Mourrain,B. Matrices in elimination theory, Journal of Symbolic Computation, 1999, v.28, 3-43 Macaulay, F.S. Algebraic theory of modular systems, Cambridge, 1916 Our main result In the case of mass-action-law model, it is always possible to reduce our polynomial algebraic system to a polynomial of only variable, steadystate reaction rate. For this purpose, an analytic technique of variable elimination is used. Computer technique of elimination is used as well. Mathematically, the obtained polynomial is a system resultant. We term it a kinetic polynomial. First publications The idea of kinetic polynomial was firstly emphasized in the paper M.Z. Lazman, G.S. Yablonskii, and V.I. Bykov, Sov. J. Chem. Phys.,2(1985)404 The mathematical apparatus was described in the paper M.Z. Lazman, and G.S. Yablonski, Kinetic Polynomial: a New Concept of Chemical Kinetics”, in “Patterns and Dynamics in Reactive Media”, IMA Proceedings,1989; Springer-Verlag, Berlin-N.Y.,1991 The Kinetic Polynomial (I) The kinetic model which corresponds to the single-route mechanism has the form: rs(z1,…,zn) - R = 0, s=1,…,n 1- zj =0; j = 1,…,n, where R is the rate of complex reaction (rate of reaction route), rs is the rate of the reaction step rs(z1,…,zn) = bszj s - b-s zjs; is the Horiuti’s number X is the concentration of the intermediate; bs ,b-s are the reaction weights, i.e. reaction rates at the unitary concentrations of intermediates Kinetic polynomial (II) In analysis of non-linear kinetic model, the concept of the limiting subsystem was introduced. The kth limiting subsystem is such system in which all steps are considered to be in an equilibrium except the kth step. Then, the set of k limiting subsystems is analyzed. The Kinetic Polynomial (III) For the linear mechanism, the kinetic polynomial has a traditional form: R = (K+ f+(C) - K- f- (C))/ ( ) , or ( ) R = Cy, or ( ) R - Cy = 0, where Cy is the cyclic characteristic; is the “Langmuir term” reflecting complexity For the typical non-linear mechanism the kinetic polynomial is represented as follows: BmRm+…+ B1R +BoCy=0 , where m are the integer numbers The Kinetic Polynomial(IV) Coefficients B have the same “Langmuir’ form as in the denominator of the traditional kinetic equation,i.e. they are concentration polynomials as well. Therefore, the kinetic polynomial can be written as follows mi , jL L mi , j1 K j L iCi R ... K j1 iCi R j L i j1 i B0 K f (C) K f (C) 0 The Kinetic Polynomial (V) The coefficients of kinetic polynomial are obtained using solutions of k limiting subsystems Properties of the Kinetic Polynomial 1) It is a non-linear polynomial, an implicit function of the reaction rate; 2) It is thermodynamically correct. At R=0 (equilibrium), K+f+(C) - K-f- (C) = 0 Properties of kinetic polynomial(II) 3)Kinetic polynomial is the generalized form which includes many typical cases: Langmuir-Hinshelwood equation, Hougen-Watson equation, equations of enzyme kinetics etc. 4)It can be used for the rigorous analysis of special cases: limiting, kinetic behavior in the domain which is close to the equilibrium. 5)Kinetic polynomial may have several physical solutions. Therefore it can be used for for description of critical phenomena, especially multiplicity of steady-states. Uniqueness of the “thermodynamic branch” Generally, the kinetic polynomial may have some physical solutions and, correspondingly, some physical branches. However, only one branch (“thermodynamic branch”) passes through the equilibrium point. Otherwise, the system will have more than one equilibrium. It is impossible. Analysis of two limiting cases: 1)LIMITATING STEPS; 2)NEIGHBORHOOD OF EQUILIBRIUM In both cases, results are obtained on the basis of solutions of limiting subsystems. Limiting Steps 1/ lim R / R [ f (C)/ f (C)] RR R Neighborhood of equilibrium K f (C) K f (C) R , Aeq where : 2 n pk '( k ) / p N ji k k A K f (C) k [(b j ) /(b j )] b k1 k i1 jk n n R+(C) / R- (C ) =[ K+f+(C) / K-f-(C) ] R(C)=R+(C) - R- (C ) eq Boreskov-Horiuti Problem: Conclusions Generally, the rate of complex reaction cannot be presented as a difference of the forward and back reaction rates. Therefore, in a general case the representation R (C) = R+(C) - R- (C) is not valid Such representation is valid only for the thermodynamic branch. 1. R+(C) / R- (C) = K+f+(C) / K-f-(C) This representation is valid for the non-linear mechanism in the domain “close to equilibrium”and for the linear mechanism in any domain 2. R+(C) / R- (C) =[ K+f+(C) / K-f-(C) ](1/lim) This representation is valid for the non-linear mechanism in the domain of limitation of some steps Bm 4 R +…+ B1R +BoC=0 , R= k3 CKO CKCO - k-3CCO2 CK2 = 0 Conditions: PO2>PCO; k3 > ki , k-i Multiplicity of steady states Low branch (1), Middle branch (2), Upper Branch(3) R1 k2 ( H (H 1))2 ;R2 k2 ( H (H 1))2 ; 2 R3 (k2Cco ) /2k1CO 2 ; (k 2Cco k2 ) H 8k2k1CO 2 2 Branches and rates (I) The first (low) and third (upper) branches are stable, the second (middle) branch is unstable. The second and middle rate expressions can not be presented in the form of differences of the forward and back reaction rates R =R + - R - , where (R + / R- ) = Keq (f+(C) / f- (C)), Branches and rates (II) The first rate expression (low branch) can be presented as a difference of values which can be termed as forward and back terms, e.g. (R+)1/2 and (R-)1/2 However, their ratio is not KeqC2CO2/C2COCO2 Reaction Rate Dependence on CO Concentration 0.7 0.6 Non-Physical Branch Reaction Rate 0.5 0.4 0.3 Physical Branch 0.2 Physical Branches 0.1 0 -0.1 Non-Physical Branches -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 CO Concentration 1.6 1.8 2 Critical Simplification Analyzing kinetic polynomial, critical simplification was found At the extinction point Rext. = k+2Cco At the ignition point Rign = k-2 Therefore, the interesting relationship is fulfilled Rext / Rign = k+2Cco /k-2 = Keq Cco It can be termed as a “Pseudo-equilibrium constant of hysteresis” Therefore, we have the similar equation for (R + / R- ) in terms of bifurcation points. Critical Phenomena in Heterogeneous Catalytic Kinetics: Multiplicity of Steady-States in Catalytic Oxidation (CO oxidation over Platinum) Reaction Rate C “Extinction” Critical Simplification: RC=k2+CCO RA=k2- B A “Ignition” D CO Concentration Experimental evidence It was found theoretically that at the point of ignition the reaction rate is equal to the constant of CO desorption. It was found experimentally, that the temperature dependence of reaction rate at this point equals to the the activation energy of the desorption process. (Wei, H.J., and Norton, P.R, J. Chem. Phys.,89(1988)1170; Ehsasi, M., Block, J.H., in Proceedings of the International Conference on Unsteady-State Processes in Catalysis, ed.by Yu.Sh. Matros, VSP-VIII, Netherlands, 1990, 47 Evolution of kinetic behavior of the complex reversible reaction with the increase of parameter (concentration) Typical scenario: from the “close to equilibrium”domain to the “far from equilibrium”- domain • “Close to Equilibrium ”- Domain. Kinetic behavior in accordance with the overall reaction •Domain of Limitation by the controlled parameter • Domain(s) of Limitation by the other concentrations (or intermediates) •Domain of Irreversibility (“Far from Equilibrium”) •Domain(s) of Critical Phenomena (“Far from Equilibrium”), e.g. multiplicity of steady states References (I) 1. 2. 3. 4. Lazman, M.Z., Yablonsky, G.S., and Bykov V.I. , Sov. J. Phys. Chem., 2(1985)404 Lazman, M.Z., and Yablonskii G.S., Kinetic Polynomial: a New Concept of Chemical Kinetics, in Patterns and Dynamics in Reactive Media, The IMA Volumes in Mathematics and its Applications, Springer-Verlag, New-York, 1991, 117-150 Yablonsky, G.S., and Lazman, Non-Linear Steady-State Kinetics of Complex Catalytic Reactions: Theory and Applications, Proceedings of International Symposium “Dynamics of Surface and Reaction Kinetics in Heterogeneous Catalysis”, Antwerp, Belgium, 1997, In Studies in Surface Science and Catalysis, 109, 371-378 Yablonsky, G.S., Mareels, M.Y., and Lazman, M.Z. The Principle of Critical Simplification in Chemical Kinetics, Chemical Engineering Science, 58 (2003) 4833-4842 References (II) 5. Yablonskii,G.S., Lazman M.Z, and Bykov V.I., Stoichiometric Number, Molecularity and Multiplicity, Reaction Kinetics and Catalysis Letters, 20 (1982)73-77 6. Aizenberg, L.A, Bykov V.I., Kytmanov A.M., and Yablonsky, G.S., Search for all steady-states of chemical kinetic equations with the modified method of elimination, Chemical Engineering Science, 38 (1983)1555-1568 7. Yablonskii G.S, Bykov V.I., Gorban A.N., and Elokhin V.I., Kinetic Models of Catalytic Reactions//Comprehensive Chemical Kinetics, V.32, Elsevier, Amsterdam, N.-Y, 1991 8. Yablonskii, G.S. and Lazman, M.Z., New Correlations to Analyze Isothermal Critical Phenomena in Heterogeneous Catalytic Reactions, Reactions Kinetics and Catalysis Letters, 59(1996)145-150 8. Bykov, V.I., Kymanov A.M. and Lazman M.Z., Elimination methods on Computational Computer Algebra, Kluwer Academic Publishers, Dordrecht, 1998