HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPREM Vol. 29. pp. 27-34 (2001) MULTIPLICITY OF STEADY STATES AND THE SHAPES OF ITS EXIDBITING IN CATALYTIC REACTIONS KINETICS N. I. KOLTSOV and F. J. KE1L1 (Department of Physical Chemistry of Chuvash State University, Moskovskii prospekt 15, 428015 Cheboksary, RUSSIA 1 Department of Chemical Engineering, Technical University of Hamburg- Harburg, Eissendorfer Strasse 38, D-21073 Hamburg, GERMANY) Received: December 28, 2000 The purpose of the presented review is the analysis of the existing approaches and mathematical methods for the investigation of the critical phenomena of a type of multiple steady states in kinetics of catalytic reactions Keywords: multiplicity; steady state; hysteresis; self-crossing; breakdown; isola; mushroom; catalytic reaction; kinetics; step scheme. Introduction One of the current problems in kinetics is the investigation of the multiplicity of steady states (MSS) and the shapes of the phase portraits in catalytic reactions. Their existence was proved by extensive research, and the results were published in many papers and at conferences. The- following sections review the development of mathematical methods and the research of MSS in catalytic processes. Well-known are Ya. Zeldovich's papers on the uniqueness of the equilibrium point [1] and the anomalous types of critical effects [2]; I. Prigozhin Brussel's school publications [3,4]; monographs by G. Jablonskii, V. Bykov, V. Elokhin, A. Gorban [5-81; and also the research work carried out at the G. Boreskov Institute of catalysis SD RAS, at theN. Zelinsky organic chemistry Institute, at the N. Semyonov chemical physics Institute,. and at some other institutions are of great importance for the development of this branch of science. At the beginning of the investigation of critical phenomena like MSS the experimental data on their existence in concrete heterogeneous catalytic reactions were collected: hydrogen oxidation [9-10], of carbon monooxide oxidation [11-13], ammonia synthesis [17] on platinum metals and others. Then researchers·tried to describe these phenomena taking into account thermal and diffusional effects [18, 19]; phase transitions [20]; changes of the catalyst surface structure [21] under the influence of components which take part in a reaction, ramified-chain [22, 23] or homogeneous-heterogeneous [24] character of reactions. Discovering MSS in the kinetic region [25, 26] for some reactions led to the third step in which MSS is described by simple model mechanisms [5-8]. The modern fourth step is characterized by the development of the theoretical methods investigation of MSS and different forms of its catalytic reaction kinetics. Some important results in this field of research is reviewed in this paper. Theory To describe existing approaches of investigation of MSS and their various shapes let us regard a general one-route catalytic reaction acting through steps + r; :~:>;;A1 + :La;x 1 <=} LhiiA1 + :La;x1 • d1 (2) l j ,.,- I j l i,j = l, ... ,S; l = l, ... ,m, 28 where A1 and X; are basic substances and intermediates; a;, b~ ;;:::: 0 are the stoichiometric coefficients ( '2tat = L«ii ); d1 > 0 are the j j stoichiometric numbers of steps, ( i = 1, ... , S ); r/ and rj are directions: rates of i~th step in forward and reverse (3) mi and m; are the frequencies of steps in forward and reverse directions, x1 are the concentrations of intermediates X 1 , C1 are the concentrations of basic substances A1 , k( are rate constants of the steps .. Under isothermal conditions in a gradientless differential reactor the scheme (2) describes the changes of concentrations of the reactants ( ~ ) and intermediate substances ( X 1 ) by the following system of the stationary equations (4) L(hi:" -ht,Jo (8) for ro_i ¢ 0 29 3i.j: V; <0ws: if for a reaction scheme of the corresponding relations {8} (or (9) ), ( lO) and { ll) are satisfied then the examined reaction is characterized by multiplicity. Otherwise there is only one stationary state. The values V1 , U~~: and W at which MSS is realized can be applied for the determination of concentration of the reactants and intermediate substances [30}. Let us illustrate an application of the criterion of existence of multiplicity in a catalytic reaction of carbon monoxide oxidation 2CO + 0 2 = 2CO~ on platinum !.31 J proceeding according to the scheme 1. O:J+2K==2KO, 2. CO+KO+K~C02+2K. (12) In the beginning we shall write down the criterion of MSS for this scheme in isothermal gradientless differential reactors nt constam concentration "* th..t reactants Ck , when nil components of a vectm· l.J ar~:: equal to zero. With respect to relations (8) and (9) MSS will occur under the following conditions: where the parameters v; and V2 correspond to substances K and KO. A simple analysis reveals that the inequalities (13) are only satisfied if all the vector components V have the same signs. In this case equation (10) is not satisfied. It specifies the impossibility of the description of MSS by the scheme (12) in the given reaction. If one takes into account changes of concentration the relations (8), (9) for reactants, the criteria take the form of The relations obtained are carried out, for example, at W=2, v;::::6, V2 =-2, U1 =-13, U 2 =-2 (the W = l:>~F1 + })t.Uk,forw_1 ::::0 j k ~ The sufficient conditions ofMSS are (9) parameters U1 and U2 correspond to the concentrations of 0 2 and CO). The given values satisfy a component of vectors V and U also the inequalities (10), (11). In Fig. 2 the dependence r = f(Cr0 ) is given which demonstrate MSS as a S-shaped "counter-clockwise" hysteresis curve. Hence the scheme (12) can be used for 30 the description of MSS in the catalytic reaction of carbon monoxide oxidation under isothermal conditions with respect to the change of concentration of the reactants. -1 r.3 4~----~=----------------- 3 2 1 Fig.2. Dependence of rate on concentration of carbon monooxide for reaction of carbon monooxide oxidation proceeding on the scheme (12) at: , , (s-1) The criterion developed was used for the investigation of both known and new mechanisms of catalytic hydrogen oxidation and carbon monoxide reactions. The most simple schemes of these reactions describing MSS in the form of S-shaped hysteresises are the following: H2 oxidation: 1. 0 2 + 2K == 2KO, 2. H 2 +KO+K-+2K+H 2 0, CO oxidation: 1. 0 2 + 2K == 2KO, 2. CO+K0-+K+C02 , 3. CO+K == KCO. Self-crossing Another shape of MSS is a kinetic curve with self- crossing (Fig. lc). This critical phenomenon is characterized by the fact that the graph of reaction rate as a function of a parameter intersects at one point. This leads to a kinetic "loop". Fig. lc shows that changing the parameter C from left to right and vice versa in the region of the loop self-crossing, kinetic dependence is realized in the form of a hysteresis - "eight" (see the direction of rate change along the CBDAC curve). It is worth noting that at the intersection point, which is characterized by various values of intermediate species concentrations on a catalyst surface, the reaction rate has the same value. This property was the basis for the mechanisms which describe self-crossing. As was shown in the papers [32, 33] in an isothermal gradientfess differential reactor at constant concentration of the basic substances (when all components of a vector U are equal to zero) any one- route~reaction consisting of steps of a kind (2) in a point of self--crossing can be characterized by the following equations between rates ( (2) I r(l) == rc 2 l, W == In :<1l ) == 0 (14) and for concentrations of intermediate substances one obtains: :::J J. . (1) "'" (2) J. -1 n ::::1 • x 1 -r x1 , - , .•• , (15) in various steady states. The equations of MSS criteria (8), (9) in case of self-crossing will be written down for reversible and irreversible steps accordingly in the following way: (17) In the case a two-step reaction consisting of steps of a type (2) it is possible if there stoichiometric coefficients satisfy relations: Hence the criterion of existence of a loop for two-step reactions is that all four inequalites (18) are fulfilled. The simplest mechanism satisfying this criterion for Ar {:::} A2 reaction is 1. A1 +X 1 +X 2 == 2X 1 + A2 , 2. X1 +2X2 =3X2 • (19) Fig. 3a illustrates kinetic dependence self-crossing for this reaction. Breakdown The kinetic dependencies with self-crossing or breakdown are shown in Fig. lc and ld. As can be observed, under certain conditions points A and B in Fig.lc with self-crossing can exist very close to each other, which leads to a kinetic curve with breakdown. The kinetic behavior of this form was obtained for reactions of carbon monoxide oxidation on palladium {34, 35} and a1so hydrogen oxidation on platinum [36]. Besides these examples the occurrence like breakdown were obtained at joint oxidation of carbon monoxide and nitrogen oxide [37]. r, s- 1 31 a b 0.3 0.2 0.2 0.1 +""'"-~~-~-~-~-~cone. CA c 0 OS CM~ A . 1.2 0 0.3 0.6 Fig. 3. Kinetic dependence r(CA1) for reaction A1 {::::} A2 proceeding via scheme (19): a)- k1 = 2, (1)_1 = 1/7, mz == 804/343, m_2 == 60/343; b)- k1 = 10, 11)_1 = 2.495, m2 = 0.668, m_2 = 0.131 (s- 1 ) 2 1.5 1 0 o~------~~~1~--~~--~2 cone. CB cone. C 8 Fig. 4. Kinetic dependencies for scheme (20) at k1 = 4, k 2 "" 1~5 , k_2 =54 (s-1): a)- monotonous branch with anisolated "specialpoint(k_1 =.!.)"; b)-isola(k_1 =0.51) 2 The treatment of breakdowns as a function of kinetic parameters for reactions of oxidation of hydrogen, carbon monoxide, ammonia on platinum on the basis of multipathway mechanisms was presented in [37, 38] for the frrst time. It was supposed that for these reactions the rate constant of interaction between adsorbed particles on a catalyst surface is rather strong, also the occupation of the catalyst by oxidizing substances is small [38]. The same approach is presented in [39] for the reaction of carbon monoxide oxidation on platinum. The breakdown of kinetic dependence was interpreted within the framework of the scheme Az+2K=2AK, B+K=BK, AK + BK-+ 2K + AB as a limiting case of the kinetic curve at k3 -+ oo • The same explanation of the occurrence of breakdowns can be found in papers {40- 42] for the reaction of oxidation of hydrogen on monocrystals of platinum. In papers [43- 461 the criterion of occurrence of a breakdown is offered at the finite values of rate constants. It is shown that the criteria relations of a breakdown are equivalent to the basic criteria of self- crossing of kinetic curves. This research shows that kinetic dependencies with self-crossing and with a breakdown are determined by an identical reaction stoichiometry, although they display different forms of the critical phenomena. Therefore, for describing breakdown it is quite enough to have reaction schemes giving self-crossing of kinetic curves. Fig. 3b illustrates breakdown for scheme (19). Breakdown and "loop" are two evolutional MSS forms. The simplest schemes of catalytic hydrogen oxidation and carbon monoxide reactions describing MSS in the form of breakdown (or with self-crossing) of kinetic curves are the following: H2 oxidation: 1. 0 2 + 2K ""2KO, 2. H2 +K=KH2 , 3. KO+KH2 -+2K +HzO, CO oxidation: 1. KO + KCO = 2K + C02 , 2. CO+K+KC0=2KCO. 3. 0 2 +2K=2KO. Isola Isola is a closed isolated branch of a stationary kinetic curve (Fig. ld). The existence conditions of this shape for a non~isothermal reaction of the first order are 32 formulated in paper [2].The basic cause of appearance of isola is a process of "rolling - unrolling" of a mushroom curve in Fig. lf. It is described in papers [47, 48}. The isola in the graph of r(C) can be obtained from a kinetic curve having "a speciaf point" (ref. £2, 49]). If one of the parameters changes slightly, this point becomes an isola. The simplest model of a kinetic curve with isola realization is a A¢=> D reaction proceeding via an intermediate, B, according to the following scheme l.A+2B+X1 =X2 +3B, 2.B+3X1 +X;. =4X1+D' (20) for which at some parameter values CB, r(CB) shows "a special point" (Fig. 4a). If the k.1 parameter slightly increases the point turns into an isola (Fig. 4b). Mushroom Mushroom behavior (Fig. lf) can be considered as a monoparameter kinetic curve which consists of two s- shaped hysteresis branches. This MSS shape rarely appears. There are a few papers in which this critical phenomenon is experimentaliy described. The kinetic behavior according to the mushroom type were experimentally reproduced in benzene [SO] and carbon monoxide [Sl} oxidation on platinum catalysts. The mushroom shape. is a transitional one between isolated and self-crossing kinetic curves. The mushroom shape is connected with the existence of "a special point". The investigation of stoichiometric conditions of the existence of a "special point" showed some interesting results. For example, the B2 ¢=> 2B1 reaction proceeding via the following scheme 20 " b \ 1 ~+X1= X2+4, 2X2=X3+4, 3. 2X1+ X3=3X1 (21) describes the kinetics with isola under certain conditions (Fig. Sa). If the conditions change, approaching and blending of the isola with a monotonous branch takes place. Thus, self-crossing appears (Fig. Sb), which finally tnrns into a mushroom form (Fig. Sc). The above-mentioned examples show that kinetic dependencies with self-crossing, breakdown, isola and mushroom are variable forms of MSS curves, they can turn into one another if the conditions of reaction change. Conclusion For all kinetic dependencies except breakdown (Fig. 1) there is a general property, namely the existence of two different steady states, which are characterized by different values of concentrations of intermediate components and reactants and also rates under the same conditions. This statement is the basis for MSS criteria [30]. The criteria developed were used for the investigation of both known and unknown mechanisms of catalytic hydrogen oxidation and carbon monoxide reactions which reveal MSS in the form of S-shape, self-crossing, breakdown kinetic dependencies, respectively. These data are given in ref. [52- 56]. The conditions of occurrence of kinetics with isola and mushroom are given in ref. [57, 58] in which the schemes of models and concrete catalytic reactions characterized by these kinetic dependencies are investigated in detail. The conditions and criteria of MSS and its different shapes are mathematical relations ·which contain stoichiometry (molecularity of basic and intermediate components in each step of the investigated reaction mechanism). concentrations of components taking part c '\ 1 ····----·~:-· --- ' l I J 6.4 0.8 U G £1.4 U 1.2 eooc .. Ca cmc. Cn Fig. 5. lsola (a}. sclf..-cmssing (b} and mushroom (c) for scheme (21) at: kJ =405. k.~ ::.iU \:>'1):a)- 11 =43;bl ~ k1 =50.3; c)· k1 :=50.33 {s" 1) 0 0.4 ' ' cone.• k_, =12, "7. =180, "-2 =144. in a reaction and kinetic parameters (rate constants and step activation energies). The solution of these algebraic equations for catalytic reactions, which proceed via two or three steps, is not difficult; it can be done by hand. However, for multiple step reactions with a great number of intermediate compounds this analysis is practically impossible without applying computers. 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ALEXEEV B.V., KOZHEVNIKOV LV. and KOLTSOV N.I.: Zhur. Phys. Chern., 1998, 72(12), 2186-2190 (in Russian) 62. ALBXEEV B.V., KOZHEVNIKOV I.V. and KOLTSOV N.I.: Comput. Chern., 1999, 23, 69-74 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35