HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPREM Vol. 29. pp. 71- 80 (2001) MECHANISMS OF AMMONIA-SYNTHESIS REACTION REVISITED WITH THE AID OF A NOVEL GRAPH-THEORETIC METHOD FOR DETERMINING CANDIDATE MECHANISMS IN DERIVING THE RATE LAW OF A CATALYTIC REACTION L. T. FAN,R BERT6K\F. FRIEDLER1,andS. SHAFIE (Department of Chemical Engineering, Kansas State University, KS-66506 Manhattan, U.S.A. 1 Department of Computer Science, University of Veszprem, Veszprem, Egyetem u. 10., P.O. Box 158, Veszprem H-8201 HUNGARY) Received: August 20,2001 Stoichiometrically exact, candidate pathways, i.e., mechanisms, for deriving the rate law of the catalytic synthesis of ammonia have been determined through the synthesis of networks of known elementary reactions constituting such pathways. This has been undertaken to reassess the validity of available mechanisms and to explore the possible existence of additional ones for the catalytic synthesis of ammonia. Synthesizing the networks of elementary reactions is exceedingly convoluted due to the combinatorial complexity arising from the fact the number of elementary reactions involved usually far exceeds that of available elementary balances, which is only 2 for the ammonia synthesis. Such a complexity can be circumvented by the rigorous and highly efficient, graph-theoretic method adopted in the present contribution. This method follows the general framework of a mathematically exact, combinatorial method eslablished for process-network synthesis. It is based on a unique graph-representation in terms of process graphs (P-graphs), a set of axioms, and a group of combinatorial algorithms. The method renders it possible to generate with dispatch all feasible independent reaction networks, i.e., pathways, only once. The pathways violating any first principle of either stoichiometry or thermodynamics are eliminated. Moreover, the method is capable of directly generating rapidly the acyclic combinations of independent pathways. Keywords:Ammonia synthesis, Reaction pathways, Mechanism, Networks, Identification, Synthesis, Algorithms, p. graph. Introduction The mechanisms or pathways of the catalytic synthesis of ammonia from nitrogen and hydrogen, N2 + 3H2 ~ 2NH3, have been investigated intensively as well as extensively because of its long history, theoretical importance, and enormous economic implication. Nevertheless, the number of stoichiometrically exact mechanisms proposed as the candidates for deriving the rate law for this reaction increases continuaUy as new computing methods are devised and additional elementary reactions are proposed [1-9]. The latter is probably attributable to the ever-enhancing sensitivity and accuracy of analytical instrumentation, in general, and spectroscopic instrumentation, in particular, for detecting active species; to the ever-enlarging variety of catalysts for increasing the yield {10]; and to the ever-expanding ranges of operating conditions for improving plant efficiency. The determination of candidate mechanisms for deriving the rate law plays a key role in the study of the kinetics of catalytic reactions. Such mechanisms must be stoichiometrically exact [7, 11-14]. A reaction pathway, comprising the steps of elementary reactio.ns, routes the precursors (starting reactants) of t!:te react1on to the targets (final products) and vice versa; in other words, a reaction pathway signifies the mechanism of the reaction. The reaction pathway per se yields no information on the rate, reversibility, equilibrium, and extent of the reaction. Any reaction pathway is in the form of a network of the steps of elementary reactions containing a .loop or loops. ln constituting a pathway, or network, each elementary reaction among plausible elementary reactions contributes the forward, reverse or no step to the network. As such, the possible combinations 41f these 3 possibilities that must be taken into account .are 72 (3 11-1) or 177,146 even if the network comprises only 11 elementary reactions. This can readily give rise to more than 100 plausible networks from which the feasible candidate pathways are to be identified; by any measure, this is a daunting task. Difficulties involved in constructing networks of chemical reactions that can be the elementary reactions leading to the mechanism of a catalytic reaction are "the combinatorial explosion of the number of resultant networks, and the complexity involved in rendering a computer program to implement the algorithm for network construction effective both synthetically (from precursors towards targets) and retrosynthetically (fr~m targets towards precursors)" [15, 16]. Substantial progress has been made to circumvent such difficulties mainly by resorting to various paradigms of linear algebra (1, 8, 11, 16-32]; nevertheless, much remains to be done. The current contribution aims at reassessing the validity of available pathways or mechanisms of the catalytic synthesis of NH3 and exploring if additional candidate pathways exist from the stoichiometric point of view. This is accomplished by resorting to a method of synthesizing a network of elementary reactions, corresponding to the pathway, i.e., mechanism, of a catalytic reaction [33]. This method, totally algorithmic in nature, has been developed by judiciously adapting the three available algorithms of the mathematically exact, graph-theoretic method for process-network synthesis. These algorithms are MSG for generating the maximal structure of the network, SSG for generating all the combinatorially feasible network structures, and ABB for determining · optimal and near optimal networks. The method is firmly rooted in a set of axioms and expressed in the parlance of process graph or P-graph, in brief [34-39]. In reality, researchers in the field of catalysis have been remarkably successful in deriving the satisfactory rate laws for reactions of interest without having a complete set of stoichiometrically exact candidate mechanisms. This is mainly achieved through the judicious identification of plausible elementary reactions on the basis of spectroscopic measurements, which is frequently aided by energetic analysis of such elementary reactions. An expression for the rate law is usually derived by postulating, for simplification, the existence of the rate-controlling and equilibrium steps among the elementary reactions proposed. The final determination of the rate law is accomplished by fitting the resultant expression to the experimentally measured rate data [40-51]. More often than not, however, a multitude of stoichiometrically exact mechanisms emerges from a single set of plausible elementary reactions, some of which clearly resemble each other. Frequently, it is nearly impossible to discriminate statistically the rate laws derived from such mechanisms in the light of the experimental data. Hence, it would be highly advantageous to obtain a complete set of stiochiometricaUy exact candidate mechanisms prior to launching an effort for the rate-law determination. In fact, to do so would greatly facilitate the execution of such an effort because it exactly defines the boundary and limits its scope. Methodology The current methodology for identifying the stoichiometrically complete mechanisms for a given overall reaction is based on the two sets of axioms, one being the set of 6 axioms of feasible reaction pathways and the other being the set of 7 axioms of combinatorially feasible networks, as well as on the graph representation of reaction steps irt terms of P- graphs. In view of formal graph-theoretic description of the reaction-pathway identification problem (see Appendix 1), these axioms and P-graph representation give rise to the 3 combinatorial algorithms, including algorithm RPIMSG for the generation of the complete reaction network; algorithm RPISSG for the generation of the combinatorially feasible pathways; and algorithm PBT for the final determination of the feasible pathways. inputs: RPI problem(}!:, M, 0); output: maximal structure (m, o ); begin comment: reduction part of the algorithm; repeat M:='¥(0); exc:=0; for allxEM begin comment: Case 1 if xe m(E) and v-(x)=0 then exc:=excuv\x); comment: Case 2 if x~ m(E) and v+(x)=0 then exc:=excuv-(x); comment: Case 3 ifxem(E) and I v-(x) I =1 and v+(x)=X(v-(x)) then exc:=excuV(x); comment: Case 4 if x~ m-(E) and ltnx) I =1 then exc:=excuX( v-(x)); comment: Case 5 if xll: m\E) and J v+(x) J =1 then exc:=excuX(v+(x)); end; 0:=0\exc; until exc=0; comment: composition part of the algorithm; m:::=al(E); ();::::0; repeat add:=qr(m)\o; o:=ouadd; m:=mu'¥-(o); until add=0; if m-(E)\m~0 or o.t(E)\m~0 then stop; comment: There is no maximal structure. write (m, o ); end. Fig. 1. Algorithm RPIMSG. Monochloroacetic acid, ammonia solution and Ca(OH)z were purchased from Aldrich (Germany) and were used as received. Axioms According to the classical chemical thermodynamics, the overall reaction and all elementary reactions in any mechanism are reversible, and each reaction step, either forward or reverse, is stoichiometrically exact [7, 11- 14]. When a pathway leading from the starting reactants (precursors) to the final products (targets) is formed by selecting one or none of the steps of each elementary reaction, the complete mechanism is naturally recovered by supplementing the opposite step to each step of the pathway. Moreover, the principle of microscopic reversibility prohibits the inclusion of any cycle in a pathway. The following set of 6 axioms of feasible reaction pathways can be formed from these first principles and conditions for any given overall reaction. (Rl) Every final product (target) is totally produced by the reaction steps represented in the pathway input: reaction pathway identification problem (E, O, M) output: all conbinatorially feasible structure (m, o) begin PRISSG(w\E), 0, 0, 0); end. procedure PRISSG(p, dp, inc, exc) begin exc:=RPIRSG(exc); if (incnexc;t0) then return; if w-(E)\'r(O\exc)¢0 or m+(E)\Y(O\exc)-#;0 then return; inc:=NX(inc, exc); for allxEp if ( v-(x)\exc\in=0) then begin dp:=dpu{x}; p:=(pu'r(v-(x)ninc)) \dp; end; ifp=0then begin o:=inc; m:=':P(o); print (m, o); return; end; letxEp; Ox:= V-(x)\exc; Oxb:= v-(x)ninc; C:=g 0}; return sol; end; end; Fig. 3. (cont'd.). function Solution( (m, o)) begin end; LP: L.vi~max e;EO if LP is not feasible then return FALSE; else begin end; SolveLP; if v(vi :::;; !) e;Eo c then begin avoid:= avoid u {(m, X(o))}; return TRUE; end; else begin o := { ei: vi > ! } ; m := 'P(o ); avoid:= avoid u {(m, o), (m, X(o))}; return FALSE; end; Fig. 3. (cont'd.). The methodology presented above has been applied to two sets of available elementary reactions, one comprising elementary reactions (1) through (10) together with elementary reaction {14) in Table 1, function pFreedom(x, inc, exc) begin if XEOF(E) or v-(x)ninc=0 end; then return I v-(x)\inc\exc I else return I v-(x)\inc\exc l-1; function cFreedom(y, inc, exc) begin ifyEot(E) or v\y)ninc=0 then return J v+(y)\inc\exc) else return I v+(y)\inc\exc l-1; end; Fig. 3. (cont'd.). Table 1 List of Plausible Elementary Reactions (1) Hz+£~H2£ (2) H2£+ .e ~H.e + H.e (3) Nz+f~Nz£ (4) N2£+£~N£+N (5) Nz£ + Hz£ ~ NzHz£ + £ (6) NzHz£ + £ ~ NH£ + NH£ (7) N.e + H.e ~ NH£ + £ (8) NH£+ H£~NH2£+ £ (9) NH£ +Hz£~ NH3£+ £ ( 10) NH2£+ H£ ~ NH3£ + £ (11) NH£+N£~N2H£ + .e (12) NHz£ + N.e ~ NzHz£ + £ (13) NzHz£ + H2f ~ N2H2£+ N£ (14) NH3£~NH3+ £ 75 Fig. 4. P-graph representation of the network comprising the forward steps of elementary reactions (1) and (2). and the other comprising elemenetary reactions (1) through (14) in the same table. For illustration, the P~ graph of a network consisting of elementary reactions (1) and (2) is depicted in Fig. 4. 76 Table 2 Independent Mechanisms Resulting from the First Set of Eleven Elementary Reactions Mechanism 1 Mechanism2 ill m (1) Hz+l~Hzl (1) Hz+l~Hz£ (2) Hz£+£~ H£ + H£ (3) Nz+i~Nz£ (3) Nz+i~Nz£ (4) Nz£ + £ ~Ni! + N (5) N2£ + Hz£~ NzHal + 1! (6) NzHz£ + £ ~ NH£ + NH£ (7) N£ +Hi!~ NH£ + 1! (8) NH£ +Hi!~ NHzi! + £ (9) NH£ + Ha£ ~ NH3i!+ 1! (10) NH2£ + H£ ~ NH3£ + 1! (14) NH3£ ~ NH3 + ££ (14) NH3£ ~ NH3 + £ Mechanism3 Mechanism4 I1J1 [8. 251 (1) Ha+f.~Hz£ (1) Hz+i~Hz£ (2) H2£ + £ ~HR. + H£ (3) Nz+i~Nz£ (3) Na+i~Nz£ (4) N2£ + f. ~Nl + N (4) N2£ + 1! ~N£ + N (5) Nzi! + Hzl ~ NaHz£ + £ (7) Nl + HR.~ NH£ + f. (7) Nf. +HR.~ NH£ + £ (8) NH£ +Hi~ NHzf. + i (8) NH£ +HR.~ NHz£ +f. (9) NH£ + H2£ ~ NH3£+ 1! (10) NHa£ +Hi~ NH3i + 1! (10) NH2i + HR.~ NH3£ + f. (14) NH3£ ~ NH3 + l {14) NH3i ~ NH3 + i MechanismS Mechanism6 [8, 251 [8. 25J (1) Ha+l~Hzl (1) Hz+i~H21! (2) H2£+ £~HR. +Hi (3) Na+I.~Na£ (3) Na+i~Nz£ (4) Nzl+i~Nl+N (5) N:at + Hzl ~ NaHz£ + i (5) Naf. + Hz£~ NaHz£ + l (6} NaHal+ l ~ NH£ + NH£ (6) ·NaHal+ l ~ NHi + NH£ (7) Nl+Hl*NHl+ t (S) NHt + Ht ~ NH2£ + l (8) NH£ + Hi~ NHal + l (10) NH2t +Hi* NH3l + l (10} NHal + Hl * NH3£ + i Table 3 Summary of Computational Results and the Corresponding Computational Requirements 1. Independent pathways Number of elementary reactions Number of LPs Computation time* Number of combinatorially independent pathways Problem#1 11 13 0.06 s 6 Problem#2 14 581 l.ls 35 2. A-cyclic pathways (independent and combined) Problem #1 Problem #2 Number of elementary 11 14 reactions Number of LPs Computation time* Number of combinatorially independent pathways 35 0.06 s 17 *Pentium II Celeron 450 MHz PC, 128 MB RAM Result and Discussion 984 1.7 s 367 When applied to the set of 11 elementary reactions in Table 1 [1-9, 25], the current method has recovered 6 independent mechanisms in 0.06 s on a PC (Pentium II Celeron 450 MHz and 128 MB RAM). All 6 mechanisms are identical with those available in the literature [1, 2, 7, 8,25], which are listed as mechanisms 1 through 6 in Table 2. Moreover, the method has generated 17 acyclic combined mechanisms in 0.6 seconds on the same PC, 11 of which have resulted from linearly combining the 6 independent mechanisms. One of the combined mechanisms corresponds to the mechanism consisting of the 11 candidate elementary reactions [8]. It is worth noting that the number of elementary reactions reported in the literature is 9 instead of 11. This increase is a result of splitting the initiation step into 2 and adding the desorption step for NII3 signifying termination. It is logical to postulate that the initiation proceeds in two steps, each involving a single active site and that the termination occurs through the desorption of NH3 formed on the active site: Any elementary reaction would be far more likely to be bi- molecular than tri-molecular £7, 9, 13, 52, 53]. An extensive review of the available literature [4-6, 9} indicates that elementary reactions (11), (12), and (13), also listed in Table 1, are highly plausible to be in the pathway of ammonia-synthesis reaction. With the addition of these 3 elementary reactions, the current method has yielded 35 independent mechanisms in 1.1 seconds and 367 acyclic mechanisms in L 7 seconds on the same PC, 332 of which have resulted from linearly combining the independent mechanisms. Table 3 summarizes the results described above. Note that there is no direct correspondence between the computational effort required for the identification of independent feasible pathways and that required for the identification 77 of acyclic feasible pathways. When the number of candidate elementary reactions is appreciable, the former tends to be greater than the latter: The two efforts resort to different sets of algorithms. With the number of elementary balances remaining invariant at 2, one for nitrogen and the other for hydrogen, each increment in the number of elementary reactions automatically translates into an exponential magnification of combinatorial complexity, thereby adding substantially to the computational time for mechanism determination. It has been amply demonstrated that the current method is capable of coping with this added burden. Nevertheless, caution should be exercised so that an excessively large number of candidate elementary reactions is not proposed in implementing the current method; otherwise, we would be confronted with a bewildering number of feasible mechanisms, thus rendering inordinately difficult the final selection of the valid mechanism through experimental reaction-rate and spectroscopic measurements in conjunction with theoretical and computational studies. 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FORMAL GRAPH-THEORETIC DESCRIPTION OF THE REACTION-PATHWAY- IDENTIFICATION PROBLEM Here, a formal description is given of the problem of reaction·pathway identification. It is couched in the parlance of graph theory in general and that of P-graph in particular [34, 35, 39]. Problem Definition Let a reaction-pathway-identification problem be defined by triplet (E, 0, M), where E is the overall reaction; 0 = { eh ez, ... , en}, the finite ordered set of elementary reactions; M == {ah a2, ••• , a1}, the finite ordered set of chemical and active species; E = [Et. E2, .•. , Ez]TE Z1, where Ej is the difference between the number of moles of the j-th chemical produced and that consumed by the overall reaction; and e,· = [e1 · e2 · · T l ·,t, ,l, ••• , e!,i] E Z , where ej,i is the difference between the number of moles the j-th chemical or active specie produced and that consumed by the i-th elementary-reaction step. Since every elementary reaction is reversible, both its forward and reverse steps are included in set 0, i.e., 'v'e; (e; E 0 => -e; E 0) In other words, for any elementary-reaction step e1 defined, its opposite step, denoted by -e1, is also defined in the problem. It is assumed that Mn0=0andE~ OuM. Representation Elementary reactions, chemicals, and active species are represented by P-graphs as follows: 79 If oi..e;) denotes the set of chemical and active species consumed or produced by the elementary-reaction step ei> we have For any chemical or active species aJ-E M, let v-(aj) and v-(aj) denote the set of elementary-reaction steps consuming and producing aj, respectively; it follows that and If v(ai) denotes the set of elementary-reaction steps consuming or producing ai, we have obviously For any set of the elementary-reaction steps, o ~ 0, let 'I'(o) and T(o) denote the set of chemical and active species consumed and produced by any element of o, respectively; it follows that 'I'(o) = U of(e1) e;EO For the overall reaction, E, let of(E) and ol(E) denote the set of starting reactants (precursors) and final products (targets), respectively; it follows that and and If oi..E) is the set of chemical species consumed or produced by the overall reaction, E, we have naturally a£.. E) = of(E) u ol(E). For any elementary-reaction step e1 E 0, let oT(e1) and al(e1) denote the set of reactants and products of e;, respectively; it follows that and T(o) = U al(e;). e;EO If l¥(o) is the set of chemical and active species consumed or produced by any element of o, we have For any set of chemical or active species m ~ M, let qF(m) and ql(m) denote the set of elementary- reaction steps producing and consuming any element of m, respectively; it follows that and qF(m} ::::: U v-(aj) OJ'?.m ql'{m) = U v""(ai). aiem 80 If (/J(m) is the set of elementary-reaction steps producing or consuming any element of m, we have For any set of elementary-reaction steps o ~ 0, let X(o) where any vertex corresponding to set m is termed M- type, and any vertex corresponding to set o is termed 0- type. The set of arcs is denote the set of opposite steps of the elementary- where reaction steps included in set o; then, X(o) ~ {e;:-eiE o}. Any P-graph representing a set of chemical or active species and elementary-reaction steps is given by pair (m, o ), where o ~ 0 is the set of the elementary- reaction steps, and m ~ M is the set of chemical and active species, where 'P(o) k;m. The set of vertices of the graph is V=oum, and In graphical representation, vertices of the 0-type are denoted by horizontal bars, and vertices of theM-type are denoted by solid circles. Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78