6676
J. Phys. Chem. 1989, 93, 6676-6681
Systematic Approach to Elucidation of Multistep Reaction Networks Friedrich G. Helfferich Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: May 25, 1988; In Final Form: January 18, 1989)
A systematic approach to elucidation of multistep reaction networks is presented, based on general equations for rates and yield ratios applicable to networks with arbitrary numbers of steps and locations of nodes and coreactant entries. Equations for specific networks are recovered by simple substitutions, cancellations, and collection of terms. The procedure saves time in deriving equations for hypothetical networks. More importantly, it reveals systematic relationships between network properties and observable kinetic behavior. This makes it possible to reject incorrect hypothetical networks by whole classes rather than one by one and provides guidance for the design of experiments that discriminate most effectively between rival networks. Simple examples are provided for illustration. The procedure as described is for reactions with all intermediates at trace level and with no steps of higher than first order in intermediates. For other types of reactions, the approach can simplify modeling and analysis but must be combined with other methods.
Introduction In order to shorten the time from inception of a new chemical process to start up of the full-scale plant, scale up directly from the laboratory bench has become highly desirable. To be safe, scale up of reactors by such large factors calls for reaction-kinetic modeling based on the differential equations that reflect the individual reaction steps at the molecular level. For multistep reactions, this requires a knowledge of the reaction network. The present contribution describes a systematic approach to network elucidation, operating with general equations that apply to networks with arbitrary numbers of steps and locations of branches and coreactant entries. Equations for given networks can be recovered from these by simple substitutions and cancellations. Apart from saving time, this approach reveals systematic relationships between network properties and observable kinetic behavior and so can provide guidance for the design of experiments with which incorrect hypothetical networks can be ruled out by whole groups instead of one by one. Here, the general equations for linear and simple types of branched and looped networks will be given and their practical application outlined and illustrated. Derivations, details, and handling of more complex types of reactions will be described elsewhere. The approach is a logical continuation of the work of Christiansen,’ who derived an implicit general equation for “closed” sequences of steps in catalysis as early as 1931 (see also ref 2 and 3). The relation to Christiansen’s formula is shown in the Appendix. Christiansen’s approach was later adapted by others to chain reactionse7 and certain cases of heterogeneous catalysis.*-” An independent, but entirely equivalent, formalism was proposed
-
( I ) Christiansen, J. A. Ein Versuch zur Anwendung der Methode der Stationaren Geschwindigkeiten auf die Reaktion CH,OH H 2 0 3H2 COz. 2.Phys. Chem., Bodenstein-Festband 1931, 69. (2) Hammett, L. P. Physical Organic Chemistry; McGraw-Hill: New York, 1940; Chapter IV. (3) Christiansen, J. A. The Elucidation of Reaction Mechanisms by the Method of Intermediate in Quasi-Stationary Concentrations. Adu. Catal. 1953, 5, 31 I . (4) Rice, F. 0.;Herzfeld, K. F. The Thermal Decomposition of Organic Compounds From the Standpoint of Radicals. VI. The Mechanism of Some Chain Reactions. J . Am. Chem. SOC.1934, 56, 284. (5) Frost, A. A.; Pearson, R. G. Kinetics and Mechanism. A Study of Organic Chemicnl Reactions, 2nd ed.; Wiley: New York, 1961; Chapter IO and Section 12.D. (6) Hill, C. G., Jr. An Introduction to Chemical Engineering Kinetics and Reactor Design; Wiley: New York, 1977; Sections 4.1.2 and 4.2. (7) Smith, J. M. Chemical Engineering Kinetics, 3rd ed.; McGraw-Hill: New York, 1981; Sections 2-4 and 2-8. (8) Temkin, M. I. Reaction Kinetics on the Surface of Solids and the Problem of Highest Catalytic Activity. Zh. Fiz. Khim. 1957, 31, 3. (9) Temkin, M. 1. Two-step Kinetics on Inhomogeneous Surfaces. Dokl. Akad. Nauk SSSR 1965, 161, 160. (10) Boudart, M. Kinetics of Chemical Processes; Prentice Hall: Englewood Cliffs, NJ, 1968; Section 3-3. (1 1) Boudart, M. Two-step Catalytic Reactions. AIChE J. 1972, 18,465.
+
+
0022-3654/89/2093-6676$01.50/0
by King and Altman for enzyme catalysis.’* Other than this, past research has focused primarily on problems such as what mechanisms, networks, and intermediates are possible in principle for a given overall reactionI3-l6 or under what conditions a conclusive experimental identification of parameters or distinction between rival mechanisms is possible.” Surprisingly, little effort has so far gone into developing general rate and yield-ratio equations for arbitrary networks without or with branches and loops. Objective and Scope The objective of this approach is to provide an easy to use mathematical framework and procedure for elucidation of complex networks. No constraints or restrictions apply to the number of steps, locations of nodes, and participation of coreactants and coproducts. However, the approach as outlined here is subject to three important limitations in that it postulates the following: (1) The reaction is homogeneous. (2) All intermediates are present at no higher than trace levels. (3) No step involves more than one molecule of intermediates. The first premise can be relaxed (e.g., with procedures as in ref 8-1 l), but the ensuing complications, while not serious, would distract here from the main line of thought. The second premise permits the Bodenstein approximation of quasi-stationary to be used, and the third restricts the mathematics to linear algebra. Both are seriously confining but essential for the approach to be manageable. To be sure, the third premise does not exclude overall reactions of higher order (e.g., aldol condensation), but interesting classes of reactions nevertheless violate one or both of these premises. For these, the approach shown here must be combined with other methods but can still significantly facilitate mathematical modeling and networks analysis. In addition, the presentation here is restricted to the establishment of differential equations for rates and product formation ratios. The problem of their integration for practical reactors is not discussed as standard procedures remain fully applicable. (12) King, E. L.; Altman, C. A Schematic Method of Deriving the Rate Laws of Enzyme-Catalyzed Reactions. J. Phys. Chem. 1956, 60, 1375. (1 3) Sinanoglu, 0. Theory of Chemical Reaction Networks. All Possible Mechanisms or Synthetic Pathways With Given Number of Reactions Steps or Species. J . Am. Chem. SOC.1975, 97, 2309. (14) Sinanoglu, 0. Finding All Possible a priori Mechanisms for a Given Type of Overall Reaction. Theor. Chim. Acta 1978, 48, 287. (15) Lee, L.-S.; Sinanoglu, 0. Reaction Mechanisms and Chemical Networks-Types of Elementary Steps and Generation of Laminar Mechanisms. 2. Phys. Chem. (Munich) 1981, 125, 129. (16) Sinanoglu, 0. I - and 2-Topology of Reaction Networks. J . Math. Phys. (N.Y.) 1981, 22, 1504. (17) Vajda, S.; Rabitz, H. Identifiability and Distinguishability of FirstOrder Reaction Systems. J . Phys. Chem. 1988, 92, 701. (18) Chapman, D. L.; Underhill, L. K. The Interaction of Chlorine and Hydrogen. The Influence of Mass. J . Chem. SOC.1913, 496. (19) Bodenstein, M. Eine Theorie der Photochemischen Reaktionsge schwindigkeiten. Z . Phys. Chem. 1913, 85, 329.
0 1989 American Chemical Society
Elucidation of Multistep Reaction Networks
The Journal of Physical Chemistry, Vol. 93, No. 18, 1989 6677
Notation A compact notation will be used for convenience. Reactants are A, B, ...; products are P, Q, ...; and intermediates are XI, X2, ..., numbered in the sequence of their occurrence in the network. Indices of rate coefficients identify reactant and product: kij is the coefficient of the step Xi Xj; kol,of the step A XI; knHl, P; etc. of the step X, Networks are shown without rate coefficients, as these are apparent from the reactants and products of the respective steps. Any coreactants are shown above the reaction arrows and any coproducts, below the arrows. Thus
-
-
-
+B
(4)
-
xi- B xj indicate steps Xi
where
xi- Q xj
Xj and Xi
-
Xj
+ Q, respectively.
x]+lJt2
Kinetic Basis In general, the rate of a molecular reaction step involving reactants Ai with arbitrary stoichiometric numbers nj is given by n
1
Aj+~j Ajtlj
xj+2~+1
~j;lj A]tlj
~j;2j+l Aj+2J+1
x]+2Jt3 Ajt2J+3
1 ~jt;,/t2 Ajt3,jt2
'*' '*'
...
Ak-2.k-l Ak-2,k-l Ak-?,k-l
Xk-l.k Ak-1.k Ak-l.k
I
Ak-l,k
*.a
'*'
Ak-l,k-2
:
(6)
1
j = 1, ..., n
-(l/nj)rA, = kn[Aj]"i i= I
(e.& see ref 20). Here, instead, pseudo-first-order coefficients X (dimension t - l ) are used, which are the products of the actual rate coefficients k and the concentrations of any coreactants. For instance, for steps Xk B C and 2A XI, respectively
+ +
-rk
-
-
= Xk,k+l [xkl
kk,k+l = kk,k+l[B] [ c ]
and -(Y2IrA = XOl[Al
A01
- -
= kOl[Al
- (1)
For a trace-level intermediate Xk in a reaction ... Xk-1 Xk Xk+l ... (coreactants and coproducts, if any, not shown), the Bodenstein approximation of quasi-stationary behavior can be ~ s e d , ~ * ~ - ' according J * - ~ ~ to which the net rate f k is small in absolute magnitude compared with the contributing terms (conversion from and to xk-1 and &+I). Provided the intermediate remains at trace level, the approximation is valid except during a short initial period in which the quasi-stationary state is established (e.g., see ref 20). The present approach is based on this approximation. Linear Networks The keystone in the present approach is a general rate equation for linear networks (or network segments) with arbitrary numbers of steps and with or without coreactants or coproducts. Consider the arbitrary linear network A
- - -- XI
X2
1 . 9
xk-1
P
(2)
with or without coreactants or coproducts participating in any steps. The full set of differential rate equations, granted our premises, is given by (I/~A)= ~A -Xoi[AI + Xio[XiI ri = Xi-1,i[Xi-11- ( h , i - I
+ Xi,i+~)[Xil+ Xi+~,i[xi+~I 0
i = 1, ..., k
( l /nP)rP = Xk-l,k[Xk-ll
X O =A,
x k =
raid =
-1
- Xk.k-l [PI P
With elimination of the unknown [Xi], this reduces to ( I / ~ P ) ~= P(KAP[A]- KPA[PI)/DAP
where ole is olefin; ald is aldehyde; cat is the catalyst, HCo(CO),; XI is the r-complex of the olefin and CO-deficient catalyst; X2 is the so-called trihydride; X3 is the cobalt carbonyl alkyl; X4 is the cobalt carbonyl acyl; and cat' is the CO-deficient catalyst, HCo(C0)3. From eq 3 to 5 [or with matrix 6 in Chart I] and with np = 1, one finds
(3) ~
(20) Ark, R. Chemical Kinetics and the Ecology of Mathematics. Am. Sci. 1970, 58, 419. (21) Wilkinson, F. Chemical Kinetics and Reaction Mechanisms; Van Nostrand Reinhold: New York, 1980; Sections 3.6 and 3.10. (22) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice-Hall: Englewood Cliffs, NJ, 1986; Section 7.1.2.
XolXl2h23X34[olel x12X23x34
+ XIOA23X34 + X10h21X34 + X10X21X32
(terms with zero factors h43and hH omitted and factor X4, common to all terms cancelled). Replacement of the X coefficients, collection of terms, and rearrangement yields ka[ole] [cat] (8) raid = 1 + kb[COI/[H21 with (23) Breslow, D. S.; Heck,R. F. Mechanism of the Hydroformylation of Olefins. Chem. Ind. (London)1960, 467. (24) Heck, R. F.; Breslow, D. S. The Reaction of Cobalttetracarbonyl With Olefins. J . Am. Chem. Soc. 1961,83, 4023. (25) Falbe, J. Synthesen mit Kohlenmonoxyd; Springer: Berlin, 1967; Section 1.2 (English translation Carbon Monoxide in Organic Synthesis, 1970).
6678 The Journal of Physical Chemistry, Vol. 93, No. 18, 1989
Helfferich A rate law that satisfactorily describes the observed dependence on concentrations is as follows:27
Equation 8 is in agreement with experimental results. The mechanism 7 is shown here with an intermediate X2 (trihydride) not included in the earliest publications. However, without it, the reaction would be zeroth order in H2 unless the carbonylation step was made reversible, in which case the rate equation would not produce the experimentally observed independence of [H,] and [CO] at constant [H2]/[CO] ratio. Rules. Equation 3 directly yields a number of useful conclusions that would otherwise require more effort to prove. ( 1 ) Steps following an irreversible step remain without effect on the rate equation. In such a case, the reverse coefficient of the step is zero and all terms containing it as a factor drop out. These are the negative term in the numerator and the last term or last several terms in the denominator. Moreover, all remaining terms now contain the forward coefficients of the steps following the irreversible one as common factors, so that these coefficients cancel. The rate equation is thus reduced to the form it would have for a network ending at the irreversible step. (For heterogeneous catalysis, this rule has been stated by Boudart.”) ( 2 ) A reaction isfirst order with respect to a reactant (with stoichiometric number I) if that reactant participates in only the first step, and it is and of order between 0 and 1 if it participates in a later step. In the first case, the reactant concentration enters into the rate equation as [A] or by way of the coefficient hi,which appears o d y in the numerator. In the second case, the concentration enters by way of a later forward X coefficient, which appears in the numerator and one or more additive terms in the denominator and so produces a nonconstant, positive order less than 1. (Of course, the reaction can approach zeroth or first order if respective terms in the denominator become dominant or negligible.) More complex situations are easily covered in the same fashion. To give only one example: (3) A reaction is of order between 1 and 2 with respect to a reactant that participates in both thefirst step and a subsequent step. A simple case in point are reactions of the type
(9) The order in H2 (between zero and one) points toward reaction of H2 in a step other than the first (rule 2). The order in C O (minus one) indicates that CO is released in an early step, to be reincorporated later in a step that is not reflected in the rate equation because it follows an irreversible step (rule 1). The simplest network with such steps is as follows:
However, for this network, eq 3 gives
The first term in the denominator must become insignificant to give the order -1 with respect to C O but must remain significant to give the order C1 with respect to H2. These demands are mutually exclusive. Moreover, eq 3 shows that, for any network, the first term in the denominator contains forward X coefficients only and so must be negligible to give the observed order -1 in C O and that this term is also the only one containing the forward coefficient XI2 of the second step, so that participation of a coreactant in that step has no effect if the term is negligible. Accordingly, no simple network with H2 entering in either the first or second step can give the observed orders with respect to both C O and H2, and an additional step must be inserted between the two:
e -
cat
ald
H
X,
x2 A
ak
x3
For this network, eq 3 gives such as aldol condensation, whose behavior has become a classical k01k12k23[aldl [H21 [cat1 textbook example (e.g., see ref 5) and which eq 3 identifies as a Talc = special case of a general rule. ki~k23[H21 kiok23[H21 [ C o l + kiok2i [ C o l (4) “One-plus” rate laws (with additive terms in the denomWith negligible first term in the denominator and with ka = inator) result from networks with at least one reverse step. The k01klZk23/k10k21 and kb = k23/k21, this is of the form of eq 9 as first term in the denominator contains only forward X coefficients. required and is indeed one of the two simplest networks producing A reverse step is needed for at least one subsequent term to be the described concentration dependence of the rate (the other being nonzero. A “one-plus” rate law results unless either the reverse identical except that the aldehyde enters after C O loss from the step is preceded by an irreversible step (and so becomes irrelevant catalyst, producing an additional factor [ald] in the first term in for the rate equation) or the respective term or terms in the the denominator, an effect that disappears when that term is denominator become negligible or all contain the same concenneglected). trations as factors. (5) A reaction is substrate-inhibited (negative reaction order Branched and Looped Networks with respect to a reactant) i f a reactant is split off as a product Network Reduction. The derivation of the general equation in an early step, to be reincorporated in later steps preceded by (3) for linear networks is independent of whether A and P are an irreversible step. Appearance as a product in an early step reactant and product of the overall reaction or are themselves lets the concentration of the species contribute a factor to one or intermediates in a larger network. The equation therefore is valid more terms in the denominator, while appearance as reactant in for the contribution of a linear network segment a step following an irreversible step remains without effect on the rate equation. A case in point is the oxo reaction, which is subxj xj+l ... Xk (10) strate-inhibited with respect to CO [see network 7 and rate eq to the rate -rj or rk. Instead, the rate contribution by the segment 81. can be written Example. A probable network is to be established for homogeneous hydrogenation of a higher aldehyde (ald) to an alcohol (1 1) (rk)Xj-Xt = Ajk[Xjl - Akj[Xkl (alc), catalyzed by cobalt phosphinohydrocarbonyl in an organic solvent (aliphatic alcohol) under synthesis gas a t m ~ s p h e r e . ~ ~ , with ~ ~ “segment coefficients”
- -
Ajk E
(26) Marko, L. The Kinetics and Mechanism of Homogeneous Aldehyde Hydrogenation With Cobalt Carbonyls as Catalysts. Proc. Chem. SOC., London 1962, 67.
Kjk/Djk,
Akj E
Kkj/Djk
(27) Van Winkle, J. L. Unpublished results, 1977.
(12)
The Journal of Physical Chemistry, Vol. 93, No. 18, 1989 6679
Elucidation of Multistep Reaction Networks which, like the Klj and Dij, are functions of the X coefficients only. The streamlined rate law (1 1) corresponds formally to a simpler, single-step segment X j X k with forward and reverse rate coefficients A j k and d k j , respectively. Every linear, multistep segment between two nodes, or between a reactant or product and a node, can be “reduced” in this way. As the simplest example, a multistep, single-node network
-
...
A-
-
/ ***
intermediate
leads to the following:
X k
For single-node, multibranch networks, this is readily extended by addition of respective terms. For instance, for the reduced network
- P
i-
... - a
Xk
with X coefficients can be reduced to, and is mathematically equivalent to, the following simpler network with A coefficients:
A-
Xk
Q
t*
R
the rate equation is as follows:
For a looped network segment
rP =
APk(AkA
the contribution to the rate -rj or r k is the sum of the contributions of the two pathways (rk)X,-Xt
=
Ljk[Xjl
- Lkj[Xkl
=
(Ajklpath I
+ (AjklpathZ
Lkj
=
(Akjlpath I
+
-
rP =
AAkAkPIAl
(15)
(In the improbable event of more than two parallel pathways, the summation is over all of these.) In this fashion, the looped segment is similarly reduced to a single-step segment xj X k with 15 coefficientsz8 Networks may contain portions that combine linear and looped segments. In the simplest case
A-
Xk
l
-
-r/j[x/l
(16)
=
LjkAk//(Ak/
+ Lkj),
r/j
=
LkjA/k/(Ak/
+ Lkj)
(17)
In this fashion, any network that meets our premises can be reduced to a simpler one with nodes only where pathways to different end products diverge, with only single reversible steps between such nodes, and with A, L, or r coefficients that are functions only of the actual rate coefficients and the concentrations of any coreactants or coproducts, not of the concentrations of the intermediates. Since loops are relatively rare, equations given from here on are written exclusively in terms of A coefficients, for which L or I’ coefficient can be substituted wherever called for. The possibility of reduction of linear networks has been pointed out by Hill,29 who gave examples of cases with one and two intermediates but no general formula. Rate Equations. For the single-node, single-branch reduced network ( 1 3), application of the Bodenstein approximation to the (28) Chern, J.-M. Diagnostic Analysis of Complex Reaction Networks, Report for Comprehensive Examination, Department of Chemical Engineering, The Pennsylvania State University, State College, PA, 1988. (29) Hill, T. L. Free Energy Transduction in Biology; Academic Press: New York, 1977; Appendix 1 .
x)
p
l
R
= [ ~ A P [+ A ~] Q P [ Q ]-
PA + ~ P Q ) [ P+] + AkQ + A k / ) ( n R P [ R l - n P R [ P l ) l / [ n k R + n k P + n/A
where the new “collective coefficients” are given by the following? rj/
(19)
the rate equation is (AkA
This can be reduced to xi X k X I with coefficient L j k , L k j , A k / , and A / k . This reduced segment is formally h e a r and so can be further reduced to a single-step segment X j X I with rate contribution
--
Q
rp
rjdxjl
+ AkP + AkQ)
which is also a useful approximation for the initial rate in a reversible network. If the segment A X k contains an irreversible step, then b k A = 0 in eq 18 or 19. For a two-node, single-branch network
-
( ~ / ) x , - x ,=
/(AkA
-
(Akjlpath 2
- -
-
However, multiple branches a t the same node are rare. If the segments X k P and X k Q contain at least one irreversible step, as will often be the case in practice, the terms involving [PI and [ Q ] do not contribute and eq 18 reduces to
(14)
where the new “loop coefficients” are given by Ljk
+ AQk[Q] + A R k [ R I ) / [ A k A + A k P + A k Q + AkRl + A k Q + AkR)
[AkP(AAk[Al
+ n / Q + ( A k A + AkQ)(A/R + A,)]
-
where IIij stands for the product of all A coefficients along a pathway i j : n ~ pA A k A k l A l p , n p E~b p / A / ~ , etC. Explicit general rate equations for networks with more than two nodes, and without or with loops, have been derivedz8but are more complex and will be published elsewhere. Yield-Ratio Equations. In practice, the rate equation even for a network with only one or two nodes tends to be too complex for effective use in network elucidation. A much better diagnostic tool is the differential yield ratios, defined as ratios of product formation rates. For the single-node network (13), the yield ratio RW rp/rQ is easily derived from eq 18
or, for an irreversible reaction, from eq 19: RPQ
=
AkP/AkQ
(21)
Usually, this simple equation also is a good approximation for the initial yield ratio of a reversible reaction. (It becomes inadequate, however, if either product remains at trace level even at high conversion of the reactant.) Note that the rules stated for linear networks are valid for both h e a r segments X k P and X k Q . This knowledge often facilitates finding the location of the node on the evidence of the dependence of the experimentally observed yield ratio on coreactant concentrations.
-
-
The Journal of Physical Chemistry, Vol. 93, No. 18, 1989
6680
In multinode networks, eq 20 and 21 are easily shown to remain valid for any two products originating from the same node, Le., for configurations
...
Xk
- P
‘ ... *.. -‘ ...
/ *** c--)
0
For a yield ratio of products originating from nodes one or two farther apart, e.g., RpR in networks with partial configurations
...
-
/ ‘**
x/
...
Xk
- P
/
- 0
- R
and
...
-
xk
and with irreversible last segments or at low conversion, one obtains
and
respectively. However, for reversible reactions with networks having widely spaced nodes, the postulates of quasi-stationary behavior and initially negligible reverse rates may become incompatible. Example. As a simple example, consider olefin hydroformylation catalyzed by cobalt phosphinohydrocarbonyl, HCo(C0)3PR3, and yielding aldehyde and paraffin. Aldehyde is formed in accordance with the Heck-Breslow mechanism (7). The problem is to locate where the pathway to paraffin branches off. The simplest possibilities are
olefin
-->
olefin
co
X,
cat
A
HZ
co
x1
GO
XZ
x2
x3
