ARTICLE IN PRESS

Chemical Engineering Science 65 (2010) 2921–2933

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ces

The steady-state kinetics of parallel reaction networks

Saurabh A. Vilekar, Ilie Fishtik, Ravindra Datta �

Fuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA

a r t i c l e i n f o

Article history:

Received 23 September 2009

Received in revised form

19 January 2010

Accepted 20 January 2010Available online 25 January 2010

Keywords:

Kinetics

Catalysis

Reaction engineering

Reaction route graphs

Quasi-steady state approximation

Parallel pathway network

Rate law

Hydrogen–bromine reaction

N2O decomposition

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.01.022

esponding author. Tel.: +1 508 831 6036.

ail address: rdatta@wpi.edu (R. Datta).

a b s t r a c t

The conventional kinetic analysis of an overall reaction (OR) is limited to a single sequential pathway of

molecular steps at a time, based either on the general quasi-steady state (QSS) approach of Bodenstein,

or on the much simpler but limited Langmuir–Hinshelwood–Hougen–Watson (LHHW) approach based

on assuming a single rate-determining step (RDS), the remaining being quasi-equilibrated (QE). We

recently described a new algebraic methodology for deriving the QSS rate expression for a reaction

sequence, which allowed interpretation of the final result in an Ohm’s law form, i.e., OR rate=OR motive

force/OR resistance of an equivalent electric circuit, where the consecutive mechanistic steps represent

resistors in series. Here, we propose a similar Ohm’s law form of QSS rate for a reaction system

involving parallel pathways, whose equivalent electrical circuit derives directly from the reaction route

(RR) Graph of its mechanism, as proposed earlier by us. The results are exact for a reaction network with

mechanistic steps linear in intermediates concentrations, while they are approximate, albeit accurate,

for non-linear step kinetics. We further show how the LHHW methodology, combined with the concept

of intermediate reaction might be utilized to obtain the step resistances involved. For illustration, we

utilize the relatively simple examples of: (1) the gas-phase hydrogen–bromine non-catalytic reaction

(non-linear kinetics), and (2) zeolite catalyzed N2O decomposition reaction (linear kinetics). However,

the approach is useful for more complex non-catalytic, catalytic and enzymatic reactions networks

as well.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Many of the industrially significant reactions involve parallelreaction pathways. A quandary for the reaction engineer has beenhow to explicitly account for these alternate pathways indescribing the overall reaction (OR) rate, rOR, in terms of thekinetics of the elementary reaction steps, i.e., in determining theso-called OR ‘‘rate law’’ in terms of the known step weights, or(Christiansen, 1953; Horiuti, 1973; Temkin, 1979; Wagner, 1970),which represent the step mass-action kinetics with the exclusionof the unknown intermediates compositions. The most generalapproach for this is the quasi-steady state (QSS) approximation ofBodenstein (Christiansen, 1953; Horiuti, 1973; Temkin, 1979;Wagner, 1970), based simply on the assumption of time-invariance of reaction intermediates. However, an explicit QSSrate expression for overall rate rOR is often unwieldy, or notpossible at all when step kinetics are non-linear in intermediatesconcentrations (Lazman and Yablonskii, 1991), only numericalresults then being possible for a given set of reaction conditions.

The Langmuir–Hinshelwood–Hougen–Watson (LHHW) meth-odology (Hougen and Watson, 1943), on the other hand, does

ll rights reserved.

generally allow the development of simple explicit expressionsfor rOR, but it is based on the often arbitrary assumption of a single

rate-determining step (RDS), the remaining being at quasi-equilibrium (QE). Further, the a priori identification of such aRDS in the mechanism, if it exists at all, is not simple. Dumesic(1999) has presented an approach based on De Donder relationsfor identification of the RDS involving the concept of stepreversibility. Thus, the RDS is defined as the step, sr, whose stepreversibility, zr ð � r

’

r= r!

r ¼ expð�ArÞ, the ratio of the step rate inthe backward, to that in the forward direction), is approximatelyequal to that of the OR, zOR � r

’

OR= r!

OR ¼ expð�AORÞ. Here, thedimensionless De Donder affinity, Ar ¼�DGr=RT for the step, andAOR ¼�DGOR=RT for the overall reaction.

Campbell (2001) has, however, pointed out that such acriterion for identifying the RDS is limited, since reversibility ofa step represents only its thermodynamic driving force, notcontaining any information, for instance, on its activation barrier,or kinetics. On the other hand, identification of RDS simply basedon activation barriers is fraught with risk as well. In fact, the netstep rate involves both kinetics and thermodynamics, i.e.,rr ¼ r

!

r� r’

r ¼ r!

rð1�zrÞ ¼ r!

rðErÞ, where Er ¼ 1�expð�ArÞ repre-sents the thermodynamic driving force (Christiansen, 1953),while 1= r

!

r represents a kinetic resistance. Campbell’s degree ofcontrol approach, on the other hand, based on sensitivity analysisto identify the step rate constant(s) that most influence rOR,

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provides a more robust approach for the identification of the RDS.However, it is based on a numerical analysis. The reaction stepresistance, Rr, discussed by us below and elsewhere is, in fact, themost convenient criterion for identifying the slow or rate-limitingsteps in a sequence. We assume, in fact, that there can, in general,be more than a single rate-limiting step (RLS), the latter beingdistinct from the rate-determining step (RDS).

Furthermore, given that there are now first-principles andsemi-theoretical methods (Heyden, 2005; Heyden et al., 2005;Shustorovich and Sellers, 1998) available that can predict thekinetics of the elementary reaction steps with increasingaccuracy, it is now increasingly important that more comprehen-sive methods for the analysis of reaction networks be developed,which is our objective here, based on an intuitively appealingelectrical analogy of reaction networks.

The electrical analogy is, in fact, commonly invoked in aqualitative discussion of reaction mechanism and kinetics (Fogler,2006), wherein reaction steps are represented by individualresistances, with the current (rate) being driven by an overallmotive force. The electrical analogy was, in fact, first proposed byNernst (1926), who suggested that the rate of a chemical reactionmight be represented in analogy to Ohm’s law, being equal to a‘‘chemical force’’ divided by a ‘‘chemical resistance.’’ Besides itsintuitive appeal, the analogy is useful in visualizing the network,and especially when rationalizing the assumption of a RLS, as onewith the highest ‘‘resistance’’ in a sequence. However, it is rarelyutilized in a quantitative analysis. This is so, because therelationships of the step resistance and the motive force to theconventional reaction kinetic and thermodynamic quantities haveremained unclear, and it is not known how one might draw anappropriate equivalent electric circuit for a complex mechanism.

We have, actually, more recently developed a Reaction Route(RR) graph approach (Fishtik et al., 2004a, b, 2005a, 2006) thatputs this analogy on a rigorous footing by: (1) providing theequivalent electrical circuit for a given mechanism adapteddirectly from its RR graph, and (2), defining the step resistancein terms of step kinetics via the relation Rr � lnð r

!

r= r’

rÞ=ð r!

r� r’

rÞ,and the dimensionless De Donder affinity, Ar, as the driving force,resulting in Ohm’s law form for step kinetics, rr ¼Ar=Rr. Thecorresponding overall rate then takes the form rOR ¼AOR=ROR,where ROR may be obtained in terms of Rr from the RR graph in amanner completely equivalent to that in electrical circuits (Fishtiket al., 2005b). However, only numerical analysis is possible in thismanner, since the step rates, r

!

r and r’

r, and, hence, the stepresistances are not known a priori, involving the unknownintermediates concentrations.

In a recent paper (Vilekar et al., 2009), however, we followedan alternate algebraic methodology for the QSS analysis of areaction sequence, in which the final result was of a form thatcould be cast into an alternate Ohm’s law form, i.e., rOR ¼ EOR=R�OR,where the OR driving force is in the conventional form,EOR ¼ 1�zOR ¼ f1�expð�AORÞg, and the OR resistance could beexpressed as a sum of the step resistances in series, R�OR ¼

PrR�r,

while the step resistances R�r could be determined a priori via theLHHW methodology, thus providing a new analytical approachbased on the electrical analogy. Here, we show that this new formcan be extended readily to parallel reaction networks as well,where R�OR relates to step resistances R�r in the usual manner ofelectrical circuits. This approach not only provides an explicit QSSrate expression for a given mechanism in terms of step kinetics,but also affords perceptive insights into the dominant pathwaysand rate-limiting steps, thus allowing rigorous network pruning.

We first describe how one might use a given molecularmechanism of an OR to construct its RR graph, which depictsreaction steps as branches interconnected at nodes such that all

possible reaction pathways for the OR are represented simply as

walks on the RR graph. Next, the RR graph is converted into anequivalent circuit by simply replacing the branches by resistorsrepresenting the steps, followed by Ohm’s law representation ofoverall rate, with the overall resistance being obtained from theresulting circuit. We show, for the case of linear step kinetics, thatthe result is exactly the same as that obtained via linear algebrafrom the conventional QSS analysis. Further, we show that, whileapproximate, the results are very accurate for non-linear kineticsas well. All the necessary details along with definitions from ourprevious work are also summarized so that the treatment below isself-contained.

For ease of comprehension, further, the application of ourapproach to parallel pathway reaction networks is illustrated herefor the relatively simple cases of: (1) gas-phase hydrogen–bromine reaction (non-linear kinetics), and (2) zeolite catalyzedN2O decomposition reaction (linear kinetics) mechanism, bothinvolving only a handful of steps. Of course, more complexmechanisms are similarly amenable to kinetic analysis.

2. Theory

2.1. Reaction Routes: basic definitions

We first consider a simple generic 4-step mechanism withparallel pathways given in Eq. (1), in order to explain theessentials of RR graphs approach, while avoiding the mathema-tical details (Fishtik et al., 2004a, b, 2005a, 2006).

ð1Þ

This mechanism admits two parallel pathways, or reaction routes(RRs), as indicated above by the stoichiometric numbers in thetwo columns. Thus, steps s1, s2, and s3, when added, result in acancellation of the intermediates I1 and I2, resulting in the OR.Similarly, s1þs4 provides the OR, which is a second RR. Moreformally, thus, we define:

Reaction Route (RR): or a reaction pathway, or a reactionsequence, is a linear combination of elementary steps,Pp

r ¼ 1 sgrsr that eliminates a specified number of intermediateand terminal species to produce a reaction, where sgr is thestoichiometric number (usually, 0, 71 or 72) of step sr in the gthRR. If all the intermediate species are eliminated the reactionroute is called a Full Route (FR).

Thus, for the above example, the two RRs may be written as

FR1 : OR¼ ðþ1Þs1þðþ1Þs2þðþ1Þs3

FR2 : OR¼ ðþ1Þs1þðþ1Þs4

)ð2Þ

On the other hand, an Empty Route (ER) or a cycle is a linearcombination of the elementary steps such that all of the species,both intermediate and terminal, are cancelled, thus producing theso-called ‘‘zero’’ OR (i.e., the stoichiometric coefficients of all thespecies are zero).

In fact, since subtracting one FR from the other, e.g., FR1�FR2,would eliminate all the species, it can provide an empty route(ER), e.g.,

ER1 : 0¼ ðþ1Þs2þðþ1Þs3þð�1Þs4 ð3Þ

A negative stoichiometric number, as above, simply indicates thatthe step in the given RR is followed in the reverse direction to thatindicated in Eq. (1). Since all elementary steps are reversible, in

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principle, although the reversibility of a step zr can vary widely,this does not present any problem.

Of course, an infinite number of such RRs may be obtained viaa linear combination of two or more RRs. However, we are onlyinterested in the so called direct RR, following Milner’s (1964)concept of directness or minimality. Such a direct FR and ERcontains no more than q+1 and q+2 steps, respectively, from agiven reaction mechanism, where q is the number of linearlyindependent intermediate species, Ik. This limitation results in afinite and unique set of RRs.

In order to construct the RR Graph, and hence the electricalcircuit, for a given mechanism, in fact, a knowledge of not eventhe entire set of direct RRs is needed. Actually, only a set ofindependent RRs is needed for this purpose, i.e. any set of m¼ p�q

RRs, which can include both FRs and ERs that among them includeall of the steps of the given mechanism. This is, in fact, related tothe fundamental RR matrix of the RR graph (Fishtik et al., 2004a, b,2005a). All the other direct RRs of the unique set can betopologically traced on the thus generated RR graph as walks, asexplained later on. A more formal approach for enumerating theentire set of (direct) FRs and ERs based on the stoichiometriccoefficient matrix of the mechanism is described by Fishtik et al.(Fishtik et al., 2004a, b, 2005a).

We will also be concerned with reaction routes that producea given intermediate Ik from terminal species. Thus, anIntermediate Reaction Route (IRR) is the RR, or pathway, inwhich all the intermediate species except the one of interestIk (often along with a reference intermediate, say the vacantsurface site, S, in case of catalytic reaction system) areeliminated.

The set of stoichiometrically distinct intermediate reactions,IRs, produced by these IRRs may be written as

IRh : gkIkþð�gkÞSþXn

i ¼ 1

ðgkiÞTi ¼ 0 ð4Þ

where gk is positive, since the intermediate Ik is considered as aproduct in the IR.

For the example considered above, i.e., Eq. (1), in fact, nosurface site is involved. Thus,

IRR1 : IR1 ¼ ðþ1Þs1 ð5Þ

is an example of an IRR for the formation of the intermediatespecies, I1.

2.2. Reaction Route Graph

The typical goals of kinetic analysis are:

1.

to determine the elementary step rates rr, by first determiningthe unknown intermediate concentrations in terms of the steprate constants and the terminal species activities (concentra-tions), and;

2.

to relate the OR rate, rOR to the step rates rr.

Both of these goals are facilitated by the availability of the RRGraph of the mechanism, which shows graphically how thevarious step rates are interconnected in a reaction networkdepicting the overall reaction system and its pathways, and howthese interconnections constrain the rates of the individual steps,and that of the OR. Of course, one could accomplish this goalwithout the help of the RR graph, just as one could solve anelectric circuit problem mathematically without drawing a circuitdiagram, but the process, as in the case of electric circuits, isassisted by the graphic visualization afforded by the graph. In fact,since RR graphs follow flow network laws, namely, Kirchhoff’s

laws, they can be directly converted into an equivalent electriccircuit by simply replacing the branches representing theelementary reactions by resistors and the OR by a battery withan EMF related to the affinity, or Gibbs free energy, of the OR.

Thus, RR Graph for an OR is a graphical representation of itsmechanism that is comprised of elementary reactions, and isdrawn in a manner so that:

1.

the directed (with arrows) branches representing the indivi-dual steps sr are interconnected at intermediate nodes (INs)such that all Reaction Routes (RRs) can be traced on it simplyas walks between terminal nodes (TNs), with the overallreaction (OR) drawn as a branch between the TNs; and;

2.

the interconnectivity of the branches at the INs is consistentwith the QSS condition for the intermediates, while theconnectivity of branches at the TNs, including the OR, isconsistent with the QSS condition of the terminal species.These are, in fact, the equivalent of Kirchhoff’s flux (or current)law (KFL or KCL) in electrical circuits.

Thus, INs interconnect only elementary steps while the TNsinterconnect elementary steps to the OR. Both set of nodes are,further, consistent with KFL and the minimality of incidence ordirectness, in the sense that eliminating a reaction step willviolate the QSS condition.

Kirchhoff’s flux (or current) law states that the step rate rr(likened to branch current Ir) of all branches incident at a node j

sum up to zero (from mass conservation, along with the fact thatVnode=0). In other words, Drj �

Ppr ¼ 1 mrjrr ¼ 0, where the

incidence coefficient mrj ¼ þ1, if a branch leaves the node j,and mrj ¼�1, if a branch is coming into the node j.

Of course, KFL is analogous to the QSS assumption for theintermediate and the terminal species.

Quasi-steady state (QSS) assumption implies the time invarianceof the species concentration, i.e., the net rate of production of a

species is zero, Dri ¼Pp

r ¼ 1 nrirr ¼ 0.

It is, thus, clear from the above that KFL and QSS approxima-tion are equivalent.

Let us apply QSS approximation to the intermediate species inthe homogeneous decomposition reaction mechanism (Eq. (1)), i.e.,

QI1: rI1

¼ ð�1Þr1þðþ1Þr2þðþ1Þr4 ¼ 0

QI2: rI2

¼ ð�1Þr2þðþ1Þr3 ¼ 0

)ð6Þ

Since these two relations are the equivalent of KFL, theconnectivity of the reaction steps at intermediate nodes impliedby these QSS approximation relations is shown in Fig. 1a. Thesecond of these relations indicates that s2 and s3 areinterconnected at one intermediate node (in red), with s2

coming in and s3 leaving, while the first relation indicates anintermediate node with s1 coming in and s2 as well as s4 leaving.These two subgraphs are depicted in Fig. 1a.

Let us next consider the QSS approximation for the 3 terminalspecies, which participate both in the elementary reactions aswell as in the OR

QA : rA ¼ ð�1Þr1þð�1ÞrOR ¼ 0

QB : rB ¼ ðþ1Þr3þðþ1Þr4þðþ1ÞrOR ¼ 0

QC : rC ¼ ðþ1Þr2þðþ1Þr4þðþ1ÞrOR ¼ 0

9>=>; ð7Þ

Because from Eq. (6), r2 ¼ r3, the QSS approximation for both Band C are the same, i.e., only one of these two equations isindependent. The remaining two independent conditions can berepresented by the KFL subgraphs shown in Fig. 1b, where bluenodes represent the terminal nodes. The only way that all thesesub-graphs (Fig. 1a and b) can be coalesced into a single RR graph

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Fig. 1. (a) KFL condition for intermediates, (b) KFL condition for terminal species, (c) RR graph.

Fig. 2. Electrical analog of the RR graph shown in Fig. 1c.

S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–29332924

is by overlaying the common branches, resulting in Fig. 1c, whichis, hence the RR graph for the mechanism in Eq. (1).

It can be seen that this RR graph is appropriate. Firstly, all ofthe RRs (FR1, FR2 and ER1) enumerated above can be traced on thisas walks. In general, a walk from a starting node to an ending nodeis an alternating sequence of nodes and branches such that agiven node may not be crossed more than once. If a walk beginsand ends at the same node, it is called a cycle, while if a walkbegins at one terminal node and ends at the other terminal node,it is a full route.

Secondly, the nodes evidently satisfy the QSS approximationfor both the intermediate (INs) and terminal species (TNs). Thus,Fig. 1c is an appropriate RR graph for the mechanism as it satisfiesboth of the basic criteria outlined above.

Moreover, RR graphs are also thermodynamically consistent,given that they concur with Kirchhoff’s potential law (KPL).

Kirchhoff’s potential law (KPL) implies that the branch affinity,i.e., negative Gibbs free energy change for a reaction step,Ar ¼�DGr (likened to branch voltage Vr) of all branches in aclosed walk (starting and ending at the same node), or a cycle,sum up to zero, i.e.,

Ppr ¼ 1 sgrAr ¼ 0, where the stoichiometric

number sgr ¼ þ1, if a branch is directed in the direction of thewalk around a cycle, and sgr ¼�1, if a branch is directed in theopposite direction.

For example, if we apply KPL to the cycle (ER1) inthe considered mechanism, Eq. (1), there results, A2þA3�A4 ¼ 0,or

k!

2

k’

2

k!

3

k’

3

k’

4

k!

4

¼ 1

Thus, the predicted (or measured) kinetics must be consistentwith this KPL condition. In other words, not all rate constantsneed to be predicted (measured), some can be obtained from KPLrelations.

2.3. Electrical analogy

Since the RR graphs follow the flow network laws (namely, KFLand KPL), they can be directly converted into an equivalent

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electric circuit by simply replacing the branches representing theelementary reactions by resistors, Rr, and the OR by a battery withan EMF, AOR. Thus, for the above 4-step decomposition examplewith the RR graph in Fig. 1c, the equivalent electrical circuit isshown in Fig. 2.

In order to complete the analogy, we define (Fishtik et al.,2004a, b, 2005a),

Reaction or step affinity Ar is negative of reaction Gibbsfree energy, Ar ¼�DGr. It is, in turn, related to the ratio of therate in the forward direction r

!

r to that in the reverse direction,r’

r, via Ar ¼ lnð r!

r= r’

rÞ, where the dimensionless affinity,Ar � Ar=RT .

Reaction or step resistance Rr is the mean value of the inverse ofthe net step rate, rr ¼ r

!

r� r’

r, between its limiting values,namely, r

!

r and r’

r, i.e.,

Rr �1

r!

r� r’

r

Z r!

r

r’r

1

rrdrr ¼

lnð r!

r= r’

rÞ

r!

r� r’

r

:

Unfortunately, unlike in electric circuits, we do not know thereaction resistance a priori, since it involves the unknownintermediates concentrations.

Ohm’s law form of step kinetics: From the definitions of branchaffinity and branch resistance for a step sr as provided above, it iseasy to see that step rate follows Ohm’s law: rr ¼Ar=Rr

It is to be noted that this does not represent a linearization ofreaction kinetics, but is simply a definition of a reactionresistance, which unlike the resistance in electrical circuits, isnot a constant, but rather changes with reaction conditions,especially temperature, as r

!

r and r’

r change with compositionand temperature.

Since the RR graphs are drawn so that they are consistent withKFL and KPL and Ohm’s law kinetics representation, there is aquantitative correspondence between RR graphs and theirelectrical circuit analogs. As a result, the overall resistance ofthe network might be calculated in terms of step resistances, withthe result that the overall rate may be written as

rOR ¼AOR

RORð8Þ

where standard electrical circuit techniques are used to evaluateROR in terms of Rr, e.g., via D�to�Y conversions, and other tools ofelectrical circuit analysis.

For instance, for the above equivalent circuit (Fig. 2) involvingseries and parallel resistors

rOR ¼AOR

ROR¼

AOR

R1þ1

1

R4þ

1

R2þR3

ð9Þ

While this rate expression is precise, it is not predictive, as thestep rates r

!

r and r’

r, and hence the step resistances Rr, are notknown a priori. A predictive approach is developed below.

2.4. New form of the electrical analogy

Next, we discuss the alternate formulation of the electricalanalogy that leads to an explicit expression for the OR rate. First,however, we define (Dumesic, 1999) reversibility of an elementarystep zr � r

’

r= r!

r, i.e., it is the ratio of the rate in the reverse to theforward direction for a step, and is related to step affinity via DeDonder relation, (De Donder and Rysselberghe, 1936; vanRysselberghe, 1958) i.e., zr ¼ expð�ArÞ.

Further, for the OR, it is

zOR ¼ r’

OR= r!

OR ¼ expð�AORÞ ¼1

KOR

Yn

i ¼ 1

ani

i ð10Þ

Furthermore, since it is a thermodynamic property, from KPL wehave

zOR ¼r’

OR

r!

OR

¼Yqþ1

r ¼ 1

r’

r

r!

r

0@

1Asr

¼Yqþ1

r ¼ 1

ðzrÞsr ð11Þ

In fact, since the intermediate species get cancelled in a FR

zOR ¼Yqþ1

r ¼ 1

o’

r

o!

r

!sr

ð12Þ

i.e., the OR reversibility, zOR, is a known quantity for a given set ofreaction conditions. Here or is the reaction step weight, whichincludes the observable (known) quantities, i.e., the rate constantsand activities of the terminal species (e.g. o

!

r ¼ k!

rabi ).Further, we formally defineQuasi-equilibrium (QE): A step is quasi-equilibrated if its

reversibility, zr-1, or alternately, if its affinity Ar-0. At trueequilibrium, of course, zr ¼ 1, and Ar ¼ 0.

Rate-limiting step (RLS) in a sequence is one, whose resistanceRr contributes significantly to ROR. There may, of course, be morethan one RLS in a sequence.

Interestingly, however, it may be noted that for a constant fluxthrough a sequence of steps, this implies that reversibility zrcontributes significantly to zOR, in agreement with (Dumesic,1999). This may be seen by noting that Rr=ROR ¼Ar=AOR in asequence with rr ¼ rOR for unit stoichiometric numbers ðsr ¼ þ1Þ.

At any rate, for the equivalent circuit shown in Fig. 2,rearranging the OR rate, Eq. (9), we have

1

rOR¼

ROR

AOR¼

R1

AORþ

1AOR

R4þ

1R2

AORþ

R3

AOR

ð13Þ

so that, we may write

1

rOR�

1

r�1þ

1

r�4þ1

1

r�2þ

1

r�3

ð14Þ

where the rate-determining step (RDS) is given by

r�rffiAOR

Rrð15Þ

where the bullet in the subscript denotes the step as the rate-determining step (RDS). Thus,

Rate-determining step (RDS): r�rðI�rÞ is the rate (current) of the

branch (resistor) srðRrÞ if all other resistors in the circuit wereshort-circuited, i.e., if the entire motive force AOR (EOR) occurredacross a chosen step (resistor) srðRrÞ, which, of course, would bethe maximum step rate (current) in the step (resistor) for the givenmotive force.

By the same token, since the driving force (Ar) drop across theremaining steps is virtually zero, they may be considered to be atquasi-equilibrium (QE). In other words, the RDS and quasi-equilibrium (QE) hypothesis (also called pseudo-equilibriumhypothesis, PEH) go hand-in-hand, as inherent in the LHHWapproach. It is to be noted, however, that while in an electriccircuit the resistance is constant and, thus, independent of thedriving force, this is not entirely correct in a reaction network. Eq.(14) is predicated on the assumption that Rr remains constant asAr changes to AOR, which is not true in general in kinetics, unlikein electrical circuits, hence the approximate sign.

In fact, r�r may be obtained via the LHHW methodology, with sras the RDS, all the other steps being at QE. Since sr is the RDS and

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all other steps are at QE, i.e., zr ¼ zOR, for sr ¼ þ1, then

r�r � r!�

rð1�zORÞ ð16Þ

From Eqs. (14) and (16), thus, we have

rOR ¼EOR

R�1þ1

1

R�4þ

1

R�2þR�3

¼EOR

R�OR

ð17Þ

where

EOR � 1�zOR ¼ f1�expð�AORÞg ð18Þ

and

R�r �1

r!�

r

ð19Þ

Here, EOR is the thermodynamic motive force term as defined byChristiansen (1953) rather than the OR affinity, AOR, and R�r is the

resistance of the step sr when it is considered as the RDS,i.e., when the entire driving force for the OR occurs across thestep, the other steps being quasi-equilibrated. As a result, thisallows R�r to be determined explicitly a priori, following the LHHW

algorithm. Hence, Eq. (17) provides an explicit predictive rateexpression for the QSS rate of the OR.

We next show how to obtain the R�r employing the LHHWapproach along with the notion of IRRs. We will, thereafter, showthat this rate expression in the form of alternate Ohm’s lawprovides exact results for the case of linear kinetics mechanisms(similar to the considered reaction mechanism, Eq. (1)). Wefurther contend that it provides an approximate, albeit, accurateresults in other cases (Vilekar et al., 2009). Furthermore, Eq. (17),is in a form that is readily amenable to comprehension as well aspruning via comparison of resistances.

2.5. LHHW methodology for reaction resistance, R�r

The resistances R�r can be obtained a priori, by treating each ofthe steps as RDS, in turn, and using the QE approximation for theremaining (Vilekar et al., 2009). The basic idea is that for a givenRDS, the q linearly independent unknown intermediate sitefractions are determined by identifying the appropriate intermedi-ate reactions, or IRs, or pathways for the formation of intermediates.As mentioned above, an IR (Eq. (4)) results from an appropriatelinear combination of steps sj that eliminates all of the intermediatespecies except that of interest, Ik, formed from terminal speciesalong with some reference intermediate, e.g., the vacant site S incase of catalytic reactions (Fishtik and Datta, 2001a, b), i.e.,XIRk

skjsj ¼ IRk ð20Þ

In analogy with KPL, the affinity of this IRkXIRk

skjAr ¼AIRkð21Þ

Using the definition of step reversibility

zr ¼ r’

r= r!

r ¼ expð�ArÞ ð22Þ

we have

zIRk¼YIRk

ðzjÞskj ¼

YIRk

r’

j

r!

j

0@

1Askj

ð23Þ

Using in this the step kinetics in terms of step weights, and notingthat, all intermediates but Ik and the reference intermediate, i.e.,vacant sites S (in case of a catalytic reaction) are eliminated by the

stoichiometric numbers chosen to produce the IR (Eq. (20))

zIRk¼

yk

y0

� �gkYIRk

o’

j

o!

j

0@

1Askj

ð24Þ

Further, if we select all the steps sj in Eq. (20), such that it doesnot include the step sr under consideration as the RDS, or in otherwords all the selected steps are among the QE steps, zIRk

¼ 1, wehave

y�k;ry�0;r¼

YIRk

o!

j

o’

j

!skj8<:

9=;

1=gk

ð25Þ

Note that we use the notation y�k;r to represent site fraction of Ik

when sr is the RDS. Finally, the site fractions thus calculated areused in the site balance, 1¼

Pqk ¼ 0 y

�

k;r, written in the form

1

y�0;r¼Xq

k ¼ 0

y�k;ry�0;r

ð26Þ

Thus, the reference site fraction y�0;r can be determined and, fromit, all the remaining site fractions y�k;r. As a result, the forward rateof the RDS, and hence the step resistance, R�r as per Eq. (19), can beevaluated a priori.

We next show that the rate expression derived via Eq. (17), i.e.rOR ¼ EOR=R�OR is the same as that obtained via QSS approximationfor the simple 4-step homogeneous (non-catalytic) reactionmechanism (Eq. (1)).

2.6. Comparison of QSS analysis of the 4-step homogeneous reaction

and the electrical analogy approach

The rates of the elementary steps of the 4-step homogeneousdecomposition reaction mechanism depicted in Eq. (1), in termsof species activities, ai

r1 ¼ r!

1� r’

1 ¼ k!

1aA|fflffl{zfflffl}o!

1

� k’

1|{z}o’

1

aI1¼o!

1�o’

1aI1

r2 ¼ r!

2� r’

2 ¼ k!

2|{z}o!

2

aI1�k

’

2aC|fflffl{zfflffl}o’

2

aI2¼o!

2aI1�o

’

2aI2

r3 ¼ r!

3� r’

3 ¼ k!

3|{z}o!

3

aI2�k

’

3aB|fflffl{zfflffl}o’

3

¼o!

3aI2�o

’

3

r4 ¼ r!

4� r’

4 ¼ k!

4|{z}o!

4

aI1�k

’

4aBaC|fflfflfflffl{zfflfflfflffl}o’

4

¼o!

4aI1�o

’

4 ð27Þ

where the corresponding step weights or (i.e., products of rateconstants and activities of the terminal species) are also defined.

The QSS approximation for the intermediates in this mechanismis provided in Eq. (6). Using the step rate kinetics in this andrearranging

ðo’

1þo!

2þo!

4ÞaI1�ðo

’

2ÞaI2¼ ðo!

1þo’

4Þ

�ðo!

2ÞaI1þðo

’

2þo!

3ÞaI2¼o

’

3 ð28Þ

This may be written in a matrix form, i.e., Wm � x¼ b, where thevarious matrices are

Wm ¼ðo’

1þo!

2þo!

4Þ �o’

2

�o!

2 ðo’

2þo!

3Þ

24

35; x¼

aI1

aI2

!; and b¼

o!

1þo’

4

o’

3

0@

1Að29Þ

ARTICLE IN PRESS

S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–2933 2927

The solution via Cramer’s rule (or by substitution) is

aI1¼o’

2o’

4þo’

2o’

3þo!

1o’

2þo!

3o’

4þo!

1o!

3

o’

1o’

2þo’

1o!

3þo!

2o!

3þo’

2o!

4þo!

3o!

4

aI2¼o’

1o’

3þo!

2o’

4þo!

2o’

3þo!

1o!

2þo’

3o!

4

o’

1o’

2þo’

1o!

3þo!

2o!

3þo’

2o!

4þo!

3o!

4

9>>>>>=>>>>>;

ð30Þ

With the unknown intermediate concentrations known interms of or, all step rates can now be found. Further, OR rate canbe found from the relation between OR and step rates, which canbe discerned from the RR graph as well. Thus, from Fig. 1c

rOR ¼ r1

rOR ¼ r3þr4 ð31Þ

Either of these relations could be used. For instance, from thefirst of these, i.e., rOR ¼ r1 ¼o

!

1�o’

1aI1and rearranging

rOR ¼

o!

1o!

2o!

3 1�o’

1o’

2o’

3

o!

1o!

2o!

3

!þðo

’

2þo!

3Þo!

1o!

4 1�o’

1o’

4

o!

1o!

4

!( )

o’

1o’

2þo’

1o!

3þo!

2o!

3þo’

2o!

4þo!

3o!

4

ð32Þ

Using Eq. (34) below for the driving force (via KPL) for the twoparallel pathways, we can factor out EOR to provide an alternateform for the OR rate

rOR ¼o!

1o!

2o!

3þðo’

2þo!

3Þo!

1o!

4

o’

1o’

2þo’

1o!

3þo!

2o!

3þo’

2o!

4þo!

3o!

4

1�1

KOR

aBaC

aA

� �ð33Þ

where from KPL (Fig. 1c)

EOR ¼ 1�o’

1o’

2o’

3

o!

1o!

2o!

3

!¼ 1�

o’

1o’

4

o!

1o!

4

!¼ 1�

1

KOR

aBaC

aA

� �ð34Þ

We next describe the methodology for deriving the rateexpression using the RR Graph approach utilizing Eq. (17). Thus,using Eqs. (17) and (34)

rOR ¼EOR

R�1þ1

1

R�4þ

1

R�2þR�3

¼1

R�1þ1

1

R�4þ

1

R�2þR�3

1�1

KOR

aBaC

aA

� �ð35Þ

In order to derive R�r, let us first consider step s1 as the RDS, theremaining steps being at QE. Thus,

R�1 ¼1

r!�

1

¼1

o!

1

ð36Þ

For step s2, similarly

R�2 ¼1

r!�

2

¼1

o!

2a�1;2

ð37Þ

where a�1;2 denotes the activity of I1 when step s2 is considered asthe RDS and all other steps are at QE. The activity a�1;2, asexplained above, is obtained by identifying IRR for the formationof I1, from reaction steps other than the RDS, s2. An appropriateIRR for I1 that does not include s2 is

IRR1 : IR1 ¼ ðþ1Þs1 ð38Þ

Thus, with no catalyst site involved in this homogeneousmechanism, using Eq. (25)

a�1;2 ¼o!

1

o’

1

ð39Þ

Using Eq. (39) in Eq. (37) we have

R�2 ¼1

o!

2o!

1

o’

1

! ¼ 1

o!

2

o’

1

o!

1

ð40Þ

Similarly, we have for the two remaining steps

R�3 ¼1

r!�

3

¼1

o!

3a�2;3

¼1

o!

3o!

1o!

2

o’

1o’

2

! ¼ 1

o!

3

o’

1o’

2

o!

1o!

2

ð41Þ

R�4 ¼1

r!�

4

¼1

o!

4a�1;4

¼1

o!

4o!

1

o’

1

! ¼ 1

o!

4

o’

1

o!

1

ð42Þ

Using Eq. (36) and (40)–(42) in Eq. (35), thus,

rOR ¼1

1

o!

1

þ1

1

1

o!

4

o’

1

o!

1

þ1

1

o!

2

o’

1

o!

1

þ1

o!

3

o’

1o’

2

o!

1o!

2

1�1

KOR

aBaC

aA

� �ð43Þ

which is just a rearranged form of Eq. (33). However, as seen herethe RR graph approach is easy to follow. Furthermore, logicalpruning of the rate expression is possible via comparison of the R�ras illustrated below.

The practical utility of the approach is next highlighted with anon-linear kinetics example first, namely, the 4-step gas-phasehydrogen–bromine reaction, followed by a linear kineticsmechanism example, i.e., zeolite catalyzed N2O decomposition.

3. Illustration: non-linear kinetics mechanism, H2–Br2

example

We first defineNon-linear kinetics mechanism as one that includes some

mechanistic steps involving more than one intermediate specieson either or both sides, so that the kinetics of these steps are non-linear in aIk

, e.g., r!

r ¼o!

ra2Ik

, and r’

r ¼o’

raIjaIl

. For example, forthe bromine decomposition step: Br2#2Br�, the rate in thereverse direction r

’

r ¼o’

1a2Br�.

It is, of course, unlikely that more than two intermediates areinvolved in an elementary reaction, so that higher than quadraticterms in aIk

are unlikely.Let us next illustrate the feasibility of the application of the above

approach to a non-linear kinetics mechanism with sr ¼ 71. Thus,we consider the classical 4-step H2–Br2 reaction mechanism

ð44Þ

Clearly, the mechanism is non-linear in intermediate concentrations.The kinetic data for the reaction mechanism is provided by Colleyand Anderson (1952).

For the non-linear case, the expression

rOR �EOR

R�OR

ð45Þ

provides only approximate, but accurate results, as shown below. Ingeneral, of course, for the non-linear case, an explicit solution via the

ARTICLE IN PRESS

S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–29332928

QSS approach is not possible. Rather, only numerical solution isgenerally obtained. Therefore, even though approximate, an explicitsolution is very useful, for instance, in reactor design and analysis.

3.1. Rate expression based on electrical analogy

For the given reaction mechanism, Eq. (44), thus, the stepkinetics are

r1 ¼ r!

1� r’

1 ¼ k!

1cBr2|fflfflffl{zfflfflffl}o!

1

� k’

1|{z}o’

1

c2Br� ¼o

!

1�o’

1c2Br�

r2 ¼ r!

2� r’

2 ¼ k!

2cH2|fflfflffl{zfflfflffl}o!

2

cBr��k’

2cHBr|fflfflfflffl{zfflfflfflffl}o’

2

cH� ¼o!

2cBr��o’

2cH�

r3 ¼ r!

3� r’

3 ¼ k!

3cBr2|fflfflffl{zfflfflffl}o!

3

cH��k’

3cHBr|fflfflfflffl{zfflfflfflffl}o’

3

cBr� ¼o!

3cH��o’

3cBr�

r4 ¼ r!

4� r’

4 ¼ k!

4|{z}o!

4

cBr�cH��k’

4cHBr|fflfflfflffl{zfflfflfflffl}o’

4

¼o!

4cH�cBr��o’

4

9>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>;

ð46Þ

Further, as can be seen from Eq. (44), this reaction mechanismhas two full routes and one empty route, similar to the genericexample, i.e., Eq. (1),

FR1 : OR¼ ðþ1Þs1þðþ1Þs2þðþ1Þs4

FR2 : OR¼ ðþ1Þs2þðþ1Þs3

ER1 : 0¼ ðþ1Þs1þð�1Þs3þðþ1Þs4

9>=>; ð47Þ

The equivalent circuit for the mechanism is shown in Fig. 3,which is obtained following a similar procedure as explained forthe academic example, and is identical to it, although withdifferent step labels.

Thus, we have from Fig. 3, as before

R�OR ¼ R�2þ1

1

R�3þ

1

R�1þR�4

ð48Þ

Let us calculate the resistances from the step kinetics. For steps1 as RDS, and all others at QE, we have

R�1 ¼1

r!�

1

¼1

o!

1

ð49Þ

Fig. 3. Equivalent electrical circuit for the 4-step HBr example along with

representative values of R�1, R�2, R�3, R�4 and the corresponding flux at conditions

reported in the text.

For step s2 as RDS, and all others at QE, we have

R�2 ¼1

r!�

2

¼1

o!

2c�Br�;2

ð50Þ

where c�Br�;2 is the concentration of Br� when step s2 is the RDS. Anappropriate IR for formation of Br� is

IRR1 : IR1 ¼ ðþ1Þs1 : Br2#2Br� ð51Þ

which gives

c�Br�;2 ¼

ffiffiffiffiffiffiffio!

1

o’

1

vuut ð52Þ

Thus, using Eq. (52) in Eq. (50), we have

R�2 ¼1

r!�

2

¼1

o!

2c�Br�;2

¼1

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuut ð53Þ

Similarly, we can obtain R�3 and R�4

R�3 ¼1

r!�

3

¼1

o!

3c�H�;3

¼1

o!

3

o’

2

o!

2c�Br�;3

¼1

o!

3

o’

2

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuut ð54Þ

R�4 ¼1

r!�

4

¼1

o!

4c�H�;4c�Br�;4

¼1

o!

4

o’

2

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuutffiffiffiffiffiffiffio’

1

o!

1

vuut ¼1

o!

4

o’

2

o!

2

o’

1

o!

1

ð55Þ

As a result

rOR ¼EOR

R�2þ1

1

R�3þ

1

R�1þR�4

¼EOR

1

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuut þ1

1

1

o!

3

o’

2

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuutþ

1

1

o!

1

þ1

o!

4

o’

2

o!

2

o’

1

o!

1

ð56Þ

where

EOR ¼ 1�zOR ¼ 1�o’

2o’

3

o!

2o!

3

¼ 1�o’

1o’

2o’

4

o!

1o!

2o!

4

ð57Þ

3.2. QSS kinetics

Let us compare the accuracy of the above expression tonumerical QSS analysis. An explicit expression for this case viaQSS procedure, of course, is not possible. For numerical compar-ison, we utilize the kinetic data (Table 1) provided by Cooley andAnderson (1952). However, before we apply the QSSapproximation, let us check the thermodynamic consistence ofthe reported kinetic data afforded by the RR graph approach.

Table 1The kinetic data for H2–Br2 reaction.

sr Reaction stepE!

r E’

r L!

r L’

r m!

r m’

r

s1 Br2#2Br� 45.25a 0 1.05�1013a 5.7�1015 1 0

s2 Br � þH2#HBrþH� 17.7 1.1 8.05�1010 3.08�1010 1 1

s3 H � þBr2#HBrþBr� 1.1 41.7 2.59�1011 9.31�1010 1 1

s4 Br � þH �#HBr 0 85.85 9�1015 5.95�1012 0 1

Activation energies in kcal/mol; pre-exponential factors are in units of mol/cm3

and s (Cooley and Anderson 1952).

kr ¼LrTm expð�Er=RTÞ.

a E!

1 changed from 45.23 and L!

1 changed from 7.18�1012, to ensure

thermodynamic consistency via KPL.

ARTICLE IN PRESS

Fig. 4. A comparison of R�1þR�4, R�3, R�4 as a function of temperature for the

conditions reported in the text.

S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–2933 2929

Applying KPL to the empty route, ER1, we have A1þA4�A3 ¼ 0.Using de Donder’s relationship we have,

k!

1

k’

1

0@

1A k

!

4

k’

4

0@

1A k

’

3

k!

3

0@

1A¼ 1

However, the reported data (Cooley and Anderson 1952) yieldsthis product as 0.68, rather than 1. Furthermore, for the emptyroute ER1, DH1þDH4�DH3 ¼ 0. The reported data (Cooley andAnderson, 1952) yields this summation as �20 cal=mol, ratherthan 0, pointing to the slight thermodynamic inconsistency of thedata available in literature. Thus, the original value of E

!

1 wasreplaced from 45230 to 45250 cal/mol and L

!

1 from 7:18� 1012 to1:05� 1013 cm3 mol�1 s�1, to ensure thermodynamic consistencyvia KPL. Such inconsistencies in the rate data become glaring viathe RR graph approach.

The conventional QSS result is obtained by algebraic solutionof the QSS equations for the two intermediates, into which themass action kinetics is substituted. Alternately, we could use thesimpler KFL equations, i.e., Eq. (58), which are evident from Fig. 3at the two intermediate nodes (red)

r4�r1 ¼ 0

r1þr3�r2 ¼ 0

)ð58Þ

Using step kinetics in this

ðo!

4cH�cBr��o’

4Þ�ðo!

1�o’

1c2Br�Þ ¼ 0 ð59Þ

ðo!

1�o’

1c2Br�Þþðo

!

3cH��o’

3cBr�Þ�ðo!

2cBr��o’

2cH�Þ ¼ 0 ð60Þ

which are two simultaneous non-linear algebraic equations whichmay be solved for the two unknown intermediate concentrationscH� and cBr�, and then substituted back into Eq. (46), or KFLequations at the terminal nodes, in order to determine the ORrate. This is not trivial, however, since substituting for cH� fromEq. (60) into Eq. (59) results in a third order equation in cBr� withthree possible roots. This clearly demonstrates the strength andintuitive power of the described electrical analogy approach.Therefore, we will use the QSS analysis below simply tonumerically verify the result obtained above, using the kineticparameters from Cooley and Anderson (1952) and experimentaldata provided by Levy (1958).

3.3. QSS vs. electrical analogy rates

Calculations based on the above QSS analysis were made for atemperature, T=473 K, and for cH2

¼ 0:428 mol=cm3, cBr2¼

0:594 mol=cm3 and cHBr ¼ 0:25 mol=cm3. Thus, the overall rate,rOR ¼ r2 ¼ r4þr3 ¼ 2:5938� 10�6 s�1 was obtained from solvingthe QSS Eqs. (59) and (60), and substituting the resultingconcentrations of the intermediate species cH� and cBr� in Eq.(46) to calculate the corresponding step rates. These QSS steprates are shown on the RR graph in Fig. 3. It can, clearly, be seenfrom Fig. 3 that KFL for the QSS step rates is followed at all thenodes. Further, the right-hand-side of Eq. (56), along with Eq. (57),was used to also directly calculate a rate of rOR ¼ 2:5938�10�6 s�1 from the derived rate expression for the given set ofconditions. Clearly, Eq. (56) provides an accurate explicit rateexpression! Further, unlike the QSS numerical approach, the rateexpression, Eq. (56), is in a form that is amenable to pruning asexplained below.

For this, R�r were calculated from the explicit Eqs. (49) and(53)–(55) for the specified conditions, as they provide anindication of the rate limiting steps, if any. These calculatedvalues of R�1, R�2, R�3, R�4 are also shown in Fig. 3. A comparison ofactivation energies or rate constants alone is not enough to

identify the RLS, since the concentrations of intermediates arealso needed to determine the step rates. However, a comparisonof R�r is more rigorous in identifying the RLSs, as discussed before.Of course, these R�r may also be used in the first equality inEq. (56), along with Eq. (57), to calculate the overall rate from theelectrical analogy, which not surprisingly, also providesrOR ¼ 2:5938� 10�6 s�1.

For the mixture composition assumed above, a comparison ofR�1þR�4, R�3, R�4 as a function of temperature is provided in Fig. 4. Itcan be seen that R�1þR�4 � R�4, and, hence, R�1 can be easilyeliminated from Eq. (56). It can be further concluded fromFig. 4, that R�35R�4, i.e., 1=R�3b1=R�4. Thus, R�4 can also beeliminated from Eq. (56), resulting in a reduced expression forthe overall resistance,

R�OR � R�2þ11

R�3

¼ R�2þR�3 ð61Þ

One could have arrived at this conclusion also from a comparisonof the branch fluxes in Fig. 3. Furthermore, the reaction isessentially irreversible at these conditions, i.e.,

zOR-0; hence EOR-1 ð62Þ

When Eq. (61) and (62) are used in Eq. (56), a substantiallysimplified expression results, i.e.,

rOR ¼EOR

1

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuut þ1

o!

3

o’

2

o!

2

ffiffiffiffiffiffiffio’

1

o!

1

vuut¼o!

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio!

1=o’

1

q1þ

o’

2

o!

3

ð63Þ

Finally, using the reaction weights in Eq. (63), we obtain

rOR ¼

k!

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik!

1=k’

1

q� �cH2

c1=2Br2

1þk’

2

k!

3

cHBr

cBr2

ð64Þ

as determined from the early experiments by Bodenstein and Lind(1907) in the temperature range of 200–300 1C. Later experimentsby Levy (1958) show that this simplified rate law is also valid athigher temperatures. Thus, only reaction steps s2 and s3 arekinetically significant for the reaction conditions mentioned

ARTICLE IN PRESS

Table 2Elementary reaction steps and their rates in the N2O decomposition on Fe-ZSM-5 (Heyden et al. 2005).

sr Reaction step rr ¼ r!

r� r’

r o!

r o’

r L!

r L’

r E!

r E’

r

s1: N2Oþ I0#I1 r1 ¼o!

1y0�o’

1y1 k!

1pN2 O k’

11.06�107 1.67�1013 0.0 6.4

s2: I1#I2þN2 r2 ¼o!

2y1�o’

2y2 k!

2 k’

2pN2

2.09�1014 4.43�108 30.7 41.9

s3: I1#I3þN2 r3 ¼o!

3y1�o’

3y3 k!

3 k’

3pN2

6.98�1013 5.82�107 30.4 50.0

s4: I2þ I3 r4 ¼o!

4y2�o’

4y3 k!

4 k’

42.15�1013 8.46�1012 14.1 22.5

s5: N2Oþ I3#I4 r5 ¼o!

5y3�o’

5y4 k!

5pN2 O k’

54.39�108a 1.67�1013 0.0 2.7

s6: I4#I5þN2 r6 ¼o!

6y4�o’

6y5 k!

6 k’

6pN2

3.12�1012a 3.19�108 20.2b 31.5

s7: I5#I0þO2 r7 ¼o!

7y5�o’

7y0 k!

7 k’

7pO2

1.67�1013 1.51�105 8.0 8.1

I0=Z�[FeO]+; I1=Z–[FeO]+(ON2); I2=Z�[OFeO]+; I3=Z�[FeO2]+; I4=Z�[FeO2]+(ON2); I5=Z�[O2FeO]+.

Activation energies in kr ¼Lr expð�Er=RTÞ are in kcal/mol; the units of the pre-exponential factors are bar–1 s–1 for adsorption/desorption reactions and s�1 for surface

reactions.

a Modified L!

r .b Modified E

!

r (Vilekar et al., 2010).

Fig. 5. Equivalent electrical circuit for the 7-step N2O decomposition reaction mechanism on Fe-ZSM-5.

S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–29332930

above. Of course, mechanistically, the other steps of initiation andtermination are also significant.

Next, we illustrate the utility of our approach by using it for apractical system with linear kinetics mechanism.

4. Illustration: linear kinetics mechanism, N2O decompositionon Fe–ZSM–5

First, we defineLinear Kinetics Mechanism: as one that includes mechanistic

steps that involve only one intermediate species on either or bothsides, so that the kinetics of these steps are linear in aIk

, e.g.,r!

r ¼o!

raIk.

Here, a 7-step mechanism is adopted from Heyden et al.,(2005) who performed quantum chemical calculations for N2Odecomposition on dehydrated mononuclear iron sites in Fe-ZSM-5,using the TURBOMOLE V5.6 suite of programs following the DFT-B3LYP approach, as an example for a linear kinetics mechanism.The step kinetics, found to be consistent with KPL requirements,are summarized in Table 2 (Vilekar et al. 2010).

The system has two full routes and one empty route

FR1 : s1þs2þs4þs5þs6þs7 ¼ OR

FR2 : s1þs3þs5þs6þs7 ¼ OR

ER1 : s2�s3þs4 ¼ 0

9>=>; ð65Þ

The corresponding RR graph/electric circuit for this systemobtained from this independent RR set, by following a proceduresimilar to that described above for the generic example, is shown

in Fig. 5. From this, thus, the overall rate can be written as

rOR ¼EOR

R�OR

¼EOR

R�1þR�5þR�6þR�7þ1

1

R�2þR�4þ

1

R�3

ð66Þ

To obtain the step resistances in this, let us first consider steps1 as the RDS, the remaining steps being at QE. Thus,

R�1 ¼1

r!�

1

¼1

o!

1y�

0;1

ð67Þ

where as explained above, the first subscript 0 in y denotes I0, andthe second subscript 1 denotes s1 as the RDS. With s1 as the RDS,thus, and all subsequent steps at QE, the appropriate IRs for theformation of five linearly independent surface intermediates, I1, I2,I3, I4, I5, from the reference intermediate I0, comprising of stepsother than s1, are

IRI1: ð�1Þs3þð�1Þs5þð�1Þs6þð�1Þs7 : O2þ2N2�N2Oþ I0#I1

IRI2: ð�1Þs4þð�1Þs5þð�1Þs6þð�1Þs7 : O2þN2�N2Oþ I0#I2

IRI3: ð�1Þs5þð�1Þs6þð�1Þs7 : O2þN2�N2Oþ I0#I3

IRI4: ð�1Þs6þð�1Þs7 : O2þN2þ I0#I4

IRI5: ð�1Þs7 : O2þ I0#I5 ð68Þ

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Then, using Eq. (25), for the QE steps, the site fraction ratios are

y�1;1y�0;1¼

o!

3

o’

3

!�1o!

5

o’

5

!�1o!

6

o’

6

!�1o!

7

o’

7

!�1

y�2;1y�0;1¼

o!

4

o’

4

!�1o!

5

o’

5

!�1o!

6

o’

6

!�1o!

7

o’

7

!�1

y�3;1y�0;1¼

o!

5

o’

5

!�1o!

6

o’

6

!�1o!

7

o’

7

!�1

y�4;1y�0;1¼

o!

6

o’

6

!�1o!

7

o’

7

!�1

y�5;1y�0;1¼

o!

7

o’

7

!�1

9>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð69Þ

Finally using these in site balance, Eq. (26),

1

y�0;1¼ 1þ

y�1;1y�0;1

!þ

y�2;1y�0;1

!þ

y�3;1y�0;1

!þ

y�4;1y�0;1

!þ

y�5;1y�0;1

!ð70Þ

Thus, we have from Eq. (67)

R�1 ¼1

o!

1

1þo’

3o’

5o’

6o’

7

o!

3o!

5o!

6o!

7

þo’

4o’

5o’

6o’

7

o!

4o!

5o!

6o!

7

þo’

5o’

6o’

7

o!

5o!

6o!

7

þo’

6o’

7

o!

6o!

7

þo’

7

o!

7

!

ð71Þ

We next consider step s2 as the RDS, and the remaining stepsat QE, to derive an explicit expression for R�2, where

R�2 ¼1

r!�

2

¼1

o!

2y�

1;2

¼1

o!

2

y�1;2y�0;2

!y�0;2

ð72Þ

Following a similar procedure as above, there result

R�2 ¼1

o!

2o!

1

o’

1

! 1þo!

1

o’

1

þo’

4o’

5o’

6o’

7

o!

4o!

5o!

6o!

7

þo’

5o’

6o’

7

o!

5o!

6o!

7

þo’

6o’

7

o!

6o!

7

þo’

7

o!

7

!

ð73Þ

R�3 ¼1

o!

3o!

1

o’

1

! 1þo!

1

o’

1

þo’

4o’

5o’

6o’

7

o!

4o!

5o!

6o!

7

þo’

5o’

6o’

7

o!

5o!

6o!

7

þo’

6o’

7

o!

6o!

7

þo’

7

o!

7

!

ð74Þ

R�4 ¼1

o!

4o!

1o!

2

o’

1o’

2

! 1þo!

1

o’

1

þo!

1o!

2

o’

1o’

2

þo’

5o’

6o’

7

o!

5o!

6o!

7

þo’

6o’

7

o!

6o!

7

þo’

7

o!

7

!

ð75Þ

R�5 ¼1

o!

5o!

1o!

3

o’

1o’

3

! 1þo!

1

o’

1

þo!

1o!

2

o’

1o’

2

þo!

1o!

3

o’

1o’

3

þo’

6o’

7

o!

6o!

7

þo’

7

o!

7

!ð76Þ

R�6 ¼1

o!

6o!

1o!

3o!

5

o’

1o’

3o’

5

! 1þo!

1

o’

1

þo!

1o!

2

o’

1o’

2

þo!

1o!

3

o’

1o’

3

þo!

1o!

3o!

5

o’

1o’

3o’

5

þo’

7

o!

7

!

ð77Þ

and

R�7 ¼1

o!

7o!

1o!

3o!

5o!

6

o’

1o’

3o’

5o’

6

!

� 1þo!

1

o’

1

þo!

1o!

2

o’

1o’

2

þo!

1o!

3

o’

1o’

3

þo!

1o!

3o!

5

o’

1o’

3o’

5

þo!

1o!

3o!

5o!

6

o’

1o’

3o’

5o’

6

!ð78Þ

Finally, these equations, i.e., Eq. (71) and Eqs. (73)–(78), areused in Eq. (66) to provide the OR rate explicitly in terms of or.The resulting expression is in fact, identical to that derived via theconventional QSS approach for this linear kinetics mechanismusing linear algebra, as illustrated above for the generic example.However, the conventional QSS approach provides a rate expres-sion that is in a considerably more complex looking and hard todiscern form.

For typical experimental conditions (Heyden, 2005), namely,with a feed of 15,000 ppm N2O in He passed over Fe-ZSM-5 atT=800 K, pN2O ¼ 0:0015 bar; pN2

¼ 0:0135 bar; pO2¼ pN2

=2 bar andusing Eq. (71) and Eq. (73)–(78) we have

R�1 ¼ 6:2893� 10�5 s

R�2 ¼ 2:1871� 101 s

R�3 ¼ 5:4224� 101 s

R�4 ¼ 3:3101� 10�10 s

R�5 ¼ 1:5216� 10�6 s

R�6 ¼ 4:9184� 10�1 s

R�7 ¼ 9:2286� 10�12 s

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð79Þ

Using Eq. (79) in Eq. (66), thus, the overall rate of the reaction is

rOR ¼ 6:2202� 10�2 s�1 ð80Þ

which, is identical to that obtained from solving the QSSequations numerically for the 7-step mechanism.

4.1. Reduced rate expression

The rate expression, Eq. (66), is in a revealing form andallows judicious network pruning. Let us first compare theresistance of the steps in series, namely, steps s1, s5, s6, and s7.As is the case with electrical circuits, the step with maximumstep resistance in series may be taken as the slowest step orthe RLS for the sequence. It can be clearly seen from Eq. (79),that R�6 is four orders of magnitude higher than R�1þR�5þR�7.Thus,

R�6bR�1þR�5þR�7 ð81Þ

Next, note that there are two parallel pathways starting atintermediate node n1 and ending at n3. For parallel pathways, thepath with the least resistance would be the dominant pathway.The total resistance of the first pathway, namely s2+s4 is R�2þR�4while that for the other pathway, step s3 is simply R�3. It is evidentfrom Eq. (79), that both the pathways have comparable resis-tances and it is imperative to retain both. Hence, the simplifiedoverall resistance is

R�OR � R�6þ1

1

R�2þR�4þ

1

R�3

ð82Þ

Further, in the series combination, s2+s4, R�4 is several orders ofmagnitude lower than R�2 and, hence, can be eliminated withoutmaterially affecting the overall flux (rate). Thus, we may furthersimplify the overall resistance

R�OR � R�6þ1

1

R�2þ

1

R�3

ð83Þ

In other words, we simply have three steps s2, s3, and s6 that arethe RLSs with resistances of similar order, so that the reduced rate

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S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–29332932

expression is

rOR ¼1

R�6þ1

1

R�2þ

1

R�3

1�o’

1o’

3o’

5o’

6o’

7

o!

1o!

3o!

5o!

6o!

7

!ð84Þ

where R�2, R�3 and R�6, given by Eqs. (73), (74) and (77) are of acomparable order and need to be retained.

R�2, R�3 and R�6 can, in fact, be further simplified based on theconcept of most abundant reactive intermediate (MARI) (Boudartand Djega-Mariadassou, 1984), i.e., by comparing the values ofy�k;r=y

�

0;r in R�r. For both, steps s2 and s3, y�k;r=y�

0;r � 0 for allintermediate species under the conditions mentioned above,which implies 1=y�0;r � 1 and, hence, the simplified expressionsfor R�2 and R�3 are

R�2 ¼o’

1

o!

2o!

1

R�3 ¼o’

1

o!

3o!

1

9>>>>>=>>>>>;

ð85Þ

Similarly, by comparing the values of y�k;r=y�

0;r in R�6 we have,

R�6 ¼1

o!

6o!

1o!

3o!

5

o’

1o’

3o’

5

! 1þo!

1o!

3

o’

1o’

3

!ð86Þ

Using Eqs. (85) and (86) in Eq. (84), the simplified rate expression,thus, is

rOR ¼1

1

o!

6o!

1o!

3o!

5

o’

1o’

3o’

5

! 1þo!

1o!

3

o’

1o’

3

!þ

11

o’

1

o!

2o!

1

þ1

o’

1

o!

3o!

1

� 1�o’

1o’

3o’

5o’

6o’

7

o!

1o!

3o!

5o!

6o!

7

!ð87Þ

Using the expressions for or (Table 2), thus, the simplified rateexpression for the aforementioned experimental conditions, inthe conventional form, is

rOR ¼1

k’

1 k’

3 k’

5pN2

k!

1 k!

3 k!

5 k!

6p2N2O

1þk!

1 k!

3pN2O

k’

1 k’

3pN2

0@

1Aþ k

’

1

k!

1ðk!

2þ k!

3ÞpN2O

1�1

KOR

p2N2

pO2

p2N2O

!

ð88Þ

Finally, since we have an essentially irreversible reaction, i.e.,

EOR � ð1�zORÞ ¼ 1�1

KOR

p2N2

pO2

p2N2O

!� 1 ð89Þ

as zOR-0, the above expression can be further simplified to

rOR ¼K1ðk!

2þ k!

3ÞpN2O

1þðk!

2þ k!

3ÞpN2

K3K5 k!

6pN2O

1þK1K3pN2O

pN2

� � ð90Þ

where Kr is the equilibrium constant for step sr.In conclusion, as a result of the systematic analysis of this

example, rate-limiting steps have been identified in an intuitivemanner, following the evaluation of R�r. Thus, steps s2, s3, and s6

i.e., the decomposition of adsorbed N2O on Z�[FeO]+ andZ�[FeO2]+ are concluded to be the rate limiting steps. We finally

wind up with a highly simplified, but accurate, rate expression forN2O decomposition on Fe-ZSM-5. Thus, the simplified rateexpression, i.e., Eq. (90) provides rOR ¼ 6:2206� 10�2 s�1 for thereaction conditions mentioned above, which is close to thatobtained from Eq. (66), or the QSS analysis, i.e., rOR ¼ 6:2202�10�2 s�1.

5. Conclusions

A simple and intuitively appealing approach is described herefor the treatment of quasi-steady state (QSS) kinetics of overallreaction (OR) mechanisms involving parallel pathways. It is basedon first determining the reaction route (RR) graph for a givenmolecular mechanism, in which reaction steps are represented bybranches meeting at the nodes such that all pathways can betraced as walks between terminal nodes. The RR graphs areconsistent with Kirchhoff’s laws of flow networks, e.g., electriccircuits, and can, thus, can be directly adapted into an equivalentelectrical circuit, in which the branches are replaced by resistorsrepresenting individual mechanistic steps. The OR rate can, hence,be expressed in Ohm’s law form, i.e., OR rate=OR motive force/ORresistance, where the overall resistance in terms of individual stepresistances is obtained following the common tools of electricalcircuit analysis.

The individual step resistance, in turn, is replaced by thevirtual maximum step rate, i.e., the rate of the step, if the entireOR affinity were available to the reaction step, with all other stepsbeing quasi-equilibrated (QE), i.e., their affinities being zero. Thiscan be ascertained following the LHHW approach involving acombination of the RDS/QE assumptions. In this manner, the QSSrate law for complex mechanisms can be determined, which maybe difficult or impossible to obtain by simply solving the QSSequations algebraically. This Ohm’s law form of rate expressionfor reaction networks with parallel pathways is proposed herefollowing our recent work, where it was shown that theconventional QSS analysis for a reaction sequence, or a singlepathway, could be cast into the Ohm’s law form. The results areexact for a reaction network with steps linear in intermediates,while they are approximate, albeit accurate, for non-linear stepkinetics.

The algorithm is illustrated here for a case of the gas-phaseH2–Br2 system (non-linear kinetics mechanism) involving4 steps and a 7-step linear zeolite catalyzed N2O decompositionmechanism, both with 2 parallel pathways. These simple systemswere chosen so as to render the process comprehensible, althoughthe approach is widely applicable. It is also shown howthe thermodynamic consistence of the kinetic data can beverified easily using the RR graph. Further, reaction flux andresistance can be used for insightful pruning and mechanisticreduction.

Notation

ai

activity of terminal species i

Ar

affinity of elementary reaction r Ar dimensionless reaction affinity of elementary reaction r AIR dimensionless affinity of intermediate reaction AOR dimensionless affinity of the overall reaction ci concentration of species i

E!

r

activation energy of the forward reaction

E’

r

activation energy of the reverse reaction

EOR

electro-motive force

DGr

Gibbs free energy change of the elementary reaction r Ik intermediate species k

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S.A. Vilekar et al. / Chemical Engineering Science 65 (2010) 2921–2933 2933

k!

r

forward rate constant of the elementary reaction r

k’

r

backward rate constant of the elementary reaction r

Kr

equilibrium constant of the elementary reaction r KOR equilibrium constant of the overall reaction n number of terminal species p number of elementary reactions q number of linearly independent intermediate species rr net rate of the elementary reaction r

r!

r

forward rate of elementary reaction r

r’

r

reverse rate of elementary reaction r

r�r

maximum rate of the elementary reaction r

r!�

r

maximum forward rate of the elementary reaction r

rOR

net rate of the overall reaction R gas constant Rr resistance of elementary reaction r, (1/rate)

R�r

resistance of elementary reaction sr, when sr is the

RDS, (1/rate)

ROR total resistance of the overall reaction network,

(1/rate)

sr elementary reaction r S unoccupied surface site T temperature Ti terminal species i

zr

reversibility of reaction sr zIR reversibility of an intermediate reaction zOR reversibility of overall reaction

Greek letters

gk

stoichiometric coefficient of intermediate species Ik inan intermediate reaction

gki

stoichiometric coefficient of terminal species i in anintermediate reaction

yk

surface coverage of intermediate species k

y�k;r

surface coverage of intermediate species k when sr is

the RDS

L!

r

forward pre-exponential factor of elementary reactionr

L’

r

reverse pre-exponential factor of elementary reactionr

m

number of linearly independent reaction routes ni stoichiometric coefficient of terminal species i in an

overall reaction

r elementary reaction sr stoichiometric number for the elementary reaction r or step weight for reaction sr

o!

r

forward step weight for reaction sr

o’

r

reverse step weight for reaction sr

Abbreviations

ER

empty route FR full route IN intermediate node IR intermediate reaction IRR intermediate reaction route KFL Kirchhoff’s flux law KPL Kirchhoff’s potential law LHHW Langmuir–Hinshelwood–Hougen–Watson MARI most abundant reactive intermediate OR overall reaction PEH pseudo-equilibrium hypothesis QE quasi-equilibrium QSS quasi-steady state

RDS

rate-determining step RLS rate limiting step RR reaction route TN terminal node

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