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Analysis of reaction schemes using maximum rates ofconstituent stepsAli Hussain Motagamwalaa and James A. Dumesica,1
aDepartment of Chemical and Biological Engineering, University of Wisconsin–Madison, Madison, WI 53706
Contributed by James A. Dumesic, April 12, 2016 (sent for review February 26, 2016; reviewed by Charles T. Campbell and Jeffrey Greeley)
We show that the steady-state kinetics of a chemical reaction can beanalyzed analytically in terms of proposed reaction schemes com-posed of series of steps with stoichiometric numbers equal to unityby calculating the maximum rates of the constituent steps, rmax,i,assuming that all of the remaining steps are quasi-equilibrated. An-alytical expressions can be derived in terms of rmax,i to calculate de-grees of rate control for each step to determine the extent to whicheach step controls the rate of the overall stoichiometric reaction. Thevalues of rmax,i can be used to predict the rate of the overall stoichio-metric reaction, making it possible to estimate the observed reactionkinetics. This approach can be used for catalytic reactions to identifytransition states and adsorbed species that are important in control-ling catalyst performance, such that detailed calculations using elec-tronic structure calculations (e.g., density functional theory) can becarried out for these species, whereas more approximate methods(e.g., scaling relations) are used for the remaining species. This ap-proach to assess the feasibility of proposed reaction schemes is exactfor reaction schemes where the stoichiometric coefficients of theconstituent steps are equal to unity and the most abundant adsorbedspecies are in quasi-equilibrium with the gas phase and can be usedin an approximatemanner to probe the performance of more generalreaction schemes, followed by more detailed analyses using fullmicrokinetic models to determine the surface coverages by adsorbedspecies and the degrees of rate control of the elementary steps.
chemical kinetics | catalysis | microkinetics
Chemical reactions take place through sequences of elemen-tary steps, and the dynamics of the overall stoichiometric reaction
are typically controlled by key steps in this sequence of steps. Forexample, in the case of a catalytic reaction, the reaction kinetics arecontrolled by the energetics of the transition states for the rate-con-trolling steps and by the energetics of species that are abundant onthe active sites (i.e., the Gibbs free energies of these transition statesand adsorbed species relative to the reactants and products of theoverall stoichiometric reaction). However, whereas the observed re-action kinetics are controlled by a limited number of transition statesand adsorbed species, it is typically required to carry out detailedmicrokinetic analyses to identify the nature of these key transitionstates and adsorbed species. Thus, a general strategy to elucidate howthe reaction kinetics are controlled by a proposed reaction mecha-nism is first to carry out density functional theory (DFT) calculationsto determine the thermodynamic properties of all adsorbed speciesand transition states, and then to build a microkinetic model to de-termine the surface coverages by all adsorbed species and the forwardand reverse rates of all elementary steps for a range of reactionconditions (1–5). Sensitivity analyses are then carried out using thismicrokinetic model to identify those transition states and adsorbedspecies that control the predicted reaction kinetics for reaction con-ditions of interest. Although this approach is effective to predict theperformance of proposed reaction schemes, it requires informationabout all of the reaction steps, whereas only a fraction of these stepsactually control catalyst performance. Moreover, this approach re-quires the numerical solution of coupled differential and/or algebraicequations, involving complexities associated with the solution of stiffdifferential equations and the formulation of initial guesses for so-lution of algebraic equations. Thus, it is desirable to develop an
analytical methodology to identify the transition states and adsorbedspecies that are likely to be important, such that detailed DFTcalculations can be carried out for these species, whereas moreapproximate methods (e.g., scaling relations) can be carried out forthe remaining species. The analysis described here provides a sim-ple approach to predict the reaction rate and the degrees of ratecontrol for all of the elementary steps of a reaction mechanism byavoiding the need to solve systems of coupled differential equationsand/or algebraic equations that constitute a full microkinetic model.This approach would be particularly useful to assess the feasibility ofproposed reaction schemes to achieve desirable reaction kinetics forchemical processes of interest and would provide both conceptualas well as practical advantages in microkinetic modeling.We present a methodology that can be used to identify the key
intermediates and transition states that control the steady-stateperformance of a catalytic reaction consisting of a series of consec-utive reaction steps passing through reactive intermediates. Reac-tions that may follow multiple pathways from reactants to productscan be analyzed along each reaction pathway separately, allowing thedominant reaction pathway to be determined. Moreover, reactionsleading to different products can be studied along each pathwayseparately, allowing the selectivity of the catalyst to be addressed.
Degree of Rate Control and Reversibility of Reaction StepsA useful tool in the analysis of reaction schemes is the degree ofrate control (XRC) (6–9), defined as
XRC,i =�∂r∂ki
�kir
����Keq,i ,kj≠ i
=�∂ln r∂ln ki
�����Keq,i ,kj≠ i
, [1]
where XRC,i is the degree of rate control for step i, r is the rate ofthe overall stoichiometric reaction, ki is the forward rate constantof step i, and the partial derivative is taken with the equilibrium
Significance
The design of active and selective catalysts is essential for a widerange of industrial applications and societal issues. A fundamentalapproach is to identify the key elementary steps and to elucidatethe predicted reaction kinetics for potential reaction mechanisms.We present a methodology to analyze analytically the perfor-mance of catalytic reaction schemes by calculation of the maxi-mum rates of the constituent steps. This proposed methodologycan be used to identify the important transition states andadsorbed species, such that more detailed calculations can becarried out for these species, whereas more approximatemethodscan be used for the remaining species, thereby substantially re-ducing the computational time required to elucidate how catalystperformance is controlled by the fundamental surface chemistry.
Author contributions: A.H.M. and J.A.D. designed research, performed research, analyzeddata, and wrote the paper.
Reviewers: C.T.C., University of Washington; and J.G., Purdue University.
The authors declare no conflict of interest.1To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1605742113/-/DCSupplemental.
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constant for step i (Keq,i) and all other rate constants (kj) heldconstant. The sum of the values of XRC,i is equal to unity for areaction scheme leading to a single stoichiometric reaction (10).Thus, the reaction scheme has a rate determining step if XRC,i = 1for step i, whereas more generally, nonzero values of XRC,i reflectthe relative contributions of these steps in controlling the rate ofthe overall stoichiometric reaction. Campbell has shown more re-cently that the degree of rate control for step i can also be repre-sented as the change in the rate of the overall stoichiometricreaction with respect to a change in the standard-state Gibbs freeenergy of the transition state for this step Go†
i (11):
XRC,i =
∂r
−∂Go†i
!RTr
�����Go
j ≠ i
=
0B@ ∂ln r
∂�−Go†
iRT
�1CA�������Go
j ≠ i
, [2]
where R is the gas constant, T is the absolute temperature, and thepartial derivative is taken with the Gibbs free energies of all otherspecies ðGo
j Þ held constant. If the values of XRC,i are known, then itis possible to estimate how changes in the values of Go†
i for thesesteps would lead to changes in the rate of the overall stoichiometricreaction (12). Specifically, the above expression is integrated start-ing from a reference catalyst, having a rate r0, transition states withGibbs free energies equal to Go†
i,0, and known values of XRC,i, to anew catalyst with rate r and Gibbs free energies equal to Go†
i :
ln�rr0
�=Xi
XRC,i
Go†
i,0 −Go†i
RT
!. [3]
This expansion to estimate changes in the rate caused by changesin the Gibbs free energies of the controlling transition statesassumes for simplicity that the degrees of rate control for thesteps remain constant.
We now consider a four-step reaction scheme for the overallstoichiometric conversion of species A to B passing through re-active intermediates I1, I2, and I3:
1. A � I12. I1 � I23. I2 � I34. I3 � Boverall : A � B.
We note the distinction between the case considered here of areaction mechanism for an overall stoichiometric reaction passingthrough a series of reactive intermediates, versus a collection ofreaction steps that produce intermediates at significant concentra-tions compared to the concentrations of the reactants (species A) orthe final product (species B). For the case considered here, theconcentrations of all reactive intermediates remain small comparedwith the concentrations of the reactants and products, such thatthe rate of consumption of the reactants is equal to the rate ofproduction of the products throughout the course of the reaction,and the time derivatives of the intermediate concentrations areapproximately zero (i.e., the steady-state approximation). Thus,
Fig. 1. Schematic showing the strategy for analyzing a reaction scheme.
Table 1. Redox mechanism for water–gas shift (case I)
Step Elementary reaction Stoichiometric coefficients, σi
1 CO + * � CO* 12 H2O + * � H2O* 13 H2O* + * � H* + OH* 14 OH* + * � H* + O* 15 CO* + O* � CO2 * + * 16 CO2* � CO2 + * 17 2H* � H2 + 2* 1
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we consider here the case where experimental results indicate thatA is converted stoichiometrically to B, and the above reactionscheme can be treated as a reaction mechanism. Accordingly, thenet rate of each reversible step is equal to the net rate of theoverall reaction, such that the stoichiometric number of each stepis equal to 1, where the stoichiometric number of a step is definedas the number of times that the step takes place for one turnoverof the overall stoichiometric reaction (more generally, the net rateof step i is equal to σi times the net rate of the overall reaction,where σi is the stoichiometric number of step i). Hereafter, wediscuss the reactions where the steady-state approximation is ap-plicable and as such we address the overall stoichiometric reactionas the overall reaction. In the following analysis we consider thecatalytic reactions in the limit of low coverage of the catalyst sites.In the later section we relax this constraint and account for thepopulation of catalyst sites by adsorbed species.We define the reversibility of step i (Zi) as the rate of the
reverse reaction divided by the rate of the forward reaction:
Z1 =k−1I1k1A
=I1
Keq,1A[4]
Z2 =k−2I2k2I1
=I2
Keq,2I1[5]
Z3 =k−3I3k3I2
=I3
Keq,3I2[6]
Z4 =k−4Bk4I3
=B
Keq,4I3[7]
Z1Z2Z3Z4 =I1
Keq,1AI2
Keq,2I1
I3Keq,3I2
BKeq,4I3
=B
KeqA= β, [8]
where ki and k−i are the forward and reverse rate constants forstep i, Keq,i is the equilibrium constant for step i, Keq is theequilibrium constant for the overall reaction, β is the reversibilityof the overall reaction, A and B are the activities of the reactant
and product, and Ii are the activities of the intermediates. Wenote that Zi can also be expressed as
Zi = exp�ΔGi
RT
�, [9]
where ΔGi is equal to the change in Gibbs free energy for stepi (e.g., ΔGi = 0 as step i becomes equilibrated). It should be notedthat ΔGi is the change in the Gibbs free energy at reaction con-dition and not the change in the standard state Gibbs free energy.Thus, the value of Zi is equal to 0 for an irreversible step andapproaches 1 for a quasi-equilibrated step. We can now writeexpressions for the net rate of each step in terms of the forwardrate and the reversibility of the step, noting that the net rates ofall of the elementary steps are equal to the net rate of the overallreaction, r :
r= r1 − r−1 = r1ð1−Z1Þ= k1Að1−Z1Þ [10]
r= r2 − r−2 = r2ð1−Z2Þ= k2I1ð1−Z2Þ [11]
r= r3 − r−3 = r3ð1−Z3Þ= k3I2ð1−Z3Þ [12]
r= r4 − r−4 = r4ð1−Z4Þ= k4I3ð1−Z4Þ. [13]
The activities of the intermediates, Ii, can be expressed in termsof the reversibilities to give
r= k1Að1−Z1Þ [14]
r=Keq,1k2 A Z1ð1−Z2Þ [15]
r=Keq,1Keq,2k3 A Z1Z2ð1−Z3Þ [16]
r=Keq,1Keq,2Keq,3k4 A Z1Z2Z3
�1−
β
Z1Z2Z3
�, [17]
where the value of Z4 has been replaced by the reversibility ofthe overall reaction (β) and the reversibilities of steps 1, 2, and 3.We note that the net rates of these steps are controlled by fourlumped parameters, Ci, defined as
r=C1Að1−Z1Þ; C1 = k1 [18]
r=C2A Z1ð1−Z2Þ; C2 =Keq,1k2 [19]
r=C3A Z1Z2ð1−Z3Þ; C3 =Keq,1Keq,2k3 [20]
r=C4A Z1Z2Z3
�1−
β
Z1Z2Z3
�; C4 =Keq,1Keq,2Keq,3k4. [21]
Table 3. Defining relations for surface coverages of adsorbed intermediates and values of equilibrium constantsfor adsorbed species (case I)
Adsorbed species, I* Lumped reaction§ θI* =Kads,i
�∏k
j=1Pνijj
�θp Kads,i Kads,i
�∏k
j=1Pνijj
�‡
CO* CO+ * � CO* Kads,COp PCO θp 2.15 × 102 1.51 × 101
H2O* H2O+ * � H2O* Kads,H2Op PH2O θp 5.93 × 10−5 1.24 × 10−5
H* 0.5 H2 + * � H* Kads,Hp P0.5H2
θp 1.49 × 10−1 9.19 × 10−2
OH* H2O+ * � OH* +0.5 H2 Kads,OHpPH2OP−0.5H2
θp 2.50 × 10−5 8.51 × 10−6
O* H2O+ * � O* +H2 Kads,Op PH2OP−1H2 θp 1.98 × 10−9 1.09 × 10−9
CO2* CO2 + * � CO2* Kads,CO2p PCO2 θp 5.21 × 10−6 4.43 × 10−7
§Lumped reactions are written to form adsorbed intermediates from gas-phase reactants and products.‡PCO, PH2O, PH2 , and PCO2 are 0.07 atm, 0.21 atm, 0.38 atm, and 0.085 atm, respectively.
Table 2. Equilibrium and rate constants for elementaryreactions (case I)
Step Elementary reaction Keq,i kfor,i, s−1
1 CO + * � CO* 2.15 × 102 1.33 × 108
2 H2O + * � H2O* 5.93 × 10−5 2.01 × 1011
3 H2O* + * � H* +OH* 6.28 × 10−2 2.64 × 106
4 OH* + * � H* +O* 1.18 × 10−5 5.24 × 101
5 CO* + O* � CO2 * + * 1.03 × 103 2.05 × 105
6 CO2* � CO2 + * 1.92 × 105 1.48 × 1012
7 2H* � H2 + 2* 4.50 × 101 5.32 × 102
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According to transition state theory, we write the rate constantfor step i in terms of a frequency factor ðν† = kBT=hÞ times theequilibrium constant for the formation of the transition statefrom the reactant of that step, where kB and h are the Boltzmannand Planck constants, respectively:
ki = ν†K†eq,i. [22]
It should be noted that in Eq. 22 both the enthalpic and theentropic changes are included in the definition of Keq and assuch the frequency factor is defined as ν† = kBT=h. Using Eq.22, the lumped parameters, Ci, are given by
C1 = ν†K†eq,1 [23]
C2 = ν†Keq,1K†eq,2 [24]
C3 = ν†Keq,1Keq,2K†eq,3 [25]
C4 = ν†Keq,1Keq,2Keq,3K†eq,4. [26]
The physical significance of Ci can be related to the equilibriumconstant for the formation of the transition state for step i fromthe reactant A ðK†
eq,A→TSi Þ:
C1 = ν†K†eq,A→TS1 [27]
C2 = ν†K†eq,A→TS2 [28]
C3 = ν†K†eq,A→TS3 [29]
C4 = ν†K†eq,A→TS4 . [30]
Thus, we can now express the net rates of the elementary steps interms of lumped equilibrium constants from the reactant A tothe transition states:
r= ν†K†eq,A→TS1
Að1−Z1Þ [31]
r= ν†K†eq,A→TS2AZ1ð1−Z2Þ [32]
r= ν†K†eq,A→TS3AZ1Z2ð1−Z3Þ [33]
r= ν†K†eq,A→TS4AZ1Z2Z3
�1−
β
Z1Z2Z3
�. [34]
A simple calculation to assess whether an elementary step maycontribute to a significant extent in a reaction scheme is toestimate the maximum rate of this step for the conditions of thecatalytic reaction of interest. If the maximum estimated rate ofthis step is much slower than the observed rate of the catalyticreaction, then this step cannot play a significant role in thereaction scheme. As seen above, the maximum rate of step icorresponds to the case where the reversibilities of all other stepsare equal to 1 and the reversibility of step i is equal to β, corre-sponding to the case where step i is rate-determining (XRC,i = 1):
r1 = rds = ν†K†eq,A→TS1
Að1− βÞ= rmax,1ð1− βÞ [35]
r2 = rds = ν†K†eq,A→TS2Að1− βÞ= rmax,2ð1− βÞ [36]
r3 = rds = ν†K†eq,A→TS3Að1− βÞ= rmax,3ð1− βÞ [37]
r4 = rds = ν†K†eq,A→TS4Að1− βÞ= rmax,4ð1− βÞ. [38]
We note that we have defined rmax,i as the maximum rate for step ifor the case where β is equal to zero. The maximum rate of an ele-mentary reaction can also be obtained using Sabatier–Gibbs analysis
Table 5. Maximum rates for elementary steps of the mechanism(case I)
Step rmax,i§,‡, s−1 rmax,i, s
−1
1 rmax,1 = ν†ðK†eq,A→TS,1Þ PCO 9.28 × 106
2 rmax,2 = ν†ðK†eq,A→TS,2Þ PH2O 4.22 × 1010
3 rmax,3 = ν†ðK†eq,A→TS,3Þ PH2O 3.28 × 101
4 rmax,4 = ν†ðK†eq,A→TS,4Þ PH2O P
−0.5H2
4.46 × 10−4
5 rmax,5 = ν†ðK†eq,A→TS,5Þ PCO PH2O P
−1H2
3.37 × 10−3
6 rmax,6 = ν†ðK†eq,A→TS,6Þ PCO2 6.55 × 105
7 rmax,7 = ν†ðK†eq,A→TS,7Þ PH2 4.49 × 100
§PCO, PH2O, PH2 , and PCO2 are 0.07 atm, 0.21 atm, 0.38 atm, and 0.085 atm, re-spectively.‡ν† = kBT=h.
Table 6. Comparison of overall rate and surface coverages ofadsorbed intermediates calculated by maximum rate analysisand using the microkinetic model (case I)
Rate/coverage Microkinetic model Maximum rate analysis
Rate of reaction, s−1 1.68 × 10−6 1.51 × 10−6
CO* 9.32 × 10−1 9.32 × 10−1
H2O* 7.21 × 10−7 7.70 × 10−7
H* 5.46 × 10−3 5.69 × 10−3
OH* 6.14 × 10−7 5.26 × 10−7
O* 1.05 × 10−11 6.76 × 10−11
CO2* 2.10 × 10−8 2.74 × 10−8
Vacant sites 6.20 × 10−2 6.18 × 10−2
Table 4. Defining relations and values of equilibrium constants for the formation of transitionstates from gas-phase reactants and products of the overall stoichiometric reaction (case I)
Step Reaction K†eq,A→TS,i K†
eq,A→TS,i
1 CO+ p�CO†p K†
eq,1 1.22 × 10−5
2 H2O+ p�H2O†p K†
eq,2 1.84 × 10−2
3 H2O+ 2p�H2O†pp Keq,2K†
eq,3 1.44 × 10−11
4 H2O+2p�OH†pp + 0.5H2 Keq,2Keq,3K0.5
eq,7K†eq,4 1.20 × 10−16
5 CO+H2O+ 2p�CO†2pp +H2 Keq,1Keq,2Keq,3Keq,4Keq,7K†
eq,5 7.99 × 10−15
6 CO2 + p�CO†2p K†
eq,6=Keq,6 7.08 × 10−7
7 H2 +2p�H†2pp K†
eq,7=Keq,7 1.08 × 10−12
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(13, 14), in which the surface coverages by the reaction intermedi-ates are first calculated assuming that the intermediates are in equi-librium with the reactants and/or the products of the overallreaction. The maximum rates of the elementary steps in the forwardand/or reverse directions are then calculated from the maximumcoverages calculated in this manner and the estimated activationbarriers. This methodology is equivalent to the approach presentedabove in which equilibrium relations are written to form the transi-tion states from the reactants and/or products of the overall reaction.An initial assessment of the feasibility of a reaction scheme is
to estimate the maximum rates of all of the steps and then verifythat all of these values are comparable to or faster than theobserved rate of the catalytic reaction under investigation.The next phase of the analysis is to estimate the degrees of
rate control for the various steps in the reaction scheme. Thisanalysis is carried out by solving for the values of Zi for the stepsby noting that the net rates of all steps are equal to the net rate ofthe overall reaction, r:
r= rmax,1ð1−Z1Þ [39]
r= rmax,2Z1ð1−Z2Þ [40]
r= rmax,3Z1Z2ð1−Z3Þ [41]
r= rmax,4Z1Z2Z3
�1−
β
Z1Z2Z3
�. [42]
The following expressions for the reversibilities are obtained:
Z1 =rmax,1rmax,2rmax,3 + βrmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2
rmax,1rmax,2rmax,3 þ rmax,2rmax,3rmax,4 þ rmax,3rmax,4rmax,1 þ rmax,4rmax,1rmax,2[43]
Z2 =rmax,1rmax,2rmax,3 + βrmax,2rmax,3rmax,4 + βrmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2
rmax,1rmax,2rmax,3 + βrmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2[44]
Z3 =rmax,1rmax,2rmax,3 + βrmax,2rmax,3rmax,4 + βrmax,3rmax,4rmax,1 + βrmax,4rmax,1rmax,2
rmax,1rmax,2rmax,3 + βrmax,2rmax,3rmax,4 + βrmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2 .
[45]
The degree of rate control for step i can be calculated from thereversibility of the step and the sensitivity, si, of the overall rateto changes in forward rate constant for the step:
si =�∂r∂ki
�kir
���kj≠ i
[46]
XRC,i = sið1−ZiÞ, [47]
where the partial derivative is taken with all other rate constants(kj) held constant. An important distinction between the sensi-tivity (Eq. 46) and the degree of rate control (Eq. 1) is that in
computing the degree of rate control, all of the equilibrium con-stants are held constant, whereas while computing the sensitivityof an elementary reaction, all of the rate constants are heldconstant except for the forward rate constant of the elementaryreaction of interest. Eq. 47 shows that the degree of rate controlapproaches zero as the reversibility of the step approaches unity(10). In a sequence of steps for which the stoichiometric coeffi-cients of the steps are equal to unity, the sensitivity for step i isrelated to the sensitivity of the preceding step i−1 (15):
si =Zi−1si−1. [48]
Accordingly, sensitivity of the rate with respect to changes in theforward rate constants is passed from one step to the followingstep for cases where the reversibility of the previous stepapproaches unity, whereas the rate is insensitive for stepsfollowing an irreversible step because the reversibility of the previousstep approaches zero. The above relation for the degree of ratecontrol XRC,i in terms of the sensitivity si, and the relation for si interms of si−1 can be written as
XRC,1 = s1ð1−Z1Þ [49]
XRC,2 = s1Z1ð1−Z2Þ [50]
XRC,3 = s1Z1Z2ð1−Z3Þ [51]
XRC,4 = s1Z1Z2Z3
�1−
β
Z1Z2Z3
�. [52]
The values of Zi can then be expressed in terms of rmax,i, and thevalue of s1 is obtained by requiring that the sum of XRC,i be equalto unity, leading to the following result:
XRC,1 =rmax,2rmax,3rmax,4
rmax,1rmax,2rmax,3 + rmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2[53]
XRC,2 =rmax,3rmax,4rmax,1
rmax,1rmax,2rmax,3 + rmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2[54]
XRC,3 =rmax,4rmax,1rmax,2
rmax,1rmax,2rmax,3 + rmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2[55]
XRC,4 =rmax,1rmax,2rmax,3
rmax,1rmax,2rmax,3 + rmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2 .
[56]
The net rate of the overall reaction is then calculated from thenet rate of any step by substitution of the appropriate values of Ziin terms of the maximum rates, rmax,i:
r=rmax,1rmax,2rmax,3rmax,4ð1− βÞ
rmax,1rmax,2rmax,3 + rmax,2rmax,3rmax,4 + rmax,3rmax,4rmax,1 + rmax,4rmax,1rmax,2 . [57]
The above relation for XRC,i can be generalized for a sequence ofn steps:
Table 7. Predicted apparent activation energy and reactionorders from the microkinetic model and maximum rateanalysis (case I)
Reaction kineticsMicrokinetic
modelMaximum
rate analysis
Apparent activation energy, kJ·mol−1 282 285
Reaction ordersH2O 1.07 1.00CO −1.84 −1.85H2 −0.59 −0.55CO2 −0.04 0.00
Table 8. Redox reaction mechanism for water–gas shift(OH disproportionation, case II)
Step Elementary reaction Stoichiometric coefficients, σi
1 CO+ * � CO* 12 H2O+ * � H2O* 13 H2O*+ * � H*+OH* 24 OH*+OH* � H2O*+O* 15 CO*+O* � CO2* + * 16 CO2* � CO2 + * 17 2H* � H2 +2* 1
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XRC,i =Πn
j=1, j≠ irmax,j
Pnk=1
�Πn
j=1, j≠ krmax,j
� , [58]
the rate of the overall reaction is then given by:
r=
�Πn
j=1rmax,j
�ð1− βÞ
Pnk=1
�Πn
j=1, j≠ krmax,j
� , [59]
the above relations show that it is possible to calculate the degreeof rate control for each step (XRC,i) and the rate of the overallreaction (r) by first calculating the maximum rate of each step(rmax,i). The expression for the rate of the overall reaction canalso be written as
r=XRC,irmax,i ð1− βÞ, [60]
indicating that the net rate of the overall reaction is equal to theproduct of the degree of rate control, the maximum rate of thestep, and the reversibility of the overall stoichiometric reaction.
Reactions on Catalyst SurfacesWe note that the results derived above are valid for a heteroge-neous, homogeneous, or biological catalytic reaction in the limit oflow coverage of the catalyst sites (16). To account for population ofcatalyst sites by adsorbed species we now modify the reactionscheme to include adsorption and reaction on active sites, *:
1. A+ *� Ip12. Ip1 � Ip23. Ip2 � Ip34. Ip3 � B+ *overall : A � B.
As a first approximation, we assume that all adsorbed species arequasi-equilibrated with the gas phase, leading to the followingrelation for conservation of active sites:
1= θp + θI1 + θI2 + θI3
1= θp +Keq,1PAθp +Keq,1Keq,2PAθp +Keq,1Keq,2Keq,3PAθp
θp =1
1+Keq,1PA +Keq,1Keq,2PA +Keq,1Keq,2Keq,3PA ,
[61]
where θp represents the fraction of the active sites that are free ofadsorbed species, and we have assumed for simplicity that thereactant A and the product B are not strongly adsorbed on the
catalyst. In general, assuming that the most abundant adsorbedspecies are in quasi-equilibrium with the gas or liquid phase, thevalue of θp can be estimated as
θp =1
1+Pni=1
Kads,i
�Πk
j=1Pνijj
� , [62]
where the sum in the denominator is over the n abundant surfacespecies, Kads,i is the lumped equilibrium constant to form a spe-cific adsorbed species from the reactants and/or products, and νijare the stoichiometric coefficients in the lumped equilibrium toproduce adsorbed species i from reactants and/or products j.It is now possible to estimate the rate of the catalytic reaction by
scaling the above expression for r, corresponding to low coverage ofthe catalyst sites, to take into account the fraction of the sites that isavailable for reaction, θp. For example, if the catalytic reaction re-quires “m” adjacent sites, then r is scaled by θmp , leading to
r=
�Πn
j=1rmax,j
�ð1− βÞ
Pnk=1
�Πn
j=1, j≠ krmax,j
��1+Pni=1
Kads,i
�Πk
j=1Pνijj
��m . [63]
Analytical Strategy for Analysis of Reaction SchemesWe now suggest an analytical strategy to assess the feasibility of aproposed reaction scheme to describe the reaction kinetics for acatalytic reaction of interest, and then to use this reactionscheme to suggest directions for research to identify promisingcatalysts for the catalytic reaction. We note that the above ap-proach is valid for heterogeneous, homogeneous, and biologicalcatalysis, and it has been used for electrochemical processes (17–19). We illustrate this strategy below for the case of a hetero-geneous catalytic reaction as illustrated in Fig. 1.The first step is to estimate the thermodynamic properties
(i.e., enthalpy of formation and absolute entropy) of the reac-tants, products, and reaction intermediates in the gas phase (orthe liquid phase). The next step is to estimate values of bindingenergies for adsorption of all species on the surface, for exampleusing experimental results from the literature, results from DFTcalculations, or scaling relations (20–24). In addition, it is nec-essary to estimate values for the entropies of the adsorbed spe-cies, for example using results from experimental studies (25),correlations in the literature (26), results from DFT calculationsof vibrational frequencies (27–34), combined with hinderedtranslator and hindered rotor models for the three modes asso-ciated with motion parallel to the surface (35, 36), or by as-suming models for the extent of mobility on the surface. It is nowpossible to estimate values for lumped equilibrium constants,Kads,i, to form the adsorbed species from the reactants and/orproducts, followed by calculation of θp. This analysis providesinformation about which species are expected to be abundant onthe surface at various reaction conditions. Accordingly, the bindingenergies of these abundant species are significant in determining
Table 9. Comparison of the rate and surface coverages ofadsorbed intermediates predicted by maximum rate analysisand the microkinetic model (case II)
Rate/coverage Microkinetic model Maximum rate analysis
Rate of reaction, s−1 1.27 × 10−5 1.29 × 10−5
CO* 9.26 × 10−1 9.32 × 10−1
H2O* 7.69 × 10−7 7.70 × 10−7
H* 6.04 × 10−3 5.69 × 10−3
OH* 6.50 × 10−7 5.26 × 10−7
O* 6.87 × 10−11 6.76 × 10−11
CO2* 2.06 × 10−8 2.74 × 10−8
Vacant sites 6.81 × 10−2 6.18 × 10−2
Table 10. Predicted apparent activation energy and reactionorders from the microkinetic model and maximum rate analysis
Reaction kineticsMicrokinetic
modelMaximum
rate analysis
Apparent activation energy, kJ·mol−1 213 221
Reaction ordersH2O 1.04 1.00CO −0.97 −0.92H2 −1.01 −1.00CO2 −0.05 0.00
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catalyst performance, and it would be desirable to obtain moreprecise values for these binding energies. In contrast, the bindingenergies of the remaining adsorbed species are kinetically in-significant, and more precise values are not required.The next step in the analysis is to estimate values for the ac-
tivation barriers for all steps in the reaction scheme. For exam-ple, the activation barrier for a step could be estimated from anexperimental value for the reaction of interest or a similar re-action, estimated from results of DFT calculations (37), or es-timated using Brønsted–Evans–Polanyi (BEP) scaling relations(38–40). These BEP relations provide estimates for the ther-modynamic properties of transition states in terms of stableadsorbed species, thereby allowing the estimation of activationenergies in terms of the binding energies used above to determinethe surface coverages by adsorbed species. It is also required in thisphase of the analysis to estimate values for changes in entropy as-sociated with formation of the transition states from the adsorbedintermediates. It is typically sufficient for initial calculations to as-sume that these entropy changes are equal to zero for reactionstaking place on the surface. It is now possible to estimate values forK†eq,A→TSi, corresponding the lumped equilibrium constants for the
formation of the transition states from the reactants and/or productsof the reaction, followed by calculation of the maximum rate foreach step of the reaction scheme.The values of rmax,i estimated above can now be compared with
the desired or measured rate of the overall stoichiometric re-action, r. If the values of rmax,i for steps in the reaction schemeare much lower than the overall rate r, then the reaction con-ditions must be changed to increase the values of these maximumrates, or a new reaction scheme must be proposed. If the valuesof rmax,i for all steps are sufficiently fast compared with the overallrate r, then it is possible to calculate values for the degrees of ratecontrol, XRC,i, to determine the extents to which each step controlsthe rate of the overall stoichiometric reaction. Accordingly, stepswith finite degrees of rate control are significant in determiningcatalyst performance, and it would be desirable to obtain moreprecise values for the properties of the transition states for thesesteps. In contrast, the transition states of the remaining steps arekinetically insignificant, and more precise values for the prop-erties of these transition states are not required.Using the values of XRC,i estimated above, it is now possible to
identify the key transition states in the reaction scheme and thenuse the expansion of the rate in terms of changes in the Gibbs
free energies of these key transition states to identify promisingcatalytic materials, as described by Campbell and coworkers (12).Optimization of catalyst performance is dictated by the Sabatierprinciple that the surface should stabilize strongly the key tran-sition states identified from reaction steps with finite values ofXRC,i, whereas the surface should not bind adsorbed species toostrongly, such that the values of Kads,i lead to values of θp that areapproximately equal to 0.5 for the optimal catalyst (41).In parallel with analysis of the reaction scheme in terms of the
values of XRC,i, the values of rmax,i can also be used to predict therate of the overall stoichiometric reaction. Thus, it is possible toestimate the reaction kinetics for the overall stoichiometric reaction,such as predicting the apparent activation energy by changing thereaction temperature, and predicting the apparent reaction ordersby changing the partial pressures of the reactants and products.These predictions can then be compared with results from experi-mental measurements to determine whether the proposed reactionscheme is consistent with the experimental data. It is also possible touse the predictions of the overall rate to determine the reactionconditions for which the proposed reaction scheme would lead tofavorable catalytic properties. More generally, it is possible to studyhow optimization of catalyst performance by changing the ther-modynamic properties of the key transition states and adsorbedspecies is affected by the nature of the reaction conditions.We note that the above analysis of proposed reaction schemes
in terms of values of lumped equilibrium constants, K†eq,A→TSi, to
form transition states from the reactants and/or products, coupledwith calculation of θp, is based on the assumptions that the stoi-chiometric coefficients of the reaction steps in the scheme are equalto unity, and the adsorbed species are in quasi-equilibrium with thegas phase. More generally, we show below that this approach can beused in an approximate manner as a preliminary method to probethe performance of more general reaction schemes, followed bymore detailed analyses using full microkinetic models to determinethe surface coverages by adsorbed species and the degrees of ratecontrol of the elementary steps.The expressions above are exact for the reversibilities (Eqs.
43–45), the degrees of rate control (Eq. 58), and the overall rate
Fig. 2. Comparison of predicted degrees of rate control using maximumrate analysis and from the full microkinetic model (case II). Blue, step 5,microkinetic model; pink, step 5, maximum rate analysis; red, step 3,microkinetic model; black, step 3, maximum rate analysis.
Table 11. Carboxyl reaction mechanism for water–gas shift(case III)
Step Elementary reactionStoichiometriccoefficients (σi)
1 CO+ * � CO* 12 H2O+ * � H2O* 13 H2O*+ * � H*+OH* 14 CO*+OH* � cCOOH* + * 15 cCOOH* � tCOOH* 16 tCOOH* � CO2* +H* 17 CO2* � CO2 + * 18 2H* � H2 + 2* 1
Table 12. Comparison of the rate and surface coverages ofadsorbed intermediates from maximum rate analysis and usingthe microkinetic model (case III)
Rate/coverage Microkinetic model Maximum rate analysis
Rate of reaction, s−1 1.72 × 10−11 1.80 × 10−11
CO* 9.34 × 10−1 9.32 × 10−1
H2O* 7.12 × 10−7 7.70 × 10−7
H* 5.35 × 10−3 5.68 × 10−3
OH* 6.08 × 10−7 5.26 × 10−7
cCOOH* 3.07 × 10−13 3.09 × 10−13
tCOOH* 4.53 × 10−12 4.57 × 10−12
CO2* 2.03 × 10−8 2.74 × 10−8
Vacant sites 6.09 × 10−2 6.18 × 10−2
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of the reaction (Eq. 59) obtained from the maximum rate anal-ysis for a reaction mechanism with all stoichiometric coefficientsequal to unity. More complex expressions are obtained for amore general reaction scheme where the stoichiometric coeffi-cients are not all equal to unity. For example, consider a two-stepreaction scheme wherein the elementary step 2 must occur twicefor each catalytic cycle:
1. A2 � 2A2. A+B � ABoverall : A2 + 2B � 2AB.
For this reaction mechanism it can be shown (SI Appendix) thatthe reversibilities, the degrees of rate control, and the overallreaction rate are
Z1 = 1+� ffiffiffi
βp2
�ξ+
18ξ2 −
18ξ�16+ 8
ffiffiffiβ
pξ+ ξ2
�1=2
Z2 =
0BB@ β
1+� ffiffi
βp2
�ξ+ 1
8ξ2 − 1
8 ξ16+ 8
ffiffiffiβ
pξ+ ξ2
�1=21CCA
1=2
XRC,1 =ξ
16+ 8ffiffiffiβ
pξ+ ξ2
�1=2 XRC,2 = 1−ξ
16+ 8ffiffiffiβ
pξ+ ξ2
�1=2
r=r28
��16+ 8
ffiffiffiβ
pξ+ ξ2
�1=2− 4
ffiffiffiβ
p− ξ
�,
[64]
where ξ= ðrmax,2=rmax,1Þ. For the case when ξ≈ 0, i.e., rmax,2 << rmax,1,the overall rate of the reaction is
r=rmax,2
2
�1−
ffiffiffiβ
p �. [65]
Far from equilibrium where β<< 1, the maximum overall rate ofreaction (written in term of the extent of the overall reaction) isrmax,2/2. Thus, maximum rate analysis can be used in an approx-imate manner for a more general reaction scheme if the maximumrates of the constituent elementary steps are defined as rmax,i/σi.We now use four case studies to examine the applicability of
our proposed approach to predict the overall rate and analyzethe reaction kinetics of stoichiometric reactions based on thecalculation of the maximum rates of the elementary steps of thereaction mechanism. In these studies, we compare the results ofmaximum rate analyses to results from full microkinetic analysesfor the water–gas shift reaction on a Cu (111) surface. The DFTvalues for the binding energies of the surface intermediates andthe activation energies of the elementary reactions were obtained
from literature (42). The goal of these case studies is to comparethe results of maximum rate analysis and the full microkineticmodel. Accordingly, the DFT parameters used in these case studieswere not optimized for predicting experimental reaction kineticsdata. For example, it is known that the binding energy of carbonmonoxide is a function of its surface coverage; also, at the experi-mental conditions the surface coverage of bidentate formate specieis high (42). However, these nuances of the mechanism were notincluded in these case studies. We present below a summary of theresults for these case studies, and the details of these case studiesare presented in SI Appendix.
Case Study I: Redox Mechanism for Water–Gas ShiftWe first consider the redox mechanism for the water–gas shiftreaction, involving the oxidation of CO by adsorbed oxygenatoms, O*. Atomic oxygen is obtained from H2O by two successiveH-atom abstraction steps (steps 3 and 4 in Table 1). The stoichio-metric coefficients of all of the elementary steps in this example areequal to unity (Table 1). This reaction is analyzed for a coppercatalyst at a temperature of 523 K and total pressure of 1 atm. Thevalues of the equilibrium constants and forward rate constants forthe elementary steps are listed in Table 2. The defining relationsand values of the equilibrium constants for the lumped reactionsthat describe the formation of adsorbed intermediates from gas-phase reactants and products of the overall stoichiometric reactionare listed in Table 3. Table 4 lists the defining relations and valuesof equilibrium constants for lumped reactions that describe theformation of the transition states from the reactants and products ofthe overall stoichiometric reaction. Defining relations and values ofthe maximum rates are shown in Table 5.The seven elementary steps and six reaction intermediates
shown in Table 1 are included in a microkinetic model to determinethe rate of the overall stoichiometric reaction and the degrees ofrate control of the various steps. The following coupled differentialequations are solved numerically to obtain the surface coverages atsteady state, the rate of the overall stoichiometric reaction, theapparent activation energy, and the apparent reaction orders:
Table 13. Predicted apparent activation energy and reactionorders from the microkinetic model and maximum rate analysis
Reaction kineticsMicrokinetic
modelMaximum
rate analysis
Apparent activation energy, kJ·mol−1 239 239
Reaction ordersH2O 1.03 1.00CO −0.91 −0.92H2 −0.52 −0.50CO2 −0.04 0.00
Table 14. Carboxyl reaction mechanism for water–gas shift(carboxyl + OH, case IV)
Step Elementary reactionStoichiometriccoefficients, σi
1 CO+* � CO* 12 H2O+* � H2O* 13 H2O*+* � H*+OH* 24 CO*+OH* � cCOOH*+* 15 cCOOH* � tCOOH* 16 tCOOH*+OHp� CO2*+ H2O* 17 CO2* � CO2 +* 18 2H* � H2 + 2* 1
Table 15. Comparison of the rate and surface coverages ofadsorbed intermediates calculated using maximum rate analysisand the microkinetic model (case IV)
Rate/coverage Microkinetic model Maximum rate analysis
Rate of reaction, s−1 3.41 × 10−7 2.86 × 10−7
CO* 9.33 × 10−1 9.32 × 10−1
H2O* 7.14 × 10−7 7.70 × 10−7
H* 5.38 × 10−3 5.68 × 10−3
OH* 6.09 × 10−7 5.26 × 10−7
cCOOH* 3.06 × 10−13 3.09 × 10−13
tCOOH* 4.50 × 10−12 4.57 × 10−12
CO2* 2.04 × 10−8 2.74 × 10−8
Vacant sites 6.12 × 10−2 6.18 × 10−2
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∂θCO∂t
= r1 − r5 = 0
∂θH2O
∂t= r2 − r3 = 0
∂θH∂t
= r3 + r4 − 2r7 = 0
∂θOH
∂t= r3 − r4 = 0
∂θO∂t
= r4 − r5 = 0
∂θCO2
∂t= r5 − r6 = 0.
These differential equations are integrated over time, t, from aknown starting point (e.g., a clean surface) to a surface operatingat steady state where the derivatives are equal to zero. Table 6shows a comparison of the overall rate of the reaction and thesurface coverages for the mechanism of the water–gas shift re-action shown in Table 1. It can be seen that the results of themicrokinetic model and the maximum rate analysis are in goodagreement. The apparent activation energy and reaction ordersobtained from the microkinetic model and the maximum rate anal-ysis are presented in Table 7. The predicted degree of rate control(SI Appendix, Fig. S1), Arrhenius plot (SI Appendix, Fig. S2), andplots of rate versus partial pressures leading to the apparent reac-tion orders (SI Appendix, Fig. S3) show good agreement between thepredictions of the microkinetic model and maximum rate analysis.
Case Study II: Redox Reaction Mechanism for Water–GasShift (OH Disproportionation)We next consider a reaction mechanism where the stoichiometriccoefficients of all of the elementary steps are not equal to unity.For this purpose we choose a redox mechanism for water–gasshift where the adsorbed atomic oxygen is obtained from thedisproportionation of two adsorbed hydroxyl intermediates, asshown in Table 8. In this reaction scheme, the dissociation ofadsorbed water must take place twice for each turnover of thereactants to products. The reaction mechanism is analyzed fora copper catalyst at a temperature of 523 K and a pressure of1 atm. Table 9 shows a comparison of the overall rate of thereaction and the surface coverages for the mechanism of thewater–gas shift reaction shown in Table 8. It can be seen that theresults from the microkinetic model and the maximum rateanalysis are in good agreement.The apparent activation energy and reaction orders obtained
from the microkinetic model and the maximum rate analysis arepresented in Table 10. The predicted degree of rate control (SIAppendix, Fig. S4), Arrhenius plot (SI Appendix, Fig. S5), and theplots of rate versus partial pressure leading to the apparent re-action orders (SI Appendix, Fig. S6) show good agreement be-tween the predictions of the microkinetic model and maximumrate analysis.
Using the values of rate and equilibrium constants taken fromthe literature, the rate of the proposed reaction mechanism(Table 8) is predominantly controlled by the elementary reactioninvolving the formation of adsorbed carbon dioxide. However, toprobe the applicability of our proposed methodology for a re-action mechanism where the rate is controlled by an elementaryreaction having a stoichiometric number that is not equal to 1,we increase the activation energy of water dissociation step (step3 in Table 8) to make it the rate-controlling step. The maximumrate of an elementary step is then obtained by the followingequation:
ri,max =1σi
0BBBB@
�Πn
j=1rmax,j
�ð1− βÞ
Pnk=1
�Πn
j=1, j≠ krmax,j
��1+
Pni=1
Kads,i
�Πk
j=1Pνijj
��21CCCCA. [66]
The activation energy for the water dissociation step was varied from1.15 eV (where the rate is predominantly controlled by the formationof adsorbed carbon dioxide) to 1.6 eV (where the rate is controlled bythe dissociation of the adsorbed water). Fig. 2 shows that the maxi-mum rate analysis approach is able to predict the degree of ratecontrol over the entire range of the activation energy studied.
Case Study III: Carboxyl Reaction Mechanism for Water–GasShiftWe now consider the “carboxyl” mechanism for the water–gasshift reaction (Table 11). The carboxyl intermediate formed instep 4 is in a cis configuration (cCOOH). In step 5, the cis-car-boxyl species isomerizes into a trans-carboxyl species (tCOOH)where the H-atom is pointing away from the surface. ThetCOOH then dissociates into adsorbed carbon dioxide andadsorbed hydrogen atom (step 6). The reaction mechanism isanalyzed for a copper catalyst at a temperature of 523 K and apressure of 1 atm. The stoichiometric coefficients of all of theelementary steps in this mechanism are equal to unity (Table 11).Table 12 shows a comparison of the overall rate of the reactionand the surface coverages for the carboxyl mechanism of thewater–gas shift reaction shown in Table 11. It can be seen thatthe results from the microkinetic model and the maximum rateanalysis are in good agreement.
Fig. 3. Comparison of predicted degrees of rate control using maximumrate analysis and from the full microkinetic model (case IV). Blue, step 6, micro-kinetic model; pink, step 6, maximum rate analysis; red, step 3, microkinetic model;black, step 3, maximum rate analysis.
Table 16. Predicted apparent activation energy and reactionorders from the microkinetic model and maximum rate analysis
Reaction kineticsMicrokinetic
modelMaximum
rate analysis
Apparent activation energy, kJ·mol−1 157 161
Reaction ordersH2O 2.03 1.94CO −0.91 −0.92H2 −1.01 −0.96CO2 −0.04 0.00
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The apparent activation energy and reaction orders obtainedfrom the microkinetic model and the maximum rate analysis arepresented in Table 13. The predicted degree of rate control (SIAppendix, Fig. S8), Arrhenius plot (SI Appendix, Fig. S9), and plotsof rate versus partial pressures leading to apparent reaction orders(SI Appendix, Fig. S10) show good agreement between the pre-dictions of the microkinetic model and maximum rate analysis.
Case Study IV: Carboxyl Reaction Mechanism for Water–GasShift (Carboxyl + OH)In this case study, we consider a “carboxyl”mechanism where thestoichiometric coefficients of all of the elementary steps are notequal to unity (Table 14). This mechanism for the water–gas shiftreaction involves the formation of adsorbed carbon dioxide fromthe reaction of adsorbed carboxyl and hydroxyl intermediates(step 6 in Table 14). Accordingly, the dissociation of adsorbedwater must take place twice for each turnover of the reactants toproducts. The reaction mechanism is analyzed for a coppercatalyst at a temperature of 523 K and a pressure of 1 atm.Table 15 shows a comparison of the overall rate of the re-
action and the surface coverages for the carboxyl mechanism ofthe water–gas shift reaction shown in Table 14. It can be seenthat results from the microkinetic model and the maximum rateanalysis are in good agreement.The apparent activation energy and reaction orders obtained
from the microkinetic model and the maximum rate analysis are
presented in Table 16. The predicted degree of rate control (SIAppendix, Fig. S11), Arrhenius plot (SI Appendix, Fig. S12), and theplots of rate versus partial pressure leading to the apparent reactionorders (SI Appendix, Fig. S13) show good agreement between theprediction of the microkinetic model and maximum rate analysis.Using the values of the rate and equilibrium constants obtained
from the literature, the rate of the proposed reaction mechanism(Table 14) is predominantly controlled by the elementary reactioninvolving the formation of adsorbed carbon dioxide. However, tostudy the applicability of our proposed methodology for a reactionmechanism where the rate is controlled by an elementary reactionhaving a stoichiometric number that is not equal to 1, we increasethe activation energy of water dissociation step (step 3 in Table 14)to make it the rate-controlling step. The activation energy for thewater dissociation step was varied from 1.0 eV (where the rate ispredominantly controlled by the formation of adsorbed carbon di-oxide) to 1.95 eV (where the rate is controlled by the dissociation ofthe adsorbed water). Fig. 3 shows that our proposed method is ableto predict the degree of rate control for the entire range of the ac-tivation energy studied.
ACKNOWLEDGMENTS.We thank Profs. ManosMavrikakis and James B. Rawlings(University of Wisconsin–Madison) and Prof. Charles T. Campbell (University ofWashington) for valuable discussions. J.A.D. thanks Prof. Michel Boudart (StanfordUniversity) for discussions regarding the identification of kinetically significantsteps in reaction mechanisms. This work was supported by US Department ofEnergy, Office of Basic Energy Sciences Grant DE-SC0014058.
1. Dumesic JA, Rudd DA, Aparicio LM, Rekoske JE, Trevino AA (1993) The microkineticsof hetrogeneous catalysis. The Microkinetics of Hetrogeneous Catalysis (AmericanChemical Soc, Washington, DC), pp 23–53.
2. Medford AJ, et al. (2015) From the Sabatier principle to a predictive theory of tran-sition-metal heterogeneous catalysis. J Catal 328:36–42.
3. Herron JA, Ferrin P, Mavrikakis M (2014) First-principles mechanistic analysis of di-methyl ether electro-oxidation on monometallic single-crystal surfaces. J Phys Chem C118(42):24199–24211.
4. Neurock M (2010) Engineering molecular transformations for sustainable energyconversion. Ind Eng Chem Res 49(21):10183–10199.
5. Lynggaard H, Andreasen A, Stegelmann C, Stoltze P (2004) Analysis of simple kineticmodels in heterogeneous catalysis. Prog Surf Sci 77(3–4):71–137.
6. Campbell CT (1994) Micro- and macro-kinetics: Their relationship in heterogeneouscatalysis. Top Catal 1:353–366.
7. Campbell CT (2001) Finding the rate-determining step in a mechanism. J Catal 204(2):520–524.
8. Kozuch S, Shaik S (2006) A combined kinetic-quantum mechanical model for assess-ment of catalytic cycles: Application to cross-coupling and Heck reactions. J Am ChemSoc 128(10):3355–3365.
9. Kozuch S, Shaik S (2008) Kinetic-quantum chemical model for catalytic cycles: The Haber-Bosch process and the effect of reagent concentration. J Phys Chem A 112(26):6032–6041.
10. Dumesic JA (2001) Reply to finding the rate-determining step in a mechanism: ComparingDeDonder relations with the “degree of rate control.” J Catal 204(2):525–529.
11. Stegelmann C, Andreasen A, Campbell CT (2009) Degree of rate control: How much the en-ergies of intermediates and transition states control rates. J Am Chem Soc 131(23):8077–8082.
12. Wolcott CA, Medford AJ, Studt F, Campbell CT (2015) Degree of rate control approachto computational catalyst screening. J Catal 330:197–207.
13. Falsig H, et al. (2007) Trends in catalytic NO decomposition over transition metalsurfaces. Top Catal 45(1–4):117–120.
14. Falsig H, Bligaard T, Christensen CH, Nørskov JK (2007) Direct NO decomposition overstepped transition-metal surfaces. Pure Appl Chem 79(11):1895–1903.
15. Cortright RD, Dumesic JA (2001) Kinetics of heterogeneous catalytic reactions:Analysis of reaction schemes. Adv Catal 46:161–264.
16. Dumesic JA (1999) Analyses of reaction schemes using De Donder relations. J Catal185(2):496–505.
17. Rossmeisl J, et al. (2012) Bifunctional anode catalysts for direct methanol fuel cells.Energy Environ Sci 5(8):8335.
18. Viswanathan V, Hansen HA, Rossmeisl J, Nørskov JK (2012) Universality in oxygenreduction electrocatalysis on metal surfaces. ACS Catal 2(8):1654–1660.
19. Montoya JH, Tsai C, Vojvodic A, Nørskov JK (2015) The challenge of electrochemicalammonia synthesis: A new perspective on the role of nitrogen scaling relations.ChemSusChem 8(13):2180–2186.
20. Abild-Pedersen F, et al. (2007) Scaling properties of adsorption energies for hydrogen-containing molecules on transition-metal surfaces. Phys Rev Lett 99(1):016105.
21. Nørskov JK, et al. (2002) Universality in heterogeneous catalysis. J Catal 209(2):275–278.22. Greeley J, Mavrikakis M (2005) Surface and subsurface hydrogen: Adsorption prop-
erties on transition metals and near-surface alloys. J Phys Chem B 109(8):3460–3471.
23. Greeley J, Mavrikakis M (2004) Alloy catalysts designed from first principles. NatMater 3(11):810–815.
24. Vorotnikov V, Vlachos DG (2015) Group additivity and modified linear scaling rela-tions for estimating surface thermochemistry on transition metal surfaces: Applica-tion to furanics. J Phys Chem C 119(19):10417–10426.
25. Campbell CT, Sellers JRV (2013) Enthalpies and entropies of adsorption on well-defined oxide surfaces: Experimental measurements. Chem Rev 113(6):4106–4135.
26. Campbell CT, Sellers JRV (2012) The entropies of adsorbed molecules. J Am Chem Soc134(43):18109–18115.
27. Herron JA, Tonelli S, Mavrikakis M (2013) Atomic and molecular adsorption onRu(0001). Surf Sci 614:64–74.
28. Santiago-Rodríguez Y, Herron JA, Curet-Arana MC, Mavrikakis M (2014) Atomic andmolecular adsorption on Au(111). Surf Sci 627:57–69.
29. Hahn K, Mavrikakis M (2013) Atomic and molecular adsorption on Re(0001). Top Catal57(1-4):54–68.
30. Herron JA, Tonelli S, Mavrikakis M (2012) Atomic and molecular adsorption onPd(111). Surf Sci 606(21–22):1670–1679.
31. Ford DC, Xu Y, Mavrikakis M (2005) Atomic and molecular adsorption on Pt(111). SurfSci 587(3):159–174.
32. Mavrikakis M, Rempel J, Greeley J, Hansen LB, Nørskov JK (2002) Atomic and mo-lecular adsorption on Rh(111). J Chem Phys 117(14):6737.
33. Krekelberg WP, Greeley J, Mavrikakis M (2004) Atomic and molecular adsorption onIr(111). J Phys Chem B 108(3):987–994.
34. Herron A, Scaranto J, Ferrin P, Li S, Mavrikakis M (2014) Trends in formic acid de-composition on model transition metal surfaces: A density functional theory study.ACS Catal 4(0001):4434–4445.
35. Sprowl LH, Campbell CT, Árnadóttir L (2016) Hindered translator and hindered rotor modelsfor adsorbates: Partition functions and entropies. J Phys Chem C, 10.1021/acs.jpcc.5b11616.
36. Campbell CT, Sprowl LH, Árnadóttir L (April 14, 2016) Calculating equilibrium and rateconstants for adsorbates: 2D ideal gas, 2D ideal lattice gas, and hindered translatormodels. J Phys Chem C, 10.1021/acs.jpcc.6b00975.
37. Hummelshøj JS, Abild-Pedersen F, Studt F, Bligaard T, Nørskov JK (2012) CatApp: Aweb application for surface chemistry and heterogeneous catalysis. Angew Chem IntEd Engl 51(1):272–274.
38. Bligaard T, et al. (2004) The Brønsted–Evans–Polanyi relation and the volcano curve inheterogeneous catalysis. J Catal 224(1):206–217.
39. Liu B, Greeley J (2011) Decomposition pathways of glycerol via C–H, O–H, and C–C bondscission on Pt(111): A density functional theory study. J Phys Chem C 115(40):19702–19709.
40. van Santen RA, Neurock M, Shetty SG (2010) Reactivity theory of transition-metalsurfaces: A Brønsted-Evans-Polanyi linear activation energy-free-energy analysis.Chem Rev 110(4):2005–2048.
41. Dumesic JA, Huber GW, Boudart M (2008) Principles of heterogeneous catalysis.Handbook of Heterogeneous Catalysis, eds Ertl G, Knözinger H, Schüth F, Weitkamp J(Wiley, Weinheim, Germany), pp 1–15.
42. Gokhale AA, Dumesic JA, Mavrikakis M (2008) On the mechanism of low-temperaturewater gas shift reaction on copper. J Am Chem Soc 130(4):1402–1414.
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