Rate‐Reactivity Model: A New Theoretical Basis for …wxjs.chinayyhg.com/upload/Files/20170114092653304/3… · · 2017-01-14Systematic Kinetic Characterization of Heterogeneous

Rate-Reactivity Model: ANew Theoretical Basis forSystematic KineticCharacterization ofHeterogeneous CatalystsGREGORY S. YABLONSKY,1 EVGENIY A. REDEKOP,2 DENIS CONSTALES,3 JOHN T. GLEAVES,4

GUY B. MARIN2

1Parks College of Engineering, Aviation and Technology, Saint Louis University, Saint Louis, MO, 63103

2Laboratory for Chemical Technology, Ghent University, B-9052, Zwijnaarde, Belgium

3Department of Mathematical Analysis, Ghent University, Krijgslaan 281, B-9000, Gent, Belgium

4Department of Energy, Environmental, and Chemical Engineering, Washington University in Saint Louis, Saint Louis,MO, 63130

Received 7 October 2015; revised 21 January 2016; accepted 17 February 2016

DOI 10.1002/kin.20988Published online 28 March 2016 in Wiley Online Library (wileyonlinelibrary.com).

ABSTRACT: Non-steady-state kinetic measurements contain a wealth of information about cat-alytic reactions and other gas–solid chemical interactions, which is extracted from experimentaldata via kinetic models. The standard mathematical framework of microkinetic models, whichare typically used in computational catalysis and for advanced modeling of steady-state data,encounters multiple challenges when applied to non-steady-state data. Robust phenomeno-logical models, such as the steady-state Langmuir–Hinshelwood–Hougen–Watson equations,are presently unavailable for non-steady-state data. Herein, a novel modeling framework isproposed to fulfill this need. The rate-reactivity model (RRM) is formulated in terms of exper-imentally observable quantities including the gaseous transformation rates, concentrations,and surface uptakes. The model is linear with respect to these quantities and their pairwiseproducts, and it is also linear in terms of its parameters (reactivities). The RRM parametershave a clear physicochemical meaning and fully characterize the kinetic behavior of a specific

Correspondence to: Evgeniy A. Redekop, Centre for MaterialsScience and Nanotechnology Chemistry, Department of Chemistry,University of Oslo, Kjemibygningen 0371 Oslo, Norway; e-mail:[email protected].

C© 2016 Wiley Periodicals, Inc.

RATE-REACTIVITY MODEL FOR KINETIC CHARACTERIZATION OF HETEROGENEOUS CATALYSTS 305

catalyst state, but unlike microkinetic models that rely on hypothetical surface intermediatesand specific reaction networks, the RRM does not require any assumptions regarding the un-derlying mechanism. The systematic RRM-based procedure outlined in this paper enables aneffective comparison of various catalysts and the construction of more detailed microkineticmodels in a rational manner. The model was applied to temporal analysis of products pulse-response data as an example, but it is more generally applicable to other non-steady-statetechniques that provide time-resolved rates and concentrations. Several numerical examplesare given to illustrate the application of the model to simple model reactions. C© 2016 WileyPeriodicals, Inc. Int J Chem Kinet 48: 304–317, 2016

INTRODUCTION

Precise kinetic characterization of chemical interac-tions between gas molecules and the surfaces of re-active materials presents a fundamental challenge inthe field of heterogeneous catalysis. Kinetic character-istics are essential for comparing the performance ofdifferent catalysts, elucidating reaction mechanisms,and constructing quantitative kinetic models [1]. Theyare also important outside catalysis, for example, forphysicochemical characterization of noncatalytic ma-terials such as adsorbents, molecular sieves, sensors,energy storage materials, and alloys. Kinetic models ofgas–solid reactions are typically postulated or derivedbased on sequences of elementary steps which, pre-sumably, compose these reactions. The intrinsic chem-ical properties of the material enter models via the ratecoefficients of the hypothetical elementary steps. Here,a novel type of kinetic models is proposed in which theemphasis is shifted to the intrinsic chemical propertiesof materials—the kinetic equations reflect the abilityof a material to transform the composition of a sur-rounding gas, with no specific assumptions about thereaction steps involved.

The kinetics of gas–solid reactions can be very con-voluted due to the complexity of underlying reactionmechanisms as well as the inherent complexity ofthe reactive materials themselves. A reaction mech-anism typically consists of many elementary steps thatcontribute to multiple concurrent reaction pathways.In addition, a number of material-related factors arelikely to complicate kinetic behavior: surface hetero-geneity, crystallographic phase transitions, species ex-change between the surface and the bulk, surface re-constructions, and lateral interactions between surfaceadsorbates. These phenomena contribute to the in situemergence of the material’s actual “working” state andmay lead to significant deviations from idealized (mi-cro)kinetic models constructed for idealized reactionsand surfaces. The active sites for many catalytic reac-tions of technological relevance have not been iden-tified or remain an object of vigorous debate. As vanSanten recently asserted, “Even when the reaction is at

a steady-state, the important consequence of the self-assembled nature of the reactive surface state is thatthe number of reactive centers, to which the rate of thecatalytic reaction is proportional, is not a well-definedconstant” [2]. Similar uncertainties persist in determin-ing which elementary steps are involved in complexreaction mechanisms such as the partial oxidation ofhydrocarbons on metal oxides. d’Alnoncourt et al. con-clude, for example, that “Unscrambling the propaneoxidation network over MoVTeNb oxides into elemen-tary steps in an ambitious and possibly unsolvable taskdue to the multitude of different and interrelated or-ganic reactions happening on the catalyst surface andperhaps also in the gas phase” [3]. Furthermore, theobserved kinetics of chemical transformations are of-ten influenced by, or in some cases even determined by,the concurrently occurring transport phenomena suchas diffusion in microporous confines [4] and withincrystal lattices [5].

Non-steady-state (transient) kinetic methods areuniquely suited for characterization of these elusive“working states” of heterogeneous catalysts and forunraveling the underlying fundamental processes af-fecting their catalytic performance [6]. In a transientexperiment, a catalyst is subjected to time-dependentperturbations, e.g., changes of the gas composition ortemperature. The population of adspecies on a ma-terial’s surface and the material itself respond dy-namically to these imposed perturbations, leading totemporal variations of gas concentrations and chem-ical rates. The resulting time-resolved data provide arich source of information about the reaction kinet-ics exhibited by industrially and academically rele-vant materials. In recent years, non-steady-state ex-perimentation has also found novel applications inspectroscopy, whereby transient processes are usedto enhance surface sensitivity and chemical speci-ficity [7]. However, the experimental and data anal-ysis workflows used in non-steady-state techniquesmust be based on an appropriate formulation of ki-netic models to harness the full potential of time-resolved data for both fundamental and practicalpurposes.

International Journal of Chemical Kinetics DOI 10.1002/kin.20988

306 YABLONSKY ET AL.

The mathematical structure of kinetic models is par-ticularly important for ensuring that the kinetic data,and the inferred kinetic parameters are not only mech-anistically informative but can also be compared ina systematic way between different materials, differ-ent experimental techniques, and different operatingconditions. To satisfy these criteria, a kinetic modelshould have a flexible and robust phenomenologicalform which expresses the relationships between ex-perimentally observable characteristics and some in-trinsic properties of the investigated materials. In gen-eral, transformation rates (R) of gaseous substancesin contact with the material’s surface are complex apriori unknown functions of gas concentrations (Cg),intrinsic descriptors of material’s properties such asrate coefficients (k), concentrations of active sites andintermediate species on the surface (Cs), concentra-tions of the material’s bulk components such as metals(CM) and metal oxides (CMO), and operating conditionsincluding temperature (T) and pressure (P):

R = f(Cg,Cs, CM,CMO, k, T , P

)(1)

Depending on the reactor type used for kinetic mea-surements, some of the quantities from Eq. (1) canbe directly measured with sufficient accuracy, whereasothers can only be assessed indirectly through a seriesof assumptions. The observable quantities in each re-actor type discussed herein will be explicitly pointedout to highlight the scope of kinetic information theycan provide. An optimal phenomenological form ofEq. (1) must be based only on the observable quanti-ties and satisfy several additional requirements. First,it must be suitable for the development of standard ex-perimental protocols for reproducible catalyst prepara-tion and characterization. Second, it must be applicablewithin a broad range of parameters to allow for knowl-edge transfer between different applications. Finally,it must also be suitable for more detailed mechanis-tic interpretation of kinetic data through, for example,mean-field microkinetics and transition state theory(TST). Contrary to steady-state kinetic data, which aretypically analyzed within the formalism of Langmuir–Hinshelwood–Hougen–Watson (LHHW) equations ormicrokinetic models [1], a general modeling frame-work suitable for the description of non-steady-statedata is presently unavailable.

To fulfill this gap, this paper introduces thephenomenological rate-reactivity model (RRM). Thequantities under observation in the RRM are thenon-steady-state rates and concentrations of gaseousspecies as well as the integral uptakes of these species

on the catalyst surface, all of which are availablefrom temporal analysis of products (TAP) measure-ments [8–10] with subsecond temporal resolution. TheRRM represents the gas transformation rates as linearfunctions of gas concentrations, gas uptakes on the sur-face, and all possible pairwise products of these quan-tities. Coefficients of this linear model (the intrinsicreactivities) can be thought of as generalizations of thesticking coefficients that are widely used for adsorp-tion characterization in surface science [11]. Similarto the sticking coefficients, the intrinsic reactivities arefunctions of the catalyst state, which, in turn, dependson the composition of the catalyst, the preparation pro-cedure, which gives the unique catalytic structures, andthe history of the catalytic process.

The proposed model provides an alternative viewof catalytic phenomena—instead of focusing on a cat-alytic reaction, it illuminates the intrinsic properties ofa catalytic material, specifically its ability to chemi-cally transform the surrounding gas environment. Themain advantage of using the RRM for interpretingkinetic data is that the ability of a specific catalyststate with known history to alter the gas composi-tion is conceptualized and measured as an empiricalquantity, i.e., a number or a set of numbers. In ouropinion, it is methodologically valuable to infer suchquantities from experimental kinetic data and to un-derstand their physicochemical meaning before resort-ing to data regression using predefined microkineticmodels. Without judicial data evaluation, model regres-sion may overlook important kinetic features or evenlead to inaccurate mechanistic conclusions, especiallywhen highly coupled physicochemical and transportphenomena occur within the same catalytic process.Contrary to combinatorial fitting of microkinetic mod-els, the RRM reflects only general physicochemicalknowledge about gas–solid reactions and does not in-volve specific mechanistic assumptions. For example,it enforces the linearity of gas rates with respect to gasconcentrations to reflect the fact that elementary gas–solid interactions are, as a rule, monomolecular withrespect to impinging gas molecules, e.g., O2 + M orO2 + 2M, but not 2O2 + M [12]. Stronger mechanisticassumptions, such as postulating a specific combina-tion of elementary steps, are more difficult to justifyand should not be accepted prematurely. We show thatthe RRM reactivities obtained under controlled reac-tion conditions can serve as a basis for rational modeldeduction (or construction), as opposed to model dis-crimination via fitting. Importantly, the RRM providesthe theoretical framework for relating intrinsic kineticproperties to incremental changes in catalyst composi-tion, and ultimately the means to guide the evolutionof catalytic properties on complex technical materials.



KINETIC CHARACTERIZATION OFHETEROGENEOUS CATALYSTS:STATE-OF-THE-ART ANDMETHODOLOGICAL CHALLENGES

This section outlines the current status of kinetic ex-perimentation in heterogeneous catalysis with an em-phasis on methodologies suitable for industrially rel-evant porous and high-surface area materials. Ratherthan being comprehensive, this brief overview aims atproviding a motivational and historical context for thedevelopment of the RRM.

Steady-State Methods

Contemporary high-throughput techniques for kineticcharacterization of heterogeneous catalysts widely em-ploy steady-state kinetic experiments. Laboratory re-actors for steady-state experiments with technologi-cally relevant porous catalysts are typically designedto approximate the behavior of either plug-flow reac-tors (PFR) or continuous stirred-tank reactors (CSTR)[13,14]. The observable quantities in CSTRs are gasconcentrations and their transformation rates, both ofwhich can be measured as a consequence of the highlyuniform composition in such devices. PFRs yield onlythe outlet gas concentrations, since the rates cannot bereadily obtained due to the spatial nonuniformity of thecatalytic bed. The macroscopic coupling of transportphenomena and kinetics in these two idealized mod-els is particularly simple, making them very attractivefor analyzing laboratory data. However, stringent cri-teria for justifying their validity in each particular casemay present significant challenges. Aside from exper-imental challenges, the theory of steady-state kineticsis well established. Usually, intrinsic steady-state dataare described within the framework of LHHW equa-tions, either postulated as phenomenological models orderived from mechanistic microkinetic models [1,15]:

R = k∏

i Cni

i

1 + ∑i KiC

ni

i

(2)

The chemical rates in Eq. (2) are usually ascribed tothe active sites, the concentrations of which are esti-mated via the chemisorption of probe molecules in aseparate titration experiment [16], determined via theuptakes of isotopically labeled species in Steady-StateIsotopic Transient Kinetic Analysis (SSITKA) [17,18],or deduced from other considerations.

There are two major limitations which hinder the es-tablishment of structure–activity relationships and thedevelopment of detailed kinetic models based on thesteady-state approach. One of these limitations stems

from the fact that steady-state rates reflect the kineticsof the slowest (rate-limiting) step in a complex reac-tion network. As a result, steady-state methods typi-cally provide only coarse kinetic characteristics, suchas overall turnover frequencies and selectivities, mak-ing it difficult to study the kinetics of different reactionsteps in detail [19–21]. For the same reason, multiplealternative reaction networks can be hypothesized thatfit steady-state data with comparable statistical signif-icance.

Another major shortcoming of the steady-state ap-proach is its inability to characterize intermediate cat-alyst states that emerge while the catalyst adapts tothe reaction environment. After the intrinsic and ex-trinsic relaxation processes have occurred [22] and thesteady-state composition/structure has developed, thekinetic behavior related to the resulting catalyst state ischaracterized. The details of catalyst evolution towardthe final steady-state remain concealed, including sur-face restructuring, changes of the oxidation state, theestablishment of steady-state coverage of surface inter-mediates, and eventual deactivation. In spectroscopy,the steady-state spectra may be dominated by kinet-ically irrelevant spectator species which are difficultto discriminate from the true reaction intermediates insteady state [23,24]. Despite these limitations, LHHWmodels derived from steady-state data are widely usedand form the basis of conventional as well as combina-torial approaches to catalyst development. The motiva-tion for introducing the RRM comes from the pressingneed to establish an equally standardized, but morepowerful modeling framework for rationalizing non-steady-state kinetic data.

Non-Steady-State (Transient) Methods

Non-steady-state approaches to kinetic characteriza-tion reveal more details about the kinetic behavior ex-hibited by a catalyst during a stimulated relaxationprocess, thus alleviating some of the limitations im-posed by steady-state experiments [6,25]. Non-steady-state experiments with porous catalytic powders werefirst proposed by Bennet et al. in the 1960s [26]based on the classical relaxation method of Eigen andDemayer [27]. In 1974, Kobayashi and Kobayashi [28]developed transient-response experiments in a differ-ential PFR (dPFR), which is more technologically sim-ple than a CSTR and also better maintains catalyst uni-formity when operated at low conversions. In principle,it is possible to measure both concentrations and ratesin such dPFRs, but transient experiments are typicallyconducted in reactors where only the outlet concentra-tions are measured. A host of in situ time-resolvedspectroscopic and scattering techniques have been



employed in recent years to simultaneously character-ize the adspecies and the chemical states of catalysts inaddition to their kinetic characteristics [23,24]. Theseoperando methods are rapidly developing; in the nearfuture, they are expected to produce increasingly re-liable and mechanistically relevant information about“working” catalyst states.

A number of practical challenges exist in system-atic acquisition, reproducibility, and analysis of non-steady-state data, preventing their widespread applica-tion. The primary experimental challenge is to organizea well-defined transport regime suitable for measur-ing intrinsic reaction rates with sufficient time reso-lution [29]. Moreover, conventional forms of kineticmodels may not always be suitable. It would seem, forexample, that the most natural theoretical frameworkfor rationalizing non-steady-state kinetic data is that ofmicrokinetic models [30] which can, in principle, cap-ture the most essential kinetic features of a catalyticprocess:

Ri(t) =∑

s

vi,srs(t)

=∑

s

vi,sks

∏j

Cnj

j (t)∏k

θnk

k (t) (3)

Despite a growing number of successful applications[31,32], developing a microkinetic model for the anal-ysis of non-steady-state data is still a time-consumingendeavor, the effectiveness of which varies from onecatalytic system to another.

Temporal Analysis of Products

TAP is an advanced methodology for precise time-resolved kinetic characterization of high-surface areamaterials, which employs very small pulses of reagentsand low background pressure to achieve a millisecondtime-resolution and superior control over the catalyststate [33]. The observable quantities in TAP experi-ments are the exit flow rates of gas species that arerecorded by a mass spectrometer. Each pulse in a se-quence does not alter the catalyst state significantly(i.e., state defining) and provides a kinetic “snapshot”of its reactive behavior, whereas long sequences of suchpulses are used to gradually alter the catalyst in a con-trolled manner (i.e., state altering) [32]. The conceptionof TAP by Gleaves et al. was originally inspired in theearly 1980s [35] by the molecular beam techniques ofsurface science [11], but unlike molecular beams, it isapplicable for high-surface area porous materials ar-ranged in a packed bed of 200–400 µm particles. The

transport of molecules through the packed bed occursvia well-defined Knudsen diffusion, which provides avery robust transport standard for measuring reactionkinetics.

An advantageous combination of transport equa-tions and boundary conditions in the TAP reactor en-ables multiple approaches to data analysis. TAP dataanalysis has developed in two complementary direc-tions. Directly observed exit-flow rates can be usedto estimate the rate constants of a preconceived mi-crokinetic model, either by moment analysis for sim-ple models [36,37] or by nonlinear regression formore complex reaction networks [38,39]. This ap-proach shares all the benefits and disadvantages ofcombinatorial model fitting. Alternatively, TAP datacan be used to obtain kinetically “model-free” cat-alyst characteristics. The key property enabling theextraction of such characteristics is the spatial uni-formity of the catalytic zone inside the microreactor.Analogous to dPFRs, the issue of sample nonunifor-mity was resolved by packing the catalytic sampleinside the reactor within a narrow zone located be-tween two longer inert zones. This packing config-uration is known as the thin-zone TAP reactor [40].A combination of well-defined Knudsen transport andthin-zone packing confers unique properties to TAP ex-periments, which arise from the mathematical status ofthe governing reaction–diffusion equations. In particu-lar, the remarkable degree of spatial uniformity withinthe catalytic zone enables the chemical rates and gasconcentrations to be derived as secondary observablesfrom the primary exit-flow rate data. Algorithms basedon these properties have recently been developed toeffectively deconvolute kinetics from transport with-out requiring assumption of the underlying reactionmechanism. The resulting “model-free” characteris-tics can be utilized for standardized characterization ofmaterials.

The first mechanistically “model-free” approachwas developed by Shekhtman et al. based on mo-ment analysis of primary TAP data [34,41]. The ze-roth, first, and second statistical moments of the exper-imental exit-flow rate curves were related to Laplacereactivities—linear coefficients of the following phe-nomenological expansion of the reaction rates in termsof gas concentrations and their temporal derivatives:

Ri(t) = ri,0Ci(t) + ri,1dCi(t)

dt+ ri,2

d2Ci(t)

dt2+ . . .

(4)

Laplace reactivities have clear physicochemicalmeanings independent of the underlying reaction



mechanism: r0 (s−1) represents an apparent reactionconstant of a given gas-phase species, dimensionlessr1 is an apparent equilibrium coefficient between a gasand some surface intermediate, and r2 (s) is an appar-ent time delay caused by processes on the surface or inthe bulk of the catalytic material in question. After theinitial analysis, these Laplace reactivities can be fur-ther interpreted in terms of detailed mechanisms andkinetic models [34]. This approach was a major stepin the development of “model-free” characterizationof catalysts and has been applied to analyze severalreactions of practical value [42,43], but it suffers fromtwo major limitations. First, primary time-resolveddata must be integrated over the entire duration of aTAP pulse to calculate the statistical moments of exit-flow rate distributions, the precursors to Laplace reac-tivities. Second, the analysis is strictly valid only forlinear kinetic dependencies due to the limitations im-posed by the integral Laplace transformations used inderivations.

Roelant et al. proposed another data analysis tech-nique [44] based on Laplace transformations, in whichexit-flow rates are used to deduce the connectivityfeatures of a general pseudo-monomolecular reac-tion network, i.e., the number of reaction interme-diates between different gases. The assumption ofpseudo-monomolecularity in their approach is neces-sary to apply the machinery of Laplace transforma-tions and transfer matrices, and this assumption im-poses the same linearity constraints as the approach ofShekhtman et al.

The Y-Procedure, an algorithm developed byYablonsky et al. [9] based on the fast Fourier trans-form technique, circumvents the limitations of otherapproaches and readily yields kinetically “model-free”time-dependent rates and gas concentrations for eachgas based on the observed exit-flow rates: F (t) →R(t), C(t). The temporal resolution of data is preservedduring this transformation, and the algorithm is suit-able for nonlinear kinetics. Transient rates and concen-trations derived via the Y-Procedure were previouslyused to elucidate the coverage-dependent rate con-stant of oxygen dissociation on polycrystalline Pt [45],to elucidate the phenomenon of momentary equilib-rium for reversible reactions [46], and to suggest ageneral mechanism discrimination strategy based onthe temporal coherency of the rate-concentration tran-sients [10]. Herein, the time-resolved quantities avail-able from TAP data via the Y-Procedure are usedto establish a novel phenomenological model of cat-alytic kinetics, which elicits the intrinsic propertiesof a material that determine how the gas concen-trations are related to the chemical transformationrates.

THEORETICAL RESULTS

The RRM is proposed in this section as the theoreticalbasis for the non-steady-state kinetic characterizationof heterogeneous catalysts. The objective of this modelis to elucidate the intrinsic ability of a solid in a givenchemical state to alter the composition of the gas mix-ture. Before the model is formally introduced and itsproperties are discussed, it is necessary to define whatis meant by a chemical state of the catalyst (catalyststate). Several additional concepts are also needed toclassify such states and to describe how these states areaffected by their kinetic measurement.

Preliminary Considerations: Catalyst Statesand Kinetic Experiments

As stated in Eq. (1), the intrinsic kinetic behavior ofsolid materials toward gas-phase species is determinedby a number of coinfluencing factors including thoserelated to the gas phase, to the chemical state of thecatalyst, and to the catalyst temperature. At a giventemperature, we are interested in the instantaneous ki-netic properties of a specific catalyst state exhibitedin response to a standard (small) perturbation of thegas composition. By a catalyst state, we understand agiven active configuration of all its parts which exhibitscertain kinetic behavior under reactive conditions. Inour view, the catalyst state is an emergent property.The relevant parts and properties defining the catalyststate include, but are not limited to, the bulk and sur-face concentrations of active components, structuralcharacteristics and surface morphology, the number ofactive sites, and, importantly, the concentrations of ad-sorbed species including reactive intermediates. Someof these catalyst characteristics, such as bulk composi-tion and structure, are determined in the early stages ofcatalyst preparation and usually do not change withina certain range of operating parameters. Other char-acteristics, such as surface composition and structure,concentration of adsorbed species, or the number ofaccessible free active sites, are determined during cat-alyst pretreatment and during the initial relaxation pe-riod when the catalyst is brought into contact with thereactive environment. It is important to note that in-dividual catalyst states can only be distinguished bytheir kinetic behavior since catalytic properties cannotbe determined from limited structural and composi-tional information. As far as kinetics are concerned,an individual catalyst state is defined by its distinct ki-netic properties that it exhibits with respect to the gascomposition.

The catalyst state and the perturbation used as akinetic probe are intricately related. Based on the



Figure 1 Classification of adspecies, catalyst states, andkinetic measurements.

measurement timescale, the rates of relevant physic-ochemical phenomena, and the size of the perturbationused as a probe, several types of adspecies, catalyststates, and kinetic experiments can be delineated (seeFig. 1). Further discussion of the RRM model requiresdefinitions and brief explanations of these types.

History-dependent compositional and structuralcharacteristics of the catalyst, which can be treatedas generalized adspecies, can be classified into “slow”and “fast,” depending on the rate of their change undergiven experimental conditions relative to the measure-ment time. The concentrations of “fast” species changesignificantly within the measurement in response to ex-ternal perturbation. The pool of such species can be rep-resented through the history of gas uptake and release,given that time-resolved information is available aboutgas transformation rates during the measurement. Ifthe catalytic surface is treated as storage with certaincapacity, the accumulation of “fast” species in the stor-age can be represented through the mass balance as adifference between the net influx (uptake of relevantgas species by the catalyst) and the net outflux (releaseof gas species from the catalyst):

Uj (t) = Uj,init +t

∫0

∑v+

i R+i (τ )dτ

−t

∫0

∑v−

i R−i (τ )dτ (5)

Here, the indices j and i are independent; j refers to dif-ferent storages which rates may be expressed throughthe rates of different gases i. Depending on a specificstoichiometry, index i will cover all or some of thegases, but only those gases that contribute to the uptakej. Instantaneous surface storages U(t) are, generally, notequivalent to the concentrations of hypothetical indi-vidual intermediates, but in many practical cases thesequantities provide convenient means to characterizethe evolution of the catalyst state on short timescales.The relationship between surface storages and distinctintermediates depends on the connectivity features ofthe underlying reaction network, as discussed in moredetail in Redekop et al [10]. The concentrations of“slow” species remain relatively intact during a sin-gle measurement, but they can be changed from onemeasurement to another by means of pretreatment of

the catalytic surface. For example, the oxidation stateof metal oxide catalysts can be controlled by exposureof the surface to either oxidative or reducing agents,depending on which state (oxidized or reduced) is de-sired. The concentration of available active sites canbe changed by partial poisoning or coking.

Kinetically distinct catalyst states can be classifiedinto “stable” and “unstable,” depending on whether ob-served kinetics change between sequential non-steady-state measurements. Although non-steady-state exper-iments are conducted to probe the kinetics of multiplecatalysts states, successive probes of the same catalyststate should, in principle, reproduce the same char-acteristics. In practice, the catalyst state may not re-main constant in between the measurements, even inthe absence of external perturbations. The stability ofcatalyst states can be tested, for example, using TAPpump–probe experiments in which one perturbation isused to prepare a material in a special state and then,after an adjustable time delay, a second perturbation isused to characterize this prepared state. Between thepump and the probe perturbations, the prepared cata-lyst state may spontaneously decay due to desorptionof intermediates, migration of active species over thesurface or into the bulk of the aterial, or morphologicalchanges of the active sites. The stability and the rateof decay of a catalyst state can be evaluated only rela-tive to the timescale and repetition frequency of kineticmeasurements.

Kinetic measurements themselves can be classifiedinto “state defining” or “state altering” with respectto the catalyst, depending on whether a single mea-surement alters the catalyst state significantly. Steady-state measurements are initially state altering while thecatalyst adapts to the reactive environment during therelaxation period: Steady-state coverages of surfaceintermediates are established, surface nanostructuresundergo morphological and compositional changes inresponse to the gas phase, and active sites are partiallyblocked by poisons. Non-steady-state measurements,on the contrary, offer more flexibility and more precisecontrol over the catalyst state. If the probing pertur-bation is small, i.e. the number of pulsed moleculesis small compared to the number of active species onthe catalyst, then a single measurement will character-ize the kinetic behavior of a given catalyst state with-out significantly altering this state. A series of suchstate-defining measurements can be used to performa controlled gradual modification of the catalyst state,such as during multipulse TAP experiments [37,42,43].SSITKA provides another example of a state-definingmeasurement in which the isotopic composition of thefeed is switched without perturbing the established cat-alyst state.



Rate-Reactivity Model

The core concept and empirical quantity in chemicalkinetics is the chemical transformation rate. The trans-formation rate of a gas substance in contact with asolid reactive material, such as a catalyst, is a com-plex function of the gas-phase composition, composi-tion and structure of the solid, temperature, and otherprocess conditions (see Eq. (1)). Our primary objec-tive is to characterize the intrinsic ability of a specificcatalyst state to alter the chemical composition of thesurrounding fluid, as sampled by imposing a composi-tional perturbation and monitoring the time-dependentresponse. To accomplish this objective, a novel type ofkinetic models is first proposed for a single gas interact-ing with a material’s surface. Then, a general form ofthe model is developed for multiple interacting gasesand more complex reactions. Finally, a more practi-cal matrix formulation is introduced and the issues ofparameter estimation are discussed.

Formalism for a Single Gas and General Proper-ties. Consider a single gas i interacting with a reac-tive material in such a way that, at least for now, thetransformation rate of this gas does not depend on anyother gases. The simplest example of such an inter-action would be chemisorption, but certain catalyticreactions also belong to this category. For example, anirreversible conversion of one gas into another via animpact mechanism. To relate the transformation rate ofgas i to some intrinsic properties of the material, wepropose the following time-dependent kinetic equa-tion, a special form of Eq. (1)

Ri(t) = ψcCi(t) + ψUUi(t) + ψCUCi(t)Ui(t) (6)

The transformation rate of ith gas species on the left-hand side of this equation is defined in a conventionalmanner as dC/dt for products and –dC/dt for reac-tants. Unlike microkinetic models, the terms on theright-hand side of this equation contain only experi-mentally observable quantities: the gas concentrationand its surface uptake. These observables are multi-plied by linear coefficients—reactivities—which arethe model parameters reflecting the intrinsic catalystproperties.

The physicochemical meaning of different terms inEq. (6) can be better understood by considering theirdimensionality and by comparing them to an equivalentmicrokinetic model for a particularly simple and well-understood example. Consider a single-site reversiblechemisorption of gas A on the surface of a solid ma-terial (A + Z = AZ). The surface uptake of A, in thiscase, coincides with the surface concentration of AZ,

and the reaction rate is given by

RA = kads (NS − CAZ) CA − kdesCAZ

= kadsNSCA − kdesCAZ − kadsCAZCA (7)

where kads and kdes are the adsorption and desorp-tion rate coefficients, respectively, and Ns is the totalconcentration of adsorption sites. From the compari-son of Eqs. (6) and (7) it follows that ψC = kadsNS ,ψU = −kdes , and ψCU = −kads . The ψC reactivity isessentially an apparent adsorption constant related tothe sticking coefficient of an empty surface, s0. TheψCU reactivity reflects the attenuation of the adsorp-tion rate by the diminishing number of available sites.As later demonstrated by numerical examples (see thesection Conclusions and Perspectives), this term canbe neglected in state-defining experiments. Finally, theψU reactivity is equivalent to the desorption rate coef-ficient.

Figure 2 depicts numerical simulations which wereperformed to exemplify the proposed model. Syntheticdata were first simulated using Eq. (7) coupled witha numerical transport model of thin-zone TAP experi-ments [47] (see the Appendix for simulation details).Realistic noise was added to the simulated data ac-cording to the TAP noise model [48]. Then, the simu-lated exit-flow rates were subjected to the Y-Procedureto reconstruct the rate-composition transients withinthe catalytic zone, i.e., R(t), C(t), and U(t) for gas A.Finally, the RRM was applied to these transients toestimate reactivities so that they might be comparedto the original microkinetic parameters. This examplewas designed to demonstrate the RRM’s performancewhen it is challenged by either a state defining (Np= 1 × 10−9 mol per pulse) or a state altering (Np =1 × 10−7 mol per pulse) perturbation. The concentra-tion of available adsorption sites was kept in both casesat NS = 1 × 10−2 mol/kg for mcat = 10 × 10−6 kg.Simulation parameter values were selected to facili-tate this state-defining vs. state-altering methodologi-cal demonstration, rather than to approximate any spe-cific experimental studies. In Fig. 2A, the simulatedrate of gas A is compared to the rate reconstructedby the Y-Procedure for a state-defining experiment.The three estimated terms of Eq. (6) are also shown,demonstrating how the observed rate can be decom-posed into its constituents using the RRM. The cor-responding reactivity values were in agreement withthe original microkinetic parameters. As expected, theRRM term corresponding to the C·U product was neg-ligible in comparison with the C and U terms. For thestate-altering experiment, on the other hand, the RRMsuccessfully captured the C·U product term, suggesting



Figure 2 The RRM analysis of simple reversible adsorption. The transformation rate of gas A is shown as simulated (blacksolid line), reconstructed via the Y-Procedure (white circles), and decomposed into its RRM terms (blue, C term; green, U term;and red, CU term; different terms are also marked by the corresponding reactivity symbols on the graph): (A) state-definingexperiment and (B) state-altering experiment.

that the model is sensitive to the coverage-inducedchanges of kinetics.

This numerical example of a simple Langmuriankinetics is given solely to illustrate the RRM principleand to show that it is practically feasible to decomposea realistic TAP curve into separate RRM terms. How-ever, such detailed analysis of even a simple single-component adsorption may provide a valuable tool forsystematic catalyst characterization, especially whenadsorption does not comply with the law of mass ac-tions due, for example, to island formation [49]. Thesame numerical example also covers an important classof catalytic reactions in which the product is formeddirectly from the impacting reagent, i.e., the impactmechanism [1] also known as Eley–Rideal. For theimpact mechanism, the rates of reagent consumptionand product formation coincide and the kinetics withineach pulse-response experiment is equivalent to theadsorption case.

The RRM offers two properties which are advanta-geous for systematic kinetic characterization: linearitywith respect to parameters (reactivities ψ) and linear-ity with respect to instantaneous gas concentrations.The linearity with respect to parameters enables robustand statistically sound inference of empirical reactivi-ties from noisy data by directly using well-establishedlinear estimation techniques, such as matrix inversionvia singular value decomposition (SVD). The linearitywith respect to gas concentrations is grounded in thecommon assumption that only one gas molecule canparticipate in an elementary act during a complex gas–solid reaction. Simultaneous interactions between twogas molecules and a single catalytic site are consideredhighly improbable. In steady-state data, the apparent

nonlinearity of rates with respect to gas concentrationstypically arises due to the changes of surface coverageswhich are induced by considerable variations of thegas composition. In a non-steady-state regime, suchchanges of surface coverages must be explicitly ac-counted for. Since the RRM operates with observableintegral surface uptakes, rather than with the coveragesof individual intermediates, the linearity is enforced apriori in the mathematical form of the model. Withinthe RRM’s range of validity, large changes of gascomposition are avoided, and the kinetic experimentsare conducted in either state-defining or moderatelystate-altering regimes. For state-defining experiments,coverages do not change significantly during the mea-surement, and nonlinearity in gas concentrations neverarises [44]. For moderately state-altering experiments,coverages change to a certain degree, but the resultingchanges are accounted for by the model through theterms proportional to surface uptakes

(ψUU (t)

). More

substantial coverage changes or changes of the cata-lyst itself do not fundamentally contradict the RRM,since reactivities ψ are allowed to be history depen-dent. However, the reactivities are not, by definition,functions of the instantaneous gas composition. Im-portantly, the initial catalyst state is considered to beknown from the pretreatment or a preceding kineticmeasurement.

Generalization for Multiple Gases. Catalytic reac-tions typically involve multiple gases which engage ina complex network of reactions on catalytic surfaces.The transformation rate of each gas may be influencedby other gases in the reacting mixture. To account forthese mutual influences, the RRM equation (6) can be



generalized as follows:

Ri(t) =N∑

k=1

ψCi,kCk(t) +

M∑j=1

ψUi,jUj (t)

+N∑

k=1

M∑j=1

ψCUi,k,jCk(t)Uj (t)

+M∑

j=1

M∑j=1

ψUUi,k,lUj (t)Ul(t) + ψi,0 (8)

Equation (8) is purely phenomenological; it is postu-lated here without any “bottom-up” justification fromthe theory of chemical reaction networks and elemen-tary reaction steps. Nevertheless, the RRM equation isbroadly applicable to virtually all chemical process in-volving gases impinging on surfaces, including thoseprocesses that are accurately described by the afore-mentioned more rigorous theories. In fact, any non-steady-state data compliant with a microkinetic modelcan also be represented in terms of the RRM.

Each term on the right-hand side in Eq. (8) corre-sponds to a compound reaction channel by which gas iis produced or consumed by the reactive material. Thefirst term represents the production or consumption ofgas i as a result of direct gas–solid interactions betweenthe material and each individual gas present in the re-action environment. This term in Eq. (8) is linear withrespect to the gas-phase concentrations C(t). There isa reactivity parameter ψC associated with each gasconcentration in this term, which has the dimensionof an apparent rate constant, 1/s. In an equivalent mi-crokinetic model, these reactivities would correspondto the rate constants of impact steps (e.g., Eley–Ridealmechanism) or the apparent rate coefficients of com-binations of steps that act like a single impact stepunder these particular reactive conditions and on theappropriate timescale.

The second term on the right-hand side of Eq. (8)represents the reaction channels through which gas i isproduced from the surface intermediates that contributeto one of the surface uptakes, in the simplest case, pro-duced from one of the intermediates without partici-pation of the gas phase. The corresponding reactivitiesψU have the dimension of a desorption rate constant.The following two terms in the equation reflect the pro-duction or consumption of gas i via complex surfaceprocesses involving either a gas and an intermediate ormultiple intermediates. Finally, the last term—the “ze-roth” reactivity ψo−—represents the ability of the solidto produce gas i without direct participation of othergases or adspecies that contribute to surface uptakes.

This term has the dimension of a reaction rate. Thecross-coupling C·U and U·U terms as well as zerothreactivity term are introduced for completeness and tobroaden the scope of the model. However, it remainsto be shown whether these terms will be regularly em-ployed in the analysis of experimental data.

The RRM equation (8) or its matrix form (10) canbe interpreted geometrically: They describe the behav-ior (trajectory) of a reactive system in a phase spacethat is spanned by an optimal set of observable tran-sients (C, U) and their products (C·U, U·U). If a givenset of coordinates is optimal, the system’s trajecto-ries in the phase space will be confined to a planarhypersurface with well-defined inclinations given byreactivities. The optimality criteria for the dimensionsof the phase space, i.e., the terms related to concen-tration transients and their products that must be in-cluded into Eq. (8), will have to be adjusted for eachparticular data set. This geometrical interpretation ofrate-composition transients was discussed by Redekopet al. for irreversible adsorption [45], reversible ad-sorption [46], as well as a multistep mechanism withadditional surface steps [10]. In the latter case, the pla-narity of the characteristic phase-space hypersurfacewas interpreted in terms of kinetic coherency betweendifferent rate-composition transients. The RRM fur-ther extends these concepts and provides a convenientmathematical framework for their analysis.

The following point about the RRM equation shouldbe stressed: only those surface species that change con-siderably within an experiment (i.e., adspecies “fast”on the measurement timescale) contribute to the dy-namic surface uptakes in Eq. (8). The concentrationsof those surface species which remain largely unper-turbed by a given experiment (i.e., adspecies “slow” onthe measurement timescale) enter Eq. (8) through thereactivity parameters (ψ), quantities which generallydepend on the catalyst state. This feature of the RRM issimilar to the reduction of nonlinear mechanisms to lin-ear pseudo-monomolecular mechanisms in the limit ofstate-defining experiments [42]. However, the RRM’slinearity range is expected to be broader than that ofpseudo-monomolecular approximation due to the factthat the reaction steps between two “fast” surface inter-mediates can be accounted for, to some degree, throughU·U terms in Eq. (8). “Slow” species that are not, ingeneral, well defined may also gradually change on alonger (global) timescale. The RRM equation locallyapproximates the behavior of such systems in the limitof infinitesimal change of these slow-varying compo-nents (see also Fig. 3).

Matrix Formalism and Practical Inference of Reac-tivities. Coupled linear RRM equations for individual



Figure 3 (A) The standard workflow of TAP data transformations: step 1, the Y-Procedure; step 2, rate integration; and step3, the RRM analysis. (B) State-by-state catalyst characterization (see explanation in the text).

gases can be conveniently written in a generalized ma-trix form for the practical purposes of parameter es-timation. Time-resolved kinetic data for a single gas(i.e., rate, concentration, and uptake) form a set of Nt

discrete temporal values which can be arranged as fol-lows according to Eq. (6)

⎡⎢⎣

R(t1)R(t2)

...

⎤⎥⎦

︸︷︷︸Nt×1

=

⎡⎢⎣

C(t1) U (t1) C(t1)U (t1)C(t2) U (t2) C(t1)U (t2)

......

...

⎤⎥⎦

︸︷︷︸Nt×3

·⎡⎣ ψC

ψU

ψCU

⎤⎦

︸︷︷︸3×1

(9)

Transient data for multiple interacting gases can beanalyzed using the matrix form of a more generalequation (8)

⎡⎢⎣

Ri(t)Ri+1(t)

...

⎤⎥⎦

︸︷︷︸(NG×Nt )×1

=

⎡⎢⎣

{1, C,U,CU,UU } (t) . . . {0}...

. . ....

{0} . . . {1, C,U,CU,UU } (t)

⎤⎥⎦

︸︷︷︸(NG×Nt )×(NG×(N+1))

·

⎡⎢⎣

ψi

ψi+1...

⎤⎥⎦

︸︷︷︸(NG×(N+1))+1

(10)

In shorthand notation, Eq. (10) can be written asR = Aψ , where R is the vector of rate values for NG

gases, � is the vector of reactivity values includingzeroth reactivity, and A is the reactivity block matrixcontaining the time-dependent values of N kineticallyrelevant composition variables plus one at each diago-nal position (corresponding to zeroth reactivities) andzeros everywhere else.

It should be noted that the rate vector R containsonly the transformation rates of gas substances, whichare in this case observable quantities, but not the trans-formation rates of surface storages. The transient stor-ages (U) are calculated by integrating certain linearcombinations of the gaseous transformation rates (seeEq. (5)). Therefore, the rates of these storages are thelinear combinations of the gaseous rates R. Additionalequations for the rates of storages would not carry anyadditional information about the system beyond what



is already contained in the equations for the gaseousrates.

Equation (10) represents a linear expansion of thetransformation rates around the point of zero gas con-centrations and surface uptakes. Another definition ofintrinsic reactivities follows from this assertion: Theyare the elements of the matrix of the first partial deriva-tives (Jacobian) of rates with respect to the basis of Nkinetically relevant composition variables (C, U, C·U,U·U); zeroth reactivities form a vector of transforma-tion rates when all composition variables are zero. Thenumber of composition variables in the basis of theJacobian, i.e., number of terms retained from the gen-eral RRM equation (8), can be optimized to providethe best data description. Beginning with essential Cand U terms, the basis can be expanded stepwise byadding one term at a time from all possible C, U, C·U,and U·U terms. The F test can be performed on eachstep to assess whether the inclusion of another term hasimproved the “goodness of fit” between the data andthe model. Alternatively, rival models characterized bydifferent degrees of refinement can be discriminatedbased on the F test to decide which composition vari-ables are more important for capturing the essentialdata features. For problems of a typical size, thesecalculations can be efficiently performed with mod-ern computers, since the essential complexity is deter-mined by the number of parameters and not by the sizeof a data set.

Standard linear estimation methods can be conve-niently applied to deduce the reactivity values from aset of noisy time-dependent measurements. Matrix Ain Eq. (10) contains all the composition transients in-cluded in the basis of the RRM expansion, includinggas concentrations, surface uptakes, and various prod-ucts thereof. This matrix can be decomposed using theSVD procedure to yield

R = U�V T ψ (11)

By inverting the decomposition, we arrive at thefollowing expression for the estimated reactivitymatrix:

ψ = V �−1UT R (12)

In practice, only the most important SVD compo-nents should be retained to filter out the noise [48]from the experimental rate-concentration transients.The criterion for rejecting or preserving componentsis the relative size of the corresponding elements inthe � matrix; typically all components above a cer-tain threshold are kept, and the remainder is dis-carded. The threshold can be defined heuristically,

based on the desired accuracy and experimental errorlevels [50].

Systematic Strategy of Catalyst CharacterizationBased on the RRM. A general three-step workflowemerges for the analysis of TAP data based onthree data transformations (Fig. 3A): (step 1) the Y-Procedure to translate exit-flow rates into thin-zonerate-concentration transients, (step 2) rate integrationto calculate instantaneous surface uptakes, and (step 3)estimation of RRM reactivities via a SVD decompo-sition of the data matrix. For comprehensive catalystcharacterization, this workflow must be repeated forall individual reactants, products, and their mixturespulsed over a broad range of prepared standard catalyststates, with an interpulse temporal spacing covering awide span of relaxation times. In Fig. 3B, the progres-sion of such comprehensive state-by-state characteri-zation is depicted schematically. The catalyst state isvaried along a path between two limiting stable statesof some natural scale which could represent the scaleof oxidation states, the scale of coking degree, or thescale of surface coverages.

Catalyst variations along the global path betweenthe limiting states can typically be ascribed to theevolution of “slow” species or other slowly vary-ing catalyst properties such as the number of avail-able active sites or surface morphology. At certainpoints along the path, the current catalyst state issampled by state-defining perturbations involving gasreagents. These perturbations, depicted in Fig. 3Bas planes intersecting the path, may create tempo-rally localized variants of the catalyst state which in-volve short-lived intermediates and other “fast” cat-alyst species. The corresponding time-resolved datawithin each intersecting plane are analyzed using theworkflow from Fig. 3A to yield a set of character-istic reactivities (�). The reactivities fully describethe kinetic behavior of specific catalyst states. Then,the lifetimes, stability, and kinetic properties of dif-ferent states can be compared by juxtaposing theirreactivities (ψ vs

∼ψ) obtained (a) for different time

delays (�t) between the pump and the probe pertur-bations, (b) for measurements of the same state onthe global path, but repeated after some time (�t′),or (c) for sequential catalyst states sampled along theglobal path with certain periodicity (�τ ). With noprior knowledge of the reaction mechanism, the re-sulting reactivity matrix reveals the degree to whichdifferent composition variables affect the rates of allparticipating gases. This information can guide therational construction of more detailed microkineticmodels.



CONCLUSIONS AND PERSPECTIVES

A novel mathematical form was proposed for kineticmodels used for non-steady-state characterization ofheterogeneous catalysts and other reactive materials.The RRM operates with only experimentally observ-able quantities; the transformation rates of all gasesin the reaction mixture are represented as linear func-tions of the gas concentrations, instantaneous gas up-takes on the catalyst, and various pairwise products ofconcentrations and uptakes. The linear coefficients ofthis phenomenological expression—the reactivities—fully describe the kinetic behavior of a given catalyststate in response to a (small) perturbation of the gascomposition. The development of this model requiredformal definition of a catalyst state—a given activeconfiguration of all its parts which exhibits certainkinetic behavior under reactive conditions. Differenttypes of catalyst states were distinguished based ontheir stability and characteristic decay rates relative tothe timescale of kinetic measurements.

Although generally applicable, the RRM was in-troduced as a framework for processing time-resolveddata originating from the TAP technique. The math-ematical advantages of the RRM and the experimen-tal advantages of TAP were combined into a system-atic strategy for comprehensive catalyst characteriza-tion based on kinetic screening of possible stable andunstable catalyst states. This strategy can be used forcomparison of transient data obtained under disparateexperimental conditions, from the high vacuum dataobtained for model catalysts by surface science to thesteady-state kinetic data obtained for industrial cataly-sis under atmospheric pressure. For steady-state data,the RRM can be applied at the limits of the reduc-tion/oxidation scale or the scale of surface coverages.In the limit of zero coverage or when the catalyst isfully reduced, the rate is nearly linear with gas concen-tration. In the limit of fully covered surface or whenthe catalyst is fully oxidized, the rate is also linear(constant).

The RRM will be further developed through paral-lel investigation of its general properties and its prac-tical applications. The general properties of the RRMwill be determined by considering a broad range ofmodel reaction mechanisms, with a special emphasison those processes that may violate microkinetic mod-els, such as the transport of active species on catalyticsurfaces or within crystal lattices. The RRM will alsobe used to characterize the kinetic behavior of complexreaction networks with multiple intermediates and re-action routes. Such mechanisms are increasingly in-voked in computational catalysis, but their complex-

ity approaches the limits of experimental falsifiabil-ity (i.e., the ability to be tested experimentally) dueto a large number of hypothetical surface intermedi-ates and elementary steps. The RRM operates usingonly observable quantities and, although it is purelyphenomenological, it may provide a convenient toolfor standardized catalyst characterization, even for re-actions that are challenging to model such as partialoxidation of hydrocarbons. However, the ultimate testfor the RRM’s utility and the primary goal for furtherinquiries is to apply the model to a wide variety of ac-tual experimental data sets and practical case studies.The same model can be used to assimilate supplemen-tary spectroscopic and structural data from operandocharacterization techniques. To conclude, the RRM es-tablishes a firm theoretical basis for systematic non-steady-state catalyst characterization. Time-resolveddata that are generated by high-throughput transientexperimentation and analyzed via the RRM have thepotential to open a new era of “big data” in the field ofinterrogative kinetics and rational catalyst design.

APPENDIX

The numerical simulations utilized a standard TAP ge-ometry with 1-mm catalytic zone containing 10 ×10−6 kgcat which was sandwitched between two14.5 mm inert zones (cross-sectional area 1.2 ×10−5 m2). The reference diffusivity of gas A (28 amu)at 298 K was assumed the same in all zones and was setto 2.5 × 10−3 m2/s. The concentration of adsorptionsites was set to 1 × 10−2 mol/kgcat, whereas the adsorp-tion and desorption constants were set to 100 m3/mol/sand 10 1/s, respectively. The pulse intensities were setto 1 × 10−9 mol per pulse for the state-defining and1 × 10−7 mol per pulse for the state-altering experi-ment. The model was solved using TAPFIT software.More details about the simulation method can be foundelsewhere [46,47].

The authors acknowledge the Long-Term StructuralMethusalem Funding by the Flemish Government and fund-ing provided by the Fund for Scientific Research Flanders(FWO). E.A.R. acknowledges the Marie Curie InternationalIncoming Fellowship granted by the European Commission(grant agreement no. 301703).

BIBLIOGRAPHY

1. Marin, G. B.; Yablonsky, G. S. Kinetics of Chemical Re-actions: Decoding Complexity; Wiley-VCH: Weinheim,Germany, 2011.



2. van Santen, R. A. Angew Chem, Int Ed 2014, 53(33),8618–8620.

3. d’Alnoncourt, N. R.; Csepei, L.-I.; Havecker, M.; Girgs-dies, F.; Schuster, M. E.; Schlogl, R.; Trunschke, A. JCatal 2014, 311, 369–385.

4. Olsbye, U.; Svelle, S.; Bjørgen, M.; Beato, P.; Janssens,T. V. W.; Joensen, F.; Bordiga, S.; Lillerud, K. P. AngewChem, Int Ed 2012, 51(24), 5810–5831.

5. Grasselli, R. K. Top Catal 2002, 21(1–3), 79–88.6. Berger, R. J.; Kapteijn, F.; Moulijn, J. A.; Marin, G. B.;

De Wilde, J.; Olea, M.; Chen, D.; Holmen, A.; Lietti,L.; Tronconi, E.; Schuurman, Y. Appl Catal A 2008,342(1–2), 3–28.

7. Urakawa, A.; Burgi, T.; Baiker, A. Chem Eng Sci 2008,63(20), 4902–4909.

8. Gleaves, J. T.; Yablonsky, G.; Zheng, X.; Fushimi, R.;Mills, P. L. J Mol Catal A: Chem 2010, 315(2), 108–134.

9. Yablonsky, G. S.; Constales, D.; Shekhtman, S. O.;Gleaves, J. T. Chem Eng Sci 2007, 62(23), 6754–6767.

10. Redekop, E. A.; Yablonsky, G. S.; Constales, D.; Ra-machandran, P. A.; Gleaves, J. T.; Marin, G. B. ChemEng Sci 2014, 110, 20–30.

11. Libuda, J.; Freund, H.-J. Surf Sci Rep 2005, 57(7–8),157–298.

12. Chorkendorff, I.; Niemantsverdriet, J. W. Concepts ofModern Catalysis and Kinetics; Wiley: Hoboken, NJ,2007.

13. Doraiswamy, L. K.; Tajbl, D. G. Catal Rev 1974, 10(1),177–219.

14. Weekman, V. W. AIChE J 1974, 20(5), 833–840.15. Hougen, O. A.; Watson, K. M. Chemical Process Princi-

ples, Part III, Kinetics and Catalysis; Wiley: New York,1947.

16. Vannice, M. A. In Kinetics of Catalytic Reactions;Springer US: New York, 2005; pp. 14–37.

17. Happel, J. Chem Eng Sci 1978, 33, 1567.18. Shannon, S. L.; Goodwin, J. G. Chem Rev 1995, 95(3),

677–695.19. Boudart, M. Chem Rev 1995, 95(3), 661–666.20. Spivey, J. J.; Sanati, M.; Kogelbauer, A.; Margitfalvi,

J. L. Turnover Frequencies in Metal Catalysis: Mean-ings, Functionalities and Relationships; Royal Societyof Chemistry: Cambridge, UK, 2004.

21. Ribeiro, F. H.; Schach von Wittenau, A. E.;Bartholomew, C. H.; Somorjai, G. A. Catal Rev 1997,39(1–2), 49–76.

22. Temkin, M. I. Kinet Catal 1976, 17, 945.23. Topsoe, H. J Catal 2003, 216(1–2), 155–164.24. Weckhuysen, B. M. Phys ChemChem Phys 2003, 5(20),

4351–4360.25. Bennett, C. O. In Advances in Catalysis; Werner O.,

Haag, B. C. G., H. K., Eds.; Academic Press, 1999;Vol. 44, pp. 329–416.

26. Bennett, C. O. AIChE J 1967, 13(5), 890–895.27. Eigen, M.; Demayer, L. In Techniques of Organic Chem-

istry; Weisberger, A., Ed.; Wiley-Interscience, NewYork; pp. 895–964.

28. Kobayashi, H.; Kobayashi, M. Catal Rev 1974, 10(1),139–176.

29. Dekker, F. H. M.; Bliek, A.; Kapteijn, F.; Moulijn, J. A.Chem Eng Sci 1995, 50(22), 3573–3580.

30. Dumesic, J. A.; Rudd, D. F.; Aparicio, L. M.; Rekoske,J. E.; Trevin, A. A. The Microkinetics of HeterogeneousCatalysis, Ed. 1; American Chemical Society: Washing-ton, DC, 1993.

31. De Wilde, J.; Das, A. K.; Heynderickx, G. H.; Marin,G. B. Ind Eng Chem Res 2001, 40(1), 119–130.

32. Schwiedernoch, R.; Tischer, S.; Correa, C.;Deutschmann, O. Chem Eng Sci 2003, 58(3–6),633–642.

33. Gleaves, J. T.; Yablonskii, G. S.; Phanawadee, P.; Schu-urman, Y. Appl Catal A 1997, 160(1), 55–88.

34. Shekhtman, S. O.; Yablonsky, G. S.; Gleaves, J. T.;Fushimi, R. Chem Eng Sci 2003, 58(21), 4843–4859.

35. Gleaves, J. T.; Ebner, J. R.; Kuechler, T. C. Catal Rev1988, 30(1), 49–116.

36. Yablonskii, G. S.; Shekhtman, S. O.; Chen, S.; Gleaves,J. T. Ind Eng Chem Res 1998, 37(6), 2193–2202.

37. Fushimi, R.; Shekhtman, S. O.; Gaffney, A.; Han, S.;Yablonsky, G. S.; Gleaves, J. T. Ind Eng Chem Res2005, 44(16), 6310–6319.

38. van der Linde, S. C.; Nijhuis, T. A.; Dekker, F. H. M.;Kapteijn, F.; Moulijn, J. A. Appl Catal A 1997, 151(1),27–57.

39. Menon, U.; Galvita, V. V.; Constales, D.; Yablonsky, G.;Marin, G. B. Catal Today 2015, 258, 214–224.

40. Shekhtman, S. O.; Yablonsky, G. S.; Chen, S.; Gleaves,J. T. Chem Eng Sci 1999, 54(20), 4371–4378.

41. Shekhtman, S. O.; Maguire, N.; Goguet, A.; Burch,R.; Hardacre, C. Catal Today 2007, 121(3–4), 255–260.

42. Shekhtman, S. O.; Goguet, A.; Burch, R.; Hardacre, C.;Maguire, N. J Catal 2008, 253(2), 303–311.

43. Morgan, K.; Inceesungvorn, B.; Goguet, A.; Hardacre,C.; Meunier, F. C.; Shekhtman, S. O. Catal Sci Technol2012, 2, 2128–2133.

44. Roelant, R.; Constales, D.; Van Keer, R.; Marin, G. B.Chem Eng Sci 2012, 83, 39–49.

45. Redekop, E. A.; Yablonsky, G. S.; Constales, D.; Ra-machandran, P. A.; Pherigo, C.; Gleaves, J. T. ChemEng Sci 2011, 66(24), 6441–6452.

46. Redekop, E. A.; Yablonsky, G. S.; Galvita, V. V.; Con-stales, D.; Fushimi, R.; Gleaves, J. T.; Marin, G. B. IndEng Chem Res 2013, 52(44), 15417–15427.

47. Roelant, R. PhD thesis, Ghent University, Gent, Bel-gium, 2011.

48. Roelant, R.; Constales, D.; Yablonsky, G. S.; Van Keer,R.; Rude, M. A.; Marin, G. B. Catal Today 2007, 121(3–4), 269–281.

49. Zambelli, T.; Barth, J. V.; Wintterlin, J.; Ertl, G. Nature1997, 390, 495–497.

50. Aster, R. C.; Borchers, B.; Thurber, C. H. In ParameterEstimation and Inverse Problems, 2nd ed.; AcademicPress: Boston, MA, 2013.


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